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! Show 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:
Connections Between the OperationsWe 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
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?
a + b = c and c – b = a are the same mathematical fact. Why It’s True, Explanation 1: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. Why It’s True, Explanation 2: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. Think / Pair / Share
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 is Commutative
Addition of whole numbers is commutative. What it Means (words):When I add two whole numbers, the order I add them doesn’t affect the sum. What it Means (symbols):For any two whole numbers a and b, a + b = b + a. Now we need a justification. Why is addition of whole numbers commutative? Why It’s True, Explanation 1:Let’s think about addition as combining two quantities of dots.
• That means a + b = b + a. Why It’s True, Explanation 2: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. Addition is AssociativeYour 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). Problem 14
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. Problem 15
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.
Properties of Multiplication and DivisionNow 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. 1 is an Identity for MultiplicationWe’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. What it Means (words):When I multiply a number by 1 (in either order), the product is that number. What it Means (symbols):For any whole number m, m × 1 = m and 1 × m = m. Examples:1 × 5 = 5, 19 × 1 = 19, and 1 × 1 = 1. Why does the number 1 act this way with multiplication? Why It’s True, Explanation 1:Let’s think first about the definition of multiplication as repeated addition:
Why It’s True, Explanation 2: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. Why It’s True, Explanation 3: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? Multiplication is CommutativeProperty: 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. Problem 18
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). Problem 19
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.
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. Properties of DivisionIt’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! Problem 22
Problem 23
Problem 24
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:
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
Four Fact FamiliesIn 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 Think / Pair / Share
Problem 26
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 |