7.RP.A.1Common CoreMathRatios and Proportional RelationshipsGrade 7
7.RP.A.1: Computing Unit Rates with Fractions
In plain English: 7.RP.A.1 is the Common Core grade 7 math standard that asks students to compute unit rates when the quantities in a ratio are fractions, such as 1/2 mile in each 1/4 hour. Students write the rate as a complex fraction and divide, for lengths, areas and other quantities, in like or different units. It builds on grade 6 unit rates and leads to proportional relationships.
Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction (1/2)/(1/4) miles per hour, equivalently 2 miles per hour.
Common Core State Standards for Mathematics · Domain: Ratios and Proportional Relationships (RP) · Cluster: Analyze proportional relationships and use them to solve real-world and mathematical problems. Also written as 7.RP.1 · Official standard
Students find unit rates when one or both amounts in a ratio are fractions. A ratio compares two quantities, such as 3 cups of flour to 2 eggs. A unit rate tells how much of the first quantity goes with exactly 1 unit of the second, such as 2 miles per 1 hour. In grade 6, students found unit rates with whole numbers. In grade 7, the amounts can be fractions, so the rate is written as a complex fraction: a fraction whose top, bottom or both are fractions, such as (1/2)/(1/4).
Students simplify a complex fraction by dividing: multiply the top by the reciprocal of the bottom (the fraction flipped over, so the reciprocal of 3/4 is 4/3). The standard names three kinds of quantities, and the lesson uses all of them: lengths (distances, map lengths), areas (square yards painted, square meters of garden) and other quantities (cups, dollars, pages). It also covers like units, where both amounts use the same unit (cups and cups), and different units, where they do not (miles and hours).
Learning Objectives
By the end of this lesson, students will be able to:
Write a unit rate from a ratio of fractions as a complex fraction and simplify it by dividing
Compute unit rates for lengths, areas and other quantities, and name the units of the answer
Tell whether a ratio uses like units or different units, and explain why a like-unit rate can be said without units
Find both unit rates of a ratio and explain why they are reciprocals
Compare two unit rates to decide which is faster, cheaper or the same
Prior Knowledge Required
Students should already be comfortable with:
Unit rates with whole numbers and rate language such as "per" and "for each" 6.RP.A.2
Dividing a fraction by a fraction, for example (2/3) ÷ (1/9) = 6 6.NS.A.1
Double number lines and converting units, such as 15 minutes = 1/4 hour 6.RP.A.3
Finding the area of a rectangle with fraction side lengths 5.NF.B.4
Write this question on the board. Students answer on scrap paper and share how they thought about it.
Warm-Up Prompt
"Leo runs 3 miles in 1/2 hour. At the same pace, how far does he run in 1 hour? What if he runs 1 1/2 miles in 1/4 hour instead?"
Many students double the first one: 3 miles in 1/2 hour means 6 miles in 1 hour. For the second, some students multiply by 4, since four quarter hours make an hour: 4 × 1 1/2 = 6. Both answers are the same speed, 6 miles per hour. Write the second one as 1 1/2 ÷ 1/4 and tell the class that today they will learn to find "per 1" amounts like this one even when both numbers are fractions. A mixed number (a whole number and a fraction, like 1 1/2) is changed to a fraction first: 1 1/2 = 3/2.
Direct Instruction20 minutes
Review the words from the overview: ratio, unit rate, complex fraction and reciprocal. Then show the three steps students use on every problem:
Decide what goes "per 1": the quantity after the word "per" goes on the bottom. For miles per hour, hours go on the bottom.
Write the complex fraction: put the amount of the other quantity on top, for example (1/2 mile)/(1/4 hour).
Divide and label: multiply the top by the reciprocal of the bottom, then write the units, such as "2 miles per hour".
Work through the examples. Start with the official example, and show it on a double number line (two number lines lined up so that matching amounts sit one above the other), as in Diagram 1.
Length per time, different units: the official example
A person walks 1/2 mile in each 1/4 hour. How far does the person walk in 1 hour?
Equation: (1/2)/(1/4) = 1/2 × 4/1 = 2. The unit rate is 2 miles per hour. Check: four quarter hours make 1 hour, and 4 × 1/2 mile = 2 miles.
Ratio of lengths, different units: a map scale
A map scale tells how much real distance a length on the map stands for. On a trail map, 3/4 inch stands for 2 1/4 miles of real trail. How many miles does 1 inch on the map stand for?
Equation: (2 1/4)/(3/4) = 9/4 × 4/3 = 3. The unit rate is 3 miles per inch. Both amounts are lengths, but the units are different, so the answer keeps both units.
Area per time, different units
Sam paints 7 1/2 square yards of fence in 3/4 hour. (A square yard is a square that is 1 yard on each side.) How many square yards does he paint per hour?
Equation: (7 1/2)/(3/4) = 15/2 × 4/3 = 10. The unit rate is 10 square yards per hour. Diagram 2 shows why: he paints 2 1/2 square yards in each 1/4 hour, and an hour has four quarter hours.
Ratio of areas, like units
In a community garden, each plot has 5/6 square meter of vegetables for every 1/3 square meter of flowers. How many square meters of vegetables are there per square meter of flowers?
Equation: (5/6)/(1/3) = 5/6 × 3/1 = 5/2 = 2 1/2. There are 2 1/2 square meters of vegetables per square meter of flowers. The units are alike, so the rate can be said without units: the vegetable area is 2 1/2 times the flower area.
Other quantities, like units, both unit rates
A lemonade recipe uses 1/3 cup of lemon juice for every 1 1/4 cups of water.
Equation: Water per cup of juice: (1 1/4)/(1/3) = 5/4 × 3/1 = 15/4 = 3 3/4 cups. Juice per cup of water: (1/3)/(1 1/4) = 1/3 × 4/5 = 4/15 cup. The two unit rates are reciprocals of each other.
Why multiply by the reciprocal? Dividing by 1/4 asks "how many quarters fit in 1?" Four quarters fit, so dividing by 1/4 is the same as multiplying by 4. Dividing by 3/4 is the same as multiplying by 4/3: first find the amount for 1/4 (divide by 3), then for 4/4 (multiply by 4). Diagram 2 shows these two steps for the fence.
Like and different units. When the units are different, such as miles and hours, the unit rate must name both: 2 miles per hour. When the units are alike, such as square meters and square meters, the units cancel, and the unit rate is a plain number that says "how many times as much." If a problem gives the same kind of quantity in two different units, such as inches and feet, students may change one unit first so that the units are alike. Ask: "Which quantity goes on the bottom if we want juice per cup of water?" (Water.)
Guided Practice15 minutes
Pairs solve four problems. For each one, Partner A says which quantity goes on the bottom and why, and Partner B writes the complex fraction. Then both simplify and compare. Circulate and ask each pair to say the units of every answer.
Guided practice problems with answers
Problem
Answer
Ana swims 3/8 mile in 1/4 hour. What is her speed in miles per hour?
(3/8)/(1/4) = 3/8 × 4 = 3/2, so 1 1/2 miles per hour
Two-thirds of a gallon of paint covers 250 square feet of wall. How many square feet does 1 gallon cover?
A copy machine enlarges a drawing so that a line 2 inches long becomes 3 1/2 inches long. How long is the copy of a 1-inch line?
(3 1/2)/2 = 7/4, so 1 3/4 inches of copy per inch of drawing (like units: the copy is 1 3/4 times as long)
Three-fourths of a pound of cheese costs $4.80. What is the price per pound?
4.80/(3/4) = 4.80 × 4/3 = 6.40, so $6.40 per pound
Listen for these errors: multiplying the two fractions instead of dividing, putting the wrong quantity on the bottom, and dropping the units. For the copy machine problem, ask: "Why is the answer bigger than 1?" (The copy is larger than the drawing.)
Independent Practice10 minutes
Students simplify each complex fraction, then solve the two word problems and label the units. Early finishers write a story that fits one of the first four.
Independent practice problems with answers
Problem
Answer
(4/5)/(1/10)
8
(2/3)/(4/9)
3/2 = 1 1/2
(1 1/4)/(1/2)
5/2 = 2 1/2
(7/8)/(1 3/4)
1/2
Kira earns $9 for 3/4 hour of dog walking. How much does she earn per hour?
$12 per hour
A bean seedling grows 3/4 inch in 1/2 day. How much does it grow per day?
1 1/2 inches per day
Closure5 minutes
Exit ticket: (1) Tomás bikes 2 miles in 1/6 hour. What is his speed in miles per hour? (Answer: 2/(1/6) = 12 miles per hour.) (2) A cook uses 3/4 teaspoon of salt for 1/2 pound of pasta. How much salt is that per pound? (Answer: (3/4)/(1/2) = 1 1/2 teaspoons per pound.) (3) In one sentence: why can a unit rate with like units be written without units?
Differentiation Strategies
For Struggling Students
Draw a double number line, like Diagram 1, before dividing, and mark the given pair and the "1" on the bottom line
Start with a whole number on top, such as 6/(1/2), then move to a fraction on top
Give a card with the sentence frame "___ per 1 ___" so students decide which quantity goes on the bottom
For Advanced Students
Ask students to find a unit rate when the time is given in minutes, such as 3/4 mile in 12 minutes, by changing minutes to a fraction of an hour first
Give two map scales in different units and ask which map shows more ground per inch
Ask students to explain why the two unit rates of any ratio multiply to 1
Assessment Guidance
What to Look For
Check that students put the "per 1" quantity on the bottom of the complex fraction before they divide, and that they divide rather than multiply the two fractions. Every answer should name its units, such as "miles per hour" or "square feet per gallon", except like-unit rates, which students should be able to say as "times as much." Look for mixed numbers changed to fractions before dividing, and for a quick reasonableness check, such as "less than an hour, so the rate per hour is bigger than the amount given."
02
Classroom Activities
3 Activities
1
Rate Card Match
15 minPairs
Pairs match each situation card to its complex fraction card and its unit rate card, then check each match by dividing. There are 12 cards: 4 situation cards, 4 complex fraction cards and 4 unit rate cards.
Situation Cards
S1: Zoe jogs 5/6 mile in 1/6 hour.
S2: A 1/3-pound bag of trail mix costs $2.50.
S3: A student paints 5 1/4 square feet of a school mural in 3/4 hour.
S4: On a house plan, 1/4 inch stands for 1 1/2 feet.
Complex Fraction and Unit Rate Cards (answer key)
S1: (5/6)/(1/6) and 5 miles per hour
S2: 2.50/(1/3) and $7.50 per pound
S3: (5 1/4)/(3/4) and 7 square feet per hour
S4: (1 1/2)/(1/4) and 6 feet per inch
Discussion Questions
S4 compares two lengths in different units. Change 1 1/2 feet to 18 inches. What is the unit rate in inches per inch? (72: the real house is 72 times as long as the plan.)
Which cards compare a length or an area with a time? Why can't you compare 5 miles per hour with 7 square feet per hour?
For S2, what is the other unit rate, pounds per dollar? (2/15 pound per dollar.)
Modification for Distance Learning
Put the 12 cards on a shared slide. Pairs drag each complex fraction card and unit rate card next to its situation and type one sentence about the units.
2
Measure the Enlargement
20 minGroups of 3-4
Each group gets a small picture card and an enlarged copy of it. Students measure both with a ruler to the nearest 1/4 inch and compute unit rates for lengths and for areas. The teacher prints the cards ahead of time: the original is 2 1/2 inches wide and 1 1/2 inches tall, and the copy is 3 3/4 inches wide and 2 1/4 inches tall.
Procedure
Measure the width and the height of both cards and record them in a table
Find the length unit rate for the width: inches of copy per inch of original, (3 3/4)/(2 1/2) = 1 1/2
Find the same unit rate for the height: (2 1/4)/(1 1/2) = 1 1/2
Find the area unit rate: (8 7/16)/(3 3/4) = 2 1/4 square inches of copy per square inch of original
Discussion Questions
The width and the height give the same length unit rate. Is the area unit rate the same number?
The area unit rate, 2 1/4, equals 1 1/2 × 1 1/2. Why does the area grow in both directions?
All the units here are alike (inches and inches, square inches and square inches). What does "1 1/2" mean without units?
Challenge Variation
Groups draw a smaller copy of the original on grid paper with a length unit rate of 1/2 (1 1/4 inches by 3/4 inch), then predict and check its area unit rate (1/4).
3
Which Is Faster or Cheaper?
15 minPairs
Pairs compare three pairs of rates. For each pair, they compute both unit rates and decide which is faster, cheaper or the same. This shows why unit rates are useful: they put two ratios on the same "per 1" footing.
Comparison Cards
Pair A (speed): Maya bikes 7/8 mile in 1/12 hour. Leo bikes 1 1/4 miles in 1/8 hour. (Maya: 10 1/2 miles per hour. Leo: 10 miles per hour. Maya is faster.)
Pair B (price): Store 1 sells 3/4 pound of almonds for $5.25. Store 2 sells 1 1/4 pounds for $6.25. (Store 1: $7 per pound. Store 2: $5 per pound. Store 2 is cheaper.)
Pair C (paint color): Mix 1 uses 1/2 cup of blue paint for 3/4 cup of white. Mix 2 uses 1/3 cup of blue for 1/2 cup of white. (Both: 2/3 cup of blue per cup of white. The two mixes are the same shade.)
Discussion Questions
Only one pair has equal unit rates. Which one, and what does that mean for the paint?
In Pair A, Leo rides farther but also for longer. Why can't you decide who is faster from the distances alone?
In Pair B, would the answer change if you compared pounds per dollar instead? Which store gives more almonds per dollar?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Double Number Line for the Official Example
The top line shows miles and the bottom line shows hours, drawn to scale so that matching amounts line up. The shaded first step is the given pair, 1/2 mile in 1/4 hour. Following the lines to 1 hour gives the unit rate, 2 miles per hour, which matches the complex fraction (1/2)/(1/4) = 2.
Diagram 2: Why Dividing by 3/4 Works
Sam paints 7 1/2 square yards of fence in 3/4 hour. Splitting the three quarter hours gives 2 1/2 square yards in each 1/4 hour, and a fourth quarter hour (dashed) completes 1 hour: 10 square yards per hour. The bars are drawn to scale, 50 pixels per square yard.
04
Homework Assignment
~30 min
7.RP.A.1 Homework: Unit Rates with Fractions
Directions: Show your work. For each unit rate, write the complex fraction first, then simplify it. Label every answer with its units, or say "times as much" when the units are alike.
A cyclist rides 5 1/4 miles in 35 minutes. Write 35 minutes as a fraction of an hour, then find the cyclist's speed in miles per hour.
Part 2: Lengths and Areas (Problems 3-4)
On a state park map, 5/8 inch stands for 2 1/2 miles. How many miles does 1 inch stand for? How long is a trail that is 1 1/2 inches long on the map?
A robot lawn mower cuts 37 1/2 square meters of grass in 3/4 hour. How many square meters does it cut per hour? How long will it take to cut a lawn of 300 square meters?
Part 3: Other Quantities in Like and Different Units (Problems 5-6)
A trail mix recipe uses 2/5 cup of raisins for every 1 1/5 cups of peanuts. Find the cups of raisins per cup of peanuts and the cups of peanuts per cup of raisins. Explain why each answer can be said as "times as much."
Store A sells 3/4 pound of grapes for $2.10. Store B sells 1 1/3 pounds of the same grapes for $3.60. Find the price per pound at each store. Which store is cheaper, and by how much per pound?
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Setup
Complex fraction written with the "per 1" quantity on the bottom
Correct division but no complex fraction shown
Fractions multiplied or quantities reversed
Computing
All fractions and mixed numbers simplified correctly
One arithmetic error
Most answers incorrect
Units
Every answer labeled, like units explained as "times as much"
Some labels missing
No units
Reasoning
Comparisons and follow-up questions answered with the unit rate
Correct answer with no reason
Missing or incorrect
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
Kai walks 3/5 mile in 1/5 hour. What is his walking speed?
Answer: C
Divide the distance by the time: (3/5)/(1/5) = 3/5 × 5 = 3 miles per hour. Choice A multiplies the fractions (3/5 × 1/5 = 3/25). Choice B divides the time by the distance, (1/5)/(3/5) = 1/3, which is hours per mile. Choice D reads 3/5 mile as 3 miles: 3 ÷ (1/5) = 15.
Question 2 of 20 · Multiple Choice
Two-thirds of a pound of strawberries costs $4. Which expression gives the price in dollars per pound?
Answer: A
Dollars per pound puts the pounds on the bottom: 4 ÷ (2/3) = 4 × 3/2 = $6 per pound. Choice B puts the dollars on the bottom and gives pounds per dollar (1/6). Choice C multiplies instead of dividing. It gives about $2.67, but a whole pound must cost more than the $4 paid for 2/3 pound. Choice D subtracts the pounds from the dollars, which mixes two different units.
Question 3 of 20 · Multiple Choice
Simplify the complex fraction (5/8)/(1/4).
Answer: D
Multiply by the reciprocal of the bottom: 5/8 × 4/1 = 20/8 = 5/2. Choice A multiplies the two fractions (5/8 × 1/4). Choice B divides the bottom by the top, (1/4)/(5/8) = 2/5. Choice C subtracts 1/4 = 2/8 from 5/8 instead of dividing.
Question 4 of 20 · Multiple Choice
Jen paints 22 1/2 square feet of fence in 3/4 hour. How many square feet does she paint per hour?
Answer: B
(22 1/2)/(3/4) = 45/2 × 4/3 = 30 square feet per hour. Choice A multiplies 45/2 × 3/4 = 135/8 = 16 7/8. Choice C divides the time by the area, which gives hours per square foot. Choice D divides by 3 only: it finds the area for one quarter hour and forgets to multiply by 4.
Question 5 of 20 · Multiple Choice
One spring day at noon, a fence post 4/5 meter tall casts a shadow 1/2 meter long. How many meters of shadow are there per meter of height?
Answer: A
Shadow per meter of height puts the height on the bottom: (1/2)/(4/5) = 1/2 × 5/4 = 5/8. Both amounts are in meters, so the shadow is 5/8 as long as the post. Choice B divides the height by the shadow (8/5). Choice C multiplies 1/2 × 4/5. Choice D subtracts 1/2 from 4/5.
Question 6 of 20 · Multiple Choice
On a hiking map, 2/3 inch stands for 5 miles. How many miles does 1 inch stand for?
Answer: D
5/(2/3) = 5 × 3/2 = 15/2 = 7 1/2 miles per inch. Choice A multiplies 5 × 2/3 = 10/3. Choice B divides the map length by the distance, (2/3)/5 = 2/15, which is really inches per mile. Choice C multiplies by 3 but forgets to divide by 2.
Question 7 of 20 · Multiple Choice
Rosa walks 2/3 mile in 10 minutes. What is her speed in miles per hour?
Answer: B
10 minutes is 10/60 = 1/6 hour, so (2/3)/(1/6) = 2/3 × 6 = 4 miles per hour. Choice A divides by 10 minutes, which gives miles per minute, not miles per hour. Choice C multiplies 2/3 × 10. Choice D multiplies 2/3 by 60 minutes and forgets to divide by 10.
Question 8 of 20 · Multiple Choice
A smoothie recipe uses 3/4 cup of yogurt for every 1 1/4 cups of fruit. How much yogurt is that per cup of fruit?
Answer: D
(3/4)/(1 1/4) = 3/4 × 4/5 = 3/5 cup of yogurt per cup of fruit. Choice A divides the fruit by the yogurt, which gives cups of fruit per cup of yogurt (5/3). Choice B multiplies 3/4 × 5/4 = 15/16. Choice C gives the yogurt for 1 1/4 cups of fruit, not for 1 cup.
Question 9 of 20 · Multiple Choice
Which unit rate compares two quantities measured in like units (the same kind of unit)?
Answer: C
Cups of water per cup of juice uses cups for both amounts, so the units are alike, and the rate means the water is 2 1/2 times the juice. Choice A compares miles with hours, choice B dollars with pounds, and choice D square meters with hours: all different units.
Question 10 of 20 · Multiple Choice
Four stores sell the same sliced turkey. Which is the best buy?
Answer: B
Find each price per pound: 3.60/(3/4) = $4.80, 2.30/(1/2) = $4.60, 7.50/(1 1/2) = $5.00 and 3.20/(2/3) = $4.80. The lowest is $4.60, choice B. Choice C is the biggest package but costs the most per pound, $5.00. Choices A and D both cost $4.80 per pound, 20 cents more than B.
Question 11 of 20 · Multiple Choice
On a poster, a photo covers 5/12 square foot, and the whole poster covers 2 1/2 square feet. How many square feet of photo are there per square foot of poster?
Answer: A
(5/12)/(2 1/2) = 5/12 × 2/5 = 1/6. The units are alike, so the photo covers 1/6 of the poster. Choice B divides the poster by the photo. Choice C multiplies 5/12 × 5/2 = 25/24. Choice D reads 2 1/2 as 1/2: (5/12)/(1/2) = 5/6.
Question 12 of 20 · Multiple Choice
Theo reads 6 1/4 pages in 1/4 hour. How many pages does he read per hour?
Answer: C
(6 1/4)/(1/4) = 25/4 × 4 = 25 pages per hour. Choice A multiplies 25/4 × 1/4 = 25/16. Choice B treats 1/4 hour as half an hour and doubles. Choice D divides the time by the pages, which gives hours per page.
Question 13 of 20 · Multiple Choice
A hybrid car uses 5/8 gallon of gas to drive 27 1/2 miles. How many miles does it go per gallon?
Answer: C
(27 1/2)/(5/8) = 55/2 × 8/5 = 44 miles per gallon. Choice A multiplies 55/2 × 5/8 = 275/16. Choice B divides the gas by the miles, which gives gallons per mile. Choice D divides by 5 only: it finds the miles for 1/8 gallon and forgets to multiply by 8.
Question 14 of 20 · Multiple Choice
A plant food label says to use 1/4 teaspoon of plant food for every 3/4 gallon of water. Which two unit rates describe this mix?
Answer: A
(1/4)/(3/4) = 1/4 × 4/3 = 1/3 teaspoon per gallon, and (3/4)/(1/4) = 3/4 × 4 = 3 gallons per teaspoon. The two rates are reciprocals. Choice B multiplies 1/4 × 3/4 = 3/16 and then takes its reciprocal, 16/3 = 5 1/3. Choice C divides the wrong way for each label. Choice D finds the first rate and repeats it instead of taking its reciprocal.
Question 15 of 20 · Short Answer
A hiker climbs 3/4 mile of steep trail in 2/5 hour. Write the unit rate as a complex fraction and find her speed in miles per hour.
(3/4)/(2/5) = 3/4 × 5/2 = 15/8, so 1 7/8 miles per hour.
Question 16 of 20 · Short Answer
An artist covers a rectangle 3/4 foot wide and 2/3 foot tall with mosaic tiles in 1/6 hour. Find the area of the rectangle, then the number of square feet she tiles per hour.
A blue ribbon is 2 1/2 yards long and a red ribbon is 5/6 yard long. How many yards of blue ribbon are there per yard of red ribbon? What does the answer mean?
(2 1/2)/(5/6) = 5/2 × 6/5 = 3 yards of blue per yard of red. The units are alike, so it means the blue ribbon is 3 times as long as the red one.
Question 18 of 20 · Short Answer
A model car is 3 3/4 inches long. The real car is 12 1/2 feet long. Find the unit rate in inches of model per foot of real car. Then change 12 1/2 feet to inches and find the unit rate in inches of model per inch of real car.
(3 3/4)/(12 1/2) = 15/4 × 2/25 = 3/10 inch per foot. 12 1/2 feet = 12 1/2 × 12 = 150 inches, so (3 3/4)/150 = 1/40 inch per inch: with like units, the model is 1/40 as long as the real car.
Question 19 of 20 · Short Answer
Ava runs 3/4 mile in 1/10 hour. Ben runs 1 1/3 miles in 1/6 hour. Who runs faster? Show both unit rates.
Ava: (3/4)/(1/10) = 30/4 = 7 1/2 miles per hour. Ben: (1 1/3)/(1/6) = 4/3 × 6 = 8 miles per hour. Ben runs faster, by 1/2 mile per hour.
Question 20 of 20 · Short Answer
Sam says a jogger who covers 3/5 mile in 1/10 hour jogs at 3/50 mile per hour. What did Sam do wrong? Find the correct unit rate.
Sam multiplied the fractions (3/5 × 1/10 = 3/50) instead of dividing. The correct rate is (3/5)/(1/10) = 3/5 × 10 = 6 miles per hour. A quick check: 1/10 hour is short, so the distance per hour must be more than 3/5 mile.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 7.RP.A.1 mean?
7.RP.A.1 means students can find a unit rate when the amounts in a ratio are fractions. For example, 1/2 mile in each 1/4 hour is a rate of 2 miles per hour. Students write the rate as a complex fraction, (1/2)/(1/4), and simplify it by dividing. The standard asks for this with lengths, areas and other quantities, in like or different units.
What grade is 7.RP.A.1, and what comes after it?
It is a grade 7 standard in the Ratios and Proportional Relationships domain. Right after it, students use unit rates to decide whether two quantities are proportional and to write equations such as y = 2x (7.RP.A.2). In grade 8, the unit rate becomes the slope of a graph (8.EE.B.5), and in Algebra I it becomes the rate of change of a linear function.
What is a complex fraction?
A complex fraction is a fraction whose top, bottom or both are fractions. For example, (2/5)/(4/5) has a fraction on top and on the bottom. To simplify it, divide the top by the bottom: 2/5 × 5/4 = 1/2. In this standard, the complex fraction is simply a unit rate that has not been simplified yet.
How do you find a unit rate with fractions?
Divide the amount of the first quantity by the amount of the second, the one that goes "per 1." For 2/3 cup of flour per 1/3 batch of muffins, divide 2/3 by 1/3: 2/3 × 3 = 2 cups per batch. Writing the division as a complex fraction helps students keep the right quantity on the bottom.
How is 7.RP.A.1 different from the grade 6 unit rate standard?
The grade 6 standard, 6.RP.A.2, uses unit rates with whole numbers and simple fractions, such as $12 for 3 notebooks. 7.RP.A.1 adds ratios of fractions, where students must divide one fraction by another, and it names areas and units that are alike or different. The idea of a rate "per 1" stays the same.
What is the difference between like units and different units?
Like units means both quantities use the same unit, such as cups and cups, and different units means they do not, such as miles and hours. With like units, the units cancel and the unit rate says how many times as much: a scale model 1/20 as long as the real object. With different units, the answer must name both units, such as square feet per gallon.
Which number goes on the bottom of the complex fraction?
The quantity that comes after the word "per" goes on the bottom. For miles per hour, hours go on the bottom; for hours per mile, miles go on the bottom. Both are correct unit rates for the same ratio, and they are reciprocals, so students should read the question carefully to see which one it asks for.
What mistakes should teachers watch for?
A common mistake is multiplying the two fractions instead of dividing, which gives an answer that is much too small. Other frequent errors are putting the wrong quantity on the bottom, forgetting to change minutes to a fraction of an hour, reading a mixed number such as 1 1/2 as 1/2, and leaving off the units. A quick check helps: if the time is less than 1 hour, the distance per hour must be larger than the distance given.
Why does dividing by a fraction give a bigger number?
Dividing by a fraction less than 1 asks how many of those small pieces fit, so the answer is larger. Dividing by 1/5 asks how many fifths fit in each whole, and five do, so it is the same as multiplying by 5. In a rate, this matches the story: if you do something in 1/5 hour, you can do it five times in 1 hour.
How can parents help with 7.RP.A.1 at home?
Parents can point out fraction rates in everyday life and ask for the "per 1" amount. For example: "This 1/2-pound block of butter costs $2.25. What is the price per pound?" ($4.50.) Or, while cooking: "The recipe uses 3/4 cup of milk for half of the batter. How much milk for the whole recipe?" Ask your child to say the units of each answer.
07
Related Standards
6 standards
These standards connect to 7.RP.A.1: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
6.RP.A.2Prerequisite
Understand a unit rate a/b for a ratio a:b and use rate language
