One Easy Addition Model for a Billion Word Problems: Change Problems
Change problems or problems dealing with some measurement that changes over time 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.
The general model for additive relationship problems is:
Small#1 + small#2 = BIG# (like 2 + 3 = 5)
Change problems involve a single quantity measured at two different times. These are like the comparison problems dealt with in http://tutorfi.com/Math/comparisonproblems, except Comparison problems look at measurements taken at the same time on two different objects. Look for descriptions of change using words like "gained" and "lost" to recognize this type of problem.
Following the additive relationship model that always has the BIG# at the right end of the equation we need two forms of the basic Change model, one for increases and one for decreases. They are:
Starting value + increase = ENDING VALUE
Cold temp in am + temp increase = HOT TEMP later in the day
Ending value + decrease = STARTING VALUE
Weight at end of after-holiday diet + weight loss = WEIGHT BEFORE DIET
Notice that time goes backwards from left to right in the decrease equation. This is because the model requires the BIG number on the right end.
SOLVING CHANGE 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. Identify the situation- Some quantity being measured has changed over time.
2. Following the appropriate generic model, use specific words to write a model for your problem.
Tip: Writing out the generic model first will help to etch the pattern deeply in your brain, especially if you say it as you write it! Follow the generic pattern to write your specific model.
3. Summarize the information given and requested. This is very easy when given numbers, trickier when given relationships. Write relationships with (semi-) algebraic expressions and test them with specific examples before proceeding. Use variables in your expressions if you like.
4. Replace the words of your model with the appropriate numbers and expressions.
5. Solve the resulting equation:
a. If you're finding the BIG amount, just add the two small numbers.
b. If you're finding the small amount or the increase or decrease, 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.
Given the change and the BIG quantity, find the small quantity.
The temperature one afternoon was 98 degrees. It had gone up 33 degrees since you first checked in the morning. What was the morning temperature?
1. The model applies: a changing quantity (temperature) is measured at two different times.
2. Use the INCREASE pattern:
AM temp + increase = PM temp
3. Given: PM temp = 98 degrees, increase = 33 degrees. Find AM temp.
4. Replace words with numbers in our model:
AM temp + increase = PM temp becomes
AM temp + 33 = 98
5. Rearrange and subtract:
AM temp = 98 - 33
AM temp = 65
6. Check: 33 + 65 = 98, yes, and this makes sense.
7. Answer: The morning temperature was 65 degrees.
Are you thinking, "Good grief! I did this in my head in a second. This took forever!"? Please be patient - illustrating a powerful process on a simple problem helps you learn the process to use on more complicated problems. "Under construction: Algebraic Thinking Skills!"
Given a BIG value and a relationship between the small value and the change, find the size of the change and the small value.
Sarah went off to college and gained a tenth of her pre-college weight during her freshman year. (Gotta watch out for that dining hall food!) If she weighed 132 pounds after freshman year, how much did she weigh before college, and how much weight did she gain?
1. The model applies: We're considering the change in Sarah’s weight at two different times.
Starting weight + increase = ENDING WEIGHT
ENDING WEIGHT = 132.
Relationship: Her weight gain was a tenth of her starting weight. That means
Increase = .1 * starting weight
Try it: if Sarah had weighed 100 pounds, she would have gained 10 more. Our expression works: 10 = .1 * 100.
4. Replace words with numbers and the relationship expression:
Starting weight + increase = ENDING WEIGHT
Starting weight + .1 * starting weight = 132
Starting weight + .1 * starting weight = 132
1.1 * starting weight = 132
Now divide both sides by 1.1:
(1.1 * starting weight)/1.1 = 132/1.1
Starting weight = 120
Using the relationship in Step 3,
Increase = .1 * starting weight = .1 * 120 = 12
Another way to find the increase would be to subtract directly:
Increase = ENDING WEIGHT – starting weight = 132 - 120 = 12
6. Check: 120 + 12 = 132, and .1 * 120 = 12. True and these numbers fit the given information.
7. Answer: Sarah weighed 120 pounds before college, and she gained 12 pounds.
Remember! Practicing the model on simple problems will help you solve change problems and form good habits to use for trickier problems. 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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