Getting Started |
| Misconception/Error The student is unable to calculate or compare the rates. |
| Examples of Student Work at this Level The student:
- Says the roadrunner is faster because it ran longer (or more miles) than the elephant.
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- Calculates unit rates in the form hours per mile but does not understand how to interpret these rates.
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| Questions Eliciting Thinking Why did you make your decision based only on the number of hours (or miles)? What else do you have to consider before making the decision?
How far can the roadrunner go in one hour? How far can the elephant go in one hour? Which runs faster?
If you gave both animals the same amount of time, which one would go farther? If both animals went the same distance, which one would get there first? |
| Instructional Implications Provide direct instruction on the meaning of rates and ratios. Describe rates and ratios as comparisons of two quantities and point out that the quantities might not contain the same units of measure. Guide the student to use ratio language, e.g., “for each,” “for every,” and “per,” when interpreting rates or describing their meaning. Use tape diagrams and double number lines to model rates and help students visualize the parts and whole. Give the student additional opportunities to write and interpret ratios in the context of a variety of problems.
Introduce the student to the concept of a unit rate and guide the student to calculate unit rates from given rates. Then work with the student in using the unit rate to find other equivalent rates. Provide additional opportunities for the student to calculate and use unit rates to make comparisons and draw conclusions in contextual problems. Consider implementing MFAS task 6.RP.1.2 Writing Unit Rates. |
Moving Forward |
| Misconception/Error The student is unable to use a given rate to find an equivalent rate. |
| Examples of Student Work at this Level The student uses the data in the table to correctly calculate and compare unit rates. When attempting to determine how long it would take the roadrunner to run 60 miles, the student:
- Subtracts two hours from four hours and says it will take two hours to go 60 miles.
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- Estimates the time and explains it will take 3 hours because 60 mph is between 50 mph and 80 mph, and 3 hours is between 2 hours and 4 hours.
- Attempts to write or solve a proportion but does so incorrectly.Â
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| Questions Eliciting Thinking How did you determine which animal is faster? Can you use this work to help you determine how long it would take the roadrunner to run 60 miles?
You made a good estimate, but can you think of a way to answer this question precisely?
Can you explain how you wrote this proportion? How did you decide where to put the values 4, 60, 80, and x? What does x represent in your proportion?
Can you explain how you solved your proportion? Can you solve it without trying to cross-multiply? Is there another way that makes sense? |
| Instructional Implications Ask the student to make a table for the roadrunner that includes the hours from 1 – 8. Have the student complete entries corresponding to the unit rate and then find the remaining corresponding distances. Guide the student to think about equivalent ratios in terms of their multiplicative relationship. Model a variety of strategies for solving problems involving equivalent ratios. Encourage the student to use ratio tables, tape diagrams, and double number lines as an alternative to writing and solving proportions.
Expose the student to the strategies of Got It students to observe alternative approaches to solving the problem. Provide the student with additional opportunities to find and compare unit rates in the context of real-world problems. |
Almost There |
| Misconception/Error The student provides incomplete explanations or justifications. |
| Examples of Student Work at this Level The student answers are correct but the supporting work is missing or the explanation is incomplete.
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| Questions Eliciting Thinking Can you explain to me what you did to get your answer? What is another way you can explain it to me?
Can you show your work mathematically? |
| Instructional Implications Assist the student in revising the written work and explanation so that it is complete and correct. Expose the student to the examples of appropriately shown work of other classmates. Provide additional opportunities to communicate and explain mathematical processes and solutions. |
Got It |
| Misconception/Error The student provides complete and correct responses to all components of the task. |
| Examples of Student Work at this Level The student says:
- The elephant is faster because it travels 25 mph, compared to 20 mph for the roadrunner.
- It will take the roadrunner three hours to run 60 miles because it can go 20 miles each hour.
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| Questions Eliciting Thinking How long will it take the roadrunner to go 30 miles? How far will the elephant travel in one day?
Is the unit rate you found an exact speed the animal traveled at every moment for the full two (or four) hours? Explain.
Is it likely that each animal will continue at the same speed for many hours? What will likely happen?
Instead of calculating miles per hour, can you calculate hours per mile? How would you interpret those unit rates? |
| Instructional Implications Ask the student to calculate unit rates as miles per hour and hours per mile for each animal and consider when one might be better to use than another. Provide the student with data from other contexts and ask the student to calculate unit rates in order to make comparisons. Have the student consider how to interpret each version of a unit rate, e.g., dollars per gallon or gallons per dollar, when making a comparison.
Provide the student with additional opportunities to write and compare ratios in which one part is not a multiple of the other part. Provide experience with values that result in fractional unit rates. |
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