Section	Teacher-facing learning goals
What are Ratios?	Lesson 1 Comprehend the word “ratio” (in written and spoken language) and the notation \(a:b\) (in written language) to refer to an association between quantities. Describe (orally and in writing) associations between quantities using the language “For every \(a\) of these, there are \(b\) of those” and “The ratio of these to those is \(a:b\) (or \(a\) to \(b\)).” Lesson 2 Coordinate discrete diagrams and multiple written sentences describing the same ratios. Draw and label discrete diagrams to represent situations involving ratios. Practice reading and writing sentences describing ratios, e.g., “The ratio of these to those is \(a:b\). The ratio of these to those is \(a\) to \(b\). For every \(a\) of these, there are \(b\) of those.”
Equivalent Ratios	Lesson 3 Draw and label a discrete diagram with circled groups to represent multiple batches of a recipe. Explain equivalent ratios (orally and in writing) in terms of different sized batches of the same recipe having the same taste. Understand that doubling or tripling a recipe involves multiplying the amount of each ingredient by the same number, yielding something that tastes the same. Lesson 4 Comprehend and respond (orally and in writing) to questions asking whether two ratios are equivalent, in the context of color mixtures. Draw and label a discrete diagram with circled groups to represent multiple batches of a color mixture. Explain equivalent ratios (orally and in writing) in terms of the amounts of each color in a mixture being multiplied by the same number to create another mixture that is the same shade. Lesson 5 Generate equivalent ratios and justify that they are equivalent. Present (in words and through other representations) a definition of equivalent ratios, including examples and non-examples.
Representing Equivalent Ratios	Lesson 6 Compare and contrast (orally and in writing) discrete diagrams and double number line diagrams representing the same situation. Explain (orally) how to use a double number line diagram to find equivalent ratios. Label and interpret a double number line diagram that represents a familiar context. Lesson 7 Comprehend and use the word “per” (orally and in writing) to mean “for each.” Draw and label a double number line diagram from scratch, with parallel lines and equally-spaced tick marks. Use double number line diagrams to find a wider range of equivalent ratios. Lesson 8 Calculate equivalent ratios between prices and quantities and present the solution method (using words and other representations). Calculate unit price and express it using the word “per” (orally and in writing). Understand the phrase “at this rate” indicates that equivalent ratios are involved. Lesson 9 Calculate the distance an object travels in 1 unit of time and express it using a phrase like “meters per second” (orally and in writing). For an object moving at a constant speed, use a double number line diagram to represent equivalent ratios between the distance traveled and elapsed time. Justify (orally and in writing) which of two objects is moving faster, by identifying that it travels more distance in the same amount of time or that it travels the same distance in less time. Lesson 10 Choose and create diagrams to help compare two situations and explain whether they happen at the same rate. Justify that two situations do not happen at the same rate by finding a ratio to describe each situation where the two ratios share one value but not the other, i.e., \(a:b\) and \(a:c\), or \(x:z\) and \(y:z\). Recognize that a question asking whether two situations happen “at the same rate” is asking whether the ratios are equivalent.
Solving Ratio and Rate Problems	Lesson 11 Comprehend the words “row” and “column” (in written and spoken language) as they are used to describe a table of equivalent ratios. Explain (orally and in writing) how to find a missing value in a table of equivalent ratios. Interpret a table of equivalent ratios that represents different sized batches of a recipe. Lesson 12 Choose multipliers strategically while solving multi-step problems involving equivalent ratios. Describe (orally and in writing) how a table of equivalent ratios was used to solve a problem about prices and quantities. Remember that dividing by a whole number is the same as multiplying by an associated unit fraction. Lesson 13 Compare and contrast (orally) double number line diagrams and tables representing the same situation. Draw and label a table of equivalent ratios from scratch to solve problems about constant speed. Lesson 14 Determine what information is needed to solve a problem involving equivalent ratios. Ask questions to elicit that information. Understand the structure of a what-why info gap activity.
Part-part-whole Ratios	Lesson 15 Comprehend the word “parts” as an unspecified unit in sentences (written and spoken) describing ratios. Draw and label a tape diagram to solve problems involving ratios and the total amount. Explain (orally) the solution method. Lesson 16 Choose and create diagrams to help solve problems involving ratios and the total amount. Compare and contrast (orally) different representations of and solution methods for the same problem.
Let's Put it to Work	Lesson 17 Apply reasoning developed throughout this unit to an unfamiliar problem. Decide what information is needed to solve a real-world problem. Make simplifying assumptions about a real-world situation.

6.2 Introducing Ratios