Ratios: Unbound
A Guide to Grade 7 Mathematics Standards
7.RP.A  Analyze proportional relationships and use them to solve realworld and mathematical problems.
Welcome to the UnboundEd Mathematics Guide series! These guides are designed to explain what new, high standards for mathematics say about what students should learn in each grade, and what they mean for curriculum and instruction. This guide, the first for Grade 7, includes three parts. The first part gives a “tour” of the standards for Ratios & Proportional Relationships using freely available online resources that you can use or adapt for your class. The second part shows how Ratios & Proportional Relationships relate to other concepts in Grade 7. And the third part explains where Ratios & Proportional Relationships are situated in the progression of learning from Grades 38.
The standards for Grade 7 contain a number of important ideas, so why begin this series with Ratios & Proportional Relationships? For starters, these standards are part of a major cluster, meaning they deserve a significant amount of class time over the course of the school year. (It’s generally a good idea to prioritize major standards within the year to make sure they get the attention they deserve.) Proportional relationships are particularly important because they’re a crucial step on the path to algebra, connecting multiplication and division from the elementary grades to linear equations, slope, and other concepts in Grade 8 and high school.^{1} In this series, major clusters and standards are denoted by a ▉. For more information on the major work of Grade 7, see the Student Achievement Partners guide Focus in Grade 7. In this series, major clusters and standards are denoted by a ▉. For more information on the major work of Grade 7, see the Student Achievement Partners guide Focus in Grade 7. In this series, major clusters and standards are denoted by a ▉. For more information on the major work of Grade 7, see the Student Achievement Partners guide Focus in Grade 7.
Proportional relationships are also a great way to start the year because they link directly to what students should learn about ratios and rates in Grade 6. Moveover, some of the other work in Grade 7 (such as the standards for Expressions & Equations and Geometry) is easier for students to understand once they have an understanding of proportional relationships. If you’re wondering where to start your year, proportional relationships are a solid bet.
In Grade 7, the standards in the Ratios & Proportional Relationships (RP) domain are grouped together into one cluster (called RP.A, since it’s the first and only cluster). Let’s take a look at precisely what the standards say, and then we’ll discuss each one more thoroughly.
7.RP.A  Analyze proportional relationships and use them to solve realworld and mathematical problems.
7.RP.A.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.
7.RP.A.2 Recognize and represent proportional relationships between quantities.
7.RP.A.2.A Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
7.RP.A.2.B Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
7.RP.A.2.C Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
7.RP.A.2.D Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
7.RP.A.3 Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. 
The order of the standards doesn’t indicate the order in which they have to be taught. (Standards are only a set of expectations for what students should know and be able to do by the end of each year; they don’t prescribe an exact sequence or curriculum.) These standards, for example, are all related, but they can be taught in the order listed, in another order, or in an integrated fashion.
As adults, we use proportional relationships all the time without naming the ideas involved. But teaching students who are new to proportional relationships requires precise knowledge of what they’re being expected to learn. Let’s pause for a moment to think about the concepts mentioned in the standards. (It’s not important that students memorize these exact definitions, but it’s good to have a clear idea of the concepts that the standards expect students to master.)
Apples (lbs) 
3 
6 
9 
12 
1 
3/2 
Cost ($) 
2 
4 
6 
8 
2/3 
1 
The standards for Grade 6 introduce students to the concept of a unit rate as the number of units of one quantity “for every 1” or “per 1” of a second quantity. (6.RP.A.2) Unit rates in Grade 6 are limited to rates of two whole numbers. So students might see examples like those described in the section above: A recipe with 1 cup of sugar for every 2 cups of flour implies a unit rate of 1/2 cup of sugar for every cup of flour. Students should also be solving problems involving unit rates of whole numbers. (6.RP.A.3) For example, if we want to make a larger batch with the same recipe—say, using 11 cups of flour—how much sugar will we need?
Now, in Grade 7, students are challenged to apply their knowledge of unit rates to situations involving fractional quantities. (7.RP.A.1) The standard gives us a nice example here: A person walks 1/2 mile in each 1/4 hour. How many miles per hour is this person walking? To do this, students will need to rely on more than just the procedures for dividing fractions; they should realize that computing unit rates of fractions is an extension of the same concepts and structures they already know. Moreover, they can use the same representations, such as tables and tape diagrams, that they’ve used in the past to solve unit rate problems.
This task might work well in an early lesson on fractional unit rates because the numbers involved aren’t too abstract. A student could represent or draw a picture of the solution to this problem without much difficulty.
Angel and Jayden were at track practice. The track is 2/5 kilometers around.
“Track Practice” by Illustrative Mathematics is licensed under CC BY 4.0.
The goal here is for students to find a unit rate for each runner: either the number of kilometers per 1 minute, or the number of minutes per 1 kilometer. Initially, they may need to start by making a table and reading the unit rate (see the solution section of the task for an example of how this might look). They might also create a tape diagram or use a number line to visualize how many minutes go into “every 1” kilometer. As students become more familiar with contexts that invite unit rate reasoning, though, they’ll naturally begin to develop more efficient arithmetic strategies for finding unit rates.
In Grade 6, students learned the concept of a ratio (6.RP.A.1) and used collections of equivalent ratios to solve problems. (6.RP.A.3) Though they probably didn’t use the term proportional relationship, this is exactly what they were working with. Now, they’ll formalize their understanding of proportional relationships and explore the properties of these relationships more deeply. (7.RP.A.2) This is the critical advancement over work in Grade 6, and an important precursor to work with linear equations in Grade 8.
The opening of standard 7.RP.A.2 gives us two charges: Students should be able to “recognize” and “represent” proportional relationships. This task uses a ratio table (a familiar representation from Grade 6) to begin investigating the characteristics of proportional relationships.
A new selfserve frozen yogurt store opened this summer that sells its yogurt at a price based upon the total weight of the yogurt and its toppings in a dish. Each member of Isabelle’s family weighed his dish, and this is what they found. Determine if the cost is proportional to the weight.
Weight (ounces) 
12.5 
10 
5 
8 
Cost ($) 
5 
4 
2 
3.20 
The cost _____________________________________ the weight.
Grade 7, Module 1, Lesson 2 (teacher version) from EngageNY.org of the New York State Education Department is licensed under CC BYNCSA 3.0.
Students should be able to see how the two rows of the table are related by the same constant of proportionality (or unit rate) and that multiplying by a certain “scale factor” between columns of the table yields equivalent ratios.
Both of these are hallmarks of proportional relationships, and students should be able to use these as tests to determine whether a relationship they’ve encountered is proportional or not.
Students should also take these sorts of tables and plot the values on the coordinate plane. They should notice that the graph is a straight line through the origin, and should be able to explain why it looks that way. To see how students might come to this conclusion by solving a problem, let’s take a look at this task:
In January, Georgia signed up for a membership at Anytime Fitness. The plan she chose cost $95 in startup fees and then $20 per month starting in February. Edwin also signed up at Anytime Fitness in January. His plan cost $35 per month starting in February, and his startup fees were waived.
“Gym Membership Plans” by Illustrative Mathematics is licensed under CC BY 4.0.
Let’s focus on parts (a) and (b) here, and the first aspect of part (c). Students are graphing two relationships here, one of them proportional and one not proportional. (The commentary and solutions for the task show what the table and graph for each relationship could look like.) In examining the table, students might notice how only one relationship has a constant of proportionality, and in examining the graph, they can observe what that proportional relationship looks like when it appears on the coordinate plane. You can take this task a few steps further by asking questions like:
Once students have plenty of experience identifying the constant of proportionality in tables and graphs, they can use these representations to write equations for proportional relationships. The lesson below shows how it might be done:
Example 1: Do We Have Enough Gas to Make It to the Gas Station?
Your mother has accelerated onto the interstate beginning a long road trip, and you notice that the low fuel light is on, indicating that there is a half a gallon left in the gas tank. The nearest gas station is 26 miles away. Your mother keeps a log where she records the mileage and the number of gallons purchased each time she fills up the tank. Use the information in the table below to determine whether you will make it to the gas station before the gas runs out. You know that if you can determine the amount of gas that her car consumes in a particular number of miles, then you can determine whether or not you can make it to the next gas station.
Mother’s Gas Record
Gallons 
Miles Driven 
8 
224 
10 
280 
4 
112 
y = 28x or m = 28g
Grade 7, Module 1, Lesson 8 (teacher version) from EngageNY.org of the New York State Education Department is licensed under CC BYNCSA 3.0.
In Example 1, we can see students interpreting the constant of proportionality in a table (for every gallon, the car goes 28 miles), and moving from there to an equation. This is an abstract concept and can be difficult for students to do for the first time, so to help them, you might consider extending the table like so:
Gallons 
Miles Driven 
8 
224 
10 
280 
4 
112 
1 

g 
It might also help to write the equation by writing unevaluated expressions in the “Miles Driven” column, once students have identified the constant of proportionality:
Gallons 
Miles Driven 
8 
8 × 28 
10 
10 × 28 
4 
4 × 28 
1 
1 × 28 
g 
As they find a pattern in the table (“the number of miles is always the number of gallons times 28”), they can write an expression, 28g, for the number of miles the car can go on any number of gallons g. From there, you can help them introduce a second variable to represent the number of miles driven (say, m), and they have their first equation.
28g = m
↑ constant of proportionality (unit rate)
After trying this in a few different situations (examining the table to find the constant of proportionality is good practice), you might ask, “What do you notice about all of these equations?” Students should be able to explain that the constant of proportionality relates the two variables in each equation.
In Grade 6, students learned the concept of a percent as a rate “per 100” and used ratiostyle reasoning to solve percent problems. (6.RP.A.3.C) (This is different from the way percent was treated in the past, when it was often introduced as a completely new idea, isolated from others. Now, however, students should view percents as a specialized application of ratios and rates.) Most of that work was singlestep, meaning that students had to interpret a situation in terms of ratios and perform a calculation or two in order to find an equivalent percent. A couple of examples:
These were good tasks for students new to percents, since they allowed them to understand percent problems as ratio situations and practice using helpful representations like double number lines and tape diagrams. Now the stakes are higher; students are going to have to solve problems that require them to calculate percentages or equivalent ratios in the midst of other procedures. (7.RP.A.3) And because of the increased complexity of problems, the level of sensemaking that students have to demonstrate is higher. Let’s take a look at an example task. (If you have a moment, look through the example solutions on the linked page as well; there are multiple methods of solving.)
The taxi fare in Gotham City is $2.40 for the first mile and additional mileage charged at the rate $0.20 for each additional 0.1 mile. You plan to give the driver a $2 tip. How many miles can you ride for $10?
“Gotham City Taxis” by Illustrative Mathematics is licensed under CC BY 4.0.
You can see there’s a lot going on here: Students need to interpret proportional “components” of the problem (such as the “additional mileage” charge), as well as nonproportional components (such as the first halfmile charge and the tip). And notice the complexity of unit rates implied here: Students won’t have seen anything like this in Grade 6, since any unit rate solution involves finding a rate of two fractions (recall that in Grade 6 unit rates were limited to noncomplex fractions). You might help students begin to tackle a problem like this by having them read it several times, and then helping them rewrite or diagram the three pieces of the problem. Which components depend on the distance traveled, and which don’t? How do you know? Once students are able to distinguish the different parts of the taxi fare, both in terms of cost and distance, they’re wellpositioned to find the solution.
This is just one example of Grade 7 work with multistep problems. To get a more comprehensive picture of what this standard looks like, we recommend browsing its entire gallery of problems from Illustrative Mathematics. One item of particular interest could be this one involving successive discounts, which is an interesting variation on percent problems:
Emily has a coupon for 20 percent off of her purchase at the store. She finds a backpack that she likes on the discount rack. Its original price is $60 but everything on the rack comes with a 30 percent discount. Emily says
Thirty percent and twenty percent make fifty percent so it will cost $30.
a. Is Emily correct? Explain.
b. What price will Emily pay for the backpack?
"Double Discounts" by Illustrative Mathematics is licensed under CC BY 4.0.
At this point, you may be wondering: Is there a way to help students take on all of these problems? While it’s not possible to teach students how to solve every single formulation of a multistep ratio or percent problem, you can give them the tools to carefully interpret a variety of situations. Here are a few tips:
The standards don’t just include knowledge and skills; they also recognize the need for students to engage in certain important practices of mathematical thinking and communication. These “Mathematical Practices” have their own set of standards, which contain the same basic objectives for Grades K12.^{3} You can read the full text of the Standards for Mathematical Practice here. You can read the full text of the Standards for Mathematical Practice here. You can read the full text of the Standards for Mathematical Practice here. (The idea is that students should cultivate the same habits of mind in increasingly sophisticated ways over the years.) But rather than being “just another thing” for teachers to incorporate into their classes, the practices are ways to help students arrive at the deep conceptual understandings required in each grade. In other words, the Practices help students get the content. The table below contains a few examples of how the Mathematical Practices might help students understand and work with ratios, proportional relationships, and percents in Grade 7.
Opportunities for Mathematical Practices: 
Teacher actions: 
Students are able to look for and make use of structureMP.7 in the context of multistep percent problems when they explain why the structure of percents often leads to multiple solution methods. For example, students should realize and articulate why they can find the final cost of an item after 5% tax by either adding 5% of the item’s price to its original price, or simply by calculating 105% of the original price. 
You can encourage students to see structure by having them model their thinking with various conceptual representations—tables, tape diagrams and double number lines. And you can have students present multiple solutions to the same problem to the rest of the class, and ask other students to explain how these methods are similar and different. (This lesson involving markup, markdown, and similar types of problems offers a number of opportunities to compare and contrast different methods.) 
Students can model with mathematics (MP.4) in order to overcome common challenges in working with multistep percent problems, such as understanding the nuances of percent increase and decrease. 
Present students with situations where conceptual representations can lend insight into a problem. For example, if an account contains $200 to start, its value decreases by 10% one month, and then increases by 10% the next month, why is the final value of the account not $200? Students might work through each part of this problem using a double number line diagram, and notice that in the first part, the “whole” is $200, while in the second part, the “whole” is only $180. 
Students can make sense of problems and persevere when solving them (MP.1) when they solve a variety of problems involving proportional relationships—particularly problems that require using several representations of a relationship. 
Consider problems that require students to gather information about a situation from one particular representation. For example, a graph of someone’s earnings over an 8hour shift might reveal a unit rate of $25 per hour. How much will this person earn in a week of fulltime work? In a month? The answers to these questions likely require students to switch to a different representation—perhaps a table or an equation. With practice, students will become more skilled at analyzing given information and planning an efficient solution process before getting to work. 
Podcast clip: Importance of the Mathematical Practices with Andrew Chen and Peter Coe (start 30:33, end 43:39)
There are lots of connections among standards in Grade 7; if you think about the standards long enough, you’ll probably start to see these relationships everywhere.^{4} The idea that standards relate strongly to one another is known as coherence, and is a distinctive feature of the Common Core State Standards for Mathematics. If you’re interested in exploring more of the connections between standards, you might want to try the Student Achievement Partners Coherence Map web app. The idea that standards relate strongly to one another is known as coherence, and is a distinctive feature of the Common Core State Standards for Mathematics. If you’re interested in exploring more of the connections between standards, you might want to try the Student Achievement Partners Coherence Map web app. The idea that standards relate strongly to one another is known as coherence, and is a distinctive feature of the Common Core State Standards for Mathematics. If you’re interested in exploring more of the connections between standards, you might want to try the Student Achievement Partners Coherence Map web app. A few are so important, though, that they deserve special attention. In this section, we’ll talk about the connections between the Ratio & Proportional Relationships domain and the Geometry and Expressions & Equations domains in Grade 7.
Let’s start with the Geometry standards, which in Grade 7 are strongly related to proportional relationships through scale drawings. (7.G.A.1) This is such a natural connection that this standard may often be taught in the same unit with proportional relationships (as opposed to a standalone geometry unit). This lesson plan is part of just such a sequence. After being introduced to scale drawings in the previous lesson, students investigate how an object and its scaled image are related in a proportional relationship.
Example 1: Jake’s Icon
Jake created a simple game on his computer and shared it with his friends to play. They were instantly hooked, and the popularity of his game spread so quickly that Jake wanted to create a distinctive icon so that players could easily identify his game. He drew a simple sketch. From the sketch, he created stickers to promote his game, but Jake wasn’t quite sure if the stickers were proportional to his original sketch.
Grade 7, Module 1, Lesson 17 (teacher version) from EngageNY.org of the New York State Education Department is licensed under CC BYNCSA 3.0.
The lesson plan includes a few questions following the problem that are worth highlighting here. Notice how they help students make the connection to prior learning about constant of proportionality.
From there, students are able to try scaling measurements for themselves. One nice thing about this sequence of problems is how it uses the language of proportional relationships to call up prior learning. Students can clearly see that this is an extension of something they already know, not a completely new topic. It also uses a familiar representation (a table) so that students can more easily identify the relationship between the two sets of measurements.
The story on Expressions & Equations starts with the work students do to solve problems with equations and inequalities. (7.EE.B.4) Recall that as students were building an understanding of proportional relationships, they had to express proportional relationships as equations. (7.RP.A.2.C) For example, in our gas mileage problem above, the number of miles, m, someone can drive on g gallons of gas when the car gets 28 miles per gallon was g = 28m. Students also had to solve multistep problems involving proportional and nonproportional aspects. This was the case in our taxi problem above, where the driver collected a charge for every fraction of a mile, but received a $2 tip regardless of the distance driven. Both of these ideas—modeling with equations and distinguishing aspects of problems (is it proportional or not?) come into play as students write and solve equations. A task like this might help students put them together.
Jonathan wants to save up enough money so that he can buy a new sports equipment set that includes a football, baseball, soccer ball, and basketball. This complete boxed set costs $50. Jonathan has $15 he saved from his birthday. In order to make more money, he plans to wash neighbors’ windows. He plans to charge $3 for each window he washes, and any extra money he makes beyond $50 he can use to buy the additional accessories that go with the sports box set.
Write and solve an inequality that represents the number of windows Jonathan can wash in order to save at least the minimum amount he needs to buy the boxed set. Graph the solutions on the number line. What is a realistic number of windows for Jonathan to wash? How would that be reflected in the graph?
“Sports Equipment Set” by Illustrative Mathematics is licensed under CC BY 4.0.
So what do we have here? Someone is earning $3 per window—a proportional relationship. But there is also the $15 already saved—a quantity not proportional to the number of windows washed. So we have to deal with that, along with the fact that we need everything (savings, earnings from window washing) to be at least $50 when all’s said and done. Representing situations in an equation like this can be tricky when students are first starting out, so it might help to think about the situation using a table and unevaluated expressions, just as they did when writing equations for proportional relationships.
Windows 
Savings ($) 
1 
(3 × 1) + 15 
2 
(3 × 2) + 15 
3 
(3 × 3) + 15 
4 
(3 × 4) + 15 
w 
Once students have thought several of these situations through, they’ll begin to recognize the proportional and nonproportional “parts” of the relationship based on context, and will need to rely on tables or other scaffolds less and less. You can help them begin to distinguish the different components in a situation through questioning. For example:
The standards for Grade 7 focus heavily on problemsolving in both the Ratio & Proportional Relationships domain, and in the Expressions & Equations domain. Two of these standards—7.RP.A.3, which deals with multistep ratio and percent problems, and 7.EE.B.4, which involves solving problems with rational numbers—are remarkably similar in terms of the situations they address. The difference is really in how students approach these problems. The good news is that experience with problems in one standard can help students master the other. For example, here’s a task aligned to standard 7.EE.B.3.
Katie and Margarita have $20.00 each to spend at Students' Choice book store, where all students receive a 20% discount. They both want to purchase a copy of the same book which normally sells for $22.50 plus 10% sales tax.
Is Katie correct? Is Margarita correct? Do they have enough money to purchase the book?
“Discounted Books” by Illustrative Mathematics is licensed under CC BY 4.0.
After studying ratio and percent problems, this situation might look somewhat familiar. Students might approach this problem using the tools of proportional reasoning—such as double number lines, tape diagrams or equations involving fractional rates—but they might also see it in terms of expressions involving rational numbers. The solution section of the task gives us two sets of calculations to think about. There’s Katie’s method:
22.50 − (0.20(22.50)) = 22.50 − 4.50 = 18.00
18.00 + (0.10(18.00)) = 18.00 + 1.80 = 19.80
And there’s Margarita’s method:
(0.80)22.50 = 18.00
(1.10)18.00 = 19.80
As a way to get students thinking about the structures involved, you might ask them to explain the similarities they see between the two solutions. They might notice, for example, that decreasing a number by 20 percent is the same as finding 80 percent of that number. You might also ask them to consider whether the two students in the problem would still calculate identical answers for any item (of any price) in the bookstore. Discussing questions like these can give students additional insight into how percents work.
Proportional relationships, while formally introduced in Grade 7, are intended to build from a careful progression of prior learning—particularly the learning that occurs around ratios in Grade 6. If your students have been following a standardsaligned program for a few years, knowing the leadup to ratios will help you leverage content from previous grades in your lessons. And if your students haven’t been exposed to the standards in a meaningful way, or they’re behind for other reasons, seeing where ratios come from will allow you to adapt your curriculum and lessons to make new ideas accessible. Let’s briefly look at the main threads that lead up to ratios in Grades K5; then we’ll see how work with proportional relationships in Grade 7 compares to work with ratios in Grade 6. After that, we’ll examine some ways that you might use this information to meet the unique needs of your students, and preview where proportional relationships are going in the next few years after Grade 7.
Podcast clip: Importance of Coherence with Andrew Chen and Peter Coe (start 9:34, end 26:19)
Starting in the elementary grades, the standards lay the groundwork for ratios and proportional relationships. Though they do this in a variety of ways, there are a few progressions of learning that contribute most directly to RP in Grades 6 and 7.
Grapes (lbs) 
2 
4 
6 
Cost ($) 
6 
12 
18 
Part 3 of the Grade 6 guide to Ratios & Proportional Relationshipscontains more about each of these progressions, including problems that illustrate each one and suggestions for students who are behind in these areas.
So what are the differences between Grade 6 and Grade 7 in terms of Ratios & Proportional Relationships? This is an important question, because knowing where students are coming from can help you start your lessons in just the right place. Throughout this guide, we’ve touched upon the advances from one grade to the next, but now we’ll see them illustrated with some problems.
In Grade 6, students should learn the concept of a unit rate (6.RP.A.2) and solve problems involving rates of whole numbers. (6.RP.A.3) Then, in Grade 7, they begin solving problems involving unit rates of fractions. (7.RP.A.1) The two problems below illustrate the difference in expectations.
Grade 6: Unit rates of whole numbers
Tickets for a baseball game cost $60 for a family of 5. Adult and youth tickets cost the same amount. Place a checkmark in either the True or False column for each of the statements below.

Grade 7: Unit rates of fractions
During their last workout, Izzy ran 2 1/4 miles in 15 minutes, and her friend Julia ran 3 3/4 miles in 25 minutes. Each girl thought she was the faster runner. Based on their last run, which girl is correct?


(Source: Ratios and Rates MiniAssessment by Student Achievement Partners is licensed under CC 0 1.0.) 
(Source: Grade 7, Module 1, Lesson 11 (teacher version) from EngageNY.org of the New York State Education Department is licensed under CC BYNCSA 3.0.)


➔ In this problem, which is fairly typical of work in Grade 6, students deal with ratio and rate situations involving whole numbers only. This allows them to focus on understanding the concept of a rate and using various models, such as tape diagrams and ratio tables, to represent problems.

➔ With a firm understanding of rate from the previous grade, students work on problems like this, which have a greater degree of computational difficulty. The representations, operations, and procedures that students used in Grade 6 are still useful in dealing with these problems. 
In Grade 6, students also learn the concept of a ratio (6.RP.A.1) and solve problems involving equivalent ratios. (6.RP.A.3) In Grade 7, they formalize sets of equivalent ratios as “proportional relationships” and represent these in different ways. (7.RP.A.2) Again, a pair of problems illustrates the advancements from one grade to the next.
Grade 6: Equivalent ratios
Javier has a new job designing websites. He is paid at a rate of $700 for every 3 pages of web content that he builds. Create a ratio table to show the total amount of money Javier has earned in ratio to the number of pages he has built.
Javier is saving up to purchase a used car that costs $4,200. How many web pages will Javier need to build before he can pay for the car?

Grade 7: Proportional relationships
The table shows the amounts of tomato sauce and cheese used to make the last 4 orders at Sara’s Pizza.
Decide whether the relationship between number of pizzas and amount of cheese is proportional. Explain your decision.


(Source: Grade 6, Module 1, Lesson 9 (teacher version) from EngageNY.org of the New York State Education Department is licensed under CC BYNCSA 3.0.)

(Source: Proportional Relationships MiniAssessment by Student Achievement Partners is licensed under CC 0 1.0.) 

➔ In this problem, we see students generating equivalent ratios from a context, and solving a problem by locating a particular ratio in a table. While this problem does involve a proportional relationship, students are only thinking in terms of equivalent ratios and are not yet using the language of proportionality.

➔ This Grade 7 problem requires students to have a formal understanding of proportional relationships, and to distinguish proportional from nonproportional relationships using, for example, a constant of proportionality. 
Lastly, in Grade 6, students solve various ratio and rate problems, many of which were singlestep. (6.RP.A.3) In Grade 7, the focus is on more complicated “multistep” problems. (7.RP.A.3) Once again, we have two problems that show how expectations advance yearoveryear.
Grade 6: Simple ratio and rate problems
Mr. Yoshi has 75 papers. He graded 60 papers, and he had a student teacher grade the rest. What percent of the papers did each person grade?

Grade 7: multistep ratio and rate problems
On Black Friday, a $300 mountain bike is discounted by 30% and then discounted an additional 10% for shoppers who arrive before 5:00 a.m. What is the cost of the bike for an earlymorning shopper on Black Friday?

(Source: Grade 6, Module 1, Lesson 27 (teacher version) from EngageNY.org of the New York State Education Department is licensed under CC BYNCSA 3.0.)

(Source: Grade 7, Module 4, Lesson 7 (teacher version) from EngageNY.org of the New York State Education Department is licensed under CC BYNCSA 3.0.) 
➔ This is a typical example of Grade 6 work with percents. Students are learning about percent as a specialized rate, and focus on solving simple problems in order to better understand the concept.

➔ Presuming that students have a firm grasp of percent as a rate “per 100,” work in Grade 7 asks students to apply their knowledge in new and more complicated ways. 
If, going into a unit on proportional relationships, you know your students don’t have a solid grasp of the ideas named above, what can you do? It’s not practical (or even desirable) to reteach Grade 6 material “from scratch”; there’s plenty of new material in Grade 7, so the focus needs to be on gradelevel standards. At the same time, there are strategic ways of wrapping up “unfinished learning” from prior grades within a unit on proportional relationships. Here are a few ideas for adapting your instruction to bridge the gaps.
Lastly, how will students use their knowledge of proportional relationships after Grade 7? This is an important question, because the answer defines the limits of instruction in Grade 7 and explains why it’s important that students thoroughly elaborate their understanding of proportional relationships through a variety of representations.
Let’s recap, then: By the end of Grade 7, students should have a firm understanding of the characteristics of proportional relationships in tables and graphs, and should be able to represent proportional relationships with equations. Moreover, they should have experience computing unit rates involving fractions and decimals, and solving all kinds of ratio and percent problems.
In Grade 8, there is no separate group of standards for ratios and proportional relationships; these ideas merge completely with the algebra content in the Expressions & Equations domain and the Functions domain. The concept of unit rate evolves into slope, (8.EE.B.5) and students will discover important properties of slope. They explore the connection between proportional relationships (i.e. can be represented by an equation y = mx) and linear equations more generally (y = mx + b). (8.EE.B.6) And after being formally introduced to the concept of a function, students will model linear relationships with functions. (8.F.B.4) Students will rely on these concepts throughout high school and, in many cases, in postsecondary work as well. One last progression of problems illustrates where students are headed.
Grade 8: Understanding slope (8.EE.B.6)
Eva, Carl, and Maria are computing the slope between pairs of points on the line shown below.
Eva finds the slope between the points (0,0) and (3,2). Carl finds the slope between the points (3,2) and (6,4). Maria finds the slope between the points (3,2) and (9,6). They have each drawn a triangle to help with their calculations (shown below).
i. Which student has drawn which triangle? Finish the slope calculation for each student. How can the differences in the x and yvalues be interpreted geometrically in the pictures they have drawn?
ii. Consider any two points (x1,y1) and (x2,y2) on the line shown above. Draw a triangle like the triangles drawn by Eva, Carl, and Maria. What is the slope between these two points? Why should this slope be the same as the slopes calculated by the three students?
(Source: “Slopes between Points on a Line” by Illustrative Mathematics is licensed under CC BY 4.0.)
➔ Having learned to recognize unit rates of proportional relationships in graphs while in Grade 7, students are poised to take on problems like this one. Here, students formalize their understanding of slope, and understand why the slope of a line is the same between any two points on the line. 
Grade 8: Comparing functions (8.F.A.2)
Maureen and Shannon decide to rent standup paddleboards while on vacation. Shop A rents paddleboards for $7.75 per hour. Shop B’s prices are shown on the poster below. Which shop offers a cheaper hourly rental rate?
(Source: Functions MiniAssessment by Student Achievement Partners is licensed under CC 0 1.0.)
➔ In this problem, we can see students comparing functions (which happen to involve proportional relationships) represented in two different ways. They’ll also have to compare nonproportional linear relationships, so time in Grade 7 examining the features of graphs, tables, and equations is wellspent. 
Grade 8: Modeling with functions (8.F.A.4)
You have $100 to spend on a barbeque where you want to serve chicken and steak. Chicken costs $1.29 per pound and steak costs $3.49 per pound.
a) Find a function that relates the amount of chicken and the amount of steak you can buy.
b) Graph the function. What is the meaning of each intercept in this context? What is the meaning of the slope in this context? Use this (and any other information represented by the equation or graph) to discuss what your options are for the amounts of chicken and amount of steak you can buy for the barbeque.
(Source: “Chicken and Steak, Variation 1" by Illustrative Mathematics is licensed under CC BY 4.0.)
➔ This problem requires students to put knowledge of unit rate to work in creating a function and interpreting the features of its graph. Without a good understanding of unit rate from Grades 6 and 7, a task like this becomes very difficult. 
If you’ve just finished this entire guide, congratulations! Hopefully it’s been informative, and you can return to it as a reference when planning lessons, creating units, or evaluating instructional materials. For more guides in this series, please visit our Enhance Instruction page. For more ideas of how you might use these guides in your daily practice, please visit our Frequently Asked Questions page. And if you’re interested in learning more about Ratios & Proportional Relationships in Grade 7, don’t forget these resources:
Student Achievement Partners: Focus in Grade 7
Draft 67 Progression on Ratios and Proportional Relationships
EngageNY: Grade 7 Module 1 Materials (proportional relationships)
EngageNY: Grade 7 Module 4 Materials (percents)
Illustrative Mathematics Grade 7 Tasks
[1] In this series, major clusters and standards are denoted by a ▉. For more information on the major work of Grade 7, see the Student Achievement Partners guide Focus in Grade 7.
[2] Much of the information in this section is taken from the Draft 67 Progression on Ratios and Proportional Relationships, one of a series of papers that describes the big ideas behind the standards and how those ideas fit together. If you’re interested in learning more about ratios and proportional relationships, it’s a good resource.
[4] The idea that standards relate strongly to one another is known as coherence, and is a distinctive feature of the Common Core State Standards for Mathematics. If you’re interested in exploring more of the connections between standards, you might want to try the Student Achievement Partners Coherence Map web app.
In our work with high academic standards, we often hear educators ask, “What does standardsaligned instruction look like?” Our Content Guides aim to answer this question by providing an indepth look at one or a few clusters of math standards at a time. The Content Guides are gradelevel and content areaspecific, and there are guides for each grade or course, from Kindergarten to Algebra II. If you want to learn more about teaching Ratios and Proportional Relationships in Grade 6, for example,our associated Content Guide will give you a comprehensive but accessible explanation about these standards, multiple Open Educational Resource (OER) examples that are aligned to the standards, and concrete suggestions to support the teaching of Grade 6 ratios and proportional reasoning.
Our goal in creating the Content Guides has been to provide busy teachers with a practical and easytoread resource on what the gradelevel math standards are saying, along with examples of instructional materials that support conceptual understanding, problemsolving, and procedural skill and fluency for students.
It’s important to note that content guides are not meant to serve as a curriculum (or any kind of studentfacing document), a guide or source material for testpreparation activities, or any kind of teacher evaluation tool.
Each Content Guide is focused on a specific group of standards. Most Content Guides follow the same threepart structure:
Part 2 explains how this group of standards is connected to other standards in the same grade. We highlight how these connections have implications for planning and teaching, and how this withingrade coherence can increase access for students. Part 2 also includes multiple student tasks from freely available online sources.
Part 3 traces selected progressions of learning leading to gradelevel content discussed in the specific Content Guide. This discussion segues into a series of concrete and practical suggestions for how teachers can leverage the progressions to teach students who may not be prepared for gradelevel mathematics. Finally, Part 3 traces the progression to content in higher grades.
Teachers who have read our Content Guides say they see benefits for all educators. Here are some suggestions for how different educators might use them.
Teachers can use the Mathematics Content Guides to:
Instructional coaches and school leaders can use the Mathematics Content Guides to:
The transition to higher standards has led teachers all over the country to make significant changes in their planning and instruction, but only onethird of teachers feel they are prepared to help their students pass the more rigorous standardsaligned assessments (Kane et.al., 2016). This is to be expected because the new high standards are a significant departure from prior standards. The standards require a deeper level of understanding of the math content they teach; a different progression of what students need to learn by which grade; as well as different pedagogy that emphasizes student conceptual understanding, problem solving and procedural fluency in equal intensity.
The support for teachers to bring high standards to their classrooms, however, has lagged behind. Research shows that teacher training in the U.S. is currently insufficient in preparing teachers to teach the demanding new standards (Center for Research in Mathematics and Science Education, 2010). And though some resources exist that “unpack” the standards, few, if any, explain and illustrate the standards. “Unpacking” the standards one by one can also result in a disjointed presentation that neglects the structure and coherence of the standards. In creating the Content Guides, we aimed to provide busy teachers with a practical, easytoread resource on their gradespecific standards and how to help all students learn them. There is ample empirical evidence that when teachers have both strong knowledge of the math content that they teach, and the pedagogical knowledge to help students master that content knowledge, their students learn more (Baumert et. al., 2010; Hill, Rowan and Ball, 2005; Rockoff et. al., 2008). With the Content Guides in hand, we hope that teachers will find more success in helping their students make progress toward college and careerreadiness.
The Progressions documents describe the gradetograde development of understanding of mathematics. These were informed by research on children’s cognitive development as well as the logical structure of mathematics. The Progressions explain why standards are sequenced the way they are. The Content Guides often highlight key ideas from the Progressions, but do not add new standards or change the expectations of what students should know and be able to do; they aim to explain and illustrate a group of standards at a time using freely available online sources. While the OER tasks and lessons in the Content Guides are one way to meet the gradelevel standards, they are not the only means for doing so.
We selected sample tasks and lessons from freely available online sources such as EngageNY, Illustrative Mathematics and Student Achievement Partners to illustrate the Standards. These sources are chosen because they are fully aligned to the new high standards based on national review of K12 curricula or are created by organizations led by the writers of the new high standards. In addition, because they are open educational resources (OER), they are freely accessible for all uses. All UnboundEd materials are also OER, as part of our commitment to make highquality, highly aligned content available to all educators.
Each Content Guide addresses a subset of the standards for the grade. The standards addressed in the first set of Content Guides for each grade usually address highpriority content; these standards are also often a good choice for teaching at the beginning of the year. More information about the selection of standards can be found in the introduction to each Content Guide. Over time, we will develop additional Content Guides for each grade and update existing ones. We plan to have four Content Guides for each grade or course, from Kindergarten to Algebra II. The guides will be published in waves, with each wave consisting of one guide for each grade. We plan to release a second set of Content Guides for each grade by the end of the 201617 school year.
If you would like to receive updates on content and events from UnboundEd, including new Content Guides, please sign up for UnboundEd announcements here.
Analyze proportional relationships and use them to solve realworld and mathematical problems. Analyze proportional relationships and use them to solve realworld and mathematical problems. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. Recognize and represent proportional relationships between quantities. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. Represent proportional relationships by equations. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. Use proportional relationships to solve multistep ratio and percent problems. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. Recognize and represent proportional relationships between quantities. Recognize and represent proportional relationships between quantities. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. Use proportional relationships to solve multistep ratio and percent problems. Model with mathematics. Make sense of problems and persevere in solving them. Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. Represent proportional relationships by equations. Use proportional relationships to solve multistep ratio and percent problems. Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. Interpret wholenumber quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. Understand division as an unknownfactor problem. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. Understand a fraction as a number on the number line; represent fractions on a number line diagram. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. Recognize and represent proportional relationships between quantities. Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. Use proportional relationships to solve multistep ratio and percent problems. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Use similar triangles to explain why the slope m is the same between any two distinct points on a nonvertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Use similar triangles to explain why the slope m is the same between any two distinct points on a nonvertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).