Getting Started |
| Misconception/Error The student does not understand the concept of a ratio. |
| Examples of Student Work at this Level The student:
- Multiplies 30 and 10 and writes 300 as the ratio.
- Divides 30 by 10 and writes 3 as the ratio.
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| Questions Eliciting Thinking What is a ratio? Can you show me how to write a ratio?
Why did you multiply (or divide) the two numbers?
What did the question ask for? |
| Instructional Implications Provide instruction on the concept of a ratio. Describe ratios as comparisons of two quantities and explain that the compared quantities may or may not contain the same units of measure. Emphasize the meaning of ratios in context and the use of ratio language (e.g., “for each,” “for every,” and “per”) when interpreting ratios. Be sure to explain the conventions typically used in writing ratios. Give the student additional opportunities to write and interpret ratios in the context of a variety of problems.
Consider using manipulatives and drawings to model ratios. Emphasize the multiplicative relationship between the parts of the ratio and ask the student to write ratios in various equivalent forms (e.g., given the ratio 2:10, write other equivalent ratios such as 1:5 and 3:15) that reflect the modeled ratio. Guide the student to observe that one part of the ratio is always a constant multiple of the other part (e.g., for equivalent ratios 2:10, 1:5, and 3:15, the second part is always five times the first part).
Consider implementing MFAS tasks Writing Ratios and Interpreting Ratios, if not done previously. |
Moving Forward |
| Misconception/Error The student writes the ratio incorrectly. |
| Examples of Student Work at this Level The student writes the ratio as 30:10, 10:30, 3:1, or 1:3. Additionally, the student is unable to explain the meaning of the ratio.
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The student writes the ratio as 40:30 and explains, “The bigger number comes first in a ratio.”
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| Questions Eliciting Thinking How did you determine the ratio?
How many minutes did Abe practice?
If Malik practiced 10 minutes longer, how many minutes did Malik actually practice?
Can you read the question aloud and tell me what mistake you made? |
| Instructional Implications Directly relate the way the ratio is described in the problem to the quantities used in writing the ratio. Ask the student to first write a description of the ratio using the exact wording given in the problem (e.g., Abe’s practice time to Malik’s practice time). Then instruct the student to find or calculate each quantity. Provide the student with a variety of contexts in which ratios are described and ask the student to write ratios including some in which quantities have to be calculated. Emphasize the meaning of ratios in context and the use of ratio language (e.g., “for each,” “for every,” and “per”) when interpreting ratios.
Consider implementing MFAS task Writing Ratios. |
Almost There |
| Misconception/Error The student is unable to explain the meaning of the ratio. |
| Examples of Student Work at this Level The student writes the ratio as 30:40 or 3:4 but is unable to interpret its meaning. The student says:
- The smallest number should always go first.
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- Abe practiced 30 minutes while Malik practiced 40 minutes.
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- Malik practiced 10 minutes longer than Abe.
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The student correctly writes and interprets the ratio, but does not understand the multiplicative relationship between the parts of the ratio.
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| Questions Eliciting Thinking What determines in what order the ratio is written?
What does the ratio 30:40 mean? What would the ratio 3:4 mean?
Suppose I said the ratio of boys to girls in our class is 3:2. Would it mean that there are only three boys and two girls in the class?
Suppose the ratio of Abe’s practice time to Malik’s practice time continues to be 30:40. How long would Malik have practiced if Abe practiced for 60 minutes? |
| Instructional Implications Model explaining the meaning of ratios in the context of problems. Use ratio language (e.g., “for each,” “for every,” and “per”) when interpreting ratios or describing their meaning.
Explore using ratios to find missing values in problem contexts (e.g., ask the student to find the number of boys in a class when the ratio of the number of girls to the number of boys is 3:2 and the number of girls is 12). Encourage the student to make drawings or use manipulatives to solve problems such as these rather than setting up and solving proportions. |
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 writes the correct ratio 30:40 (or 3:4) and explains that this ratio means that for every 30 (or 3) minutes that Abe practices, Malik practices 40 (or 4) minutes.
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| Questions Eliciting Thinking How do you know 10 minutes is not part of the ratio?
How many times longer did Malik practice than Abe?
If the ratio of Abe’s practice time to Malik’s practice time is 2:3, than how long would Malik practice when Abe practices 30 minutes? |
| Instructional Implications Introduce the student to the concept of equivalent ratios. Give the student a ratio and ask him or her to write a number of equivalent ratios. Ask the student to describe how to form ratios equivalent to a given ratio.
Ask the student to explain how the numbers in equivalent ratios can change but the relationship between the numbers remains the same.
Explore using ratios to find missing values in problem contexts (e.g., ask the student to find the number of boys in a class when the ratio of the number of girls to the number of boys is 3:2 and the number of girls is 12). Encourage the student to make drawings or use manipulatives to solve problems such as these rather than setting up and solving proportions. |
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