What do gambling, driving a car, and cooking have in common? Ratios. And 6th-grade math just happens to emphasize 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.”
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. So the ratio standards start with conceptual understanding. In order to promote understanding, the other standards refer to daily-life[1] 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 the 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. The standards for future years address ratios of more than two quantities.
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.)
The second standard has the students solving problems with ratios yet includes understanding ratio concepts. 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 compare zero items in daily life. The students will need to calculate unit rates when one rate is a multiple of the other.
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.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.
The third standard is about calculating and mathematical techniques to apply ratios. The standard lists specific skills to use. Standard 3 also states that students should be able to work with percents as ratios and convert units of measurement.
The ratio standards are the backbone of the sixth-grade standards. The ratios can be incorporated into all the other standards and the other standards can be incorporated into the ratios standard.
[1] School is not “imaginary life” so I do not use the phrase “real-life examples”.