How to Use One Easy Addition Model to Solve a Billion Different Word Problems

Wouldn’t it be wonderful to have one addition model that would help you solve a ton of different problems? Read on to discover a “power tool” that works for simple grade-school type problems through more complicated algebraic problems.

Many “word problems” involve two quantities that add up to a third quantity. Two of the numbers are given, or some relationship between them is given, and you are asked to find the missing number(s). For example:

Males + females = total population
Price + tax = total cost
Beginning value + appreciation = new value

Being able to recognize these problems and apply the appropriate addition model is key to solving them.

Recall (or be introduced to) the elementary school concept of “fact families”: The numbers 2, 3, and 5 form an addition “fact family”; they are in an additive relationship with each other: 2 + 3 = 5

There are three other ways to express this very same relationship:
3 + 2 = 5 and 5 - 3 = 2 and 5 - 2 = 3

If you understand how numbers work, you don’t need to memorize these ways separately - the original implies the rest. There’s really just one relationship and four ways to express it. In general, whenever you have a situation where two small numbers add up to a BIG number, you’re dealing with this basic model:

Small#1 + small#2 = BIG# (just like 2 + 3 = 5)

This implies:

Small#1 = BIG# - small#2 (like 2 = 5 -3)

Small#2 = BIG# - small#1 (like 3 = 5 - 2)

[It’s also true that small#2 + small#1 = BIG# (like 3 + 2 = 5), but this fact isn’t helpful here, so we’ll drop it.]

In words, if you’re asked to find the BIG number, you simply add the two small ones. To find one of the small numbers, you subtract the other small number from the BIG number.

I recommend that you read and learn this addition model with words, like this: “Additive Relationship: One small number plus the other small number equals the BIG number.” Use words you can easily remember and quote to yourself and which appeal to your common sense. As part of the model, remember the simple example, 2 + 3 = 5, and its various forms.

My teaching experience has shown repeatedly that “speed kills in math.” Therefore, I give you this solemn recommendation: whenever you encounter a problem like this, no matter which number you’re being asked to find, begin by identifying the BIG number - the other two must be the small ones. Then write the general addition model, using appropriate words to capture the main idea, the key relationship. The more complicated the problem, the more important this step is.

Writing an equation with words first helps you use your common sense, express the logic of the problem, and get the model (equation) correct. (It also shows your teacher that you know what you’re doing - good for partial credit even if you flub the arithmetic or algebraic solution later.) This practice trains you to think algebraically which is worth big bucks in the long run, but that’s another topic...

After you write the main idea equation with words, you can fill in the numbers or relationships given and you’re on your way to a correct solution.

Now, let’s apply this addition model to four common problem types - Classification, Chunking, Comparison, and Change. Watch for the key pattern: three quantities, two small ones adding up to a BIG one. Your first job is to determine which is the BIG quantity and that should be a matter of common sense. Then you write your problem’s version of the model:

Small#1 + small#2 = BIG#

And “Do the math!”


Type 1: Classification (sorting)

In Classification Problems, you have a group of objects sorted into two subgroups:

# In group1 + # in group2 = TOTAL #

For example:
# Girls + # boys = TOTAL # students


Type 2: Chunking

Chunking problems are much like Classification problems, but deal with one whole quantity that’s divided into two chunks:

chunk1 + chunk2 = TOTAL

For example:
Price of item + sales tax = TOTAL cost


Type 3: Comparison

Comparison problems look at two quantities and the difference between them:

Small quantity + difference = BIG QUANTITY

For example:
Peewee’s weight + weight difference = BRUISER’S weight


Type 4: Change

Change problems consider how a single quantity changes over time. There are two types, INCREASE and DECREASE problems. The models are:

Starting value + increase = ENDING VALUE

For example:
Temp in am + temp increase = WARMER TEMP later in the day


Ending value + decrease = STARTING VALUE

For example:
Weight at end of after-holiday diet + weight loss = WEIGHT BEFORE DIET

Notice that time goes forward from left to right in the increase equation, but it goes from right to left in the decrease equation. This is because we always want to have the BIG number on the right end for our addition model. Math is patterns!


CAUTION: If your problem has the word “each” in it, it is probably not one of these types, at least, not a simple one. “Each” signals a multiplicative (not additive) relationship. E.g. if there are 3 packs of gum and each pack has 15 sticks, there must be 45 sticks of gum, since 3 * 15 = 45.

Practice using this power tool! Be on the lookout for additive relationships as you go through your day. Quiz yourself or your child on addition models: “What two small numbers here add up to a BIG number? What is the basic equation?” You should be able to spot many of these every day. Notice the pattern and think like a mathematician!

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