Mathematics for Elementary Teachers by Michelle Manes is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License, except where otherwise noted.
Image used in cover by Robin Férand on 500px.com / CC BY 3.0.
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This book will help you to understand elementary mathematics more deeply, gain facility with creating and using mathematical notation, develop a habit of looking for reasons and creating mathematical explanations, and become more comfortable exploring unfamiliar mathematical situations.
The primary goal of this book is to help you learn to think like a mathematician in some very specific ways. You will:
• Make sense of problems and persevere in solving them. You will develop and demonstrate this skill by working on difficult problems, making incremental progress, and revising solutions to problems as you learn more.
• Reason abstractly and quantitatively. You will demonstrate this skill by learning to represent situations using mathematical notation (abstraction) as well as creating and testing examples (making situations more concrete).
• Construct viable arguments and critique the reasoning of others. You will be expected to create both written and verbal explanations for your solutions to problems. The most important questions in this class are “Why?” and “How do you know you’re right?” Practice asking these questions of yourself, of your professor, and of your fellow students.
• Model with mathematics. You will demonstrate this skill by inventing mathematical notation and drawings to represent physical situations and solve problems.
• Use appropriate tools strategically. You will be expected to use computers, calculators, measuring devices, and other mathematical tools when they are helpful.
• Attend to precision. You will write and express mathematical ideas clearly, using mathematical terms properly, providing clear definitions and descriptions of your ideas, and distinguishing between similar ideas (for example “factor” versus “multiple”.)
• Look for and make use of mathematical structure. You will find, describe, and most importantly explain patterns that come up in various situations including problems, tables of numbers, and algebraic expressions.
• Look for and express regularity in repeated reasoning. You will demonstrate this skill by recognizing (and expressing) when calculations or ideas are repeated, and how that can be used to draw mathematical conclusions (for example why a decimal must repeat) or develop shortcuts to calculations.
Throughout the book, you will learn how to learn mathematics on you own by reading, working on problems, and making sense of new ideas on your own and in collaboration with other students in the class.
This book was developed at the University of Hawai`i at Mānoa for the Math 111 and 112 (Mathematics for Elementary Teachers I and II) courses. The materials were written by Prof. Michelle Manes with tremendous assistance from lots of people.
I owe a huge debt to Dr. Tristan Holmes, who has taught the courses for years and assisted greatly on the revision and current format of the textbook. I also thank the graduate students who helped to design and develop the original iBook version of these materials: Amy Brandenberg, Jon Brown, Jessica Delgado, Paul Nguyen, Geoff Patterson, and especially Ryan Felix. Thanks to Monique Chyba, PI of the SUPER-M project (NSF grant DGE-0841223), for supporting this work, and to the UH Mānoa College of Natural Sciences and College of Education for their support as well.
Thanks also to the hundreds of Math 111 and 112 students I’ve taught over the past ten years. Your enthusiasm, energy, joy, and humor is what keeps me going.
I am grateful to all of my colleagues and professors, past and present, from whom I have learned so much about mathematics and about education. Special thanks to Dr. Carol Findell and Dr. Suzanne Chapin at Boston University, who gave me an entirely new perspective on mathematics teaching and learning.
I can never thank Dr. Al Cuoco enough for his support and intellectual leadership. I owe him more than I can say.
Unless otherwise noted, images were created by Michelle Manes using LaTeX, Mathematica, or Geometer’s Sketchpad.
Michelle Manes
Honolulu, HI
December, 2017
I
Solving a problem for which you know there’s an answer is like climbing a mountain with a guide, along a trail someone else has laid. In mathematics, the truth is somewhere out there in a place no one knows, beyond all the beaten paths.
– Yoko Ogawa
1
The Common Core State Standards for Mathematics (http://www.corestandards.org/Math/Practice) identify eight “Mathematical Practices” — the kinds of expertise that all teachers should try to foster in their students, but they go far beyond any particular piece of mathematics content. They describe what mathematics is really about, and why it is so valuable for students to master. The very first Mathematical Practice is:
Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary.
This chapter will help you develop these very important mathematical skills, so that you will be better prepared to help your future students develop them. Let’s start with solving a problem!
Draw curves connecting A to A, B to B, and C to C. Your curves cannot cross or even touch each other,they cannot cross through any of the lettered boxes, and they cannot go outside the large box or even touch it’s sides.
After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it).
Problem Solving Strategy 1 (Wishful Thinking). Do you wish something in the problem was different? Would it then be easier to solve the problem?
For example, what if ABC problem had a picture like this:
Can you solve this case and use it to help you solve the original case? Think about moving the boxes around once the lines are already drawn.
Here is one possible solution.
2
The main activity of mathematics is solving problems. However, what most people experience in most mathematics classrooms is practice exercises. An exercise is different from a problem.
In a problem, you probably don’t know at first how to approach solving it. You don’t know what mathematical ideas might be used in the solution. Part of solving a problem is understanding what is being asked, and knowing what a solution should look like. Problems often involve false starts, making mistakes, and lots of scratch paper!
In an exercise, you are often practicing a skill. You may have seen a teacher demonstrate a technique, or you may have read a worked example in the book. You then practice on very similar assignments, with the goal of mastering that skill.
Note: What is a problem for some people may be an exercise for other people who have more background knowledge! For a young student just learning addition, this might be a problem:
Fill in the blank to make a true statement: .
But for you, that is an exercise!
Both problems and exercises are important in mathematics learning. But we should never forget that the ultimate goal is to develop more and better skills (through exercises) so that we can solve harder and more interesting problems.
Learning math is a bit like learning to play a sport. You can practice a lot of skills:
But the point of the sport is to play the game. You practice the skills so that you are better at playing the game. In mathematics, solving problems is playing the game!
For each question below, decide if it is a problem or an exercise. (You do not need to solve the problems! Just decide which category it fits for you.) After you have labeled each one, compare your answers with a partner.
1. This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers.(Note: Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15. )
Can you break another clock into a different number of pieces so that the sums are consecutive numbers? Assume that each piece has at least two numbers and that no number is damaged (e.g. 12 isn’t split into two digits 1 and 2).
2. A soccer coach began the year with a $500 budget. By the end of December, the coach spent $450. How much money in the budget was not spent?
3. What is the product of 4,500 and 27?
4. Arrange the digits 1–6 into a “difference triangle” where each number in the row below is the difference of the two numbers above it.
5. Simplify the following expression:
6. What is the sum of and ?
7. You have eight coins and a balance scale. The coins look alike, but one of them is a counterfeit. The counterfeit coin is lighter than the others. You may only use the balance scale two times. How can you find the counterfeit coin?
8. How many squares, of any possible size, are on a standard 8 × 8 chess board?
9. What number is 3 more than half of 20?
10. Find the largest eight-digit number made up of the digits 1, 1, 2, 2, 3, 3, 4, and 4 such that the 1’s are separated by one digit, the 2’s are separated by two digits, the 3’s by three digits, and the 4’s by four digits.
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Think back to the first problem in this chapter, the ABC Problem. What did you do to solve it? Even if you did not figure it out completely by yourself, you probably worked towards a solution and figured out some things that did not work.
Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.
George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985.Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY 2.0 (http://creativecommons.org/licenses/by/2.0)], via Wikimedia Commons
In 1945, Pólya published the short book How to Solve It, which gave a four-step method for solving mathematical problems:
This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.
Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!
We have already seen one problem solving strategy, which we call “Wishful Thinking.” Do not be afraid to change the problem! Ask yourself “what if” questions:
You need to be sure to go back to the original problem at the end, but wishful thinking can be a powerful strategy for getting started.
This brings us to the most important problem solving strategy of all:
Problem Solving Strategy 2 (Try Something!). If you are really trying to solve a problem, the whole point is that you do not know what to do right out of the starting gate. You need to just try something! Put pencil to paper (or stylus to screen or chalk to board or whatever!) and try something. This is often an important step in understanding the problem; just mess around with it a bit to understand the situation and figure out what is going on.
And equally important: If what you tried first does not work, try something else! Play around with the problem until you have a feel for what is going on.
Last week, Alex borrowed money from several of his friends. He finally got paid at work, so he brought cash to school to pay back his debts. First he saw Brianna, and he gave her 1/4 of the money he had brought to school. Then Alex saw Chris and gave him 1/3 of what he had left after paying Brianna. Finally, Alex saw David and gave him 1/2 of what he had remaining. Who got the most money from Alex?
After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem?
This problem lends itself to two particular strategies. Did you try either of these as you worked on the problem? If not, read about the strategy and then try it out before watching the solution.
Problem Solving Strategy 3 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric, like this one, thinking visually can help! Can you represent something in the situation by a picture?
Draw a square to represent all of Alex’s money. Then shade 1/4 of the square — that’s what he gave away to Brianna. How can the picture help you finish the problem?
After you have worked on the problem yourself using this strategy (or if you are completely stuck), you can watch someone else’s solution.
Problem Solving Strategy 4 (Make Up Numbers). Part of what makes this problem difficult is that it is about money, but there are no numbers given. That means the numbers must not be important. So just make them up!
You can work forwards: Assume Alex had some specific amount of money when he showed up at school, say $100. Then figure out how much he gives to each person. Or you can work backwards: suppose he has some specific amount left at the end, like $10. Since he gave Chris half of what he had left, that means he had $20 before running into Chris. Now, work backwards and figure out how much each person got.
Watch the solution only after you tried this strategy for yourself.
If you use the “Make Up Numbers” strategy, it is really important to remember what the original problem was asking! You do not want to answer something like “Everyone got $10.” That is not true in the original problem; that is an artifact of the numbers you made up. So after you work everything out, be sure to re-read the problem and answer what was asked!
How many squares, of any possible size, are on a 8 × 8 chess board? (The answer is not 64… It’s a lot bigger!)
Remember Pólya’s first step is to understand the problem. If you are not sure what is being asked, or why the answer is not just 64, be sure to ask someone!
After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem, even if you have not solved it completely?
It is clear that you want to draw a picture for this problem, but even with the picture it can be hard to know if you have found the correct answer. The numbers get big, and it can be hard to keep track of your work. Your goal at the end is to be absolutely positive that you found the right answer. You should never ask the teacher, “Is this right?” Instead, you should declare, “Here’s my answer, and here is why I know it is correct!”
Problem Solving Strategy 5 (Try a Simpler Problem). Pólya suggested this strategy: “If you can’t solve a problem, then there is an easier problem you can solve: find it.” He also said: “If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?” In this case, an 8 × 8 chess board is pretty big. Can you solve the problem for smaller boards? Like 1 × 1? 2 × 2? 3 × 3?
Of course the ultimate goal is to solve the original problem. But working with smaller boards might give you some insight and help you devise your plan (that is Pólya’s step (2)).
Problem Solving Strategy 6 (Work Systematically). If you are working on simpler problems, it is useful to keep track of what you have figured out and what changes as the problem gets more complicated.
For example, in this problem you might keep track of how many 1 × 1 squares are on each board, how many 2 × 2 squares on are each board, how many 3 × 3 squares are on each board, and so on. You could keep track of the information in a table:
size of board | # of 1 × 1 squares | # of 2 × 2 squares | # of 3 × 3 squares | # of 4 × 4 squares | … |
1 by 1 | 1 | 0 | 0 | 0 | |
2 by 2 | 4 | 1 | 0 | 0 | |
3 by 3 | 9 | 4 | 1 | 0 | |
… |
Problem Solving Strategy 7 (Use Manipulatives to Help You Investigate). Sometimes even drawing a picture may not be enough to help you investigate a problem. Having actual materials that you move around can sometimes help a lot!
For example, in this problem it can be difficult to keep track of which squares you have already counted. You might want to cut out 1 × 1 squares, 2 × 2 squares, 3 × 3 squares, and so on. You can actually move the smaller squares across the chess board in a systematic way, making sure that you count everything once and do not count anything twice.
Problem Solving Strategy 8 (Look for and Explain Patterns). Sometimes the numbers in a problem are so big, there is no way you will actually count everything up by hand. For example, if the problem in this section were about a 100 × 100 chess board, you would not want to go through counting all the squares by hand! It would be much more appealing to find a pattern in the smaller boards and then extend that pattern to solve the problem for a 100 × 100 chess board just with a calculation.
If you have not done so already, extend the table above all the way to an 8 × 8 chess board, filling in all the rows and columns. Use your table to find the total number of squares in an 8 × 8 chess board. Then:
(We will come back to this question soon. So if you are not sure right now how to explain and justify the patterns you found, that is OK.)
This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers. (Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15.)
Can you break another clock into a different number of pieces so that the sums are consecutive numbers? Assume that each piece has at least two numbers and that no number is damaged (e.g. 12 isn’t split into two digits 1 and 2.)
Remember that your first step is to understand the problem. Work out what is going on here. What are the sums of the numbers on each piece? Are they consecutive?
After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What progress have you made?
Problem Solving Strategy 9 (Find the Math, Remove the Context). Sometimes the problem has a lot of details in it that are unimportant, or at least unimportant for getting started. The goal is to find the underlying math problem, then come back to the original question and see if you can solve it using the math.
In this case, worrying about the clock and exactly how the pieces break is less important than worrying about finding consecutive numbers that sum to the correct total. Ask yourself:
Of course, solving the question about consecutive numbers is not the same as solving the original problem. You have to go back and see if the clock can actually break apart so that each piece gives you one of those consecutive numbers. Maybe you can solve the math problem, but it does not translate into solving the clock problem.
Problem Solving Strategy 10 (Check Your Assumptions). When solving problems, it is easy to limit your thinking by adding extra assumptions that are not in the problem. Be sure you ask yourself: Am I constraining my thinking too much?
In the clock problem, because the first solution has the clock broken radially (all three pieces meet at the center, so it looks like slicing a pie), many people assume that is how the clock must break. But the problem does not require the clock to break radially. It might break into pieces like this:
Were you assuming the clock would break in a specific way? Try to solve the problem now, if you have not already.
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The “Look for Patterns” strategy can be particularly appealing, but you have to be careful! Do not forget the “and Explain” part of the strategy. Not all patterns are obvious, and not all of them will continue.
Start with a circle.
If I put two dots on the circle and connect them, the line divides the circle into two pieces.
If I put three dots on the circle and connect each pair of dots, the lines divides the circle into four pieces.
Suppose you put one hundred dots on a circle and connect each pair of dots, meaning every dot is connected to 99 other dots. How many pieces will you get? Lines may cross each other, but assume the points are chosen so that three or more lines never meet at a single point.
After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What strategies did you try? What did you figure out? What questions do you still have?
The natural way to work on this problem is to use smaller numbers of dots and look for a pattern, right? If you have not already, try it. How many pieces when you have four dots? Five dots? How would you describe the pattern?
Now try six dots. You will want to draw a big circle and space out the six dots to make your counting easier. Then carefully count up how many pieces you get. It is probably a good idea to work with a partner so you can check each other’s work. Make sure you count every piece once and do not count any piece twice. How can you be sure that you do that?
Were you surprised? For the first several steps, it seems to be the case that when you add a dot you double the number of pieces. But that would mean that for six dots, you should get 32 pieces, and you only get 30 or 31, depending on how the dots are arranged. No matter what you do, you cannot get 32 pieces. The pattern simply does not hold up.
Mathematicians love looking for patterns and finding them. We get excited by patterns. But we are also very skeptical of patterns! If we cannot explain why a pattern would occur, then we are not willing to just believe it.
For example, if my number pattern starts out: 2, 4, 8, … I can find lots of ways to continue the pattern, each of which makes sense in some contexts. Here are some possibilities:
This is a a repeating pattern, cycling through the numbers 2, 4, 8 and then starting over with 2.
To get the next number, multiply the previous two numbers together.
So how can you be sure your pattern fits the problem? You have to tie them together! Remember the “Squares on a Chess Board” problem? You might have noticed a pattern like this one:
If the chess board has 5 squares on a side, then there are
So there are a total of
squares on a 5 × 5 chess board. You can probably guess how to continue the pattern to any size board, but how can you be absolutely sure the pattern continues in this way? What if this is like “Dots on a Circle,” and the obvious pattern breaks down after a few steps? You have to tie the pattern to the problem, so that it is clear why the pattern must continue in that way.
The first step in explaining a pattern is writing it down clearly. This brings us to another problem solving strategy.
Problem Solving Strategy 11 (Use a Variable!). One of the most powerful tools we have is the use of a variable. If you find yourself doing calculations on things like “the number of squares,” or “the number of dots,” give those quantities a name! They become much easier to work with.
For now, just work on describing the pattern with variables.
Now comes the tough part: explaining the pattern. Let us focus on an 8 × 8 board. Since it measures 8 squares on each side, we can see that we get 8 × 8 = 64 squares of size 1 × 1. And since there is just a single board, we get just one square of size 8 × 8. But what about all the sizes in-between?
Using the Chess Board video in the previous chapter as a model, work with a partner to carefully explain why the number of 3 × 3 squares will be 6 · 6 = 36, and why the number of 4 × 4 squares will be 5 · 5 = 25.
There are many different explanations other than what is found in the video. Try to find your own explanation.
Here is what a final justification might look like (watch the Chess Board video as a concrete example of this solution):
Solution (Chess Board Pattern). Let n be the side of the chess board and let k be the side of the square. If the square is going to fit on the chess board at all, it must be true that k ≤ n. Otherwise, the square is too big.
If I put the k × k square in the upper left corner of the chess board, it takes up k spaces across and there are (n – k) spaces to the right of it. So I can slide the k × k square to the right (n – k) times, until it hits the top right corner of the chess board. The square is in (n – k + 1) different positions, counting the starting position.
If I move the k × k square back to the upper left corner, I can shift it down one row and repeat the whole process again. Since there are (n – k) rows below the square, I can shift it down (n – k) times until it hits the bottom row. This makes (n – k + 1) total rows that the square moves across, counting the top row.
So, there are (n – k + 1) rows with (n – k + 1) squares in each row. That makes (n – k + 1)^{2} total squares.
Thus, the solution is the sum of (n – k + 1)^{2} for all k ≤ n. In symbols:
Once we are sure the pattern continues, we can use it to solve the problem. So go ahead!
There is a number pattern that describes the number of pieces you get from the “Dots on a Circle” problem. If you want to solve the problem, go for it! Think about all of your problem solving strategies. But be sure that when you find a pattern, you can explain why it is the right pattern for this problem, and not just another pattern that seems to work but might not continue.
5
You have several problem solving strategies to work with. Here are the ones we have described so far (and you probably came up with even more of your own strategies as you worked on problems).
Try your hand at some of these problems, keeping these strategies in mind. If you are stuck on a problem, come back to this list and ask yourself which of the strategies might help you make some progress.
You have eight coins and a balance scale. The coins look alike, but one of them is a counterfeit. The counterfeit coin is lighter than the others. You may only use the balance scale two times. How can you find the counterfeit coin?
You have five coins, no two of which weigh the same. In seven weighings on a balance scale, can you put the coins in order from lightest to heaviest? That is, can you determine which coin is the lightest, next lightest, . . . , heaviest.
You have ten bags of coins. Nine of the bags contain good coins weighing one ounce each. One bag contains counterfeit coins weighing 1.1 ounces each. You have a regular (digital) scale, not a balance scale. The scale is correct to one-tenth of an ounce. In one weighing, can you determine which bag contains the bad coins?
Suppose you have a balance scale. You have three different weights, and you are able to weigh every whole number from 1 gram to 13 grams using just those three weights. What are the three weights?
There are a bunch of coins on a table in front of you. Your friend tells you how many of the coins are heads-up. You are blindfolded and cannot see a thing, but you can move the coins around, and you can flip them over. However, you cannot tell just by feeling them if the coins are showing heads or tails. Your job: separate the coins into two piles so that the same number of heads are showing in each pile.
The digital root of a number is the number obtained by repeatedly adding the digits of the number. If the answer is not a one-digit number, add those digits. Continue until a one-digit sum is reached. This one digit is the digital root of the number.
For example, the digital root of 98 is 8, since 9 + 8 = 17 and 1 + 7 = 8.
Record the digital roots of the first 30 integers and find as many patterns as you can. Can you explain any of the patterns?
If this lattice were continued, what number would be directly to the right of 98? How can you be sure you’re right?
3 | 6 | 9 | 12 | … | |||||
1 | 2 | 4 | 5 | 7 | 8 | 10 | 11 | 13 | … |
Arrange the digits 0 through 9 so that the first digit is divisible by 1, the first two digits are divisible by 2, the first three digits are divisible by 3, and continuing until you have the first 9 digits divisible by 9 and the whole 10-digit number divisible by 10.
There are 25 students and one teacher in class. After an exam, everyone high-fives everyone else to celebrate how well they did. How many high- fives were there?
In cleaning out your old desk, you find a whole bunch of 3¢ and 7¢ stamps. Can you make exactly 11¢ of postage? Can you make exactly 19¢ of postage? What is the largest amount of postage you cannot make?
Find the largest eight-digit number made up of the digits 1, 1, 2, 2, 3, 3, 4, and 4 such that the 1’s are separated by one digit, the 2’s are separated by two digits, the 3’s by three digits, and the 4’s by four digits.
Kami has ten pockets and 44 dollar bills. She wants to have a different amount of money in each pocket. Can she do it?
How many triangles of all possible sizes and shapes are in this picture?
Arrange the digits 1–6 into a “difference triangle” where each number in the row below is the difference of the two numbers above it.
Example: Below is a difference triangle, but it does not work because it uses 1 twice and does not have a 6:
4 | 5 | 3 | ||
1 | 2 | |||
1 |
Certain pipes are sold in lengths of 6 inches, 8 inches, and 10 inches. How many different lengths can you form by attaching three sections of pipe together?
Place the digits 1, 2, 3, 4, 5, 6 in the circles so that the sum on each side of the triangle is 12. Each circle gets one digit, and each digit is used exactly once.
Find a way to cut a circular pizza into 11 pieces using just four straight cuts.
6
This section might seem like a bit of a sidetrack from the idea of problem solving, but in fact it is not. Mathematics is a social endeavor. We do not just solve problems and then put them aside. Problem solving has (at least) three components:
If you are not able to do that last step, then you have not really solved the problem. We will talk more about how to write up a solution soon. Before we do that, we have to think about how mathematicians use language (which is, it turns out, a bit different from how language is used in the rest of life).
A mathematical statement is a complete sentence that is either true or false, but not both at once.
So a “statement” in mathematics cannot be a question, a command, or a matter of opinion. It is a complete, grammatically correct sentence (with a subject, verb, and usually an object). It is important that the statement is either true or false, though you may not know which! (Part of the work of a mathematician is figuring out which sentences are true and which are false.)
For each English sentence below, decide if it is a mathematical statement or not. If it is, is the statement true or false (or are you unsure)? If it is not a mathematical statement, in what way does it fail?
Now write three mathematical statements and three English sentences that fail to be mathematical statements.
Notice that “1/2 = 2/4” is a perfectly good mathematical statement. It does not look like an English sentence, but read it out loud. The subject is “1/2.” The verb is “equals.” And the object is “2/4.” This is a very good test when you write mathematics: try to read it out loud. Even the equations should read naturally, like English sentences.
Statement (5) is different from the others. It is called a paradox: a statement that is self-contradictory. If it is true, then we conclude that it is false. (Why?) If it is false, then we conclude that it is true. (Why?) Paradoxes are no good as mathematical statements, because it cannot be true and it cannot be false.
Consider this sentence:
After work, I will go to the beach, or I will do my grocery shopping.
In everyday English, that probably means that if I go to the beach, I will not go shopping. I will do one or the other, but not both activities. This is called an “exclusive or.”
We can usually tell from context whether a speaker means “either one or the other or both,” or whether he means “either one or the other but not both.” (Some people use the awkward phrase “and/or” to describe the first option.)
Remember that in mathematical communication, though, we have to be very precise. We cannot rely on context or assumptions about what is implied or understood.
In mathematics, the word “or” always means “one or the other or both.”
The word “and” always means “both are true.”
For each sentence below:
You are handed an envelope filled with money, and you are told “Every bill in this envelope is a $100 bill.”
Suppose you were given a different sentence: “There is a $100 bill in this envelope.”
What is the difference between the two sentences? How does that difference affect your method to decide if the statement is true or false?
Some mathematical statements have this form:
These are universal statements. Such statements claim that something is always true, no matter what.
Some mathematical statements have this form:
These are existential statements. Such statements claim there is some example where the statement is true, but it may not always be true.
For each statement below, do the following:
Look back over your work. you will probably find that some of your arguments are sound and convincing while others are less so. In some cases you may “know” the answer but be unable to justify it. That is okay for now! Divide your answers into four categories:
You have a deck of cards where each card has a letter on one side and a number on the other side. Your friend claims: “If a card has a vowel on one side, then it has an even number on the other side.”
These cards are on a table.
Which cards must you flip over to be certain that your friend is telling the truth?
After you have thought about the problem on your own for a while, discuss your ideas with a partner. Do you agree on which cards you must check? Try to come to agreement on an answer you both believe.
Here is another very similar problem, yet people seem to have an easier time solving this one:
You are in charge of a party where there are young people. Some are drinking alcohol, others soft drinks. Some are old enough to drink alcohol legally, others are under age. You are responsible for ensuring that the drinking laws are not broken, so you have asked each person to put his or her photo ID on the table. At one table, there are four young people:
Which IDs and/or drinks do you need to check to make sure that no one is breaking the law?
After you have thought about the problem on your own for a while, discuss your ideas with a partner. Do you agree on which cards you must check? Compare these two problems. Which question is easier and why?
A conditional statement can be written in the form
If some statement then some statement.
Where the first statement is the hypothesis and the second statement is the conclusion.
These are each conditional statements, though they are not all stated in “if/then” form. Identify the hypothesis of each statement. (You may want to rewrite the sentence as an equivalent “if/then” statement.)
Remember that a mathematical statement must have a definite truth value. It is either true or false, with no gray area (even though we may not be sure which is the case). How can you tell if a conditional statement is true or false? Surely, it depends on whether the hypothesis and the conclusion are true or false. But how, exactly, can you decide?
The key is to think of a conditional statement like a promise, and ask yourself: under what condition(s) will I have broken my promise?
If I win the lottery, then I’ll give each of my students $1,000.
There are four things that can happen:
What can we conclude from this? A conditional statement is false only when the hypothesis is true and the conclusion is false. In every other instance, the promise (as it were) has not been broken. If a mathematical statement is not false, it must be true.
If you live in Honolulu, then you live in Hawaii.
Is this statement true or false? It seems like it should depend on who the pronoun “you” refers to, and whether that person lives in Honolulu or not. Let us think it through:
So in fact it does not matter! The statement is true either way. The right way to understand such a statement is as a universal statement: “Everyone who lives in Honolulu lives in Hawaii.”
This statement is true, and here is how you might justify it: “Pick a random person who lives in Honolulu. That person lives in Hawaii (since Honolulu is in Hawaii), so the statement is true for that person. I do not need to consider people who do not live in Honolulu. The statement is automatically true for those people, because the hypothesis is false!”
You need to give a specific instance where the hypothesis is true and the conclusion is false. For example:
If you are a good swimmer, then you are a good surfer.
Do you know someone for whom the hypothesis is true (that person is a good swimmer) but the conclusion is false (the person is not a good surfer)? Then the statement is false!
For each conditional statement, decide if it is true or false. Justify your answer.
On your own, come up with two conditional statements that are true and one that is false. Share your three statements with a partner, but do not say which are true and which is false. See if your partner can figure it out!
7
At its heart, mathematics is a social endeavor. Even if you work on problems all by yourself, you have not really solved the problem until you have explained your work to someone else, and they sign off on it. Professional mathematicians write journal articles, books, and grant proposals. Teachers explain mathematical ideas to their students both in writing and orally. Explaining your work is really an essential part of the problem-solving process, and probably should have been Pólya’s step 5.
Writing in mathematics is different from writing poetry or an English paper. The goal of mathematical writing is not florid description, but clarity. If your reader does not understand, then you have not done a good job. Here are some tips for good mathematical writing.
Do Not Turn in Scratch Work: When you are solving problems and not exercises, you are going to have a lot of false starts. You are going to try a lot of things that do not work. You are going to make a lot of mistakes. You are going to use scratch paper. At some point (hopefully!) you will scribble down an idea that actually solves the problem. Hooray! That paper is not what you want to turn in or share with the world. Take that idea, and write it up carefully, neatly, and clearly. (The rest of these tips apply to that write-up.)
(Re)state the Problem: Do not assume your reader knows what problem you are solving. (Even if it is the teacher who assigned the problem!) If the problem has a very long description, you can summarize it. You do not have rewrite it word-for-word or give all of the details, but make sure the question is clear.
Clearly Give the Answer: It is not a bad idea to state the answer right up front, then show the work to justify your answer. That way, the reader knows what you are trying to justify as they read. It makes their job much easier, and making the reader’s job easier should be one of your primary goals! In any case, the answer should be clearly stated somewhere in the writeup, and it should be easy to find.
Be Correct: Of course, everyone makes mistakes as they are working on a problem. But we are talking about after you have solved the problem, when you are writing up your solution to share with someone else. The best writing in the world cannot save a wrong approach and a wrong answer. Check your work carefully. Ask someone else to read your solution with a critical eye.
Justify Your Answer: You cannot simply give an answer and expect your reader to “take your word for it.” You have to explain how you know your answer is correct. This means “showing your work,” explaining your reasoning, and justifying what you say. You need to answer the question, “How do you know your answer is right?”
Be Concise: There is no bonus prize for writing a lot in math class. Think clearly and write clearly. If you find yourself going on and on, stop, think about what you really want to say, and start over.
Use Variables and Equations: An equation can be much easier to read and understand than a long paragraph of text describing a calculation. Mathematical writing often has a lot fewer words (and a lot more equations) than other kinds of writing.
Define your Variables: If you use variables in the solution of your problem, always say what a variable stands for before you use it. If you use an equation, say where it comes from and why it applies to this situation. Do not make your reader guess!
Use Pictures: If pictures helped you solve the problem, include nice versions of those pictures in your final solution. Even if you did not draw a picture to solve the problem, it still might help your reader understand the solution. And that is your goal!
Use Correct Spelling and Grammar: Proofread your work. A good test is to read your work aloud (this includes reading the equations and calculations aloud). There should be complete, natural-sounding sentences. Be especially careful with pronouns. Avoid using “it” and “they” for mathematical objects; use the names of the objects (or variables) instead.
Format Clearly: Do not write one long paragraph. Separate your thoughts. Put complicated equations on a single displayed line rather than in the middle of a paragraph. Do not write too small. Do not make your reader struggle to read and understand your work.
Acknowledge Collaborators: If you worked with someone else on solving the problem, give them credit!
Here is a problem you’ve already seen:
Find the largest eight-digit number made up of the digits 1, 1, 2, 2, 3, 3, 4, and 4 such that the 1’s are separated by one digit, the 2’s are separated by two digits, the 3’s by three digits, and the 4’s by four digits.
Below you will find several solutions that were turned in by students. Using the criteria above, how would you score these solutions on a scale of 1 to 5? Give reasons for your answers.
Solution (Solution 1).
41312432
This is the largest eight-digit b/c the #s 1, 2, 3, 4 & all separated by the given amount of spaces.
Solution (Solution 2).
41312432
You have to have the 4 in the highest place and work down from there. However unable to follow the rules the 2 and the 1 in the 10k and 100k place must switch.
Solution (Solution 3).
41312432
First, I had to start with the #4 because that is the largest digit I could start with to get the largest #. Then I had to place the next 4 five spaces away because I knew there had to be four digits separating the two 4’s. Next, I place 1 in the second digit spot because 2 or 3 would interfere with the rule of how many digits could separate them, which allowed me to also place where the next 1 should be. I then placed the 3 because opening spaces showed me that I could fit three digits in between the two 3’s. Lastly, I had to input the final 2’s, which worked out because there were two digits separating them.
Solution (Solution 4).
1×1
2xx2
3xxx3
4xxxx4
Answer: 41312432
Solution (Solution 5).
4 3 2 4 3 2
4 2 2 4
4 1 3 1 4 3
*4 1 3 1 2 4 3 2
4 needs to be the first # to make it the biggest. Then check going down from next largest to smallest. Ex:
4 3 __________
4 2 __________
4 1 __________
Solution (Solution 6).
41312432
I put 4 at the 10,000,000 place because the largest # should be placed at the highest value. Numbers 2 & 3 could not be placed in the 1,000,000 place because I wasn’t able to separate the digits properly. So I ended up placing the #1 there. In the 100,000 place I put the #3 because it was the second highest number.
Solution (Solution 7).
41312432
Since the problem asks you for the largest 8 digit #, I knew 4 had to be the first # since it’s the greatest # of the set. To solve the rest of the problem, I used the guess and test method. I tried many different combinations. First using the #3 as the second digit in the sequence, but came to no answer. Then the #2, but no combination I found correctly finished the sequence.I then finished with the #1 in the second digit in the sequence and was able to successfully fill out the entire #.
Solution (Solution 8).
4 _ _ _ _ 4 _ _
4 has to be the first digit, for the number to be the largest possible. That means the other 4 has to be the 6th digit in the number, because 4’s have to be separated by four digits.
4 _ 3 _ _ 4 3 _
3 must be the third digit, in order for the number to be largest possible. 3 cannot be the second digit because the other 3 would have to be the 6th digit in the number, but 4 is already there.
4 1 3 1 _ 4 3 _
1’s must be separated by one digit, so the 1’s can only be the 2nd and 4th digit in the number.
4 1 3 1 2 4 3 2
This leaves the 2s to be the 5th and 8th digits.
Solution (Solution 9).
With the active rules, I tried putting the highest numbers as far left as possible. Through trying different combinations, I figured out that no two consecutive numbers can be touching in the first two digits. So I instead tried starting with the 4 then 1 then 3, since I’m going for the highest # possible.
My answer: 41312432
8
A lot of people — from Polya to the writers of the Common Core State Standards and a lot of people in between — talk about problem solving in mathematics. One fact is rarely acknowledged, except by many professional mathematicians: Asking good questions is as valuable (and as difficult) as solving mathematical problems.
After solving a mathematical problem and explaining your solution to someone else, it is a very good mathematical habit to ask yourself: What other questions can I ask?
Recall Problem 3, “Squares on a Chess Board”:
How many squares of any possible size are on a standard 8 × 8 chess board? (The answer is not 64! It’s a lot bigger!)
We have already talked about some obvious follow-up questions like “What about a 10 × 10 chess board? Or 100 × 100? Or ?”
But there are a lot of interesting (and less obvious . . . and harder) questions you might ask:
Recall Problem 4, “Broken Clock”:
This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers. Can you break another clock into a different number of pieces so that the sums are consecutive numbers?
The original problem only asks if you can find one other way. The obvious follow-up question: “Find every possibly way to break the clock into some number of pieces so that the sums of the numbers on each piece are consecutive numbers. Justify that you have found every possibility.”
Choose a problem from the Problem Bank (preferably a problem you have worked on, but that is not strictly necessary). What follow-up or similar questions could you ask?
II
Images and Videos on Pixabay are released under Creative Commons CC0.
-Laplace
The “Dots and Boxes” approach to place value used in this part (and throughout this book) comes from James Tanton, and is used with his permission. See his development of these and other ideas at http://gdaymath.com/.
9
Here are some dots; in fact there are nine of them:
We’re going to a play an “exploding dots” game. Here’s the only rule for the game:
Whenever there are two dots in single box, they “explode,” disappear, and become one dot in the box to the left.
We start by placing nine dots in the rightmost box.
Two dots in that box explode and become one dot in the box to the left.
Once again, two dots in that box explode and become one dot in the box to the left.
We do it again!
Hey, now we have more than two dots in the second box, so those can explode and move!
And the rightmost box still has more than two dots.
Keep going, until no box has two dots.
After all this, reading from left to right we are left with one dot, followed by zero dots, zero dots, and one final dot.
Solution: The 1←2 code for nine dots is: 1001.
On Your Own. Here’s a diagram showing what happens for seven dots in a 1←2 box. Trace through the diagram, and circle the pairs of dots that “exploded” at each step.
Solution: The 1←2 code for seven dots is: 111.
Note: In solving this problem, you don’t need to draw on paper; that can get tedious! Maybe you could use buttons or pennies for dots and do this by hand.
After you worked on the problem, compare your answer with a partner. Did you both get the same code? Did you have the same process?
10
Let’s play the dots and boxes game, but change the rule.
Whenever there are three dots in single box, they “explode,” disappear, and become one dot in the box to the left.
Here’s what happens with fifteen dots:
Solution: The 1←3 code for fifteen dots is: 120.
After you have worked on the problems on your own, compare your ideas with a partner. Can you describe what’s going on in Problem 6 and why?
11
Let’s go back to the 1←2 rule and examine what’s really going on.
Whenever there are two dots in single box, they “explode,” disappear, and become one dot in the box to the left.
Two dots in the right-most box is worth one dot in the next box to the left.
If each of the original dots is worth “one,” then the single dot on the left must be worth two.
But we also have two dots in the box of value 2 is worth one dot in the box just to the left…
So that next box must be worth two 2’s, which is four!
And two of these fours make eight.
We said earlier that the 1←2 code for nine dots was 1001. Let’s check:
so this works!
You should have found that ten dots has 1←2 code 1010.
Yep! .
Numbers written in the 1←2 code are called binary numbers or base two numbers. (The prefix “bi” means “two.”)
From now on, when we want to indicate that a number is written in base two, we will write a subscript “two” on the number.
So means “the number of dots that has 1←2 code 1001,” which we already saw was nine.
Important! When we read we say “one zero zero one base two.” We don’t say “one thousand and one,” because “thousand” is not a binary number.
two dots = ____{two} seventeen dots = ____{two}
sixty-three dots = ____{two} one hundred dots = ____{two}
You probably realize by now that a number is an abstract concept with many representations. The standard decimal representation of a number is only one of these. For computers, numbers are always represented in binary. The basic units are transistors which are either on (1) or off (0).
A transistorImage of circuit board from http://www.publicdomainpictures.net/, licensed under CC0 Public Domain. is said to store one bit of information. Eight bits make a byte and a typical home computer’s central processing unit performs computations on registries that are each 8 bytes (64-bits).
Using the 1←2 rule we can represent the numbers 0 through 18,446,744,073,709,551,615 with 64 bits.
12
In the 1←3 system, three dots in one box is worth one dot in the box one spot to the left. This gives a new picture:
Each dot in the second box from the left is worth three ones. Each dot in the third box is worth three 3’s, which is nine, and so on.
We said that the 1←3 code for fifteen is 120. We see that this is correct because:
Answer these questions about the 1←3 system.
In the 1←4 system, four dots in one box are worth one dot in the box one place to the left.
Answer these questions about the 1←4 system.
In the 1←10 system, ten dots in one box are worth one dot in the box one place to the left.
Recall that numbers written in the 1←2 system are called binary or base two numbers.
Numbers written in the 1←3 system are called base three numbers.
Numbers written in the 1←4 system are called base four numbers.
Numbers written in the 1←10 system are called base ten numbers.
In general, numbers written in the 1←b system are called base b numbers.
In a base b number system, each place represents a power of b, which means for some whole number n. Remember this means b multiplied by itself n times:
Whenever we’re dealing with numbers written in different bases, we use a subscript to indicate the base so that there can be no confusion. So:
If the base is not written, we assume it’s base ten.
Remember: when you see the subscript, you are seeing the code for some number of dots.
We’re now going to describe some general methods for converting from base b to base ten, where b can represent any whole number bigger than one.
If the base is b, that means we’re in a 1←b system. A dot in the right-most box is worth 1. A dot in the second box is worth b. A dot in the third box is worth , and so on.
So, for example, the number represents
because we imagine three dots in the right-most box (each worth one), two dots in the second box (each representing b dots), one dot in the third box (representing dots), and so on. That means we can just do a short calculation to find the total number of dots, without going through all the trouble of drawing the picture and “unexploding” the dots.
This represents the number
We’re now going to describe some general methods for converting from base ten to base b, where b can represent any whole number bigger than one.
There are two general methods for doing these conversions. For each method, we’ll work out an example, and then describe the general method. The first method we describe fills in the boxes from left to right.
To convert to a base five number (without actually going through the tedious process of exploding dots in groups of five).
Find the largest power of five that is smaller than 321. We’ll just list powers of five:
So we know that the left-most box we’ll use is the 125 box, because 625 is too big.
How many dots will be in the 125 box? That’s the same as asking how many 125’s are in 321. Since
we should put two dots in the 125 box. Three dots would be too much.
How many dots are left unaccounted for? dots are left.
Now repeat the process: The largest power of five that’s less than 71 is . If we put two in the 25 box, that takes care of 50 dots. (Three dots would be 75, which is too much.)
So far we have two dots in the box and two dots in the box, so that’s a total of
We have dots left to account for.
Repeat the process again: The biggest power of 5 that’s less than 21 is 5. How many dots can go in the 5 box? , so we can put four dots in the 5 box.
We have one dot left to account for. If we put one dot in the 1 box, we’re done.
So .
The general algorithm to convert from base ten to base :
The method is a little tricky to describe in complete generality. It’s probably better to try a few examples on your own to get the hang of it.
Use the method above to convert to base three, to base four, and to base five.
Here’s another method to convert base ten numbers to another base, and this method fills in the digits from right to left. Again, we’ll start with an example and then describe the general method.
To convert to a base seven number, imagine there are 712 dots in the ones box. We’ll write the number, but imagine it as dots.
Find out how many groups of 7 you can make, and how many dots would be left over.
That means we have 101 groups of 7 dots, with 5 dots left over.
“Explode” the groups of 7 one box to the left, and leave the 5 dots behind.
Now repeat the process: How many groups of 7 can you make from the 101 dots?
“Explode” the groups of 7 one box to the left, and leave the 3 dots behind.
Repeat:
“Explode” the groups of 7 one box to the left, and leave 0 dots behind.
Since there are fewer than 7 dots in each box, we’re done.
Of course, we can (and should!) check our calculation by converting the answer back to base ten:
So here’s a second general method for converting base ten numbers to an arbitrary base b:
Again, the method probably makes more sense if you try it out a few times.
Use the method described above to convert to base three, four, five, and six.
13
Our number system is a western adaptation of the Hindu-Arabic numeral system developed somewhere between the first and fourth centuries AD. However, numbers have been recorded with tally marks throughout history. The Ishango BoneImage of Ishango bone by Ben2 (Own work), CC-BY-SA-3.0 or CC BY-SA 2.5-2.0-1.0], via Wikimedia Commons. from Africa is about 25,000 years old. It’s the lower leg bone from a baboon, and contains tally marks. We know the marks were used for counting because they appear in distinct groups.
This reindeer antlerImage of antler By Ryan Somma from Occoquan, USA [CC BY-SA 2.0], via Wikimedia Commons from France is about 15,000 years old, and also shows clearly grouped tally marks.
Of course, we still use tally marks today!Image of Hanakapiai beach warning sign by God of War at the English language Wikipedia [GFDL or CC-BY-SA-3.0], via Wikimedia Commons.
Base ten numbers (the ones you have probably been using your whole life), and base b numbers (the ones you’ve been learning about in this chapter) are both positional number systems.
A positional number system is one way of writing numbers. It has unique symbols for 1 through b – 1, where b is the base of the system. Modern positional number systems also include a symbol for 0.
The positional value of each symbol depends on its position in the number:
The value of a number is the sum of the positional values of its digits.
In an additive number system, the value of a written number is the sum of the face values of the symbols that make up the number. The only symbol necessary for an additive number system is a symbol for 1, however many additive number systems contain other symbols.
The ancient Romans used a version of an additive number systems. The Romans represented numbers this way:
number | Roman Numeral |
1 | I |
5 | V |
10 | X |
50 | L |
100 | C |
500 | D |
1,000 | M |
So the number 2013 would be represented as MMXIII. This is read as 2,000 (two M’s), one ten (one X), and three ones (three I’s).
For any additive number system very large numbers become impractical to write. To represent the number one million in Roman numerals it would take one thousand M’s!
However, the Roman numerals did have one efficiency advantage: The order of the symbols mattered. If a symbol to the left was smaller than the symbol to the right, it would be subtracted instead of added. So for example nine is represented as IX rather than VIIII.
If you don’t already know how to use Roman numerals, research it a little bit. Then answer these questions.
The earliest positional number systems are attributed to the Babylonians (base 60) and the Mayans (base 20). These positional systems were both developed before they had a symbol or a clear concept for zero. Instead of using 0, a blank space was used to indicate skipping a particular place value. This could lead to ambiguity.
Suppose we didn’t have a symbol for 0, and someone wrote the number
It would be impossible to tell if they mean 23, 203, 2003, or maybe two separate numbers (two and three).
At the time, Roman Numerals dominated Europe, and the official means of calculations was the abacus. Muḥammad ibn Mūsā al-KhwārizmīImage of al-Khwarizmi statue by M. Tomczak [CC BY-SA 3.0], via Wikimedia Commons. described the use of Hindu-Arabic system in his book On the Calculation with Hindu Numerals in 825 CE, but it was not well-known in Europe.
Fibonacci’s book Liber Abaci described the Hindu-Arabic system and its business applications for a European readership. His book was well-received throughout Europe, and it marked the beginning of a reawakening of European mathematics.
The Hindu-Arabic number system is now used nearly exclusively throughout the globe. But many cultures had their own number systems before contact and trade with other countries spread the work of al-Khwārizmī throughout the world.
There is evidence that pre-contact Hawaiians actually used two different number systems. Depending on what they were counting, they might use base 4 instead (or a mixed base-10 and base-4 system). One theory is that certain objects (fish, taro, etc.) were often put in bundles of 4, so were more natural to count by 4’s than by 10’s. The number four also had spiritual significance in Hawaiian culture.
Humans have 5 fingers on each handImage of hand from https://pixabay.com, licensed under CC0 Creative Commons., making base ten a natural choice for counting. But there are 4 gaps between the fingers, meaning that a hand can carry four fish or taro plants or similar objects, making base four a natural choice for some cultures. |
In the mixed base system, instead of powers of 10, numbers are broken down into sums of numbers that look like 4 times a power of 10 (40, 400, 4000, etc.).
1 | ‘ekahi |
2 | ‘elua |
3 | ‘ekolu |
4 | ‘ehā (or kauna) |
5 | ‘elima |
6 | ‘eono |
7 | ‘ehiku |
8 | ‘ewalu |
9 | ‘eiwa |
10 | ‘umi |
11–19 | ‘umi kumamā {kahi, lua, kolu, hā, etc.} |
20 | iwakālua |
21–29 | Iwakālua kumamā {kahi, lua, kolu, hā, etc.} |
30 | kanakolu |
31–39 | kanakolu kumamā {kahi, lua, etc.} |
40 | kanahā |
400 | lau |
4,000 | mano |
40,000 | kini |
400,000 | lehu |
Here are a few examples (refer to the table above for the Hawaiian names of the numbers):
‘ekolu kini, ‘ewalu lau me ‘ekahi
translates to three 40,000’s, eight 400’s, and one;
3 ⋅ 40000 + 8 ⋅ 400 + 1 = 123201
5207 = 1 ⋅ 4000 + 3 ⋅ 400 + 7
would be ‘ekahi mano, ‘ekolu lau me ‘ehiku
Work on the following exercises on your own or with a partner.
1. Translate this Hawaiian number to English and then write it in base ten.
‘ekahi kanahā me kanakolu kumamāiwa
2. Translate this base‐ten number to Hawaiian.
1,573
How is learning about different number systems (including representing numbers in different bases) valuable to you as a future teacher?
14
How do we know if a number is even? What does it mean?
Some number of dots is even if I can divide the dots into pairs, and every dot has a partner.
Some number of dots is odd if, when I try to pair up the dots, I always have a single dot left over with no partner.
The number of dots is either even or odd. It’s a property of the quantity and is doesn’t change when you represent that quantity in different bases.
Which of these numbers represent an even number of dots? Explain how you decide.
Compare your answers to problem 13 with a partner. Then try these together:
You know that you can tell if a base ten number is even just by looking at the ones place. But why is that true? That’s not the definition of an even number. There are a few key ideas behind this handy trick:
(some multiple of ten) + (ones digit)
and two times a whole number is always even.
(some multiple of ten) + (ones digit)
(even number) + (ones digit),
1. Write the numbers zero through fifteen in base seven:
base ten | base seven |
---|---|
0 | |
1 | |
2 | |
3 | |
4 | |
5 | |
6 | |
7 | |
8 | |
9 | |
10 | |
11 | |
12 | |
13 | |
14 | |
15 |
2. Circle all of the even numbers in your list. How do you know they are even?
3. Find a rule: how can you tell if a number is even when it’s written in base seven?
1. Write the numbers zero through fifteen in base four:
base ten | base four |
---|---|
0 | |
1 | |
2 | |
3 | |
4 | |
5 | |
6 | |
7 | |
8 | |
9 | |
10 | |
11 | |
12 | |
13 | |
14 | |
15 |
2. Circle all of the even numbers in your list. How do you know they are even?
3. Find a rule: how can you tell if a number is even when it’s written in base four?
15
Explain what is wrong with writing or .
Convert each base ten number to a base four number. Explain how you did it.
Challenges:
In order to use base sixteen, we need sixteen digits — they will represent the numbers zero through fifteen. We can use our usual digits 0–9, but we need new symbols to represent the digits ten, eleven, twelve, thirteen, fourteen, and fifteen. Here’s one standard convention:
base ten | base sixteen |
---|---|
7 | |
8 | |
9 | |
10 | |
11 | |
12 | |
13 | |
14 | |
15 | |
16 |
How many different symbols would you need for a base twenty-five system? Justify your answer.
All of the following numbers are multiples of three.
All of the following numbers are multiples of five.
Convert each number to the given base.
What bases makes theses equations true? Justify your answers.
What bases makes theses equations true? Justify your answers.
Convert each base four number to a base ten number. Explain how you did it.
Challenges:
Consider this base ten number (I got this by writing the numbers from 1 to 60 in order next to one another):
Can you find two different numbers (not necessarily single digits!) and so that ? Can you find more than one solution? Justify your answers.
16
Jay decides to play with a system that follows a 1←1 rule. He puts one dot into the right-most box. What happens?
Poindexter decides to play with a system that follows the rule 2←3.
1. Describe what this rule does when there are three dots in the right-most box.
2. Draw diagrams or use buttons or pennies to find the codes for the following numbers:
3. Can you find (and explain) any patterns?
Repeat problem 45 for your own rule. Choose two numbers and . For each of the numbers
figure out the code. Look for patterns, and explain them if you can!
III
-S. Gudder
The “Dots and Boxes” approach to understanding operations used in this part (and throughout this book) comes from James Tanton, and is used with his permission. See his development of these and other ideas at http://gdaymath.com/.
17
When learning and teaching about arithmetic, it helps to have mental and physical models for what the operations mean. That way, when you are presented with an unfamiliar problem or a question about why something is true, you can often work it out using the model — this might mean drawing pictures, using physical materials (manipulatives), or just thinking about the model to help you reason out the answer.
Write down your mental models for each of the four basic operations. What do they actually mean? How would you explain them to a second grader? What pictures could you draw for each operation? Think about each one separately, as well as how they relate to each other:
After writing down you own ideas, share them with a partner. Do you and your partner have the same models for each of the operations or do you think about them differently?
Teachers should have lots of mental models — lots of ways to explain the same concept. In this chapter, we’ll look at some different ways to understand the four basic arithmetic operations. First, let’s define some terms:
Counting numbers are literally the numbers we use for counting: 1, 2, 3, 4, 5… These are sometimes called the natural numbers by mathematicians, and they are represented by the symbol .
Whole numbers are the counting numbers together with zero.
Integers include the positive and negative whole numbers, and mathematicians represent these with the symbol . (This comes from German, where the word for “number” is “zählen.”)
We already have a natural model for thinking about counting numbers: a number is a quantity of dots. Depending on which number system you use — Roman numerals, base ten, binary, etc. — you might write down the number in different ways. But the quantity of dots is a counting number, however you write it down.
18
For now, we’ll focus on the base-10 system. Here’s how we think about the number 273 in that system:
And here is the number 512:
We can add these in the natural way: just combine the piles of dots. Since they’re already in place-value columns, we can combine dots from the two numbers that are in the same place-value box.
We can count up the answer: there are 7 dots in the hundreds box, 8 dots in the tens box, and 5 dots in the ones box.
And saying out the long way we have:
This gives the answer: 785.
Let’s do another one. Consider 163+489.
And this is absolutely correct:
The answer is 5 | 14 | 12, which we might try to pronounce as “five hundred and fourteeny-tenty twelvety.” The trouble with this answer is that most of the rest of the world wouldn’t understand what we are talking about.
Since this is a base 10 system, we can do some explosions.
The answer is “six hundred fifty two.” Okay, the world can understand this one!
Solve the following exercises by thinking about the dots and boxes. (You can draw the pictures, or just imagine them.) Then translate the answer into something the rest of the world can understand.
Use the dots and boxes technique to solve these problems. Do not covert to base 10! Try to work directly in the base given. It might help to actually draw the pictures.
Let’s go back to the example 163+489. Some teachers don’t like writing:
They prefer to teach their students to start with the 3 and 9 at the end and sum those to get 12. This is of course correct — we got 12 as well.
But they don’t want students to write or think “twelvety,” so they have their students write something like this:
This can seem completely mysterious. What’s really going on? They are exploding ten dots, of course!
Now we carry on with the problem and add the tens. Students are taught to write:
But what this means is better shown in this next picture. Notice the “exploded” (or regrouped) dot at the very top, which is added to the tens box in the answer.
And now we finish the problem by combining the dots in the hundreds boxes:
In the standard algorithm, we work from right to left, doing the “explosions” as we go along. This means that we start adding at the ones place and work towards the left-most place value, “carrying” digits that come from the explosions. (This is really not carrying; a better term for it is regrouping. Ten ones become one ten. Ten tens become one hundred. And so on.)
In the dots and boxes method, we add in any direction or order we like and then we do the explosions at the end.
19
To model addition, we started with two collections of dots (two numbers), and we combined them to form one bigger collection. That’s pretty much the definition of addition: combining two collections of objects. In subtraction, we start with one collection of dots (one number), and we take some dots away.
Suppose we want to find 376-125 in the dots and boxes model. We start with the representation of 376:
Since we want to “take away” 125, that means:
So the answer is:
And saying it out the long way we have:
Let’s try a somewhat harder example: 921-551. We start with the representation of 921:
Since we want to “take away” 551, that means we take away five dots from the hundreds box, leaving four dots.
Now we want to take away five dots from the tens box, but we can’t do it! There are only two dots there. What can we do? Well, we still have some hundreds, so we can “unexplode” a hundreds dot, and put ten dots in the tens box instead. Then we’ll be able to take five of them away, leaving seven.
(Notice that we also have one less dot in the hundreds box; there’s only three dots there now.)
Now we want to take one dot from the ones box, and that leaves no dots there.
So the answer is:
Solve the following exercises by thinking about dots and boxes. (You can draw pictures, or just imagine them.)
Use the dots and boxes technique to solve these problems. Do not covert to base 10! Try to work directly in the base given. It might help to actually draw the pictures.
Just like in addition, the standard algorithm for subtraction requires you to work from right to left, and “borrow” (this is really regrouping!) whenever necessary. Notice that in the dots and boxes approach, you don’t need to go in any particular order when you do the subtraction. You just “unexplode” the dots as necessary when computing.
Here’s how the standard algorithm looks with the dots and boxes model for 921 – 551: Start with 921 dots.
Then take away one dot from the ones box.
Now we want to take away five dots from the tens box. But there aren’t five dots there. So we “unexplode” one of the hundreds dots to get more tens:
In the standard algorithm, we show the unxplosion as a regrouping, subtracting one from the hundreds place of 921 and adding ten to the tens place. So we are rewriting
Finally, we want to take away five from the eight dots left in the hundreds column.
20
Jenny was asked to compute . She wrote:
Can you adapt Jenny’s method to solve these problems? Write your answers in base eight. Try to work directly in base eight rather than converting to base 10 and back again!
Jenny might have been thinking about multiplication as repeated addition. If we have some number and we multiply that number by 4, what we mean is:
If we take the number 243192 and add it to itself four times using the “combining method,” we get
Notice that we have used both × and · to represent multiplication. It’s a bit awkward to use × when you’re also using variables. Is it the letter x? Or the multiplication symbol ×? It can be hard to tell! In this case, the symbol · is more clear.
We can even simplify the notation further, writing 4N instead of 4 · N. But of course we only do that when we are multiplying variables by some quantity. (We wouldn’t want 34 to mean 3 · 4, would we?)
Here is a strange addition table. Use it to solve the following problems. Important: Don’t try to assign numbers to A, B, and C. Solve the problems just using what you know about the operations!
A + B B + C 2A 5C 3A + 4B
How does an addition table help you solve multiplication problems like 5C?
21
Suppose you are asked to compute 3906 ÷ 3. One way to interpret this question (there are others) is:
“How many groups of 3 fit into 3906?”
In the quotative model of division, you are given a dividend (here it is 3906), and you are asked to split it into equal-sized groups, where the size of the group is given by the divisor (here it is 3).
In our dots and boxes model, the dividend 3906 looks like this:
and three dots looks like this:
So we are really asking:
“How many groups of fit into the picture of 3906?”
There is one group of 3 at the thousands level, and three at the hundreds level, none at the tens level, and two at the ones level.
Notice what we have in the picture:
• One group of 3 in the thousands box.
• Three groups of 3 in the hundreds box.
• Zero groups of 3 in the tens box.
• Two groups of 3 in the ones box.
This shows that 3 goes into 3906 one thousand, three hundreds and two ones times. That is,
Let’s try a harder one! Consider 402 ÷ 3. Here’s the picture:
We are still looking for groups of three dots:
There is certainly one group at the 100’s level.
and now it seems we are stuck there are no more groups of three!
What can we do now? Are we really stuck? Can you finish the division problem?
Here are the details worked out for 402 ÷ 3. But don’t read this until you’ve thought about it yourself!
Since each dot is worth ten dots in the box to the right we can write:
Now we can find more groups of three:
There is still a troublesome extra dot. Let’s unexplode it too
This gives us more groups of three:
In the picture we have:
• One group of 3 in the hundreds box.
• Three groups of 3 in the tens box.
• Four groups of 3 in the ones box.
Finally we have the answer!
Solve each of these exercises using the dots and boxes method:
62124 ÷ 3 61230 ÷ 5
Let’s turn up the difficulty a notch. Consider 156 ÷ 12. Here we are looking for groups of 12 in this picture:
What does 12 look like? It can be twelve dots in a single box:
But most often we would write 12 this way, as a ten and 2 ones:
We certainly see some of these in the picture. There is certainly one at the tens level:
Note: With an unexplosion this would be twelve dots in the tens box, so we mark one group of 12 above the tens box.
We also see three groups of twelve ones:
So in the picture we have:
• One group of 12 dots in the tens box.
• Three groups of 12 dots in the ones box.
That means
156 ÷ 12 = 13.
Use the dots and boxes model to compute each of the following:
13453 ÷ 11
4853 ÷ 23
214506 ÷ 102
Remember that base five numbers are in a 1 ← 5 dots-and-boxes system. What are the place values in the 1 ← 5 system? Fill in the blanks:
2130 ÷ 10
41300 ÷ 100
We used dots and boxes to show that 402 ÷ 3 = 134.
In elementary school, you might have learned to solve this division problem by using a diagram like the following:
At first glance this seems very mysterious, but it is really no different from the dots and boxes method. Here is what the table means.
To compute 402 ÷ 3, we first make a big estimation as to how many groups of 3 there are in 402. Let’s guess that there are 100 groups of three.
How much is left over after taking away 100 groups of 3? We subtract to find that there is 102 left.
How many groups of 3 are in 102? Let’s try 30:
How many are left? There are 12 left and there are four groups of 3 in 12.
That accounts for entire number 402. And where do we find the final answer? Just add the total count of groups of three that we tallied:
402 ÷ 3 = 100 + 30 + 4 = 134.
We saw that 402 is evenly divisible by 3: 402 ÷ 3 = 134. This means that 403, one more, shouldn’t be divisible by three. It should be one dot too big.
Do we see the extra dot if we compute 402 ÷ 3 with dots and boxes?
Yes we do! We have one dot left at the end that can’t be divided. This is how it looks in the standard algorithm.
In school, we say that we have a remainder of one and sometimes write:
But what does that really mean? It means that we have 134 groups of three with one dot left over. So
Let’s try another one: 263 ÷ 12. Here’s what we have:
And we are looking for groups like this:
Here goes!
Unexploding won’t help any further and we are indeed left with one remaining dot in the tens position and a dot in the ones position. We have 21 groups of twelve, and a remainder of eleven.
5210 ÷ 4
4857 ÷ 23
31533 ÷ 101
22
Another way we often think about numbers is as abstract quantities that can be measured: length, area, and volume are all examples.
In a measurement model, you have to pick a basic unit. The basic unit is a quantity — length, area, or volume — that you assign to the number one. You can then assign numbers to other quantities based on how many of your basic unit fit inside.
For now, we’ll focus on the quantity length, and we’ll work with a number line where the basic unit is already marked off.
Imagine a person — we’ll call him Zed — who can stand on the number line. We’ll say that the distance Zed walks when he takes a step is exactly one unit.
When Zed wants to add or subtract with whole numbers on the number line, he always starts at 0 and faces the positive direction (towards 1). Then what he does depends on the calculation.
If Zed wants to add two numbers, he walks forward (to the right of the number line) however many steps are indicated by the first number (the first addend). Then he walks forward (to your right on the number line) the number of steps indicated by the second number (the second addend). Where he lands is the sum of the two numbers.
If Zed wants to add 3 + 4, he starts at 0 and faces towards the positive numbers. He walks forward 3 steps, then he walks forward 4 more steps.
Zed ends at the number 7, so the sum of 3 and 4 is 7. 3 + 4 = 7. (But you knew that of course! The point right now is to make sense of the number line model.)
When Zed wants to subtract two numbers, he he walks forward (to the right on the number line) however many steps are indicated by the first number (the minuend). Then he walks backwards (to the left on the number line) the number of steps indicated by the second number (the subtrahend). Where he lands is the difference of the two numbers.
If Zed wants to subtract 11 – 3, he starts at 0 and faces the positive numbers (the right side of the number line). He walks forward 11 steps on the number line, then he walks backwards 3 steps.
Zed ends at the number 8, so the difference of 11 and 3 is 8. 11 – 3 = 8. (But you knew that!)
4 + 5 6 + 9 10 – 7 8 – 1
6 – 9 1 – 7 4 – 11 0 – 1
Since multiplication is really repeated addition, we can adapt our addition model to become a multiplication model as well. Let’s think about 3 × 4. This means to add four to itself three times (that’s simply the definition of multiplication!):
3 × 4 = 4 + 4 + 4.
So to multiply on the number line, we do the process for addition several times.
To multiply two numbers, Zed starts at 0 as always, and he faces the positive direction. He walks forward the number of steps given by the second number (the second factor). He repeats that process the number of times given by the first number (the first factor). Where he lands is the product of the two numbers.
If Zed wants to multiply 3 × 4, he can think of it this way:
Zed starts at 0, facing the positive direction. The he repeats this three times: take four steps forward.
He ends at the number 12, so the product of 3 and 4 is 12. That is, 3 × 4 = 12.
Remember our quotative model of division: One way to interpret 15 ÷ 5 is:
How many groups of 5 fit into 15?
Thinking on the number line, we can ask it this way:
Zed takes 5 steps at a time. If Zed lands at the number 15, how many times did he take 5 steps?
To calculate a division problem on the number line, Zed starts at 0, facing the positive direction. He walks forward the number of steps given by the second number (the divisor). He repeats that process until he lands at the first number (the dividend). The number of times he repeated the process gives the quotient of the two numbers.
If Zed wants to compute 15 ÷ 5, he can think of it this way:
He starts at 0, facing the positive direction.
Since he repeated the process three times, we see there are 3 groups of 5 in 15. So the quotient of 15 and 5 is 3. That is, 15 ÷ 5 = 3.
2 × 5 7 × 1 10 ÷ 2 6 ÷ 1
4 × 0 0 × 5 3 × (-2) 2 × (-1)
0 ÷ 2 0 ÷ 10 3 ÷ 0 5 ÷ 0
23
So far we have focused on a linear measurement model, using the number line. But there’s another common way to think about multiplication: using area.
For example, suppose our basic unit is one square:
We can picture 4 × 3 as 4 groups, with 3 squares in each group, all lined up:
But we can also picture them stacked up instead of lined up. We would have 4 rows, with 3 squares in each row, like this:
So we can think about 4 × 3 as a rectangle that has length 3 and width 4. The product, 12, is the total number of squares in that rectangle. (That is also the area of the rectangle, since each square was one unit!)
Vera drew this picture as a model for 15 × 17. Use her picture to help you compute 15 × 17. Explain your work.
Draw pictures like Vera’s for each of these multiplication exercises. Use your pictures to find the products without using a calculator or the standard algorithm.
23 × 37 8 × 43 371 × 42
How were you taught to compute 83 × 27 in school? Were you taught to write something like the following?
Or maybe you were taught to put in the extra zeros rather than leaving them out?
This is really no different than drawing the rectangle and using Vera’s picture for calculating!
23 × 14 106 × 21 213 × 31
Here’s an unusual way to perform multiplication. To compute 22 × 13, for example, draw two sets of vertical lines, the left set containing two lines and the right set two lines (for the digits in 22) and two sets of horizontal lines, the upper set containing one line and the lower set three (for the digits in 13).
There are four sets of intersection points. Count the number of intersections in each and add the results diagonally as shown:
The answer 286 appears!
There is one possible glitch as illustrated by the computation 246 × 32:
Although the answer 6 thousands, 16 hundreds, 26 tens, and 12 ones is absolutely correct, one needs to carry digits and translate this as 7,872.
In the 1500s in England, students were taught to compute multiplication using following galley method, now more commonly known as the lattice method.
To multiply 43 and 218, for example, draw a 2 × 3 grid of squares. Write the digits of the first number along the right side of the grid and the digits of the second number along the top.
Divide each cell of the grid diagonally and write in the product of the column digit and row digit of that cell, separating the tens from the units across the diagonal of that cell. (If the product is a one digit answer, place a 0 in the tens place.)
To get the answer, add the entries in each diagonal, carrying tens digits over to the next diagonal if necessary. In our example, we have
24
So far, you have seen a couple of different models for the operations: addition, subtraction, multiplication, and division. But we haven’t talked much about the operations themselves — how they relate to each other, what properties they have that make computing easier, and how some special numbers behave. There’s lots to think about!
The goal in this section is to use the models to understand why the operations behave according to the rules you learned back in elementary school. We’re going to keep asking ourselves “Why does it work this way?”
Each of these models lends itself to thinking about the operation in a slightly different way. Before we really dig in to thinking about the operations, discuss with a partner:
We defined addition as combining two quantities and subtraction as “taking away.” But in fact, these two operations are intimately tied together. These two questions are exactly the same:
27 – 13 = ____ 27 = 13 + _____.
More generally, for any three whole numbers a, b, and c, these two equations express the same fact. (So either both equations are true or both are false. Which is the case depends on the values you choose for a, b, and c!)
c – b = a c = a + b.
In other words, we can think of every subtraction problem as a “missing addend” addition problem. Try it out!
Here is a strange addition table. Use it to solve the following problems. Justify your answers. Important: Don’t try to assign numbers to A, B, and C. Solve the problems just using what you know about the operations!
A + C B + C A – C C – A A – A B – C
How does an addition table help you solve subtraction problems?
We defined multiplication as repeated addition and division as forming groups of equal size. But in fact, these two operations are also tied together. These two questions are exactly the same:
27 ÷ 3 = _____ 27 = _____ × 3.
More generally, for any three whole numbers a, b, and c, these two equations express the same fact. (So either both equations are true or both are false. Which is the case depends on the values you choose for a, b, and c!)
c ÷ b = a c = a · b.
In other words, we can think of every division problem as a “missing factor” multiplication problem. Try it out!
Rewrite each of these division questions as a “missing factor” multiplication question. Which ones can you solve and which can you not solve? Explain your answers.
9 ÷ 3 100 ÷ 25 0 ÷ 3 9 ÷ 0 0 ÷ 0
Here’s a multiplication table.
C × D C × A A × A C ÷ D D ÷ C D ÷ E
A ÷ C A ÷ D D ÷ A A ÷ A
How does a multiplication table help you solve division (and exponentiation) problems?
Throughout this course, our focus is on explanation and justification. As teachers, you need to know what is true in mathematics, but you also need to know why it is true. And you will need lots of ways to explain why, since different explanations will make sense to different students.
Arithmetic Fact: a + b = c and c – b = a are the same mathematical fact.
Why is this not a good explanation?
“I can check that this is true! For example, 2+3 = 5 and 5 – 3 = 2. And 3 + 7 = 10 and 10 – 7 = 3. It works for whatever numbers you try.”
a + b = c and c – b = a are the same mathematical fact.
First we’ll use the definition of the operations.
Suppose we know c – b = a is true. Subtraction means “take away.” So
c – b = a
means we start with quantity c and take away quantity b, and we end up with quantity a. Start with this equation, and imagine adding quantity b to both sides.
On the left, that mans we started with quantity c, took away b things, and then put those b things right back! Since we took away some quantity and then added back the exact same quantity, there’s no overall change. We’re left with quantity c.
On the right, we would be combining (adding) quantity a with quantity b. So we end up with: c = a + b.
On the other hand, suppose we know the equation a + b = c is true. Imagine taking away (subtracting) quantity b from both sides of this equation: a + b = c.
On the left, we started with a things and combined that with b things, but then we immediately take away those b things. So we’re left with just our original quantity of a.
On the right, we start with quantity c and take away b things. That’s the very definition of c – b. So we have the equation:
a = c – b.
Let’s use the measurement model to come up with another explanation.
The equation a + b = c means Zed starts at 0, walks forward a steps, and then walks forward b steps, and he ends at c.
If Zed wants to compute c – b, he starts at 0, walks forward c steps, and then walks backwards b steps. But we know that to walk forward c steps, he can first walk forward a steps and then walk forward b steps. So Zed can compute c – b this way:
The last two sets of steps cancel each other out, so Zed lands back at a. That means c – b = a.
On the other hand, the equation c – b = a means that Zed starts at 0, walks forward c steps, then walks backwards b steps, and he ends up at a.
If Zed wants to compute a + b, he starts at 0, walks forward a steps, and then walks forwards b additional steps. But we know that to walk forward a steps, he can first walk forward c steps and then walk backwards b steps. So Zed can compute a + b this way:
The last two sets of steps cancel each other out, so Zed lands back at c. That means a + b = c.
c ÷ b = a is the same fact as c = a × b.
You probably know several properties of addition, but you may never have stopped to wonder: Why is that true?! Now’s your chance! In this section, you’ll use the definition of the operations of addition and subtraction and the models you’ve learned to explain why these properties are always true.
Here are the three properties you’ll think about:
For each of the properties, we don’t want to confuse these three ideas:
Notice that examples and explanations are not the same! It’s also very important not to confuse the definition of a property with the reason it is true!
These properties are all universal statements — statements of the form “for all,” “every time,” “always,” etc. That means that to show they are true, you either have to check every case or find a reason why it must be so.
Since there are infinitely many whole numbers, it’s impossible to check every case. You’d never finish! Our only hope is to look for general explanations. We’ll work out the explanation for the first of these facts, and you will work on the others.
Addition of whole numbers is commutative.
When I add two whole numbers, the order I add them doesn’t affect the sum.
For any two whole numbers a and b,
a + b = b + a.
Now we need a justification. Why is addition of whole numbers commutative?
Let’s think about addition as combining two quantities of dots.
• That means a + b = b + a.
We can also use the measurement model to explain why a + b = b + a no matter what numbers we choose for a and b. Imagine taking a segment of length a and combining it linearly with a segment of length b. That’s how we get a length of a + b.
But if we just rotate that segment so it’s upside down, we see that we have a segment of length b combined with a segment of length a, which makes a length of b + a.
But of course it’s the same segment! We just turned it upside down! So the lengths must be the same. That is, a + b = b + a.
Your turn! You’ll answer the question, “Why is addition of whole numbers associative?”
Property: Addition of whole numbers is associative.
What it Means (words): When I add three whole numbers in a given order, the way I group them (to add two at a time) doesn’t affect the sum.
What it Means (symbols): For any three whole numbers a, b, and c,
(a + b) + c = a + (b + c).
Property: The number 0 is an identity for addition of whole numbers.
What it Means (words): When I add any whole number to 0 (in either order), the sum is the very same whole number I added to 0.
What it Means (symbols): For any whole numbers n,
n + 0 = n and 0 + n = n.
Since addition and subtraction are so closely linked, it’s natural to wonder if subtraction has some of the same properties as addition, like commutativity and associativity.
Justin asked if the operation of subtraction is commutative. That would mean that the difference of two whole numbers doesn’t depend on the order in which you subtract them.
In symbols: for every choice of whole numbers a and b we would have a – b = b – a.
Jared says that subtraction is not commutative since 4 – 3 = 1, but 3 – 4 ≠ 1. (In fact, 3 – 4 = -1.)
Since the statement “subtraction is commutative” is a universal statement, one counterexample is enough to show it’s not true. So Jared’s counterexample lets us say with confidence:
Subtraction is not commutative.
Can you find any examples of whole numbers a and b where a – b = b – a is true? Explain your answer.
Lyle asked if the operation of subtraction is associative.
Jess asked if the number 0 is an identity for subtraction.
Now we’re going to turn our attention to familiar properties of multiplication and division, with the focus still on explaining why these properties are always true.
Here are the four properties you’ll think about:
For each of the properties, remember to keep straight:
Once again, it’s important to distinguish between examples and explanations. They are not the same! Since there are infinitely many whole numbers, it’s impossible to check every case, so examples will never be enough to explain why these properties hold. You have to figure out reasons for these properties to hold, based on what you know about the operations.
We’ll work out the explanation for the last of these facts, and you will work on the others.
The number 1 is an identity for multiplication of whole numbers.
When I multiply a number by 1 (in either order), the product is that number.
For any whole number m,
m × 1 = m and 1 × m = m.
1 × 5 = 5, 19 × 1 = 19, and 1 × 1 = 1.
Why does the number 1 act this way with multiplication?
Let’s think first about the definition of multiplication as repeated addition:
So we see that m × 1 = m for any whole number m.
We can also use the number line model to create a justification. If Zed calculates 1×m, he will start at 0 and face the positive direction. He will then take m steps forward, and he will do it just one time. So he lands at m, which means 1 × m = m.
If Zed calculates m × 1, he starts at 0 and faces the positive direction. Then he takes one step forward, and he repeats that m times. So he lands at m. We see that m × 1 = m.
In the area model, m × 1 represents m rows with one square in each row. That makes a total of m squares. So m × 1 = m.
Similarly, 1 × m represents one row of m squares. That’s also a total of m squares. So 1 × m = m.
The example presented several different explanations. Do you think one is more convincing than the others? Or more clear and easier to understand?
Property: Multiplication whole numbers is commutative.
What it Means (words): When I multiply two whole numbers, switching the order in which I multiply them does not affect the product.
What it Means (symbols): For any two whole numbers a and b,
a · b = b · a.
Property: Multiplication of whole numbers is associative.
What it Means (words): When I multiply three whole numbers in a given order, the way I group them (to multiply two at a time) doesn’t affect the product.
What it Means (symbols): For any three whole numbers a, b, and c,
(a · b) · c = a · (b · c).
Property: Multiplication distributes over addition.
What it means: The distributive law for multiplication over addition is a little hard to state in words, so we’ll jump straight to the symbols. For any three whole numbers x, y, and z:
x · (y + z) = x · y + x · z.
Examples: We actually did calculations very much like the examples above, when we looked at the area model for multiplication.
8 · (23) = 8 · (20 + 3) = 8 · 20 + 8 · 3 = 160 + 24 = 184
5 · (108) = 5 · (100 + 8) = 5 · 100 + 5 · 8 = 500 + 40 = 540
Which of the following pictures best represents the distributive law in the equation
Explain your choice.
(a) | |
(b) | |
(c) | |
(d) | |
(e) |
Use the distributive law to easily compute each of these in your head (no calculators!). Explain your solutions.
Use one of our models for multiplication and addition to explain why the distributive rule works every time.
It’s natural to wonder which, if any, of these properties also hold for division (since you know that the operations of multiplication and division are connected).
If division were associative, then for any choice of three whole numbers a, b, and c, we would have
a ÷ (b ÷ c) = (a ÷ b) ÷ c.
Remember, the parentheses tell you which two numbers to divide first.
Let’s try the example a = 9, b = 3, and c = 1. Then we have:
9 ÷ (3 ÷ 1) = 9 ÷ 3 = 3
and
(9 ÷ 3) ÷ 1 = 3 ÷ 1 = 3.
So is it true? Is division associative? Well, we can’t be sure. This is just one example. But “division is associative” is a universal statement. If it’s true, it has to work for every possible example. Maybe we just stumbled on a good choice of numbers, but it won’t always work.
Let’s keep looking. Try a = 16, b = 4, and c = 2.
16 ÷ (4 ÷ 2) = 16 ÷ 2 = 8
and
(16 ÷ 4) ÷ 2 = 4 ÷ 2 = 2.
That’s all we need! A single counterexample lets us conclude:
Division is not associative.
What about the other properties? It’s your turn to decide!
You probably know another property of multiplication that hasn’t been mentioned yet:
If I multiply any number times 0 (in either order), the product is 0. This is sometimes called the zero property of multiplication. Notice that the zero property is very different from the property of being an identity!
1. Write what the zero property means using both words and symbols:
For every whole number n . . .
2. Give at least three examples of the zero property for multiplication.
3. Use one of our models of multiplication to explain why the zero property holds.
5 ÷ 0 0 ÷ 5 7 ÷ 0 0 ÷ 7 0 ÷ 0
In elementary school, students are often encouraged to memorize “four fact families,” for example:
2 + 3 = 5 5 – 3 = 2
3 + 2 = 5 5 – 2 = 3
Here’s a different “four fact family”:
2 · 3 = 6 6 ÷ 3 = 2
3 · 2 = 6 6 ÷ 2 = 3
So far we’ve been thinking about division in what’s called the quotative model. In the quotative model, we want to make groups of equal size. We know the size of the group, and we ask how many groups. For example, we think of 20 ÷ 4 as:
How many groups of 4 are there in a group of 20?
Thinking about four fact families, however, we realize we can turn the question around a bit. We could think about the partitive model of division. In the partitive model, we want to make an equal number of groups. We know how many groups, and we ask the size of the group. In the partitive model, we think of 20 ÷ 4 as:
20 is 4 groups of what size?
When we know the original amount and the number of parts, we use partitive division to find the size of each part.
When we know the original amount and the size of each part, we use quotative division to find the number of parts.
Here are some examples in word problems:
For each word problem below:
Write your own word problems: Write one partitive division problem and one quotative division problem. Choose your numbers carefully so that the answer works out nicely. Be sure to solve your problems!
Why think about these two models for division? You won’t be teaching the words partitive and quotative to your students. But recognizing the two kinds of division problems (and being able to come up with examples of each) will make you a better teacher.
It’s important that your students are exposed to both ways of thinking about division, and to problems of both types. Otherwise, they may think about division too narrowly and not really understand what’s going on. If you understand the two kinds of problems, you can more easily diagnose and remedy students’ difficulties.
Most of the division problems we’ve looked at so far have come out evenly, with no remainder. But of course, that doesn’t always happen! Sometimes, a whole number answer makes sense, and the context of the problem should tell you which whole number is the right one to choose.
What is 43 ÷ 4?
We can think about division with remainder in terms of some of our models for operations. For example, we can calculate that 23 ÷ 4 = 5 R3. We can picture it this way:
23 ÷ 4 = 5 R3 and 23 = 5 · 4 + 3.
40 ÷ 12 59 ÷ 10 91 ÷ 16
25
Anu refuses to tell anyone if she is working in a 1←10 system, or a 1←5 system, or any other system. She makes everyone call it a 1 ← x system but won’t tell anyone what x stands for.
We know that boxes in a 1←10 have values that are powers of ten: 1, 10, 100, 1000, 10000…
And boxes in a 1←5 system are powers of five: 1, 5, 25, 125, 625…
So Anu’s system, whatever it is, must be powers of x:
When Anu writes she must mean:
And when she writes she means:
Anu decides to compute .
She obtains:
Did it work?
Anu later tells use that she really was thinking of a 1←10 system so that x does equal ten. Then her number 2556 really was two thousand, five hundred and fifty six and 12 really was twelve. Her statement:
is actually 2556 ÷ 12 = 213.
Uh Oh! Anu has changed her mind. She now says she was thinking of a 1←11 system.
Now means .
Similarly, means , and means .
So Anu’s computation is actually the (base 10) statement:
3328 ÷ 13 = 256.
26
Compute the following using dots and boxes:
64212 ÷ 3
44793 ÷ 21
6182 ÷ 11
99916131 ÷ 31
637824 ÷ 302
2125122 ÷ 1011
1. Make a base six addition table.
2. Use the table to solve these subtraction problems.
Do these calculations in base four. Don’t translate to base 10 and then calculate there — try to work in base four.
1. Make a base five multiplication table.
2. Use the table to solve these division problems.
Write the rest of this four fact family.
Write the rest of this four fact family.
Notes: In part 2, “O” represents the letter “oh,” not the digit zero.
Here’s another AlphaMath problem.
Find all solutions to this AlphaMath problem in base 9.
Notes: Even though this is two calculations, it is a single problem. All T’s in both calculations represent the same digit, all B’s represent the same digit, and so on.
Remember that “O” represents the letter “oh” and not the digit zero, and that two and three digit numbers never start with the digit zero
This is a single AlphaMath problem. (So all G’s represent the same digit. All A’s represent the same digit. And so on.)
Solve the problem in base 6.
A perfect square is a number that can be written as or (some number times itself).
Geoff spilled coffee on his homework. The answers were correct. Can you determine the missing digits and the bases?
Which of the following models represent the same multiplication problem? Explain your answer.
(a) | |
(b) | |
(c) | |
(d) |
Show an area model for each of these multiplication problems. Write down the standard computation next to the area model and see how it compares.
20 × 33 24 × 13 17 × 11
Suppose the 2 key on your calculator is broken. How could you still use the calculator compute these products? Think about what properties of multiplication might be helpful. (Write out the calculation you would do on the calculator, not just the answer.)
1592 × 3344 2008 × 999 655 × 525
Today is Jennifer’s birthday, and she’s twice as old as her brother. When will she be twice as old as him again? Choose the best answer and justify your choice.
Identify each problem as either partitive or quotative division and say why you made that choice. Then solve the problem.
Use the digits 1 through 9. Use each digit exactly once. Fill in the squares to make all of the equations true.
IV
—Leo Tolstoy
The “Pies Per ChildPie image by Claus Ableiter (Own work) [GFDL, CC-BY-SA-3.0 or CC BY-SA 2.5-2.0-1.0], via Wikimedia Commons” approach to fractions used in this part comes from James Tanton, and is used with his permission. See his development of these and other ideas at http://gdaymath.com/.
27
Fractions are one of the hardest topics to teach (and learn!) in elementary school. What is the reason for this? In this part of the book, will try to provide you with some insight about this (as well as some better ways for understanding, teaching, and learning about fractions). But for now, think about what makes this topic so hard.
You may have struggled learning about fractions in elementary school. Maybe you still find them confusing. Even if you were one of the lucky ones who did not struggle when learning about fractions, you probably had friends who did struggle.
With a partner, talk about why this is. What is so difficult about understanding fractions? Why is the topic harder than other ones we tackle in elementary schools?
Remember that teachers should have lots of mental models — lots of ways to explain the same concept. In this chapter, we will look at some different ways to understand the idea of fractions as well as basic operations on them.
28
One of the things that makes fractions such a difficult concept to teach and to learn is that you have to think about them in a lot of different ways, depending on the problem at hand. For now, we are going to think of a fraction as the answer to a division problem.
Suppose 6 pies are to be shared equally among 3 children. This yields 2 pies per kid. We write
The fraction is equivalent to the division problem . It represents the number of pies one whole child receives when three kids share six pies equally.
In the same way …
This final example is actually saying something! It also represents how fractions are usually taught to students:
If one pie is shared equally between two kids, then each child receives a portion of a pie which we choose to call “half.”
Thus students are taught to associate the number “ ” to the picture .
In the same way, the picture is said to represent “one-third,” that is, . (And this is indeed the amount of pie an individual child would receive if one pie is shared among three.)
The picture is called “one-fifth” and is indeed , the amount of pie an individual receives if three pies are shared among five children.
And the picture is called “three-fifths” to represent , the amount of pie an individual receives if three pies are shared among five children.
Carefully explain why this is true: If five kids share three pies equally, each child receives an amount that looks like this: .
Your explanation will probably require both words and pictures.
Work on the following exercises on your own or with a partner.
1. Draw a picture associated with the fraction .
2. Draw a picture associated with the fraction . Is your picture really the amount of pie an individual would receive if three pies are shared among seven kids? Be very clear on this!
3. Let’s work backwards! Here’s the answer to a division problem:
This represents the amount of pie an individual kid receives if some number of pies is shared among some number of children. How many pies? How many children? How can you justify your answers?
4. Here’s another answer to a division problem:
How many pies? How many children? How can you justify your answers?
5. Here is another answer to a division problem:
How many pies? How many children? How can you justify your answers?
6. Leigh says that “ is three times as big as .” Is this right? Explain your answer.
7. Draw a picture for the answer to the division problem . Describe what you notice about the answer.
8. Draw a picture for the answer to the division problem . Describe what you notice about the answer.
9. What does the division problem represent? How much pie does an individual child receive?
10. What does the division problem represent? How much pie does an individual child receive?
11. What does the division problem represent? How much pie does an individual child receive?
12. Here is the answer to another division problem. This is the amount of pie an individual child receives:
How many pies were in the division problem? How many kids were in the division problem? Justify your answers.
13. Here is the answer to another division problem. This is the amount of pie an individual child receives:
How many pies were in the division problem? How many kids were in the division problem? Justify your answers
14. Many teachers have young students divide differently shaped pies into fractions. For example, a hexagonal pie is good for illustrating the fractions:
Some rectangular pies are distributed to some number of kids. This picture represents the amount of pie an individual child receives. The large rectangle represents one whole pie.
How many pies? How many kids? Carefully justify your answers!
In our model, a fraction represents the amount of pie an individual child receives when pies are shared equally by kids.
For a fraction , the top number (which, for us, is the number of pies) is called the numerator of the fraction, and the bottom number (the number of kids), is called the denominator of the fraction.
Most people insist that the numerator and denominator each be whole numbers, but they do not have to be.
To understand why the numerator and denominator need not be whole numbers, we must first be a little gruesome. Instead of dividing pies, let’s divide kids! Here is one child:
represent?
This means assigning one pie to each “group” of half a child. So how much would a whole child receive? Well, we would have a picture like this:
The whole child gets two pies, so we have:
Draw pictures for these problems if it helps!
represent? Justify your answer using the “Pies Per Child Model.”
Justify your answer.
represents the number 10. (How much pie is given to half a kid? To a whole kid?)
Justify your answer.
A fraction with a numerator smaller than its denominator is called (in school math jargon) a proper fraction. For example, is “proper.”
A fraction with numerator larger than its denominator is called (in school math jargon) an improper fraction. For example, is “improper.” (In the 1800’s, these fractions were called vulgar fractions.)
For some reason, improper fractions are considered, well, improper by some teachers. So students are often asked to write improper fractions as a combination of a whole number and a proper fraction (often called “mixed numbers”). Despite their name and these prejudices, improper fractions are useful nonetheless!
With a mixed number, you have a good sense of the overall size of the number: “a little more than five,” or “a bit less than 17.” But it is often easier to do calculations with improper fractions (why do you think that is?).
If seven pies are shared among three kids, then each kid will certainly receive two whole pies, leaving one pie to share among the three children.
Thus, equals plus . People write:
and call the result a mixed number. One can also write:
which is what really means. But most people choose to omit the plus sign.
If 4 children share 23 pies, we can give them each 5 whole pies. That uses 20 pies, and there are 3 pies left over.
Those three pies are still to be shared equally by the four kids. We have:
For fun, let us write the number 2 as a fraction with denominator 5:
So:
We have written the mixed number as the improper fraction .
Students are often asked to memorize the names “proper fractions,” “improper fractions,” and “mixed number” so that they can follow directions on tests and problem sets.
But, to a mathematician, these names are not at all important! There is no “correct” way to express an answer (assuming, that the answer is mathematically the right number). We often wish to express our answer in a simpler form, but sometimes the context will tell you what form is “simple” and what form is more complicated.
As you work on problems in this chapter, decide for yourself which type of fraction would be best to work with as you do your task.
29
We know that is the answer to a division problem:
represents the amount of pie an individual child receives when pies are shared equally by children.
What happens if we double the number of pie and double the number of kids? Nothing! The amount of pie per child is still the same:
For example, as the picture shows, and both give two pies for each child.
And tripling the number of pies and the number of children also does not change the final amount of pies per child, nor does quadrupling each number, or one trillion-billion-tupling the numbers!
This leads us to want to believe:
(at least for positive whole numbers ).
We say that the fractions and are equivalent.
For example,
yields the same result as
and as
Write down a lot of equivalent fractions for , for , and for 1.
is the same problem as:
Most people say we have cancelled or taken a common factor 4 from the numerator and denominator.
Mathematicians call this process reducing the fraction to lowest terms. (We have made the numerator and denominator smaller, in fact as small as we can make them!)
Teachers tend to say that we are simplifying the fraction. (You have to admit that does look simpler than .)
As another example, can certainly be simplified by noticing that there is a common factor of 10 in both the numerator and the denominator:
We can go further as 28 and 35 are both multiples of 7:
Thus, sharing 280 pies among 350 children gives the same result as sharing 4 pies among 5 children!
Since 4 and 5 share no common factors, this is as far as we can go with this example (while staying with whole numbers).
Mix and Match: On the top are some fractions that have not been simplified. On the bottom are the simplified answers, but in random order. Which simplified answer goes with which fraction? (Notice that there are fewer answers than questions!)
Use the “Pies Per Child Model” to explain why the key fraction rule holds. That is, explain why each individual child gets the same amount of pie in these two situations:
30
Here are two very similar fractions: and . What might it mean to add them? It might seem reasonable to say:
So maybe represents 5 pies among 14 kids, giving the answer . It is very tempting to say that “adding fractions” means “adding pies and adding kids.”
The trouble is that a fraction is not a pie, and a fraction is not a child. So adding pies and adding children is not actually adding fractions. A fraction is something different. It is related to pies and kids, but something more subtle. A fraction is an amount of pie per child.
One cannot add pies, one cannot add children. One must add instead the amounts individual kids receive.
Let us take it slowly. Consider the fraction . Here is a picture of the amount an individual child receives when two pies are given to seven kids:
Consider the fraction . Here is the picture of the amount an individual child receives when three pies are given to seven children:
The sum corresponds to the sum:
The answer, from the picture, is .
Remember that means “the amount of pie that one child gets when five pies are shared by seven children.” Carefully explain why that is the same as the picture given by the sum above:
Your explanation should use both words and pictures!
Most people read this as “two sevenths plus three sevenths gives five sevenths” and think that the problem is just as easy as saying “two apples plus three apples gives five apples.” And, in the end, they are right!
This is how the addition of fractions is first taught to students: Adding fractions with the same denominator seems just as easy as adding apples:
4 tenths + 3 tenths + 8 tenths = 15 tenths.
(And, if you like, .)
82 sixty-fifths + 91 sixty-fifths = 173 sixty-fifths:
We are really adding amounts per child not amounts, but the answers match the same way.
We can use the “Pies Per Child Model” to explain why adding fractions with like denominators works in this way.
Think about the addition problem :
Since in both cases we have 7 kids sharing the pies, we can imagine that it is the same 7 kids in both cases. First, they share 2 pies. Then they share 3 more pies. The total each child gets by the time all the pie-sharing is done is the same as if the 7 kids had just shared 5 pies to begin with. That is:
Now let us think about the general case. Our claim is that
Translating into our model, we have kids. First, they share pies between them, and represents the amount each child gets. Then they share more pies, so the additional amount of pie each child gets is . The total each kid gets is .
But it does not really matter that the kids first share pies and then share pies. The amount each child gets is the same as if they had started with all of the pies — all of them — and shared them equally. That amount of pie is represented by .
This approach to adding fractions suddenly becomes tricky if the denominators involved are not the same common value. For example, what is ?
Let us phrase this question in terms of pies and kids:
Talk about these questions with a partner before reading on. It is actually a very difficult problem! What might a student say, if they do not already know about adding fractions? Write down any of your thoughts.
One way to think about answering this addition question is to write in a series of alternative forms using our key fraction rule (that is, multiply the numerator and denominator each by 2, and then each by 3, and then each by 4, and so on) and to do the same for :
We see that the problem is actually the same as . So we can find the answer using the same-denominator method:
Here is another example of adding fractions with unlike denominators: . In this case, Valerie is part of a group of 8 kids who share 3 pies. Later she is part of a group of 10 kids who share 3 different pies. How much total pie did Valerie get?
Of course, you do not need to list all of the equivalent forms of each fraction in order to find a common denominator. If you can see a denominator right away (or think of a faster method that always works), go for it!
Cassie suggests the following method for the example above:
When the denominators are the same, we just add the numerators. So when the numerators are the same, shouldn’t we just add the denominators? Like this:
What do you think of Cassie’s suggestion? Does it make sense? What would you say if you were Cassie’s teacher?
Try these exercises on your own. For each addition exercise, also write down a “Pies Per Child” interpretation of the problem. You might also want to draw a picture.
Now try these subtraction exercises.
31
So far, we have been thinking about a fraction as the answer to a division problem. For example, is the result of sharing two pies among three children.
Of course, pies do not have to be round. We can have square pies, or triangular pies or squiggly pies or any shape you please.
This “Pies Per Child Model” has served us perfectly well in thinking about the meaning of fractions, equivalent fractions, and even adding and subtracting fractions.
However, there is no way to use this model to make sense of multiplying fractions! What would this mean?
So what are fractions, if we are asked to multiply them? We are forced to switch models and think about fractions in a new way.
This switch is fundamentally perturbing. Think about students learning this for the first time. We keep switching concepts and models, and speak of fractions in each case as though all is naturally linked and obvious. None of this is obvious, it is all absolutely confusing. This is just one of the reasons that fractions can be such a difficult concept to teach and to learn in elementary school!
For each of the following visual representations of fractions, there is a corresponding incorrect symbolic expression.
In thinking about fractions, it is important to remember that there are always units attached to a fraction, even if the units are hidden. If you see the number in a problem, you should ask yourself “half of what?” The answer to that question is your unit, the amount that equals 1.
So far, our units have been consistent: the “whole” (or unit) was a whole pie, and fractions were represented by pies cut into equal-sized pieces. But this is just a model, and we can take anything, cut it into equal-sized pieces, and talk about fractions of that whole.
One thing that can make fraction problems so difficult is that the fractions in the problem may be given in different units (they may be “parts” of different “wholes”).
Mr. Li shows this picture to his class and asks what number is shown by the shaded region.
Mr. Li exclaims, “Everyone is right!”
This picture
represents . The whole segment (the unit) is split into three equal pieces by the tick marks, and two of those three equal pieces are shaded.
For each picture below, say what fraction it represents and how you know you are right.
If we think about fractions as “portions of a segment,” then we can talk about their locations on a number line. We can start to treat fractions like numbers. In the back of our minds, we should remember that fractions are always relative to some unit. But on a number line, the unit is clear: it is the distance between 0 and 1.
This measurement model makes it much easier to tackle questions about the relative size of fractions based on where they appear on the number line. We can mark off different fractions as parts of the unit segment. Just as with whole numbers, fractions that appear farther to the right are larger.
You probably came up with benchmarks and intuitive methods to think about the relative sizes of fractions. Here are some of these methods. (Did you come up with others?)
Greater than 1: A fraction is greater than 1 if its numerator is greater than its denominator. How can we see this? Well, the denominator represents how many pieces in one whole (one unit). The numerator represents how many pieces in your portion. So if the numerator is bigger, that means you have more than the number of pieces needed to make one whole.
Greater than : A fraction is greater than if the numerator is more than half the denominator. Another way to check (which might be an easier calculation): a fraction is greater than if twice the numerator is bigger than the denominator.
Why? Well, if we double the fraction and get something bigger than 1, then the original fraction must be bigger than .
Same denominators: If two fractions have the same denominator, just compare the numerators. The fractions will be in the same order as the numerators. For example, . Why? Well, the pieces are the same size since the denominators are the same. If you have more pieces of the same size, you have a bigger number.
Same numerators: If the numerators of two fractions are the same, just compare the denominators. The fractions should be in the reverse order of the denominators. For example, . The justification for this one is a little trickier: The denominator tells you how many pieces make up one whole. If there are more pieces in a whole (if the denominator is bigger), then the pieces must be smaller. And if you take the same number of pieces (same numerator), then the bigger piece wins.
Numerator = denominator: You can easily compare two fractions whose numerators are both one less than their denominators. The fractions will be in the same order as the denominators. Think of each fraction as a pie with one piece missing. The greater the denominator, the smaller the missing piece, so the greater the amount remaining. For example, , since and .
Numerator = denominator − constant: You can extend the test above to fractions whose numerators are both the same amount less than their denominators. The fractions will again be in the same order as the denominators, for exactly the same reason. For example, , because both are four “pieces” less than one whole, and the pieces are smaller than the pieces.
Equivalent fractions: Find equivalent fractions that lets you compare numerators or denominators, and then use one of the above rules.
Consider the patterns below.
Pattern 1:
Pattern 2:
Pattern 3:
Answer these questions about each of the patterns.
The patterns above are called arithmetic sequences: a sequence of numbers where the difference between consecutive terms is a constant. Here are some other examples:
Pattern A:
Pattern B:
Pattern C:
If you have not done so already, find the common difference between terms for Patterns 1, 2, and 3. Are they really arithmetic sequences?
Then make up your own arithmetic sequence using whole numbers. Exchange sequences with a partner, and check if your partner’s sequence is really an arithmetic sequence.
Here are several more number patterns:
Pattern 4:
Pattern 5:
Pattern 6:
Pattern 7:
For each of the sequences above, decide if it is an arithmetic sequence or not. Justify your answers.
Find two fractions between and so the resulting four numbers are in an arithmetic sequence.
Find three fractions between and so the resulting four numbers are in an arithmetic sequence.
Make up two fraction sequences of your own, one that is an arithmetic sequence and one that is not an arithmetic sequence.
Exchange your sequences with a partner, but do not tell your partner which is which.
When you get your partner’s sequences: decide which is an arithmetic sequence and which is not. Check if you and your partner agree.
32
One of our models for multiplying whole numbers was an area model. For example, the product is the area (number of 1 × 1 squares) of a 23-by-37 rectangle:
So the product of two fractions, say, should also correspond to an area problem.
Let us start with a segment of some length that we call 1 unit:
Now, build a square that has one unit on each side:
The area of the square, of course, is square unit.
Now, let us divide the segment on top into three equal-sized pieces. (So each piece is .) And we will divide the segment on the side into seven equal-sized pieces. (So each piece is .)
We can use those marks to divide the whole square into small, equal-sized rectangles. (Each rectangle has one side that measures and another side that measures .)
We can now mark off four sevenths on one side and two thirds on the other side.
The result of the multiplication should be the area of the rectangle with on one side and on the other. What is that area?
Remember, the whole square was one unit. That one-unit square is divided into 21 equal-sized pieces, and our rectangle (the one with sides and ) contains eight of those rectangles. Since the shaded area is the answer to our multiplication problem we conclude that
How can you extend the area model for fractions greater than 1? Try to draw a picture for each of these:
Work on the following exercises on your own or with a partner.
How are these two problems different? Draw a picture of each.
When a problem includes a phrase like “ of …,” students are taught to treat “of” as multiplication, and to use that to solve the problem. As the above problems show, in some cases this makes sense, and in some cases it does not. It is important to read carefully and understand what a problem is asking, not memorize rules about “translating” word problems.
You probably simplified your work in the exercises above by using a multiplication rule like the following.
Of course, you may then choose to simplify the final answer, but the answer is always equivalent to this one. Why? The area model can help us explain what is going on.
First, let us clearly write out how the area model says to multiply . We want to build a rectangle where one side has length and the other side has length . We start with a square, one unit on each side.
If the answer is , that means there are total equal-sized pieces in the square, and of them are shaded. We can see from the model why this is the case:
Stick with the general multiplication rule
Write a clear explanation for why of the small rectangles will be shaded.
Often, elementary students are taught to multiply fractions by whole numbers using the fraction rule.
For example, to multiply , we think of “2” as , and compute this way
We can also think in terms of our original “Pies Per Child” model to answer questions like this.
We know that means the amount of pie each child gets when 7 children evenly share 3 pies.
If we compute that means we double the amount of pie each kid gets. We can do this by doubling the number of pies. So the answer is the same as : the amount of pie each child gets when 7 children evenly share 6 pies.
Finally, we can think in terms of units and unitizing.
The fraction means that I have 7 equal pieces (of something), and I take 3 of them.
So means do that twice. If I take 3 pieces and then 3 pieces again, I get a total of 6 pieces. There are still 7 equal pieces in the whole, so the answer is .
Roy says that the fraction rule
is “obvious” if you think in terms of multiplying fractions. He reasons as follows:
We know multiplying anything by 1 does not change a number:
So, in general,
Now, , so that means that
which means
By the same reasoning, , so that means that
which means
What do you think about Roy’s reasoning? Does it make sense? How would Roy explain the general rule for positive whole numbers :
33
Dividing fractions is one of the hardest ideas in elementary school mathematics. By now, you are used to the rule: to divide by a fraction, multiply by its reciprocal. (“invert and multiply”). But ask yourself: Why does this rule work? Does it really make sense to you? Can you explain why it makes sense to a third grader?
We are going to build up to the “invert and multiply” rule, but along the way, we’ll find some more meaningful ways to understand division of fractions. So please play along: pretend that you don’t already know the “invert and multiply” rule, and solve the problems in this chapter with other methods.
Remember the quotative model for division: means:
How many groups of 3 can I find in 18?
We start with 18 dots (or candy bars or molecules), and we make groups of 3 dots (or 3 whatevers). We ask: how many groups can we make?
18 dots, split into groups of 3 dots. Since there are 6 groups, we have 18 ÷ 3 = 6. |
This same idea applies when we divide fractions. For example, means:
How many groups of can I find in 6?
Let’s draw a picture of 6 pies, and see how many groups of we can find:
We found nine equal groups of size , so we conclude that
Unfortunately, it’s not always quite so straightforward to find the equal groups. For example, asks the question:
How many groups of can I find in ?
Let’s draw a picture of of a pie, and see how many groups of we can find:
The first pictures shows of a pie. The second picture shows two equal groups of inside of , but there’s a little bit left over. We conclude
But how much more? Can we figure it out exactly?
Here’s a method that will let you do the computation exactly. We’ll use rectangular pies, and divide them up into rows and columns based on the denominators of the numbers we’re dividing.
Start by drawing two identical rectangles, each with 4 rows (from the denominator of and 3 columns (from the denominator of ).
Shade of the first rectangle (this is exactly three rows), and shade of the second rectangle (so that’s one column).
Now ask: how many copies of can I find in ? Well, is equal to four of the smaller squares. So we find groups equal to that:
In the picture of , we can find:
We conclude:
Use either method above to find the following quotients. Remember, pretend that you don’t know any method to divide fractions except finding equal-sized groups.
Solve each of the following fraction division problems using the “groups of equal size” method:
What do you notice?
This leads to our first fraction division method:
If two fractions have the same denominator, then when you divide them, you can just divide the numerators. In symbols,
We know that we can always turn a division problem into a “missing factor” multiplication problem. Can that help us compute fraction division? Sometimes!
For each division problem, rewrite it as a missing factor multiplication question. Then find the quotient using what you know about multiplying fractions.
Unfortunately, the missing factor method doesn’t always work out so nicely. For example,
can be rewritten as
There isn’t a nice ratio of whole numbers that obviously fills in the blank, but we’ll come back to this idea and resolve it soon.
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The missing factor method is a particularly nice way to understand fraction division. It builds on what we know about multiplication and division, reinforcing that these operations have the same relationship whether the numbers are whole number, fractions, or anything else. It makes sense. But we’ve seen that it doesn’t always work out nicely. For example,
can be rewritten as
You want to ask:
So we have:
You learned about fractions like
back in the “What is a Fraction?” chapter. This means that each of a kid gets 3 pies. So how much does an individual kid (one whole kid) get? You could draw a picture to help you figure it out. But we can also use the key fraction rule to help us out.
This process is going to be key to understanding why the “invert and multiply” rule for fraction division actually makes sense.
pies are shared equally by children. How much pie does each child get?
Technically, we could just write down the answer as
and be done! The answer is equivalent to this fraction, so why not?
Is there a way to make this look friendlier? Well, if we change those mixed numbers to “improper” fractions, it helps a little:
That’s a bit better, but it’s still not clear how much pie each kid gets. Let’s use the key fraction rule to make the fraction even friendlier. Let’s multiply the numerator and denominator each by 3. (Why three?) Remember, this means we’re multiplying the fraction by , which is just a special form of 1, so we don’t change its value.
Now multiply numerator and denominator each by 4. (Why four?)
We now see that the answer is . That means that sharing pies among children is the same as sharing 92 pies among 63 children. (In both situations, the individual child get exactly the same amount of pie.)
Let’s forget the context now and just focus on the calculations so that we can see what is going on more clearly. Try this one:
Multiplying the numerator and denominator each by 5 (why did we choose 5?) gives
Now multiply the numerator and denominator each by 3 (why did we choose 3?):
Is her solution correct, or is she misunderstanding something? Carefully explain what is going on with her solution, and what you would do as Jessica’s teacher.
Is his solution correct, or is he misunderstanding something? Carefully explain what is going on with his solution, and what you would do as Isaac’s teacher.
Perhaps without realizing it, you have just found another method to divide fractions.
Consider . We know that a fraction is the answer to a division problem, meaning
And now we know how to simplify ugly fractions like this one! Multiply the numerator and denominator each by 5:
Now multiply them each by 7:
Done! So
Let’s do another! Consider :
Let’s multiply numerator and denominator each by 9 and by 11 at the same time. (Why not?)
(Do you see what happened here?)
So we have
Compute each of the following, using the simplification technique in the examples above.
Consider the problem . Janine wrote:
She stopped before completing her final step and exclaimed: “Dividing one fraction by another is the same as multiplying the first fraction with the second fraction upside down!”
First check each step of Janine’s work here and make sure that she is correct in what she did up to this point. Then answer these questions:
We now have several methods for solving problems that require dividing fractions:
Dividing fractions:
Discuss your opinions about our four methods for solving fraction division problems with a partner:
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We’ve spent the last couple of chapters talking about dividing fractions: how to make sense of the operation, how to picture what’s going on, and how to do the computations. But all of this kind of begs the question: When would you ever want to divide fractions, anyway? How does that even come up?
It’s important that teachers are able to come up with situations and problems that model particular operations, which means you have to really understand what the operations mean and when they are used.
A common answer to
Come up with a situation where you would want to compute.
Is something like this:
My recipe calls for cups of flour, but I only want to make half a recipe. How much flour should I use?
But that problem doesn’t ask you to divide fractions. It asks you to cut your recipe in half, which means dividing by 2 or multiplying by .
Why is it so hard to come up with division problems that use fractions? Maybe it’s because fractions are already the answer to a division problem, so you’re dividing and then dividing some more. Maybe it’s because they just make it look so complicated. In any case, it’s worth spending some time thinking about division problems that involve fractions and how to recognize and solve them.
One handy trick: Write a problem that involves division of whole numbers, and then see if you can change the numbers to fractions in a sensible way.
Here are some division problems involving whole numbers:
Here are some very similar problems, rewritten to use fractions instead:
For each one of the fraction division questions, we can understand why it’s a division problem:
Recall what partitive division asks: For , we ask 20 is 4 groups of what size?
So for , we ask: is half a group of what size?
You try it.
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Does the fraction make sense?
It seems pretty clear that zero pies among eleven kids gives zero pies per child:
The same reasoning would lead us to say:
The “Pies Per Child Model” offers one explanation: If there are no pies for us to share, no one gets any pie. It does not matter how many children there are. No pie is no pie is no pie.
We can also justify this claim by thinking about a missing factor multiplication problem:
The only way to fill that in and make a true statement is with 0, so .
What happens if things are flipped the other way round?
Does the fraction make sense?
Students often learn in school that “dividing by 0 is undefined.” But they learn this as a rule, rather than thinking about why it makes sense or how it connects to other ideas in mathematics. In this case, the most natural connection is to a multiplication fact, the zero property for multiplication:
That says we can never find solutions to problems like
Using the connection between fractions and division, and the connection between division and multiplication, that means there is no number . There is no number . And there is no number . They are all “undefined” because they are not equal to any number at all.
Can we give meaning to at least? After all, a zero would appear on both sides of that equation!
Who is right? Can they all be correct? What do you think?
Cyril says that , and he believes he is correct because it passes the check: .
But 17 also passes the check, and so does 887231243. In fact, I can choose any number for , and will pass the check!
The trouble with the expression (with not zero) is that there is no meaningful value to assign to it. The trouble with is different: There are too many possible values to give it!
Dividing by zero is simply too problematic to be done! It is best to avoid doing so and never will we allow zero as the denominator of a fraction. (But all is fine with 0 as a numerator.)
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Harriet is part of a group of five children who share four pies. Jeff is part of a group of seven children who share four pies. Jean is part of a group of seven children who share six pies.
Yesterday was Zoe’s birthday, and she had a big rectangular cake. Today, of the cake is left. The leftover cake is shown here.
Draw a picture of the original (whole) cake and explain your work.
Use benchmarks and intuitive methods to arrange the fractions below in ascending order. Explain how you decided. (The point of this problem is to think more and compute less!):
Which of these fractions has the larger value? Justify your choice.
Solve each division problem. Look for a shortcut, and explain your work.
Yoko says
because she cancels the sixes:
But note:
So is Yoko right? Does her cancelation rule always work? If it does not always work, can you find any other example where it works? Can you find every example where it works?
Jimmy says that a fraction does not change in value if you add the same amount to the numerator and the denominator. Is he right? If you were Jimmy’s teacher, how would you respond?
Jill, her brother, and another partner own a pizza restaurant. If Jill owns of the restaurant and her brother owns of the restaurant, what fraction does the third partner own?
John spent a quarter of his life as a boy growing up, one-sixth of his life in college, and one-half of his life as a teacher. He spent his last six years in retirement. How old was he when he died?
Nana was planning to make a red, white, and blue quilt. One-third was to be red and two-fifths was to be white. If the area of the quilt was to be 30 square feet, how many square feet would be blue?Image used under Creative Commons CC0 1.0 Universal Public Domain Dedication.
Rafael ate one-fourth of a pizza and Rocco ate one-third of it. What fraction of the pizza did they eat?
Problem 18 (Tangrams). TangramsTangram image from Wikimedia Commons, public domain. are a seven-piece puzzle, and the seven pieces can be assembled into a big square.
Mikiko said her family made two square pizzas at home. One of the pizzas was 8 inches on each side, and the other was 12 inches on each side. Mikiko ate of the small pizza and of the large pizza. So she said that she ate
of a pizza. Do you agree with Mikiko’s calculation? Did she eat of a whole pizza? Carefully justify your answer. (This question is tricky. It’s probably a good idea to draw a picture!)
Look at the triangle of numbers. There are lots of patterns here! Find as many as you can. In particular, try to answer these questions:
Marie made a sheet cake at home, but she saved some to bring to work and share with her co-workers the next day. Answer these questions about Marie’s cake. (Draw a picture!)
An elementary school held a “Family Math Night” event, and 405 students showed up. Two-thirds of the students who showed up won a door prize. How many students won prizes?
For each picture shown:
For each problem, use only the digits 0, 1, 2,. . . , 9 at most once each in place of the variables. Find the value closest to 1. Note that can be a different value in each of the three problems. Justify your answer: How do you know it is the closest to 1?
A town plans to build a community garden that will cover of a square mile on an old farm. One side of the garden area will be along an existing fence that is of a mile long. If the garden is a rectangle, how long is the other side?
Nate used pounds of seed to plant acres of wheat. How many pounds of seed did he use per acre?
The family-sized box of laundry detergent contains 35 cups of detergent. Your family’s machine requires cup per load. How many loads of laundry can your family do with one box of detergent?
Jessica bikes to campus every day. When she is one-third of the way between her home and campus, she passes a grocery store. When she is halfway to school, she passes a Subway sandwich shop. This morning, Jessica passed the grocery store at 8:30am, and she passed Subway at 8:35am. What time did she get to campus?
If you place a full container of flour on a balance scale and place on the other side a pound weight plus a container of flour (the same size) that is full, then the scale balances. How much does the full container of flour weigh?
Geoff spent of his allowance on a movie. He spent of what was left on snacks at school. He also spent $3 on a magazine, and that left him with of his total allowance, which he put into his savings account. How much money did Geoff save that week?
Lily was flying to San Francisco from Honolulu. Halfway there, she fell asleep. When she woke up, the distance remaining was half the distance traveled while she slept. For what fraction of the trip was Lily asleep?
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Consider the problem: Share 7 pies equally among 12 kids. Of course, given our model for fractions, each child is to receive the quantity “” But this answer has little intuitive feel.
Suppose we took this task as a very practical problem. Here are the seven pies:
Is it possible to give each of the kids a whole pie? No.
How about the next best thing — can each child get half a pie? Yes! There are certainly 12 half pies to dole out. There is also one pie left over yet to be shared among the 12 kids. Divide this into twelfths and hand each kid an extra piece.
So each child gets of a pie, and it is indeed true that
(Check that calculation. . . don’t just believe it!)
This seems quite reasonable. Instead of seven pieces each of size , each kid gets a piece that is and a piece that is . It’s a lot less cutting, and a lot less messy!
1. Suppose you want to share five pies among six children, but you want each child to get a small number of (relatively) large pieces rather than five pieces of size . Following the example above, how could you do it?
2. Using similar ideas, how could you share 4 pies among 7 kids?
The Egyptians (probably) were not particularly concerned with splitting up pies. But in fact, they did have a very strange (to us) way of expressing fractions. We know this by examining the Rhind Papyrus. This ancient document indicates that fractions were in use as many as four thousand years ago in Egypt, but the Egyptians seem to have worked primarily with unit fractions. They insisted on writing all of their fractions as sums of fractions with numerators equal to 1, and they insisted that the denominators of the fractions were all different.
Accurate reckoning for inquiring into things, and the knowledge of all things, mysteries…all secrets.
The Rhind Papyrus is an ancient account of Egyptian mathematics named after Alexander Henry Rhind. Rhind was a Scotsman who acquired the ancient papyrus in 1858 in Luxor, Egypt.
The papyrus dates back to around 1650 B.C. It was copied by a scribe named Ahmes (the earliest known contributor to the field of mathematics!) from a lost text written during the reign of king Amenehat III. The opening quote is taken from Ahmes introduction to the Rhind PapyrusImage of Rhind Papyrus from Wikimedia Commons, public domain.. The papyrus covers topics relating to fractions, volume, area, pyramids, and more.
To write a fraction as an Egyptian fraction, you must rewrite the fraction as:
The Egyptians would not write , and they would not even write . Instead, they wrote
The Egyptians would not write , and they would not even write . Instead, they wrote
(You should check that the sums above give the correct resulting fractions!)
Write the following as a sum of two different unit fractions. Be sure to check your answers.
Can you find a general rule for how to write as an Egyptian fraction? (Assume is an odd number.)
Write the following as a sum of distinct unit fractions. (“Distinct” means the fractions must have different denominators.) Note that you may need to use more than two unit fractions in some of the sums. Be sure to check your answers.
Can you find a general process for fractions bigger than ?
Write the following fractions as Egyptian fractions.
Can you find a general algorithm that will turn any fraction at all into an Egyptian fraction?
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In an advanced algebra course students are often asked to work with complicated expressions like:
We can make it look friendlier by using the key fraction rule, exactly the same technique we used in the chapter on “Dividing Fractions: Invert and Multiply.” In this example, let us multiply the numerator and denominator each by . (Do you see why this is a good choice?) We obtain:
and is much less scary.
Notic that expressions like
cannot be rewritten as a decimal. Expressions like this arise in numerous applications, so it is important for math and science students to be able to work with fractions in fraction form, without always resorting to converting to decimals.
As another example, given:
one might find it helpful to multiply the numerator and the denominator each by and then each by :
For
it might be good to multiply numerator and denominator each by . (Why?)
Can you make each of these expressions look less scary?
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So far, we have no single model that makes sense of fractions in all contexts. Sometimes a fraction is an action (“Cut this in half.”) Sometimes it is a quantity (“We each get 2/3 of a pie!”) And sometimes we want to treat fractions like numbers, like ticks on the number line in-between whole numbers.
We could say that a fraction is just a pair of numbers and , where we require that . We just happen to write the pair as .
But again this is not quite right, since a whole infinite collection of pairs of numbers represent the same fraction! For example:
So a single fraction is actually a whole infinite class of pairs of numbers that we consider “equivalent.”
How do mathematicians think about fractions? Well, in exactly this way. They think of pairs of numbers written as , where we remember two important facts:
This is a hefty shift of thinking: The notion of a “number” has changed from being a specific combination of symbols to a whole class of combinations of symbols that are deemed equivalent.
Mathematicians then define the addition of fractions to be given by the daunting rule:
This is obviously motivated by something like the “Pies Per Child Model.” But if we just define things this way, we must worry about proving that choosing different representations for and lead to the same final answer.
For example, it is not immediately obvious that
give answers that are equivalent. (Check that they do!)
They also define the product of fractions as:
Again, if we start from here, we have to prove that you get equivalent answers for different choices of fractions equivalent to and .
Then mathematicians establish that the axioms of an arithmetic system hold with these definitions and carry on from there! (That is, they check that addition and multiplication are both commutative and associative, that the distributive law holds, that all representations of 0 act like an additive identity, and so on.)
This is abstract, dry, and not at all the best first encounter to offer students on the topic of fractions. And, moreover, this approach completely avoids the question as to what a fraction really means in the “real world.” But it is the best one can do if one is to be completely honest.
So… what is a fraction, really? How do you think about them? And what is the best way to talk about them with elementary school students?
V
Image used under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
-Paul Lockhart
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Algebra skills are essential for your future students. Why? Here are just a few reasons:
You might wonder why future elementary teachers should master algebra, a topic usually studied (by that name, anyway) in 8th grade and beyond. But the Common Core Standards for School Mathematics has standards in “Operations and Algebraic Thinking” beginning in kindergarten!
Everyone who shows up to school has already learned a lot about abstraction and generalization — the fundamental ideas in algebra. They are all capable of learning to formalize these ideas. Your job as an elementary school teacher will be to provide your students with even more experiences in abstraction and generalization in a mathematical context, so that these ideas will seem quite natural when they get to a class with the name “Algebra.”
Let’s start with a problem:
I can use four 4’s to make 0:
I can also use four 4’s to make the number 10:
Your challenge: Use four 4’s to make all of the numbers between 0 and 20. (Try to find different solutions for 0 and 10 than the ones provided.) You can use any mathematical operations, but you can’t use any digits other than the four 4’s.
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Here’s another problem:
Here is a large square made up of 100 smaller unit squares. The unit squares along the border of the large square are colored red. Without counting one-by-one, can you figure out how many red squares there are in the picture?
Clearly describe how you figured out the number of red squares, and how you know your answer is correct.
Justin calculated the number of squares as . He justified his answer this way:
Since the dimensions of the big square are , there are 10 squares along each of the four sides. So that gives me 40 red squares. But then each corner is part of two different sides. I’ve counted each of the corners twice. So I need to make up for that by subtracting 4 at the end.
Justin showed this picture to justify his work:
There are lots of different ways to calculate the number of colored squares along the border of a square. Below are the calculations several other students did. For each calculation, write a justification and draw a picture to show why it calculates the number of squares correctly. Think about using color in your picture to make your work more clear.
Now suppose that you have a large square with the unit squares along the border colored red. Adapt two of the techniques above to calculate the number of red unit squares.
For each technique you used, write an explanation and include a picture. Think about how to use colors or other methods to make your picture and explanation more clear.
Now suppose that you have a large square with the unit squares along the border colored red. Adapt two of the techniques above to calculate the number of red unit squares.
For each technique you used, write an explanation and include a picture. Think about how to use colors or other methods to make your picture and explanation more clear.
Describe some general rules:
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The notion of equality is fundamental in mathematics, and especially in algebra and algebraic thinking. The symbol “=”’ expresses a relationship. It is not an operation in the way that + and are × operations. It should not be read left-to-right, and it definitely does not mean “… and the answer is …”.
For your work to be clear and easily understood by others, it is essential that you use the symbol = appropriately. And for your future students to understand the meaning of the = symbol and use it correctly, it is essential that you are clear and precise in your use of it.
Let’s start by working on some problems.
Akira went to visit his grandmother, and she gave him $1.50 to buy a treat.
He went to the store and bought a book for $3.20. After that, he had $2.30 left.
How much money did Akira have before he visited his grandmother?
Examine the following equations. Decide: Is the statement always true, sometimes true, or never true? Justify your answers.
Consider the equation
If someone asked you to solve the equations in Problem 8, what would you do in each case and why?
Kim solved Problem 7 this way this way:
Let’s see:
so the answer is 4.
What do you think about Kim’s solution? Did she get the correct answer? Is her solution clear? How could it be better?
Although Kim found the correct numerical answer, her calculation really doesn’t make any sense. It is true that
But it is definitely not true that
She is incorrectly using the symbol “=”, and that makes her calculation hard to understand.
This picture shows a (very simplistic) two-pan balance scale. Such a scale allows you to compare the weight of two objects. Place one object in each pan. If one side is lower than the other, then that side holds heavier objects. If the two sides are balanced, then the objects on each side weigh the same.
In the pictures below:
1. In the picture below, what do you know about the weights of the triangles and the circles? How do you know it?
2. In the picture below, what do you know about the weights of the circles and the stars? How do you know it?
3. In the picture below, what do you know about the weights of the stars and the squares? How do you know it?
In the pictures below:
How many purple squares will balance with one circle? Justify your answer.
In the pictures below:
How many purple squares will balance the scale in each case? Justify your answers.
(a)
(b)
(c)
In the pictures below:
What will balance the last scale? Can you find more than one answer?
In the pictures below:
What do Problems 11–14 above have to do with the “=” symbol?
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Here is a pattern made from square tiles.
Here are some pictures that students drew to describe how the pattern was growing.
Ali’s picture
Michael’s picture
Kelli’s picture
Describe in words how each student saw the pattern growing. Use the students’ pictures above (or your own method of seeing the growing pattern) to answer the following questions:
Hy saw the pattern in a different way from everyone else in class. Here’s what he drew:
Hy’s picture.
The next few problems present several growing patterns made with tiles. For each problem you work on, do the following:
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Below, you’ll find patterns described in various ways: through visual representations, algebraic expressions, in tables of numbers, and in words. Your job is to match these up in a way that makes sense.
Note: there may be more than one algebraic expression to match a given pattern, or more than one pattern to match a given description. So be ready to justify your answers.
(a) | (b) | (c) |
(d) | (e) | (f) |
(g) | (h) | (i) |
(j) | (k) | (l) |
(m) | (n) | (o) |
Pattern 1
Pattern 2
Pattern 3
Pattern 4
Pattern 5
Pattern 6
Pattern 7
Table A | ||||
Input | 1 | 2 | 3 | 4 |
Output | 1 | 4 | 9 | 16 |
Table B | ||||
Input | 1 | 2 | 3 | 4 |
Output | 10 | 15 | 20 | 25 |
Table C | ||||
Input | 1 | 2 | 3 | 4 |
Output | 1 | 3 | 5 | 7 |
Table D | ||||
Input | 1 | 2 | 3 | 4 |
Output | 3 | 5 | 7 | 9 |
Table E | ||||
Input | 1 | 2 | 3 | 4 |
Output | 4 | 7 | 10 | 13 |
Table F | ||||
Input | 1 | 2 | 3 | 4 |
Output | 4 | 10 | 16 | 22 |
Table G | ||||
Input | 1 | 2 | 3 | 4 |
Output | 2 | 4 | 6 | 8 |
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When most people think about algebra from school, they think about “solving for .” They imagine lots of equations with varying levels of complexity, but the goal is always the same: find the unknown quantity. This is a procedural view of algebra.
Even elementary students can be exposed to ideas in procedural algebra. This happens any time they think about unknown quantities and try to solve for them. For example, when first grade students learn to add and subtract numbers “within 10,’” they should frequently tackle problems like these:
Although procedural algebra is important, it’s not the most important skill, and it’s certainly not the whole story.
You also need to foster thinking about structural algebra in your students: using symbols to express meaning in a situation. If there is an x on your page, you should be able to answer, “what does the x mean? What does it represent?”
Most of what you’ve done so far in this chapter is structural algebra. You’ve used letters and symbols not to represent a single unknown quantity, but a varying quantity. For example, in Section 4 you used letters to represent the “figure number’” or “case number” in a growing pattern. The letters could take on different values, and the expressions gave you information: how many tiles or toothpicks or stars you needed to build that particular figure in that particular pattern.
Krystal was looking at this pattern, which may be familiar to you from the Problem Bank:
She wrote down the equation
In Krystal’s equation, what does represent? What does represent? How do you know?
Candice was thinking about this problem:
Today is Jennifer’s birthday, and she’s twice as old as her brother. When will she be twice as old as him again?
She wrote down the equation . In Candice’s equation, what does represent? What does represent? How do you know?
Sarah and David collect old coins. Suppose the variable stands for the number of coins Sarah has in her collection, and stands for the number of coins David has in his collection. What would each of these equations say about their coin collections?
The pictures below show balance scales containing bags and blocks. The bags are marked with a “?”’ because they contain some unknown number of blocks. In each picture:
For each picture, determine how many blocks are in each bag. Justify your answers.
(a)
(b)
(c)
When he was working on Problem 24, Kyle wrote down these three equations.
(i) .
(ii) .
(iii) .
Match each equation to a picture, and justify your choices. Then solve the equations, and say (in a sentence) what the solution represents.
Draw a balance puzzle that represents the equation
Now solve the balance puzzle. Where is the “” in your puzzle? What does it represent?
Draw a balance puzzle that represents the equation
Now solve the equation. Explain what happens.
Which equation below is most like the one in Problem 27 above? Justify your choice.
Draw a balance puzzle that represents the equation
Now solve the equation. Explain what happens.
Which equation below is most like the one in Problem 29 above? Justify your choice.
Create a balance puzzle where the solution is not a whole number of blocks. Can you solve it? Explain your answer.
There are three piles of rocks: pile A, pile B, and pile C. Pile B has two more rocks than pile A. Pile C has four times as many rocks as pile A. The total number of rocks in all three piles is 14.
Look back at Problems 21–32. Which of them felt like structural algebraic thinking? Which felt like procedural algebraic thinking? Did any of the problems feel like they involved both kinds of thinking?
You have seen that in algebra, letters and symbols can have different meanings depending on the context.
In much the same way, equations can represent different things.
is always true, for every value of . There is nothing to solve for, and no relationship between varying quantities. (If you do try to “solve for ,” you will get the equation , much like you saw in Problem 29. Not very satisfying!)
Give an example of each type of equation. Be sure to say what the symbols in the equations represent.
Answer the following questions about the equation
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Problems 34-36 ask you to solve problems about a strange veterinarian who created three mystifying machines.
Cat Machine: Place a cat in the input bin of this machine, press the button, and out jump two dogs and a mouse.
Dog Machine: This machine converts a dog into a cat and a mouse.
Mouse Machine: This machine can convert a mouse into a cat and three dogs.
Each machine can also operate in reverse. For example, if you have two dogs and a mouse, you can use the first machine to convert them into a cat.
The veterinarian hands you two cats, and asks you to convert them into exactly three dogs (no extra dogs and no other animals). Can you do it? If yes, say what process you would use. If no, say why not.
The veterinarian hands you one dog. He says he only wants cats, but he doesn’t care how many. Can you help him? How?
The veterinarian hands you one cat. He says he only wants dogs, but he doesn’t care how many. Can you help him? How?
Problems 37-40 present several growing patterns made with toothpicks. For each problem you work on, do the following:
In a mobile, the arms must be perfectly balanced for it to hang properly. The artist Alexander Calder was famous for his artistic mobiles.You can view some of his amazing work here. Click “Explore Works.”
Problems 41-42 present you with mobile puzzles. In these puzzles:
In this puzzle:
Find the weight of each piece. Is there more than one answer? How do you know you are right?
In this puzzle, the total weight is 54 grams.
Find the weight of each piece. Is there more than one answer? How do you know you are right?
VI
I always say when you see that old black-and-white footage of the rocket on the launch pad and it falls over and explodes, that’s because people had slide rules. Not having the decimal point is a real drawback. You want the decimal point, take it from me.
-Bill Nye
The “Dots and Boxes” approach to place value used in this part (and throughout this book) and the “pies per child” approach to fractions comes from James Tanton, and are used with his permission. See his development of these and other ideas at http://gdaymath.com/.
48
Let’s start with a quick review of place value, different bases, and our “Dots & Boxes” model for thinking about these ideas.
Whenever there are two dots in single box, they “explode,” disappear, and become one dot in the box to the left.
We start by placing nine dots in the rightmost box.
Two dots in that box explode and become one dot in the box to the left.
Once again, two dots in that box explode and become one dot in the box to the left.
We do it again!
Hey, now we have more than two dots in the second box, so those can explode and move!
And the rightmost box still has more than two dots.
Keep going, until no box has two dots.
After all this, reading from left to right we are left with one dot, followed by zero dots, zero dots, and one final dot.
Solution: The 2←1 code for nine dots is: 1001.
Whenever there are three dots in single box, they “explode,” disappear, and become one dot in the box to the left.
Here’s what happens with fifteen dots:
Solution: The 1←3 code for fifteen dots is: 120.
Recall that numbers written in the 1←2 system are called binary or base two numbers.
Numbers written in the 1←3 system are called base three numbers.
Numbers written in the 1←4 system are called base four numbers.
Numbers written in the 1←10 system are called base ten numbers.
In general, numbers written in the 1←b system are called base b numbers.
In a base b number system, each place represents a power of b, which means for some whole number n. Remember this means b multiplied by itself n times:
Whenever we’re dealing with numbers written in different bases, we use a subscript to indicate the base so that there can be no confusion. So:
If the base is not written, we assume it’s base ten.
Remember: when you see the subscript, you are seeing the code for some number of dots.
Work through the two examples above carefully to be sure you remember and understand how the “Dots & Boxes” model works. Then answer these questions:
Work on the following exercises on your own or with a partner.
Quickly compute each of the following. Write your answer in the same base as the problem.
49
Up to now our “Dots & Boxes” model has consisted of a row of boxes extending infinitely far to the left. Why not have boxes extending to the right as well?
Let’s work specifically with a 1←10 rule and see what boxes to the right could mean.
It has become convention to separate boxes to the right of the ones place with a decimal point. (At least, this is what the point is called in the base ten world… “dec” means “ten” after all!)
What is the value of the first box to the right of the decimal point? If we denote its value as , we have that ten ’s is equivalent to 1. (Remember, we are using a 1 ← 10 rule.)
From we get that .
Call the value of the next box to the right .
From we get .
If we keep doing this, we see that the boxes to the right of the decimal point represent the reciprocals of the powers of ten.
The decimal is represented by the picture:
It represents three groups of , that is:
The decimal is represented by the picture:
It represents seven groups of .
Of course, some decimals represent fractions that can simplify further. For example:
Similarly, if a fraction can be rewritten to have a denominator that is a power of ten, then it is easy to convert it to a decimal. For example, is equivalent to , and so we have:
Can you write as a decimal? Well,
We can write the denominator as a power of ten using the key fraction rule:
So we see that:
Here is a more interesting question: What fraction is represented by the decimal ?
There are two ways to think about this.
Approach 1:
From the picture of the “Dots & Boxes” model we see:
We can add these fractions by finding a common denominator:
So
Approach 2:
Let’s unexplode the three dots in the position to produce an additional 30 dots in the position.
So we can see right away that
Work on the following exercises on your own or with a partner.
1. Brian is having difficulty seeing that represents the fraction . Describe the two approaches you could use to explain this to him.
2. A teacher asked his students to each draw a “Dots & Boxes” picture of the fraction .
Jin drew this:
Sonia drew this:
The teacher marked both students as correct.
3. Choose the best answer and justify your choice. The decimal equals:
4. Choose the best answer and justify your choice. The decimal equals:
5. Choose the best answer and justify your choice. The decimal equals:
6. Choose the best answer and justify your choice. The decimal equals:
7. What fraction is represented by each of the following decimals?
8. Write each of the following fractions as decimals. Don’t use a calculator!
9. Write each of the following fractions as decimals. Don’t use a calculator!
10. Write each of the following as a fraction (or mixed number).
11. Write each of the following numbers in decimal notation.
Do and represent the same number or different numbers?
Here are two dots and boxes pictures for the decimal .
And here are two dots and boxes picture for the decimal .
50
Just like in base 10, we can add boxes to the right of the decimal point in other bases, like base 5.
However, the prefix “dec” in “decimal point” means ten. So we really shouldn’t call it a decimal point anymore. Maybe a “pentimal point”? (In fact, the general term is radix point.)
In general, in a base- system, the boxes to the left of the ones place represent positive powers of the base . Boxes to the right of the ones place represent reciprocals of those powers.
Work on the following exercises on your own or with a partner.
Tami and Courtney were working on converting to a familiar base-10 fraction. Courtney said this:
The places in base five to the right of the point are like and then . Since this has two places, the answer should be .
Tami thought about what Courtney said and replied:
I don’t know what the right answer is, but I know that can’t be right. The number is less than one, since there are no numbers in the ones place and no explosions that we can do. But the fraction is more than one. It’s almost two. So they can’t be the same number.
Find the “decimal” representation of in each of the following bases. Be sure that you can justify your answer. (You might want to review the example of in the previous chapter.)
51
When you studied fractions, you had lots of different ways to think about them. But the first way, and the one we keep coming back to, is to think of a fraction as the answer to a division problem.
Suppose 6 pies are to be shared equally among 3 children. This yields 2 pies per child. We write:
The fraction is equivalent to the answer to the division problem . It represents the number of pies one whole child receives.
In the same way…
sharing 10 pies among 2 kids yields pies per kid,
sharing 8 pies among 2 kids yields pies per kid,