Getting Started 
Misconception/Error The student attempts to use multiplicative reasoning but makes significant errors. 
Examples of Student Work at this Level The student indicates using multiplication and division but is unable to produce equivalent ratios.
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Questions Eliciting Thinking How can you tell if two ratios are equivalent?
Can you tell me exactly what you did when you rewrote 3:12 as 2:6?
Can you write a ratio or a fraction that is equivalent to ? How can you check your answer? 
Instructional Implications Review what it means for ratios to be equivalent. Be sure the student understands the multiplicative relationship between equivalent ratios. Assist the student in devising strategies for determining when ratios are equivalent such as converting each ratio to a unit rate or testing for a constant of proportionality.
Model generating ratios equivalent to a given ratio by multiplying or dividing both parts of the ratio by the same value. Then have the student test the ratios to determine if they are equivalent. Provide additional opportunities to write ratios equivalent to a given ratio and to determine if a set of given ratios is equivalent. 
Moving Forward 
Misconception/Error The student uses additive reasoning to write equivalent ratios. 
Examples of Student Work at this Level The student repeatedly adds three to the previous number of tablespoons of coffee and 12 to the previous number of ounces of water to generate equivalent ratios.
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Questions Eliciting Thinking How is the number of ounces of water related to the number of tablespoons of coffee?
Can you use multiplication to generate equivalent ratios?
Can you use division to generate equivalent ratios?
Can you determine how many ounces of water are needed for 25 tablespoons of coffee?
Can you write the given ratio as a unit rate? 
Instructional Implications Transition the student thinking about equivalent ratios in multiplicative terms. Show the student that for each of the entries in the ratio table, the amount of water is always four times the amount of coffee. Explain that this indicates that each ratio is equivalent to the given ratio. Guide the student to identify this value, four, as the constant of proportionality and to use it to generate more equivalent ratios by selecting additional amounts of coffee and multiplying each one by four to calculate the associated amount of water.
Guide the student to write an equation (e.g., y = 4x) to represent the relationship between the amount of water and the amount of coffee in the table. Emphasize that each ratio in the table satisfies this equation. Provide additional ratios and ask the student to write an equation that models the relationship between the parts of the ratio and to use the equation to generate equivalent ratios. 
Almost There 
Misconception/Error The studentâ€™s explanation is incomplete or contains a misuse of mathematical terminology. 
Examples of Student Work at this Level The student:
 Simplifies to and then says equivalent ratios were generated by â€śmultiplying the denominator by 4,5,6,7 and 8.â€ť
 Explains how to find the constant of proportionality but does not explain how other ratios were calculated.
 Explains how to rewrite the given ratio as a unit rate but does not explain how other ratios were calculated.

Questions Eliciting Thinking Did you only multiply the denominator of by 4, 5, 6, 7 and 8 to generate equivalent ratios?
How did you use the constant of proportionality to write equivalent ratios?
How did you use the unit rate to write equivalent ratios? 
Instructional Implications Review related terminology such as constant of proportionality and unit rate. Assist the student in revising the explanation so that it is complete and correct. Provide additional opportunities to explain and justify mathematical processes. 
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 completes the table with correctly written equivalent ratios. The student indicates calculating the constant of proportionality and using it to generate equivalent ratios.
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Questions Eliciting Thinking I see that you calculated the ratio of 1:4. What is this ratio called and how is it useful?
What if you had n tablespoons, what expression would you write for water?
How are the unit rate and the constant of proportionality related? 
Instructional Implications Encourage the student to write an equation that shows the relationship between the amount of coffee and the amount of water for each entry. Ask the student to use the equation to find the amount of one quantity given an amount of the other.
Challenge the student to write an equivalent ratio in which one quantity is given as a fraction. For example, ask the student how much water is needed for 1 tablespoons of coffee.
Provide the student with a fraction of a part of a quantity (e.g., a quantity of chocolate milk is chocolate syrup and the remainder is milk). Challenge the student to determine the ratio of chocolate syrup to milk and to use it to write equivalent ratios. 