Unit 2, Lesson 9
Solving Problems about Proportional Relationships
Grade 7
9.1
Warm-up
A person is running a distance race at a constant rate. What time will they finish the race?
What specific information do you need to be able to solve the problem?
9.2
Activity
Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.
If your teacher gives you the problem card:
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Silently read your card and think about what information you need to answer the question.
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Ask your partner for the specific information that you need. “Can you tell me _____?”
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Explain to your partner how you are using the information to solve the problem. “I need to know _____ because . . . .”
Continue to ask questions until you have enough information to solve the problem.
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Once you have enough information, share the problem card with your partner, and solve the problem independently.
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Read the data card, and discuss your reasoning.
If your teacher gives you the data card:
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Silently read your card. Wait for your partner to ask for information.
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Before telling your partner any information, ask, “Why do you need to know _____?”
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Listen to your partner’s reasoning and ask clarifying questions. Only give information that is on your card. Do not figure out anything for your partner!
These steps may be repeated.
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Once your partner says they have enough information to solve the problem, read the problem card, and solve the problem independently.
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Share the data card, and discuss your reasoning.
9.3
Activity
A company is hiring people to read through all the comments posted on their website to make sure they are appropriate. Four people applied for the job and were given one day to show how quickly they could check comments.
- Person 1 worked for 210 minutes and checked a total of 50,000 comments.
- Person 2 worked for 200 minutes and checked 1,325 comments every 5 minutes.
- Person 3 worked for 120 minutes, at a rate represented by ,
where is the number of comments checked and is the time in minutes. - Person 4 worked for 150 minutes, at a rate represented by .
- Order the people from greatest to least in terms of total number of comments checked.
- Order the people from greatest to least in terms of how fast they checked the comments.
Student Lesson Summary
Whenever we have a situation involving constant rates, we are likely to have a proportional relationship between quantities of interest.
- When a bird is flying at a constant speed, then there is a proportional relationship between the flying time and distance flown.
- If water is filling a tub at a constant rate, then there is a proportional relationship between the amount of water in the tub and the time the tub has been filling up.
- If an aardvark is eating termites at a constant rate, then there is a proportional relationship between the number of termites the aardvark has eaten and the time since it started eating.
Sometimes we are presented with a situation, and it is not so clear whether a proportional relationship is a good model. How can we decide if a proportional relationship is a good representation of a particular situation?
- If you aren’t sure where to start, look at the quotients of corresponding values. If they are not always the same, then the relationship is definitely not a proportional relationship.
- If you can see that there is a single value that we always multiply one quantity by to get the other quantity, it is definitely a proportional relationship.
After establishing that it is a proportional relationship, setting up an equation is often the most efficient way to solve problems related to the situation.
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