Understand ratio and unit rate concepts and use them to solve problems.

Given a real-world context, write and interpret ratios to show the relative sizes of two quantities using appropriate notation: a/b, a to b, or a:b where b ≠ 0.

Given a real-world context, determine a rate for a ratio of quantities with different units. Calculate and interpret the corresponding unit rate.

Extend previous understanding of fractions and numerical patterns to generate or complete a two-or three-column table to display equivalent part-to-part ratios and part-to-part-to-whole ratios.

Apply ratio relationships to solve mathematical and real-world problems involving percentages using the relationship between two quantities.

Solve mathematical and real-world problems involving ratios, rates and unit rates, including comparisons, mixtures, ratios of lengths and conversions within the same measurement system.

A ratio shows a proportional relationship between two numbers.

Ratios can be written in three different ways: with words, as a fraction, or with a colon.

Equivalent ratios can be expressed in a table. This table represents a part to part ratio, which means that it represents a proportional relationship between two parts. For every 1 cup of quinoa you are cooking, you will need 2 cups of water.

This table shows a second relationship, the whole (total cups of ingredients.) In addition to expressing the relationship between cups of quinoa and cups of water, the relationship of cups of quinoa to total cups of ingredients can be expressed, for example.

Percentages are a special type of part to whole ratio in which the whole is always 100. Students should first relate percents to their decimal and fraction equivalents.

Percentages are fractional parts that are expressed with a denominator of 100, so if the fractional part does not have a denominator of 100, students need to create equivalent fractions.

When the denominators are factors of 100, it is simple to create equivalent fractions. However, when the denominators are not factors of 100, students will need to employ other strategies to determine decimal and percent equivalents.

One helpful strategy is to consider fractions as division. When looking at the division symbol, as shown, students should think of a fraction bar with two dots to represent unknown values.

To determine this decimal/percent equivalent, perform the division as indicated in the fraction bar. Students must have a solid understanding of decimal division prior to performing this operation. If students are getting confused as to which number is the dividend and which is the divisor, they should consider if they should expect a quotient greater than 1 or less than 1. This simple step can help students set up their division problem appropriately.

A rate is a special type of ratio in which two different units are being compared; for example: miles per hour. This table can be used to find a rate of miles per hour over the course of a trip.

Eventually, students will be asked to fill in missing values in a rate table. This skill is closely related to finding equivalent fractions.

Let's look at a real-world situation. I paid $13.99 for a 6-lb bag of gummy bears. I can represent the rate of dollars to pounds as shown below.

A unit rate is a special type of rate in which the denominator is 1. In this situation, I am looking for amount of dollars that each 1 pound of gummy bears costs. In order to find this unit rate, I divide the two numbers.

One helpful strategy is to consider fractions as division, as described in the section about percentages.

Some students may struggle with this concept, so it can be helpful for them to think of creating an equivalent fraction with a denominator of 1, by dividing by a form of 1. In this case, the form of 1 that we divide by is 6/6 because this will yield a denominator of 1.