What do gambling, driving a car, and cooking have in common? Ratios. And 6th-grade math emphasizes ratios. There are three ratio and proportional relationship standards for 6th grade.
CC.6.RP.1 Understand ratio concepts and use ratio reasoning to solve problems. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC:
CC.6.RP.2 Understand ratio concepts and use ratio reasoning to solve problems. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0 (b not equal to zero), and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” (Expectations for unit rates in this grade are limited to non-complex fractions.)
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC:
These standards start with “understand”. Understanding is only the second level in Bloom’s Taxonomy. Yet in math, it is possible to apply a skill (the third level of Bloom’s) without understanding. Many students are great at robot-like, following the algorithm without any understanding. The ratio standards start with conceptual understanding. In order to promote understanding, the other standards refer to daily-life[ii] examples. This implies that to understand math you must be able to see and experience the math around you.
So, what are the students supposed to understand? First, the students must understand the concept of a ratio. What is it? Where is it? Why is it? When is it? How is it? Then the students must specifically understand the concept of a unit ratio. Again, with what, where, why, when, and how.
Ratios are quantifying and standardizing comparisons of amount. The students have learned to compare amounts by telling if one amount is less than, more than, or equal to another. Ratios go farther. Ratios tell to what extent the numbers or amounts are greater than or less than. Ratios are scalable comparisons – 1 head to 4 legs has the same ratio value as 3 heads to 12 legs.
The standards for 6th-grade are for ratios comparing only two quantities. There are ratios of more than two quantities (compound ratios).
The second standard is understanding a unit rate can be associated with an equivalent ratio. This can be hard to understand until you convert unit rates to equivalent ratios and equivalent ratios to unit rates. The students will be using only the counting numbers in the ratios. The students will need to understand that b ≠ 0. Most of the time, the concept of b ≠ 0 will not even come up because the ratios are based on daily life, and we do not usually compare zero items in daily life.
The third standard (quoted below) is a multipart standard about calculating and mathematical techniques to apply ratios. The standard lists specific skills that can be used – make tables, use tape diagrams, use double number lines, plot on a coordinate grid, or equations. Only making tables and plotting on the coordinate grid are required.
This is where the standards can be helpful. If you know that your assessments are truly aligned to the standards with precision, you can teach only what is in the standards. There are many techniques that have become old standbys that are included in units simply because they have always been taught that way. If it is not in the standard, and your assessments are truly standards-based, and your students understand the concept without the old standby, you can move on. Of course, if you find the students understand better with the old standbys, use them.
Standard 3 also states that students should be able to work with percents as ratios and convert units of measurement.
CC.6.RP.3 Understand ratio concepts and use ratio reasoning to solve problems. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
CC.6.RP.3a Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
CC.6.RP.3b Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
CC.6.RP.3c 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.
CC.6.RP.3d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC:
The organization of CC.6.RP.3 is interesting. We have the initial, all-encompassing standard, then the 4 sub-standards. The overarching concepts are understand ratios concepts and solve problems. Based on the “parent standard” of CC.6.RP.3 I would expect all the “child standards” to have a rigor of either comprehension or application but only one does. 3b has a rigor of application but the other three focus on procedural skills and fluency.
Math.CC.6.RP.3 lists 4 examples of ratio reasoning.
- tables of equivalent ratios
- tape diagrams
- double number line diagrams
- equations.
However, the child standards list different types of ratio reasoning.
- tables of equivalent ratios
- plotting on a coordinate grid
- unit pricing
- constant speed
- percents
- convert measurement units
So do we need to teach all of these? No. The first 4 are examples: tables and equations are in other standards but tape diagrams and line diagrams are not requirements. Again, it comes down to knowing your students and your assessments. If the assessments go outside of the standards, you have to teach what is on the test. If the students are not understanding, the techniques listed for CC.6.RP.3 can be useful as well as any other old standby.
The ratio standards are the backbone of the sixth-grade standards. This is one area where I definitely disagree with the standards. Ratios are something many students already have experience with during primary grades. They color 3 out of 5 items or color 4 items red and 3 items blue. Ratios should be taught before fractions (except 1/2, 1/4, 3/4, 1/3, 2/3 since the students use these in daily life). If the students already have a conceptual understanding of equivalent ratios, equivalent fractions should be easy. But we are stuck with the standards we have so students learn fractions early. Primary teachers can add understanding of ratios if they have time. Then fractions, hopefully, will be less terrifying.
[ii] School is not “imaginary life” so I do not use the phrase “real-life examples”.