Solve Word Problems with One Easy Addition Model: Chunking Problems
To effectively solve word problems you’ll need to learn about chunking problems. Chunking problems are a sub-type of additive relationship problems common in school math and daily life, so it's good to recognize them quickly and apply the appropriate model to solve them.
Brief review of Part 1, Overview:
Example addition "fact family": 2, 3, and 5.
Key concepts for our model:
- The sum of the two small numbers is the big number. 2 + 3 = 5
- The difference between the big number and either of the small numbers (BIG - small) is the other small number.
5 - 3 = 2
The model for additive relationship problems is:
Small#1 + small#2 = BIG#
(Please see http://tutorfi.com/Math/additionmodel for more details.)
Chunking problems fit this pattern: there's a total made up of two separate parts. Chunking problems are very much like the Classification problems discussed in http://tutorfi.com/Math/classificationproblems. The difference is that Classification problems deal with groups of countable items, while chunking problems involve one whole thing made of two separate parts. The basic chunking model is:
part1 + part2 = TOTAL
Price of item + sales tax = TOTAL cost
Solve Word Problems: Chunking 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- some total quantity, divided into two separate parts.
2. Following the generic model, use specific words to write a model for your problem.
Recommendation: While you're learning, write out the generic model first. The repetition will etch the pattern deeply in your brain, especially if you say it as you write it! Then follow the pattern of the generic model 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. 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 in order to solve word problems following the process.
Given a small number and the TOTAL, find the other small number.
You went shopping and spent $13.08. If the tax was $0.74, what was the pre-tax cost of the items you bought?
1. The model applies: there's a total (the amount you spent) and two separate parts (cost of items, tax).
2. Our specific model: cost of items + tax = TOTAL spent
3. Given: TOTAL spent = $13.08, tax = $0.74. Find cost of items.
4. Replace words with numbers in our model:
Cost of items + tax = TOTAL spent becomes
Cost of items + $0.74 = $13.08
5. Rearrange and subtract:
Cost of items = $13.08 - $0.74
Cost of items = $12.34
6. Check: $12.34 + $0.74 = $13.08, yes, and this makes sense.
7. Answer: The pre-tax price of the items bought was $12.34.
Are you thinking, "Good grief! I knew the answer in a few 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 in order to solve word problems. Then apply the same process to more complicated problems. Doing this builds your algebraic thinking skills.
Given a TOTAL and a relationship between the small numbers, find both small numbers.
You're planning a trip to your favorite amusement park, and you're going to go through the town where your cousins live to take them with you. The distance from your cousins' house to the park is twice the distance from home to their house, and the whole trip is 240 miles. How far is it from home to your cousins' house and from there to the park?
1. The model applies: TOTAL distance of the trip, two separate parts (home to cousins, cousins to park).
2. Model: home to cousins + cousins to park = TOTAL distance
TOTAL distance = 240.
Relationship: It's twice as far from cousins to park as it is from home to cousins, so
Cousins to park = 2 * home to cousins
Try it: if it were 50 miles from home to cousins, it would be 100 (twice as many) miles from cousins to park. Our expression works: 100 = 2 * 50.
4. Replace words with numbers and the relationship expression:
Home to cousins + cousins to park = TOTAL distance becomes
Home to cousins + 2 * home to cousins = 240
Home to cousins + 2 * home to cousins = 240
3 * home to cousins = 240
3 * home to cousins = 3 * 8Home to cousins = 80
Using the relationship in Step 3, since its 80 miles from home to cousins, Cousins to park = 2 * 80 = 160 miles
6. Check: 80 + 160 = 240, and 80 * 2 = 160. True and these numbers fit the given information.
7. Answer: It is 80 miles from home to your cousins, and another 160 miles from their home to the park.
Remember! Practicing the model on simple problems will help you form good habits to help you solve word problems that are trickier. This kind of algebraic thinking -model-based, linear and precise - is key to successful completion of a college degree and highly valued in the best-paying jobs.
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