Getting Started 
Misconception/Error The student does not use proportional reasoning. 
Examples of Student Work at this Level The student:
 Rewrites the numbers in the given ratio as improper fractions or decimals (e.g., as or as 2.25:6.75).
 Changes the ratio to decimal form (correctly or not) and writes each number in the ratio as and .
 Uses the given numbers to calculate a single value.

Questions Eliciting Thinking What is a ratio? What two quantities are being compared in this problem? How are they being compared? Is the order important?
Do you know what a unit rate is? A unit rate compares a number to what? 
Instructional Implications Review the concept of ratio. Describe ratios as comparisons of two quantities and point out that the quantities may or may not contain the same units of measure. Then, provide instruction on finding unit rates with associated whole number quantities. Describe unit rates as a comparison of one quantity to one unit of another quantity. Compare and contrast ratios, rates, and unit rates. Model how to determine unit rates from given rates.
Consider implementing CPALMS lesson plan Itâ€™s Carnival Time! (ID 47394), and then assessing the studentâ€™s progress with any of the following MFAS tasks: Writing Unit Rates (6.RP.1.2), Identifying Unit Rates (6.RP.1.2), Explaining Rates (6.RP.1.2), and/or Book Rates (6.RP.1.2).
Next, model finding unit rates with quantities that include fractions and mixed numbers. Review operations with fractions and mixed numbers as needed. Provide the student with additional opportunities to determine unit rates. Emphasize the meaning of the unit rate in context and model the use of ratio language when describing the meaning.
Consider implementing MFAS task Unit Rate Length (7.RP.1.1). 
Moving Forward 
Misconception/Error The student is unable to transform the ratio into a unit rate. 
Examples of Student Work at this Level The student:
 Writes the ratio rather than :1.
 Reverses the unit rate, comparing the new patio to the old patio, writing 3 to 1 for the unit rate.
 Performs rational number operations incorrectly.

Questions Eliciting Thinking Can you tell me what you mean by as the unit rate? Can you rewrite this fraction as a rate compared to one so that it tells us how many times smaller the old patio is compared to the new one?
Do ratios and rates have to contain only whole numbers? How can you change 1:3 to a ratio of x:1?
Is the order of the numbers in a ratio important? How could you write the original ratio as the area of the new patio to the area of the old patio? 
Instructional Implications Clarify the definition of unit rate as a comparison of one quantity to one unit of another quantity. Emphasize that the unit of one has to be the second part of the comparison. Reinforce the meaning of unit rates in context and encourage the student to use unit rate language (e.g., â€śfor every one,â€ť â€śfor each one,â€ť â€śper oneâ€ť) when describing the meaning of unit rates in context. Using a table, tape diagram or double number line, model how to find the second unit rate. Be sure to point out that the two parts of the ratio cannot simply be reversed from a:1 to 1:a. Make it clear that rates and ratios can contain fractions.
Consider implementing MFAS task Explaining Rates (6.RP.1.2), Computing Unit Rates (7.RP.1.1), and Comparing Unit Rates (7.RP.1.1). 
Almost There 
Misconception/Error The student makes errors in interpreting the unit rate in the context of the problem. 
Examples of Student Work at this Level The student correctly converts the ratio to a unit rate but when interpreting its meaning, the student:
 Explains that it indicates how much area needs to be added on to the old patio.
 Gives a vague explanation saying, â€śThe new patio is bigger than the old one.â€ť
 Gives an unclear or mathematically incorrect explanation of the unit rate saying, â€śOne unit of the old patio is .â€ť
 Makes a mathematical error in writing the unit rate as :1 and provides an unclear explanation.

Questions Eliciting Thinking What does the original ratio tell you? Would this help you interpret the unit rate?
What do you mean that the patio is bigger (or needs something added)?
What is the multiplicative relationship between the areas of the old and new patios?
What does :1 mean in the context of this problem? How are the areas of the patios related? 
Instructional Implications Model explaining the meaning of rates in the context of problems. Use unit rate language (e.g., â€śfor each oneâ€ť, â€śfor every oneâ€ť, and â€śper oneâ€ť) when interpreting unit rates or describing their meaning. Have the student practice writing descriptions of rates using rate and unit rate language.
Help the student to identify and correct any mathematical errors. When the student is ready, consider implementing this task again but with different rational numbers. 
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 a unit rate of to 1, :1 or 0.3333â€¦ to 1 and explains, â€śThe area of the old patio is the area of the new patio.â€ť

Questions Eliciting Thinking How did you determine the unit rate?
If a student said the correct unit rate is 1:3 or that the new patio is three times bigger than the old patio, what did that student do wrong?
Why does the second part of the unit rate have to be one?
What are other ratios that are equivalent to the unit rate of :1? 
Instructional Implications Have the student use the unit rate to solve a problem. For example, ask the student to determine:
 The size of the old patio if the new patio is to have an area of 12.
 The size of each patio if their combined areas are 40 sq. ft.
Pair the student with a Moving Forward student. Have the student explain to the Moving Forward partner how to find the unit rates and what each means. 