FutureStepsA quick starting check.
- Divide whole numbers, including answers that are decimals.
- Read units such as dollars, notebooks, miles and hours.
- Understand that equivalent ratios describe the same proportional relationship.
Build the method, one step at a time.
Let the units tell you what to divide
A rate compares quantities measured in different units. A unit rate gives the amount for one unit, such as dollars per notebook or miles per hour. The word per means for each.
For dollars per notebook, divide the number of dollars by the number of notebooks. Reversing that division gives notebooks per dollar. Both rates can be meaningful, but they answer different questions, so write the units beside the calculation.
Divide both quantities by the same amount
If six identical notebooks cost 9 dollars, divide both quantities by six. Nine dollars becomes 1.50 dollars and six notebooks becomes one notebook. The equivalent rate is 1.50 dollars per notebook.
You can use a ratio table to show the same idea: six notebooks cost 9 dollars, two notebooks cost 3 dollars, and one notebook costs 1.50 dollars. Every row describes the same price relationship. Multiplying the unit price by the original quantity should recover the original total.
Make a fair comparison
Before comparing, make the units match. Dollars per ounce and dollars per pound are not directly comparable. The examples here use matching units and comparable items, with no extra fees or discounts.
For a price comparison, the smaller cost per item is cheaper per item. For average speed, a larger number of miles per hour means more distance covered per hour. An average rate describes the whole interval; it does not prove that the rate stayed constant at every moment.
Two worked examples.
Compare two notebook packs
- Pack A contains six identical notebooks for 9 dollars. Calculate , so its unit price is 1.50 dollars per notebook.
- Pack B contains four comparable notebooks for 7 dollars. Calculate , so its unit price is 1.75 dollars per notebook.
- Pack A costs 0.25 dollars less per notebook. Pack B has the smaller total price, but Pack A has the smaller unit price.
Check your reasoning: 1.50 dollars × dollars and 1.75 dollars × dollars. Both unit prices rebuild their pack totals.
Find an average cycling speed
- A cyclist covers 18 miles in 1.5 hours. The question asks for miles per hour, so divide distance by time.
- Calculate . The average speed is 12 miles per hour.
- At that constant rate, 2 hours would correspond to miles. This extension assumes the rate stays the same.
Check your reasoning: At 12 miles per hour, one hour accounts for 12 miles and half an hour for 6 miles. Together that is 18 miles in 1.5 hours.
Try it before you reveal it.
Write down your thinking. Open a hint if you get stuck, then compare your method with the explanation.
Practice 1
Eight identical pens cost 12 dollars. What is the price per pen?
Show a hint
Put dollars before the division sign and pens after it.
Show the answer and explanation
1.50 dollars per pen
. Multiplying 1.50 dollars by eight pens returns the 12-dollar total.
Practice 2
Comparable juice bottles contain 2 liters for 5 dollars and 3 liters for 6 dollars. Which has the lower price per liter?
Show a hint
Find dollars per liter for each bottle before comparing.
Show the answer and explanation
The 3-liter bottle: 2 dollars per liter
The first bottle costs 2.50 dollars per liter because . The second costs 2 dollars per liter because . The second unit price is lower.
Practice 3
A machine makes 45 labels in 3 minutes. At a constant rate, how many labels would it make in 7 minutes?
Show a hint
Find labels per minute, then scale up to seven minutes.
Show the answer and explanation
105 labels
labels per minute. Then labels. This prediction uses the stated constant-rate assumption.
What to take with you.
Write the target units first. Divide to find the amount per one unit, then compare or scale while keeping those units consistent.
Try explaining one example without looking at the steps. If a step is hard to explain, revisit that part before moving on.
Give me two prices for different-sized packs of comparable items. Ask which units I need and let me calculate each unit price before you check my comparison.
Eligible students ages 13 and older with Study access can sign in and use this prompt for more practice. Check AI explanations against your class materials.
Continue in StudyWritten by FutureSteps for study and review. This lesson expands a topic in our Texas starter collection. Course reference: Texas TEKS · Grade 6 Mathematics (§111.26(b)(4)(B); §111.26(b)(4)(D)). These references do not imply endorsement, complete standard coverage or assessed mastery. Follow your school’s materials and course sequence.
Explore more in the 6th grade learning guide.