5.1: Addition and Subtraction

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Page ID: 132886

Amy Lagusker
College of the Canyons

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Additive Problems

Any problem that involves addition or subtraction can be called an "additive problem". Both addition and subtraction represented with an equation involving a sum, and just because a problem might be written in a way that refers to addition (or subtraction) doesn't mean everyone will solve the problem with addition (or subtraction). For example, in the following problem, the situation is presented as an addition problem. But most people solve this problem with subtraction.

Example $\PageIndex{1}$

Viktor has a box of marbles. His friend gives him 9 marbles. Afterward, Viktor counts the total of all the marbles and finds he now has 42 marbles. How may marbles did Viktor have before he got more marbles from his friend?

Solution

One way to answer this is to first note that the problem can be summarized as follows.

Viktor's original marbles + 9 marbles from his friend = 42 marbles total

Since we already know the relationship between addition and subtraction, we can rewrite this as follows.

Viktor's original marbles = 42 marbles total – 9 marbles from his friend

Since 42 – 9 = 33, Viktor had 33 marbles before he got more marbles from his friend.

It is tempting to assume that all problems like the one in the previous example "should" be solved this way. But consider if we simply changed the sizes of the values.

Example $\PageIndex{1}$

Wally has a box of dice. His friend gives him 52 dice. Afterward, Wally counts the total of all the dice and finds he now has 59 dice. How many dice did Wally have before he got more dice from his friend?

Solution

One way to answer this is to imagine acting it out. Pretend you are Wally and have a box of dice. Next to the box of dice, you have the 52 dice from the friend. When you count them out afterward, suppose you started with the 52 new dice and counted on from there to count the box of dice. Since you know that the final result is 59, there can't be that many dice in the box to begin with. In fact, even if you only knew how to count on your fingers, you could quickly count from 52 to 59 and get that there had to be 7 dice in the box.

So, Wally had 7 dice before he got more dice from his friend.

What these two examples highlight is that if someone is comfortable with multiple mathematical strategies for solving additive problems, they can pick the strategy that works best for each given problem. If a child learns only one strategy for solving additive problems (or even specific types of additive problems), then they are likely to encounter problems for which that one strategy is quite inefficient and prone to producing errors.

Before continuing, it will help if we have a few definitions.

Definitions: Addend, Sum, Subtrahend, Minuend, and Difference

Two values added together are each called addends. The total of those values is called the sum. For example, in $5+9=14$, 5 and 9 are addends and 14 is the sum.

When one value is subtracted from another, the value being subtracted is called the subtrahend, and the value the subtrahend is subtracted from is the minuend. The result of the subtraction is called the difference. For example, in $7-3=4$, 7 is the minuend, 3 is the subtrahend, and 4 is the difference.

Types of Addition & Subtraction Exercises

For now, let's focus on exercises rather than word problems or story problems. Even by keeping things simple, we have a variety of ways of thinking of addition and subtraction. It is important to note that we will use "?" to represent what is unknown (i.e., what we want to find). This is the foundation of algebra! So, even if you work with very young children, you are helping them lay the foundations of what they will be learning many years later in algebra courses.

Three types of addition exercises:

Missing Sum
1. $7+3 =$ ?
2. Story problem example: "Nestor ate 7 pieces of candy and Oscar ate 3 pieces of candy. How many pieces of candy did they eat in total?
Missing Start (first addend)
1. ? $+3=10$
2. Story problem example: Kaitlyn walked some dogs on Saturday and then three more dogs on Sunday. In totall, she walked 10 dogs on Saturday and Sunday. How many dogs had she walked on Saturday?
Missing Change (second addend)
1. $7+$ ? $=10$
2. Story problem example: Deon rode his bike 7 miles to the park. Afterward, he rode further until his GPS showed he had ridded a total of 10 miles. How much further did he ride after he rode to the park?

There are many sub-types of addition problems, but notice that these three main types really depend on where the unknown value is in an equation with a single addition operation, which has three values: two addends and a sum. Similarly, there are three possibilities for the unknown value in an equation with a single subtraction operation: a minuend, a subtrahend, and a difference.

Three types of subtraction exercises:

Missing Minuend
1. ? $- 3 = 7$
2. Story problem example: “Mary has seven bananas, but she started with more than that and then ate three. How many bananas did she start with?”
Missing Subtrahend
1. $10 - $ ? $=7$
2. Example: “Jennifer has ten oranges. She sold some of them and has 7 left. How many did she sell?”
Missing Difference
1. $10-3=$ ?
2. Example: “Cory has ten oranges and three bananas. How many more oranges does Cory have than bananas?”

Perform the operation $354-89$. Do not use a calculator. Then continue reading.

What kind of exercise did you just do? It's likely that when you think of "subtraction", you think of this type of exercise. We now know there are many kinds of situations that involve subtraction, or involve addition that are solved by performing a subtraction operation.

Additionally, we must consider that there are many ways of performing addition and subtraction. How did you subtract 89 from 354? After you have thought about this, find someone else and ask them to do it. Or maybe try to do it a different way. Maybe pretend that you didn't know whatever method you used to perform the subtraction. Then compare the methods.

Methods of Subtraction

Consider the work of nine 2^nd graders, all subtracting 354-89, just like you did. Grade each student, as if you were their teacher, using a scale from 1 – 5, 5 being the best. Having the correct answer is only one point out of five. The other four points come from the students’ procedure and thoughts. Remember, even if you do not understand HOW they arrived at their correct answer, that does not make their procedure incorrect.

Work of nine children calculating a difference — Figure 3.2.2

Now, ask yourself these questions. Were you surprised by any of the children's methods in the work above? Did you figure out how each of them did their work? Did the students who got the wrong answer use a poor method or just make a mistake while using a find method?

Interestingly, all of the children performing $354-89$ used methods that involved actually subtracting. They could have used an "adding up" strategy. Consider the following example.

Example $\PageIndex{2}$

Lance buys some supplies totaling $7.32. He hands the cashier a ten-dollar bill. His change is $2.68. If the cashier counts back his change, how might the cashier do this?

Solution

The cashier will count up in the following say if they use the fewest coins as possible without using a 50¢ coin:

“$7.32 + $1 + $1 + 25¢ + 25¢ + 10¢ + 5¢ + 1¢ + 1¢ + 1¢ = $10.00”

Even if we are only dealing with a subtraction exercise without a context like money, we can still use an adding up strategy.

Example $\PageIndex{3}$

Subtract 342 – 186 using a number line and adding up.

Solution

One way to do this is to start at 186, add 4 to get to 190, add 10 to get to 200, add 100 to get to 300, and add 42 to get to 342. Since the total added up is 156, we know $186+156=342$. See the number line below.

This means $342-186=156$.

Notice that one advantage to using an adding up strategy is that you do not have to worry about whether a digit of the subtrahend is smaller than a digit of the minuend. In terms of the standard way children are taught an algorithm for subtracting in the United States, this means there is no need to "borrow".

Practice Problems

Perform the following subtractions by adding up.

753 – 345 = ________
421 – 175 = ________

Search

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Example \(\PageIndex{1}\)

Solution

Example \(\PageIndex{1}\)

Solution

Definitions: Addend, Sum, Subtrahend, Minuend, and Difference

Example \(\PageIndex{2}\)

Example \(\PageIndex{3}\)