What is the difference between a ratio and a proportion?

A ratio compares two quantities, such as 4 pencils to 3 pens. A proportion is a statement that two ratios are equal, such as 4/3 = 12/9. Students usually meet ratios and unit rates in 6th grade and then use proportions and the constant of proportionality in 7th grade.

Why does my child get unit rate questions wrong when they can divide fine?

Most errors are about which number goes on top, not about the division itself. Dollars per pound and pounds per dollar both come from the same two numbers, so sessions spend time on reading the phrase 'per' and deciding what quantity is being measured against one unit.

Do students need to know cross-multiplication?

It is one valid method and it is covered, but ratio tables, scaling, and unit rates are introduced first so the learner can check whether an answer is reasonable. Students who rely only on cross-multiplication often cannot tell when they have set the proportion up backwards.

How does this connect to what comes next in algebra?

The constant of proportionality in y = kx becomes slope in 8th grade, and a proportional relationship is a linear function with a y-intercept of zero. Students who are comfortable finding k from a table or graph tend to find slope and linear equations more familiar.

Ratios & Proportions

Middle School (6-8) · Mathematics

Ratios and proportions is where middle school math shifts from counting and arithmetic to reasoning about how two quantities change together. Students learn to describe relationships like 3 cups of flour to 2 cups of water, find unit rates such as miles per hour or dollars per ounce, and recognise when a relationship is proportional — meaning one quantity is always the same number of times the other. This topic underlies percent problems, scale drawings, unit conversion, and the slope work that begins in 8th grade, so gaps here tend to resurface in algebra. Evelyn Tutor works through these ideas by voice, asking the learner to explain what each number in a ratio actually counts before any calculation happens.

Start a session on Ratios & Proportions

What this covers

Writing and reading ratios in all three forms (3:2, 3 to 2, 3/2) and distinguishing part-to-part from part-to-whole comparisons
Finding unit rates, including rates from complex fractions such as 3/4 mile in 1/2 hour, and using them to compare prices or speeds
Building equivalent ratios with ratio tables, tape diagrams, and double number lines, then scaling up or down to answer questions
Identifying the constant of proportionality k from a table, graph, equation, or verbal description, and writing y = kx
Testing whether a relationship is proportional — checking that all ratios y/x are equal and that the graph is a straight line through the origin
Applying proportional reasoning to percent increase and decrease, tax, tip, discount, simple interest, scale drawings, and unit conversions

Where learners get stuck

Adding instead of multiplying to make equivalent ratios — turning 2:3 into 3:4 by adding one to each part: Years of practice with additive patterns make 'do the same thing to both' feel correct. Students need to see that 2:3 and 3:4 give different amounts of the same mixture, usually by scaling to a common value or comparing unit rates side by side.
Losing track of which quantity is which when setting up a proportion, so units end up crossed (miles over hours on one side, hours over miles on the other): Cross-multiplication is often taught as a procedure before the meaning of the equal ratios is secure, so students focus on getting an equation to solve rather than on keeping the same quantity in the same position in both ratios.
Using the wrong base in percent problems — for example, adding 20% then subtracting 20% and expecting to return to the original price: Percent is a ratio to a specific whole, and the whole changes after the first step. Students who treat percents as fixed amounts rather than multipliers miss this, which also makes percent increase and percent of a percent confusing.
Assuming any straight-line graph or any table with a steady pattern shows a proportional relationship: Tables like (1, 5), (2, 7), (3, 9) increase steadily, so they look regular, but y/x is not constant and the line misses the origin. The distinction only becomes clear when students check the ratio at every row rather than the difference between rows.

What a session looks like

A typical 25-minute session starts with the tutor giving a short scenario — a recipe, a price per pound, a map scale — and asking the learner to state the ratio in words before writing anything. From there the learner works through two or three problems out loud, building a ratio table or naming the unit rate, and the tutor asks what each number represents rather than just checking the answer. When a mistake appears, the tutor usually poses a smaller related question (for example, doubling one quantity to see what happens to the other) instead of correcting directly. Sessions often end with one problem that reverses the direction: given the unit rate or the constant of proportionality, find the missing quantity.

Helpful to know first

Fluent multiplication and division with whole numbers and decimals
Simplifying fractions and recognising equivalent fractions
Multiplying and dividing fractions, including dividing by a fraction
Converting between fractions, decimals, and percents
Plotting ordered pairs in the first quadrant of the coordinate plane

Questions

What is the difference between a ratio and a proportion?: A ratio compares two quantities, such as 4 pencils to 3 pens. A proportion is a statement that two ratios are equal, such as 4/3 = 12/9. Students usually meet ratios and unit rates in 6th grade and then use proportions and the constant of proportionality in 7th grade.
Why does my child get unit rate questions wrong when they can divide fine?: Most errors are about which number goes on top, not about the division itself. Dollars per pound and pounds per dollar both come from the same two numbers, so sessions spend time on reading the phrase 'per' and deciding what quantity is being measured against one unit.
Do students need to know cross-multiplication?: It is one valid method and it is covered, but ratio tables, scaling, and unit rates are introduced first so the learner can check whether an answer is reasonable. Students who rely only on cross-multiplication often cannot tell when they have set the proportion up backwards.
How does this connect to what comes next in algebra?: The constant of proportionality in y = kx becomes slope in 8th grade, and a proportional relationship is a linear function with a y-intercept of zero. Students who are comfortable finding k from a table or graph tend to find slope and linear equations more familiar.