3.1: Addition and Subtraction

Addition and Subtraction

Addition and subtraction are fundamental building blocks of mathematical understanding, forming the basis for more advanced numerical reasoning. Young children first encounter these operations through real-world experiences, often using fingers, objects, or verbal counting strategies to find sums and differences. At this stage, addition is seen as putting together, while subtraction is commonly understood as taking away. As their understanding deepens, children begin to recognize the relationship between the two operations, using known addition facts to solve subtraction problems and vice versa. Encouraging students to explore various ways to compose and decompose numbers fosters a flexible understanding of number relationships, laying the groundwork for number sense and mental computation (Feike et al., 2018).

As future teachers—or even as mentors to younger family members—you have the opportunity to shape how children experience mathematics from the very beginning. By providing meaningful experiences with addition and subtraction, you can help young learners move beyond memorization to develop a deep understanding of numbers and operations. Encouraging strategies such as choral counting, subitizing, and using real-world contexts will empower children to see math as a sense-making activity rather than a set of disconnected rules. (Franke et al., 2018; NCTM 2020) Helping students develop confidence and fluency in these early concepts will not only set them up for success in later mathematics but will also nurture a positive relationship with math that lasts a lifetime.

Think About and Discuss

Here are some more moments of brilliance from our younger mathematicians. Before you take a look at the videos by famed math educator Marylin Burns, think about how an elementary school student would solve 90-75.

Task 1: Discuss with peers how you think they may solve the problem and try to explain it using math reasoning or math expressions.

Task 2: View the videos.

Meeyah 90-__=75 https://youtu.be/FVfF7n2FeiQ?si=oMIOGwS6D2cWYVD1

Rocco 90-__=75 https://youtu.be/zHeqBZrT-lo?si=IU9pPx-gG6-vaSRC

Task 3: Imagine you were Rocco’s teacher, what feedback would you give him about his mathematical thinking and reasoning.

Let’s take this mathematical brilliance we witnessed to explore this section’s content on addition and subtraction.

Addition

Whole numbers are made up of zero and the counting numbers 0, 1, 2, 3, 4, 5, . . . We begin our work by exploring how to add two whole numbers together. To add 3 + 4, we find a set with three elements and a different set with four elements (having no elements in common with the first set). We combine the two sets and then count how many elements are in the union.

So to add two numbers, 3 and 4, we must find a set A such that n(A) = 3 and a set B such that n(B) = 4. Then, 3 + 4 is how many elements are in \(A \cup B\).

Definition: Set Theory Addition of Whole Numbers

Let a and b be any two whole numbers. If A and B are disjoint sets with a = n(A) and b = n(B), then a + b = n(A ∪ B).

The number a + b, read “a plus b,” is called the sum of a and b, and a and b are called addends or summands of a + b.

Addition is called a binary operation because two (“bi”) numbers are combined to produce a unique (one and only one) number. Multiplication is another example of a binary operation with numbers. Intersection, union, and set difference are binary operations using sets.

Example \(\PageIndex{1}\)

a) Use the set theory definition of addition to show that 3 + 2 = 5.

b) Use the set theory definition of addition to show that 0 + 4 = 4.

Answer

a) Let A = {x, y, z} and B = {m, r} Since n(A) = 3, n(B) =2, and \(A \cap B\) = Ø then

3+2 = n(A)+n(B)	by substituting n(A) for 3 and n(B) for 2
= n(\(A \cup B\))	by the set theory definition of addition
= n({x, y, z, m, r})	by computing A
= 5	by counting the elements in A

Therefore, 3 +2 = 5.

b) Let A = { } and B = {a, b, c, d}. Since n(A) = 0, n(B) = 4 and \(A \cup B\) = Ø, then

0 + 4 = n(A) + n(B)	by substituting n(A) for 0 and n(B) for 4
= n(\(A \cup B\))	by the set theory definition of addition
= n({a, b, c, d})	by computing \(A \cup B\)
= 4	by counting the elements in \(A \cup B\)

Therefore, 0 + 4 = 4

🧠Let's Listen to Learn❤️

 

Measurement Model

Addition can also be represented on the whole-number line. In the measurement model, addition of whole numbers is represented by directed arrows of whole number lengths along the whole number line. Example 2 + 5. Figure: 3.1.1

Place an arrow length 2 starting at 0. Then place an arrow length 5 starting at the end of the first arrow. The sum is the total distance from 0.

Figure \(\PageIndex{1}\)

Learning the addition properties of whole numbers helps children develop number sense, construct relationships among facts, and build fluency in mental math and problem-solving. These properties also lay the foundation for algebraic reasoning by allowing students to generalize operations and make sense of number relationships (NCTM, 2020).

Properties of Whole-Number Addition

Commutative Property for Whole-Number Addition

Let a and b be any whole numbers. Then a + b = b + a.

Associative Property for Whole-Number Addition

Let a, b, and c be any whole numbers. Then (a + b) + c = a + (b + c).

Identity Property for Whole-Number Addition

There is a unique whole number, namely 0, such that for all whole numbers a, a + 0 = a = 0 + a. Because of this property, zero is called the additive identity or the identity for addition.

Most of us have been adding numbers together for years and years and so this property may seem obvious — it is second nature that the order in which we add two numbers is irrelevant. But if you take the opportunity to ask a child who is just learning to add the following two questions, even one right after the other, you may notice the child can do the first one quickly and easily but then struggles a little longer at the second one. That is usually the case if the child hasn't yet discovered the commutative property of addition. As adults, we take this property for granted.

🧠Let's Listen to Learn❤️ 

 

First question: What is 7 + 2? Second question: What is 2 + 7?

Think About and Discuss

How do you think a child might figure out the answer to the first question and how might he or she figure out how to do the second question? Be specific and assume the answers haven't been memorized yet! Why might the second question be perceived as a harder problem?

Answer

When adding two numbers together, some people start with the first number and count on the second number. So if you think about 7 + 2, start with 7 and count two more in your head — eight, nine. For 2 + 7, start with 2 and count seven more in your head — three, four, five, six, seven, eight, nine. Although the answer is the same, 7 + 2 was quicker and easier to keep track of. Just think about the difference between 1000 + 1 and 1 + 1000 using this counting on method! Thank goodness for the commutative property of addition!

The Associative Property of Addition states that if a , b and c are any three numbers, then (a + b) + c = a + ( b + c )

The root in the word "associative" is associate. Think about whether the middle number will associate with the first number or the last number for the first computation. As for adding numbers in your head, it may be a lot easier to do it one way instead of the other. For instance, to add 58 + 39 + 41, you could think (58 + 39) + 41 = 97 + 41 = 138 or you could think 58 + (39 + 41) = 58 + 80 = 138. 97 + 41 and 58 + 80 both equal 138, but that isn't obvious until after you add. I prefer to add the 39 and 41 together first. The associative property allows that. But again, the associative property is something that most adults take for granted and use but they don't really think about it.

Think About and Discuss

Here is another example. Although the sum of the three numbers is the same for each problem, the way you do the computations are different.

(84 + 56) + 73 140 + 73 213

84 + (56 + 73) 84 + 129 213

Task 1: Use "Thinking Strategies" to compute the sums.

a) 9 + 7 b) 32 + 51 c) 125 + 46 d) 37 + (42 + 13)

Task 2: Make up an example of your own showing an application of the associative property, then share it with others to see how they solve your problem.

Example \(\PageIndex{2}\)

State which property (Commutative or Associative Property of Addition) is being used in each equation. Ask yourself: Is the difference between the left and right side due to order (commutative property) or parentheses (associative property)?

a. _____	(99 + 76) + 38 = (76 + 99) + 38
b. _____	(65 + 22) + 56 = 56 + (65 + 22)
c. _____	(57 + 88) + 43 = 57 + (88 + 43)
d. _____	(a + b) + (c + d) = ((a + b) + c) + d

Answer

a) commutative b) commutative c) associative d) associative

Closure Property for Whole Numbers

The sum of any two whole numbers is a whole number.

The closure property of addition for whole numbers. We say the set of whole numbers is closed under addition because when you add any two whole numbers together, you get another unique whole number.

A set is closed under addition if the sum of any two elements in the set (these could be the same elements or two different elements) produces a unique element in the same set. In general, a set is closed under an operation if when you perform the operation on any two elements in the set, the answer is an element of the same set.

To show that a set is not closed under addition, you must give a counterexample showing that when you add two numbers in the set, you get a sum that is not in the set. The two numbers you choose can be the same number or different numbers.

Example \(\PageIndex{3}\)

a) Is the set {1, 2, 3, 4, 5} closed under addition?

b) Is {4} closed under addition?

c) Is {5, 10, 15, 20, . . .} closed under addition?

Answer

a) Imagine you had two hats, each containing these five numbers in the set written on pieces of paper. So each hat contains the set {1, 2, 3, 4, 5}. If you pull a number out of each hat, you might get the same number twice or you might get different numbers. Some of the sums are 1 + 1 = 2, 2 + 3 = 5, 1 + 3 = 4, 3 + 3 = 6, etc. Note that the first three sums 2, 5 and 4 are in the set {1, 2, 3, 4, 5}. But for a set to be closed, the sum of every possible two numbers in the set must yield a number in the original set. Since 3 + 3 = 6, (where the two addends, 3 and 3, are in the original set, but the sum 6 is NOT in the original set), then the set {1, 2, 3, 4, 5} is not closed. There are any number of counterexamples you can use to show it is not closed. 3 + 3 = 6 will do. Or someone else might write 4 + 5 = 9. What are two other counterexamples you could use to show {1, 2, 3, 4, 5} is not closed under addition?

b) The only possibility to try is 4 + 4, which is 8, and 8 is not in the set. So the set is not closed. The counterexample to show it is not closed is 4 + 4 = 8.

c) Start by trying a few examples: 5 + 5 = 10 which is in the set; 10 + 15 = 25 which is in the set (note the three dots which means the next few numbers are 25, 30 and 35. It looks like it is closed. But how can you convince someone that the sum of any two numbers in the set will always be in the original set. Note that the set is composed of multiples of 5. Note any two multiples of 5 can be written as 5 times something, like 5x and 5y. Let 5x and 5y denote two arbitrary elements in the original set. Adding them together, we get 5x + 5y = 5(x+y) which is another multiple of 5 (since it is 5 times something). So, you should be convinced that the sum of any two elements in the set is a multiple of 5, which is what the original set consisted of. Therefore, {5, 10, 15, 20, . . .} is closed under addition.

The Magic of Numbers: A Journey of Discovery (Inspired by Where Mathematics Comes From, Lakoff and Nunez, 2000

Imagine a world where numbers are like puzzle pieces, and every time we try a new math operation—like adding, subtracting, or dividing—we want the pieces to fit perfectly. But sometimes, they don’t!

Take natural numbers, for example: 1, 2, 3, 4, and so on. If we add any two natural numbers, we always get another natural number. That’s what we call closure—when we perform an operation, we stay within the same number set.

But what happens when we subtract? Suddenly, we hit a roadblock: 5 - 5 = 0—oops! Zero wasn’t in our original set. So, we had to expand our number system to include it. Then, we tried 3 - 5 and found we needed negative numbers! Later, we realized that division was another challenge: 3 ÷ 5 isn’t a natural number, so we created fractions.

This pattern of discovery—finding gaps and expanding the number system—is the heart of mathematics. Closure isn’t just a math rule; it’s a powerful force that drives new ideas and deeper understanding. It’s like opening new doors in a never-ending mathematical adventure.

By exploring closure, we help students see math as an exciting journey rather than just a set of rules. When they ask, "Why do we need negative numbers?" or "Where did fractions come from?"—they’re stepping into the same process mathematicians have followed for centuries.

Let’s encourage that curiosity and show students how math grows to fit the needs of our thinking!

Subtraction

Think About and Discuss

What Was Rebecca Thinking?

Marylin Burns is back, think about how a young elementary school student would solve 7 - 3.

Task 1: View the video of Rebecca solving the problem.

https://youtu.be/TC7I_SdKWtA?si=yjHkTt1zW1d_Wb7H

Task 2: Discuss with your peers Rebecca's strategy.

Task 3: Use Rebecca's strategy to solve two other subtraction problems: 31 – 15 and 52 – 25. Are there other ways ("thinking strategies") to solve those problems? Discuss.

🧠Let's Listen to Learn❤️ 

 

There are two distinct approaches to subtraction. The one most of us are familiar with is the Take-Away Method. A typical way someone might introduce the idea of subtraction is by saying "If I put five apples in the bowl and then take away two of the apples, how many are left in the bowl?" The illustration below shows this as a subtraction problem where after two apples are removed from the bowl, there are three apples remaining Figure 3.1.1. Also, the problem “If you walk 7 miles from home and turn back to walk 4 miles toward home, how many miles are you from home?” can be solved with a measurement model using the takeaway approach Figure 3.1.2.

Figure \(\PageIndex{1}\): Take-Away Approach Set Model

Figure \(\PageIndex{2}\): Take-Away Approach Measurement Model

Definition: Subtraction of Whole Numbers using Set Theory

If B is a subset of A, then n(A) – n(B) = n(A – B)

Subtraction Vocabulary: For x – y = z, x is called the minuend, y is called the subtrahend, and z (the answer) is called the difference.

Example \(\PageIndex{4}\)

Use the set theory definition of subtraction to show that 5 – 2 = 3.

Let A = {v, w, x, y, z} and B = {w, z}. Since n(A) = 5, n(B) = 2 and B \(\subseteq\) A,

\[\begin{aligned} 5 – 2 &= n(A) – n(B) && \text{ by substituting } n(A) \text{ for 5 and } n(B) \text{ for 2} \\ &= n(A – B) && \text{ by the set theory definition of subtraction} \\ &= n(\{v,x,y\}) && \text{ by computing }A – B \\ &= 3 && \text{ by counting the elements in } A – B \end{aligned} \nonumber \]

Therefore, 5 – 2 = 3.

Definition: Subtraction Missing-Addend Approach

Let a and b be any two whole numbers. a – b is the whole number c such that a = b + c. In other words, if c is added to the subtrahend, b, the sum is the minuend, a. The answer, c, is called the missing addend.

The missing-addend approach can be modeled using both set and measurement models. The number line diagram above illustrates the measurement model, showing how subtraction can be visualized by identifying the missing distance (or addend) between two numbers. For example, if a person starts at 3 and needs to reach 7, the missing addend is 4, reinforcing the idea that 7−3 can be thought of as "What do I add to 3 to get 7?". Additionally, children sometimes struggle with missing-addend problems when they are presented in isolation rather than in word problem contexts. Despite these challenges, the missing-addend approach remains a powerful way to deepen students' understanding of subtraction, particularly through real-world applications and the use of visual models like the number line.

Wonder, Play, Grow

Solve each problem and identify which approaches (SET, MEASUREMENT, TAKE-AWAY, MISSING ADDEND) you used.

a) Charles has 23 rocks that he found at the park. On his way home he lost 8 of them. How many does he have now?

b) Nancy helped Mario on his paper route. Nancy delivered 8 papers and Mario delivered 23 papers. How many more papers did Mario deliver than Nancy?

c) Emma wants to run 12 miles this week. So far, she has run 3 miles. How many more miles does she need to run to reach her goal?

d) Make up two word problems that would require the subtraction problem 8 – 3 to be computed. The first should use the take-away approach and the second should use the missing addend approach. Explain and show how you would solve each word problem using the given approach.

Answer

a) SET Model with Take-Away Approach b) SET Model, and you can use either the Take-Away or Missing Addend Approach

c) Measurement Model, and you can use either the Take-Away or Missing Addend Approach

If you think of subtraction in terms of the missing addend approach, then we say that the statements a – b = c and a = b + c are equivalent to each other. Consider the statement, 8 – 2 = 6. It is equivalent to the statement 8 = 2 + 6. We also know that 2 + 6 = 6 + 2 because of the commutative property of addition. Therefore, 8 = 6 + 2, which in turn is equivalent to the statement 8 – 6 = 2. This gives us four facts about how to relate the numbers 2, 6 and 8 using addition and subtraction

8 – 2 = 6

8 = 2 + 6

8 – 6 = 2

8 = 6 + 2

Some teachers relate subtraction and addition by using the idea of "fact families" such as the four facts above, which is one fact family.

It is important to note that each addition statement gives us two subtraction statements, which is how many people learn their subtraction facts. Because of this relationship between addition and subtraction, once a child learns basic addition facts, the subtraction facts naturally follow.

Example \(\PageIndex{5}\)

Write down two subtraction statements that are equivalent to an addition statement.

Answer

4 - 1 = 3, 4 - 3 = 1, 3 + 1 = 4

Think About and Discuss

Is there a whole-number answer for every whole-number subtraction problem? In other words, is subtraction of whole numbers closed? Explain your answer and provide a counterexample if it is not closed.

Example \(\PageIndex{6}\)

Determine which sets, if any, are closed under subtraction. Provide a counterexample if a set is not closed.

a. {0}

b. {0, 2, 4, 6, ...}

Answer

a) Closed - since 0 - 0 = 0. b) Not Closed since 6-8 = -2 or 12-20 = -8, you get negative numbers so those values are not in the original set.