One Easy Addition Model for a Billion Word Problems: Classification Problems

Classification problems are a sub-type of additive relationship problems common in school math and daily life, so it's good to be able to recognize them and quickly apply the appropriate model to solve them.

Brief review of Part 1: A sample addition "fact family" is 2, 3, and 5. Four different equations express the relationship among the numbers, but you don't need to memorize them all. You just need to know the fact family (2, 3, 5) and the key concepts:

- The sum of the two small numbers is the big number.

- The difference between the big number and either of the small numbers (BIG - small) is the other small number.

The four equations are all the ways to write those facts.

The model for additive relationship problems is:

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

This implies:

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

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


Classification problems fit this pattern. They involve a collection of objects sorted into two separate (not overlapping) groups:

# In group1 + # in group2 = TOTAL #

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


SOLVING CLASSIFICATION PROBLEMS

When you are given any two of the numbers, or one of the numbers and a relationship between two of the numbers, you can find the missing number(s). The process is:

1. Recognize the situation- A collection of items, sorted into two separate groups.

2. Following the generic model, use specific words to write a model for your problem.

Note: Until you're good at these problems, I recommend writing out the generic model first. Doing that repeatedly is a good way to etch the pattern deeply in your brain, especially if you say it as you write it! Then use the generic model as a pattern to write the specific model for your problem.

3. Summarize the information given and requested- This is very easy when given numbers, trickier when given relationships between numbers. You need to write the relationships with (semi-) algebraic expressions and test them with specific examples to make sure they're right. You may use variables in your expressions if you're comfortable doing this.

4. Replace the words of your model with the appropriate numbers and expressions.

5. Solve the resulting equation
a. If you're finding the TOTAL #, just add the two small numbers.
b. If you're finding one of the small numbers, rearrange the equation and do a subtraction.
c. If there's a relationship involved, you'll need to do some algebraic-type work to find one number and go back to the relationship from Step 3 to find the other.

6. Check your answers to be sure they fit the information given and make sense. Redo if they don't.

7. When you're confident your solution is correct, write it in a complete sentence.


Here are two examples following the process.


Example 1
Given a small number and the TOTAL, find the other small number.

A vet is caring for 32 pets. There are 18 dogs. How many of the pets are not dogs?

1. The model applies: there's a collection (animals at the vet) and two separate subgroups (dogs and not-dogs).

2. Our specific model: # dogs + # not-dogs = TOTAL # animals

3. Given: # dogs = 18, TOTAL # animals = 32. Find the # not-dogs.

4. Replace words with numbers in our model:
# dogs + # not-dogs = TOTAL # animals becomes
18 + # not-dogs = 32

5. Rearrange and subtract:
# not-dogs = 32 - 18
# not-dogs = 14

6. Check: 14 + 18 = 32, yes, and this makes sense.

7. Answer: There are 14 animals that are not dogs.

Are you thinking, "Good grief! I knew the answer in two seconds. This took forever!”? I understand. But this is how we do things in math - illustrate a powerful process on a simple problem to learn the process. Then apply the same process to more complicated problems.


Example 2
Given a TOTAL and a relationship between the small numbers, find both small numbers.

A tray of hot dogs and hamburgers has 36 items on it. There are twice as many hot dogs as hamburgers. How many of each is there?

1. The model applies: collection of food items, two separate groups (hot dogs, hamburgers).

2. # hot dogs + # hamburgers = TOTAL # items

3. Given:
TOTAL = 36.
Relationship: There are twice as many hot dogs as hamburgers, so
# hot dogs = 2 * (# hamburgers)
Try it: if there were 3 burgers, there would be 6 (twice as many) hot dogs. Our expression works: 6 = 2 * 3.

4. Replace words with numbers and the relationship expression:
# hot dogs + # hamburgers = TOTAL # items becomes
2 * (# hamburgers) + # hamburgers = 36

5. Solve:
3 * (# hamburgers) = 36
3 * (# hamburgers) = 3 * 12
# hamburgers = 12

From the relationship in Step 3, since there are 12 burgers,
# hot dogs = 2 * 12 = 24

6. Check: 24 + 12 = 36, and 12 * 2 = 24. True and these numbers fit the given information.

7. Answer: There are 12 hamburgers and 24 hot dogs.


Practice using this model when solving classification problems even if the problems are simple and you'll develop good habits to lean on for trickier problems. You'll be way ahead in learning to use algebraic thinking. That kind of thinking ability - model-based, linear and precise - is key to successful completion of a college degree and is highly valued in the best-paying jobs.

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