6th Grade Ratios Unit
Learning Module
- Creator(s):
Learning Module Objectives:
6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.
6.RP.2 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.
6.RP.3 Solve problems involving finding the whole, part, or percent given the other two.
These objectives come directly from the Common Core Standards for sixth grade math.
This unit is intended for sixth graders, but could also be used for seventh graders as an introduction to their own proportinal relationships Common Core standards. Teachers looking for extension opportunities for fifth graders could use this module as well.
Prerequisite knowledge includes: simplifying/reducing fractions, dividing fractions, multiplying fractions, dividing decimals, multiplying decimals, and finding least common multiple. Proficiency of the 5th grade Common Core standards is assumed.
In seventh grade, students will extend what they have learned in this unit to applied percent problems, such as finding tax/tip, discounts, markups, etc. The ideas of ratios, rates, and proportions also leads into linear functions and equations.
Learning Objective: 6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.
A ratio is defined as a comparison between numbers. We use them to efficiently describe relationships and maintain equivalency.
There are three different notations that can be used when writing a ratio. All three notations are read aloud the same way.
| Notation | Example | Read aloud as... |
|---|---|---|
| Fraction | 4/9 | "Four to nine" |
| Colon | 4:9 | "Four to nine" |
| Words | 4 to 9 | "Four to nine" |
As you watch the video below, pay attention to the different ways to write ratios and how to simplify/reduce ratios.
Next, play this game to test your skills! Can you correctly describe each relationship pictured as a ratio?
Lastly, read through this lesson on ratios and try the activities at the bottom.
Work for the Week
Comment: In what aspects of your life have you heard people comparing numbers or using ratios?
Update: Create a ratio relationship using a picture or video of your own so that your classmates can leave a reply with the ratio you have shown. For example, take a picture of 3 shoes and 6 socks or a video of 2 adults and 3 children.
Replies: Reply to three of your classmates' updates. State the ratio that your classmate has shown through a picture or video. You must state the ratio in a different format (1:2, 1/2, 1 to 2) than any replies above yours. Once an update has its three replies, it is off-limits for any additional replies.
Purpose of this update: This update provides the foundation knowledge for the rest of the unit. Here, students learn what ratios are -- a comparison. Next, they will learn about proportions, which are two equivalent ratios, and two special types of ratios: rates and percents. This update is necessary to provide the foundation knowledge.
Teaching tips: Students often struggle with differentiating between ratios and fractions. During your discussion session, reinforce that ratios are pronounced "four to nine" instead of "four ninths." Also, touch on how a ratio must have two parts because it is a comparison. For example, though the fraction 5/1 would be simplified to 5, the ratio 5/1 must remain written in that exact way. Similarly, though the fraction 9/2 would be simplified to 4_1/2, it would be incorrect to rewrite a ratio into a mixed number. A ratio must always have two parts, rather than a single number.
Extensions: An appropriate extension would be to have students work with multi-part ratios and/or to write ratios for measurements with different units. Multi-part ratios are not much different than two-part ratios, but they usually need to be written using colon-notation or in words, because a multi-part fraction notation would look unusual. Ratios with different units must be converted to the same unit before they are reduced. They differ from rates (which are introduced in update 5) because they have the potential to be converted into the same unit.
Three-part ratios:
Ratios with different units:
Learning Objective: 6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.
In the last update, we introduced the fact that ratios are used to maintain equivalency. This is important when we want to do things like double a recipe so it feeds more people. Here are some questions that ratios can help you solve:
• If a recipe calls for 3 cups of sugar for every 5 cups of flour, how many cups of sugar should you use if you only have 2 cups of flour?
• There are 30 students in the class. If there are 6 boys for every 4 girls, how many students are boys and how many are girls?
There are many different tools you can use to solve ratio problems. We will take a look at a few of them.
Strategy #1: Grouping (the video begins with an introduction to ratios and the method is introduced 7 minutes into the video)
Strategy #2: Tape Diagram
Strategy #3: Tables
All three of these strategies can be used interchangeably to solve the same problems. It really just depends on your personal preference!
Work for the Week
Comment: Which strategy for solving ratio problems do you prefer? Why?
Update: Post a video lesson of yourself teaching how to use the strategy of your choice to solve the problem below. Also, create your own problem and teach how to solve it.
There are 4 blueberries for every raspberry in the fruit salad. If there are 50 berries in the fruit salad, how many are blueberries and how many are raspberries?
Replies: Reply to three of your classmates' updates. Give your classmate feedback on their video lesson in the following areas: mathematical precision, organization, and clarity.
Purpose of this update: In solving ratio problems, students begin to use proportional reasoning. This is an important skill for ratio problems in general, but it also carries over to linear functions and other topics in math. Here, students are introduced to several different strategies for solving ratio problems.
Teaching tips: Typically, students for whom math comes easily will be drawn to strategy #1, while spatial learners are drawn to strategy #2. Strategy #3 draws in a wide range of students because it is simple, yet provides structure to help organize work. Students should be encouraged to practice all three strategies so they can decide what works best for them as individuals. When it comes time for an assessment, students should be allowed to choose whatever strategy they want to use. These strategies are simply math tools, and should not be the focus of the assessment.
Emphasize to students the importance of making their answer clear in their work. If they just use one of the strategies, but never make clear which number in the tool is their answer, they are not doing a good job of communicating their thinking. I usually insist on my students writing a clear answer off to the side of the tool with units, then boxed or circled.
Extensions: Similarly to in the last update, the best extension to this topic is giving students problems with multi-part ratios. In the last update, they were just writing these multi-part ratios, but in this update, they will be solving problems using them.
Other Resources: Here is the first video that goes along with the video in strategy #2. Usually students don't need to see this video, but here it is, just in case they do.
Learning Objective: 6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.
A proportion is defined as two equivalent ratios. In this update, we are going to take a look at how to write proportions and how to test if two ratios form a proportion.
Watch this video to see how to set up a proportion from a word problem.
If you are given two ratios, you can test out if they form a proportion by setting them equal to each other, and then using cross products. In the example below, you can see that we have multiplied along each diagonal (product of red numbers and then product of blue numbers) to test if the products are equal. In this case, they are equal, so 4:6 does form a proportion with 16:24. This means the ratios are equivalent.
Watch this video to see the method in action!
One more thing before you go! Take this quiz to see if you can determine whether the relationships are proportional.
Work for the Week
Comment: In what situations would you use a ratio, and in what situations would you use a proportion?
Update: On your iPad or on paper, show how you could use proportions to prove that two ratios are equivalent. Post a photograph or screen shot of your work.
Replies: Reply to three of your classmates' updates. Name one additional ratio that is also equivalent to the ratios used in your classmate's proportion.
Purpose of this update: Proportions are simply two ratios that are equivalent, so proportional reasoning is a natural next step for this learning target. In this update, students will learn how to set up proportions and how to determine if two ratios form a proportion using cross multiplication. Cross multiplication is a strategy used not only for solving proportions, but also for solving equations. It is an important skill for the students to have mastered for their future studies.
Teaching tips: During your discussion, it may be helpful to spend some time going over why cross multiplication works. It's always better to offer students the opportunity to see the "why" behind what mathematicians do, rather than asking them to use the strategies "just because."
The idea of "fairness" seems to really resonate with the students and help them think using proportional reasoning. Consider sharing some examples, such as, "Would it be fair for 3 students to share one pizza when 6 other students are sharing 3 pizzas?" Loophole-seekers may need to be reassured that all students in the problem have the same appetites and nutritional needs.
Other resources: These videos give some more background information about cross multiplication and why it works.
Learning Objective: 6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.
The most common reason proportions are used is to figure out an unknown value in a proportional relationship. There are many different ways to solve proportions.
Strategy 1: Cross Product
Strategy #2: Equivalency (also shows cross product)
Complete this activity to practice solving for missing values in a proportion.
This game will help you practice setting up proportions from word problems.
Work for the Week
Comment: How are the strategies presented related to one another?
Update: Post a video lesson of yourself teaching how to use the strategy of your choice to solve the problem below. Also, create your own problem and teach how to solve it as well.
Skyler is 1.2 meters tall and her shadow is 4.8 meters. If Jaden's shadow is 3.9 meters, how tall is he?
Replies: Reply to three of your classmates' updates. Give your classmate feedback on their video lesson in the following areas: mathematical precision, organization, and clarity.
Purpose of this update: Solving proportions is a fairly ubiquitous topic in math. In this update, students will learn how to do just that.
Teaching tips: Some proportion problems can be solved using the ratio problem methods in update #2. Students should be allowed to use whatever method makes the most sense to them. A good rule of thumb is that if there are only two quantities being compared in the problem, use a proportion, and if there is a third quantity or a "total," in the problem, use a ratio.
Easier to solve with a proportion: An animal shelter is home to 4 dogs for every 5 cats. If there are 35 cats, how many dogs are there?
Easier to solve with a ratio: An animal shelter is home to 4 dogs for every 5 cats. If there are 81 total animals, how many dogs are there? [Reason: Students will try to set up a proportion with 4/5=x/81, but 81 is the total number of animals, and 5 is the total number of cats. To solve with a proportion, students must realize that they need to change the 5 to a 9, because that is the total of cats and dogs in the ratio. This is very hard for students to understand.]
Both of these problems can be solved using any proportion or ratio method, but it is worthwhile to have a discussion and share this rule of thumb to help students make the best choice for them.
Extensions: Topic C in this Engage New York unit has some interesting scale drawing problems that would connect nicely with this update.
If students have a foundation of Algebra, they can try these multi-step proportion problems.
For students without a foundation of Algebra, problems such as the ones in this video would make a good extension.
Learning Objective: 6.RP.2 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.
Rates are a special type of ratio where quantities with different units are compared.
Go through this lesson to learn about rates and see what you know!
Oftentimes, it is more helpful to rewrite a given rate into a unit rate. A unit rate is a rate "per one." Watch this video to see how to find unit rates.
Work for the Week
Comment: What are some differences and similarites between rates and ratios?
Update: Take a picture or a video of a rate you come across out in the world this week. Post the picture in your update.
Replies: Reply to three of your classmates' updates. Using the rate in the picture or video, state a rate that is equivalent.
Purpose of this update: For the rest of the learning module, the students will be learning about applications of ratios in the forms of rates and percents. This update introduces rates and provdes the distinction between rates and other types of ratios.
Teaching tips: Students have a lot of trouble remembering that ratios cannot have units, while rates must have units. One trick to remembering that is to look at the end of each word: ratio (no! units), rates (yes! units).
Another distinction comes in the way that rates are pronounced aloud. While ratios use the word "to," rates use the word "per." A common example of this is "miles per hour." In your discussion session, be sure to use precise language when saying ratios and rates aloud.
Extensions: This is a great time to introduce unit analysis to students who are ready for something more challenging. Unit analysis is used throughout mathematics and in science classes.
This unit analysis lesson provides an overview of how it works.
Here is another video that shows examples of unit analysis.
Learning Objective: 6.RP.2 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.
We compare rates all the time! Whether we're comparing pitcher's ERAs or deciding which bag of chips is a better deal, comparing rates happens frequently in so many aspects of our lives.
Watch this video to learn how to compare rates.
Now, try this practice to see if you can compare rates.
Work for the Week
Comment: When would you need to compare rates in real life?
Update: Post a video lesson of yourself teaching how to compare the rates below. Also, create your own problem and teach how to solve it as well.
1. Two calculators cost $75 at Walmart, while nine calculators cost $275 at Target. Which is a better deal?
2. Jessie read 35 pages in 40 minutes, while Jordan spent 15 minutes reading 14 pages. Who read faster?
Replies: Reply to three of your classmates' updates. Give your classmate feedback on their video lesson in the following areas: mathematical precision, organization, and clarity.
Purpose of this update: Students will be comparing rates in and out of school for the rest of their lives. They likely already have some experience with comparing rates while shopping or using sports statistics. This update provides a clear explanation of the skills necessary to compare rates.
Teaching tips: This is historically the most challenging part of the unit, even though it seems straightforward. Here's an example with the problem areas explained:
Target is selling 3 plain black t-shirts for $7.99, while Walmart is charing $4.99 for 2 of the same plain black t-shirts. Which is a better deal?
Problem area #1: To solve this problem, we need to divide the numbers in the same order. We cannot do 3 shirts/$7.99 for Target and $4.99/2 shirts for Walmart. Students have trouble understanding that they must flip one of the division problems.
Problem area #2: If students solve this problem by dividing cost by number of shirts, it's pretty straightforward. Target is $2.66 per shirt and Walmart is $2.50 per shirt. Students are able to get this answer easily if they divide in this order. They understand they are looking for the smaller number (the cheaper price).
However, if students divide shirts by cost, Target is 0.38 shirts per dollar, and Walmart is 0.4 shirts per dollar. Here, students must find the larger number because we are looking for the store that will give us more shirts for the same dollar.
With problems involving money, it is easy to tell students to always divide money first, but in problems that involve speed or other topics, they will run into this issue as well. Spend a significant amount of time in your discussion sharing examples of these problems and having the class practice interpreting their answers. Sometimes the larger number will be the answer, and sometimes the smaller number will be the answer.
Extensions: An appropriate extension would be to give students a multi-step problem in which they must compare rates. This unit has some challenging rate problems.
Learning Objective: 6.RP.3 Solve problems involving finding the whole, part, or percent given the other two.
Percents are a special type of ratio that compare quantities to 100. To learn more about percents, go through this module. Be sure to click around to each part of the module using the menu below the video.
This video reviews how to convert between fractions, decimals, and percents.
Work for the Week
Comment: In what contexts do you hear people talking about percents outside of school?
Update: Find an example of percents mentioned in an article, video, book, newspaper, magazine, etc. Post a picture, video, screenshot, or link to what you found and describe the comparison that the percents make. Be sure that your multimedia has at least 3 examples of percents.
Replies: Reply to three of your classmates' updates. Rewrite one of the percent your classmate posted as a fraction and a decimal. Once a percent has been chosen in a reply, no other students cna use the same percent in their own replies.
Purpose of this update: This update provides some foundational knowledge of percents and helps students learn how to convert between fractions, decimals, and percents. The conversions are an important part of a student's number sense and math fluency.
Teaching tips: It is helpful to emphasize the fact that percents are always ratios compared to 100, or "per 100." If students are having trouble with the conversions, consider making them a cheat sheet that shows how to convert each type of number to the others.
Extensions: It's a bit harder to work with really large or really small percents.
Other resources: Here is a Quizlet I made to help students memorize the basic fraction/decimal/percent conversions.
Learning Objective: 6.RP.3 Solve problems involving finding the whole, part, or percent given the other two.
We can use percents and proportions together to solve mathematical problems. There are three different types of problems you will encounter: finding the part, finding the whole, and finding the percent.
Next, we will take a look at some strategies for finding these missing pieces. There are many other strategies out there that you can use as well. If you know other strategies, feel free to use them in your update for this week.
Finding Any Missing Number - This strategy can be used to find any missing piece of the percent proportion.
The following videos show alternate strategies for finding one missing piece of the percent proportion.
Finding the Percent
Finding the Part
Finding the Whole (or "Total")
Work for the Week
Comment: Which strategy was the easiest? Which is the most challenging? Give one tip to your classmates on how they can make the strategy you found the easiest make sense to them, too.
Update: Post a video lesson of yourself teaching how to solve the problems below. Also, make up your own problem where you must solve for the whole, and teach how to solve it as well.
1. A survey in the student newspaper stated that 32% of the students polled walk to school. If there are 675 students in the school, about how many walk to school?
2. Jason correctly answered 18 questions out of 20 on his quiz. What is his percent grade?
Replies: Reply to three of your classmates' updates. Give your classmate feedback on their video lesson in the following areas: mathematical precision, organization, and clarity.
Purpose of this update: In this update, students use proportions to solve percent problems. This helps show the relationship between proportions and percents, and also reinforces the idea that percents are a special type of ratio. Students who had trouble solving proportions in the earlier updates also get to revisit those skills here after taking a little break from them, which helps hone their skills.
Teaching tips: There are too many strategies for solving these types of problems to include in this update. Encourage students to always use the strategy shown in the first video, because then they only need to know one strategy to find any missing number. However, students are welcome to use the other strategies presented in the update, or any strategy they may have known before.
Extensions: Any percent application problem is an excellent extension here. Some great choices would be percent of change and percent increase/decrease problems (markup, discount, sale, tax, tip, comission). Students with a foundation of Algebra can also work on simple and compound interest problems.
Topic B in this Engage New York unit has some challenging percent application problems.
This project is similar to the updates you have been doing throughout the unit. For you project, you will be creating problems to show your knowledge of the topics in this unit, then film yourself solving them in a video lesson. You must create one problem per topic that is different than the one you created for your updates.
Use the feedback your peers left you in the comments of each update to make your video lessons more precise and helpful.
Use the structure tool to create an introduction, a page for each topic, and a conclusion. On each topic page, you must include your video lesson and 1-2 paragraphs explaining the topic in writing. In your explanation, be sure to define important vocabulary and explain what you know about the topic.
Topics:
• Ratios
• Proportions
• Rates
• Percents
Rubric:
This peer-reviewed work serves as the final assessment. Rather than a traditonal test where students must answer teacher-created questions, this final assessment allows students to demonstrate their knowledge using contexts than interest them. The resulting video lessons can be compiled in a bank for students to come back to as the year goes on. This product of Collaborative Intelligence is a resource that will be useful for current and future students. The peer-reviewed work should be released by the fifth update to ensure ample time for students to work on it.
This brief survey can be used for teachers to quickly check in with the students. This survey does not need to be used for a grade, but rather can give the teacher formative feedback on what topics need to be reviewed in the weekly discussions. The teacher can release this survey as a pre-test before the learning module begins, midway through the module, or at the end of the module.