Ratios and Rates
Lessons in this chapter
About Ratios and Rates
Ratios and Rates introduces proportional reasoning, which is the single most useful idea in middle school math and the one that carries furthest into daily life.
Write and Identify Equivalent Ratios establishes that a ratio describes a relationship, not an amount. Two to three and four to six are the same ratio, which is the same logic as equivalent fractions with a different meaning attached.
Use Ratio Tables to Solve Problems and Graph and Compare Ratio Relationships give two representations. A ratio table scales both quantities together; graphing the pairs produces a straight line through the origin, which is the visual signature of a proportional relationship.
Find and Apply Unit Rates is the most practical lesson in 6th grade. Reducing a ratio to “per one” is what makes comparison possible, so $4.50 for 3 liters becomes $1.50 per liter and can be compared with any other price.
Convert Units Using Rates and Conversion Factors reframes unit conversion as multiplying by a ratio equal to 1, which explains why conversion works rather than leaving it as a remembered rule. Solve Complex Real-World Problems with Ratios and Rates then combines all of it.
All six lessons in Ratios and Rates open as 6th grade math worksheets inside the page. Use Ratio Tables to Solve Problems, Find and Apply Unit Rates and Convert Units Using Rates and Conversion Factors each give a set of math problems worked one at a time and scored as you go. They are free, and nothing here needs an account.
Frequently asked questions
Write and Identify Equivalent Ratios sets up the idea that a ratio describes a relationship rather than an amount, so 2 to 3 and 4 to 6 say the same thing. Use Ratio Tables to Solve Problems and Graph and Compare Ratio Relationships then give that relationship two representations, a table and a straight line through the origin. Find and Apply Unit Rates reduces a ratio to one unit, which is what makes prices and speeds comparable, Convert Units Using Rates and Conversion Factors turns that into unit conversion, and Solve Complex Real-World Problems with Ratios and Rates puts the whole chapter to work. It sits in the Common Core domain Ratios and Proportional Relationships (6.RP).
A ratio compares two quantities of any kind. A rate compares quantities with different units, like miles per hour or dollars per liter.
A rate reduced to one unit of the second quantity. $4.50 for 3 liters is a unit rate of $1.50 per liter, which is what makes two prices comparable.
Both quantities scale together. If 2 cups of flour need 3 eggs, then 4 cups need 6, and each column of the table keeps the same relationship.
Because the two quantities change at a constant rate, and zero of one means zero of the other. That line through the origin is the signature of a proportional relationship.
A ratio equal to 1, like 100 cm over 1 m. Multiplying by it changes the units without changing the amount, which is why unit conversion works.