Getting Started 
Misconception/Error The student is unable to recognize equivalent ratios. 
Examples of Student Work at this Level The student determines that the ratios within each table are not equivalent because:
 There are different numbers in each row of the table.
 There is less cranberry juice than ginger ale in each row of the table.
 The same number cannot be added each time to the numbers in the left column to get the numbers in the right column.
 All the ratios look different and [the numbers] keep getting farther apart.
The student uses a faulty strategy for comparing the ratios, e.g., by forming ratios within the same columns, saying Â and Â for Chrisâ€™s table and Â and Â for Jennyâ€™s table.
The student says the ratios in Chrisâ€™s table are proportional because you can multiply each amount of cranberry juice by four to get the related amount of ginger ale, but the ratios in Jennyâ€™s table are not proportional because there is no number to multiply to get all the numbers to equal.

Questions Eliciting Thinking What is a ratio? Using Chrisâ€™s table, can you write an example of a ratio of cups of cranberry juice to cups of ginger ale?
What is the proportion of cranberry juice in Chrisâ€™s first recipe?
How can you tell if two ratios are equal? 
Instructional Implications Review the concept of a ratio and emphasize the meaning of specific ratios in context. For example, explain that if the ratio of cranberry juice to ginger ale is 1:4 then there is one cup of cranberry juice for each four cups of ginger ale in a recipe. Guide the student to use ratio language, e.g., â€śfor eachâ€ť, â€śfor everyâ€ť, and â€śper,â€ť when interpreting ratios or describing their meaning. Use models or drawings to illustrate the ratio. Then review what it means for ratios to be equivalent. Assist the student in devising strategies for determining when ratios are equivalent such as converting each ratio to a unit rate or writing each ratio as a fraction in lowest terms.
Be sure the student understands the multiplicative relationship between the parts of a ratio. Show the student that for each of Chrisâ€™s recipes, the amount of ginger ale is always four times the amount of cranberry juice. Explain that this also indicates that the ratio of cranberry juice to ginger ale is the same for each of Chrisâ€™s recipes. Provide an additional table entry such as nine cups of cranberry juice and ask the student to determine the amount of ginger ale needed to maintain the 1:4 ratio. Provide additional tables of ratios and ask the student to determine which ratios, if any, are equivalent and to explain why.
Provide continued instruction on rates and ratios. Provide realworld context that is accessible to the student and ask comparative questions in context. Encourage the student to use tape diagrams and double number lines to model and explore ratios. 
Moving Forward 
Misconception/Error The student is unable to compare two different ratios. 
Examples of Student Work at this Level The student answers that the two ratios represented by each table are the same because:
 They both have more ginger ale than cranberry juice.
 When you add each row of cranberry juice and ginger ale in the two tables they are equal, e.g., 1 + 4 = 2 + 3 or 2 + 8 = 4 + 6.
 If you add up all the numbers in each table they will both equal 55.
 The student answers that Chrisâ€™s recipe is stronger because he uses more ginger ale.

Questions Eliciting Thinking How can you compare two ratios?
What proportion of Chrisâ€™s punch is cranberry juice? What proportion of Jennyâ€™s punch is cranberry juice?
Can you convert each ratio to a unit rate? How can that help you find the punch with a higher proportion of cranberry juice? 
Instructional Implications Review the meaning of ratios and strategies for comparing ratios. Guide the student to convert each ratio, i.e., 1:4 and 2:3, to a unit rate and interpret the meaning of each unit rate in the context of the problem, e.g., â€śIn Chrisâ€™s recipe there is Â cup of cranberry juice per cup of ginger ale.â€ť Have the student compare the unit rates to determine which punch is stronger.
Additionally, assist the student in using each ratio to express the fraction of the punch that is cranberry juice. For example, the ratio 1:4 indicates that Chrisâ€™s punch consists of one part cranberry juice for each four parts of ginger ale so that the cranberry juice represents one of five total parts. Therefore Chrisâ€™s punch is Â cranberry juice. Use a visual model to make this conversion clearer. Ask the student to calculate the fraction of Jennyâ€™s punch that is cranberry juice and to compare the fractions to determine which punch is stronger.
Provide additional opportunities to compare ratios that arise in realworld contexts. 
Almost There 
Misconception/Error The student is unable to explain the comparison of the ratios using proportional reasoning. 
Examples of Student Work at this Level The student concludes that Jennyâ€™s punch contains a higher proportion of cranberry juice but offers an incomplete explanation. The student says:
 She uses more cranberry juice.
 Chris didnâ€™t add as much as Jenny.
 Jennyâ€™s recipe would be stronger because itâ€™s 2:3 (without further explanation).
The student offers an incorrect explanation for Jennyâ€™s punch having the stronger cranberry taste:Â
 Chris only started with one cup of cranberry juice compared to Jenny starting with two cups.
 Chris has to add three cups but Jenny only has to add one cup of ginger ale.
 All of Jennyâ€™s numbers are higher.
 Jennyâ€™s amounts of cranberry juice are always double Chrisâ€™s cranberry juice amounts.

Questions Eliciting Thinking How do you know that Jennyâ€™s recipe is stronger? Can you explain how you determined this?
Suppose each recipe contained one cup of cranberry juice and Chrisâ€™s recipe also contained two cups of ginger ale. How much ginger ale could Jennyâ€™s recipe contain so that Jennyâ€™s recipe is stronger? Weaker? 
Instructional Implications Model explaining the meaning of rates and ratios in the context of problems. Use ratio language, e.g., â€śfor eachâ€ť, â€śfor everyâ€ť, and â€śper,â€ť when interpreting or describing the meaning of ratios and unit rates. Model using and explaining unit rates to compare ratios. Expose the student to the explanations of his or her peers at the Got It level. Provide more opportunities for the student to justify his or her reasoning in comparing ratios. 
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 concludes that:
 Each of the ratios in Chrisâ€™s recipes is equivalent to 1:4 (or you can always multiply the amount of cranberry juice by four to get the amount of ginger ale).
 Each of the ratios in Jennyâ€™s recipes is equivalent to 2:3 (or you can always multiply the amount of cranberry juice by 1.5 to get the amount of ginger ale).
 Jennyâ€™s recipe is stronger because:
 She has a greater proportion of cranberry juice to ginger ale, compared to Chrisâ€™s recipe.
 For two cups of cranberry juice, Jenny will use three cups of ginger ale but Chris will use eight cups of ginger ale, so his will taste weaker.
 When Chris uses one cup of cranberry juice he uses four cups of ginger ale, but when Jenny uses one cup of cranberry juice she will only use 1.5 cups of ginger ale.
 Jennyâ€™s is stronger because Â of it is cranberry juice but in Chrisâ€™s only Â is cranberry juice.

Questions Eliciting Thinking What would the unit rate for each recipe be? If you want to make a recipe using only one cup of ginger ale, how much cranberry juice would you need for each recipe?
What amount of each ingredient would you need to use to result in a total of one cup (or a total of 20 cups) of each recipe? 
Instructional Implications Encourage the student to write an equation that shows the relationship between the amount of cranberry juice and the amount of ginger ale for each punch, e.g., g = 4c describes the relationship between these quantities for Chrisâ€™s punch where c is the amount of cranberry juice and g is the amount of ginger ale. Ask the student to use the equation to find the amount of one quantity given an amount of the other.
Provide the student with a fraction of a part of a twocomponent quantity, e.g., a quantity of chocolate milk is Â chocolate syrup and the remainder is milk. Challenge the student to determine the ratio of chocolate syrup to chocolate milk. 