Grade: 6
Unit #1: Ratios and Unit Rates
Time: 35 Days
Unit Overview
Representing and reasoning about ratios and collections of equivalent ratios: Because the multiplication table is familiar to sixth
graders, situations that give rise to columns or rows of a multiplication table can provide good initial contexts when ratios and proportional
relationships are introduced. Pairs of quantities in equivalent ratios arising from whole number measurements such as “3 lemons for every
$1” or “for every 5 cups grape juice, mix in 2 cups peach juice” lend themselves to being recorded in a table. Initially, when students make
tables of quantities in equivalent ratios, they may focus only on iterating the related quantities by repeated addition to generate equivalent
ratios. As students work with tables of quantities in equivalent ratios (also called ratio tables), they should practice using and understanding
ratio and rate language. It is important for students to focus on the meaning of the terms “for every,” “for each,” “for each 1,” and “per”
because these equivalent ways of stating ratios and rates are at the heart of understanding the structure in these tables, providing a
foundation for learning about proportional relationships in Grade 7.
Students graph the pairs of values displayed in ratio tables on coordinate axes. The graph of such a collection of equivalent ratios lies on a
line through the origin, and the pattern of increases in the table can be seen in the graph as coordinated horizontal and vertical increases.
By reasoning about ratio tables to compare ratios, students can deepen their understanding of what a ratio describes in a context and what
quantities in equivalent ratios have in common.
For example, suppose Abby’s orange paint is made by mixing 1 cup red paint for every 3 cups yellow paint and Zack’s orange paint is made
by mixing 3 cups red for every 5 cups yellow. Students could discuss that all the mixtures within a single ratio table for one of the paint
mixtures are the same shade of orange because they are multiple batches of the same mixture. For example, 2 cups red and 6 cups yellow
is two batches of 1 cup red and 3 cups yellow; each batch is the same color, so when the two batches are combined, the shade of orange
doesn’t change. Therefore, to compare the colors of the two paint mixtures, any entry within a ratio table for one mixture can be compared
with any entry from the ratio table for the other mixture.
It is important for students to focus on the rows (or columns) of a ratio table as multiples of each other. If this is not done, a common error
when working with ratios is to make additive comparisons. For example, students may think incorrectly that the ratios 1: 3 and 3: 5 of red to
yellow in Abby’s and Zack’s paints are equivalent because the difference between the number of cups of red and yellow in both paints is the
same, or because Zack’s paint could be made from Abby’s by adding 2 cups red and 2 cups yellow. The margin shows several ways
students could reason correctly to compare the paint mixtures.
Strategies for solving problems: Although it is traditional to move students quickly to solving proportions by setting up an equation, the
Standards do not require this method in Grade 6. There are a number of strategies for solving problems that involve ratios. As students b
strategies become increasingly abbreviated and efficient. For example, suppose grape juice and peach juice are mixed in a ratio of 5 to 2
and we want to know how many cups of grape juice to mix with 12 cups of peach juice so that the mixture will still be in the same ratio.
Students could make a ratio table as shown in the margin, and they could use the table to find the grape juice entry that pairs with 12 cups of
peach juice in the table. This perspective allows students to begin to reason about proportions by starting with their knowledge about
multiplication tables and by building on this knowledge.
As students generate equivalent ratios and record them in tables, their attention should be drawn to the important role of multiplication and
division in how entries are related to each other. Initially, students may fill ratio tables with columns or rows of the multiplication table by skip
counting, using only whole number entries, and placing these entries in numerical order. Gradually, students should consider entries in ratio
tables beyond those they find by skip counting, including larger entries and fraction or decimal entries. Finding these other entries will
require the explicit use of multiplication and division, not just repeated addition or skip counting. For example, if Seth runs 5 meters every 2
seconds, then Seth will run 2.5 meters in 1 second because in half the time he will go half as far. In other words, when the elapsed time is
divided by 2, the distance traveled should also be divided by 2. More generally, if the elapsed time is multiplied (or divided) by N, the
distance traveled should also be multiplied (or divided) by N. Double number lines can be useful in representing ratios that involve fractions
and decimals.
As students become comfortable with fractional and decimal entries in tables of quantities in equivalent ratios, they should learn to
appreciate that unit rates are especially useful for finding entries. A unit rate gives the number of units of one quantity per 1 unit of the other
quantity. The amount for N units of the other quantity is then found by multiplying by N. Once students feel comfortable doing so, they may
wish to work with abbreviated tables instead of working with long tables that have many values. The most abbreviated tables= consist of only
two columns or two rows; solving a proportion is a matter of finding one unknown entry in the table.
Measurement conversion provides other opportunities for students to use relationships given by unit rates. For example, recognizing “12
inches in a foot,” “1000 grams in a kilogram,” or “one kilometer is 5/8 of a mile” as rates, can help to connect concepts and methods
developed for other contexts with measurement conversion.
Major Cluster Standards
Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The
ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A
received, candidate C received nearly three votes.”
6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠0, and use rate language in the context of a ratio
relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.”
“We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”
6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios,
tape diagrams, double number line diagrams, or equations.
a) Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs
of values on the coordinate plane. Use tables to compare ratios.
b) Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at
that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
c) Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding
the whole, given a part and the percent.
d) Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
Major Cluster Standards Unpacked
6.RP.1 A ratio is the comparison of two quantities or measures. The comparison can be part-to-whole (ratio of guppies to all fish in an
aquarium) or part-to-part (ratio of guppies to goldfish).
Example 1:
A comparison of 6 guppies and 9 goldfish could be expressed in any of the following forms:
6
, 6 to 9 or 6:9. If the number of guppies is
9
represented by black circles and the number of goldfish is represented by white circles, this ratio could be modeled as

These values can be regrouped into 2 black circles (guppies) to 3 white circles (goldfish), which would reduce the ratio to,
2
, 2 to 3 or 2:3.
3

Students should be able to identify and describe any ratio using “For every _____ ,there are _____” In the example above, the ratio could
be expressed saying, “For every 2 guppies, there are 3 goldfish”.
NOTE: Ratios are often expressed in fraction notation, although ratios and fractions do not have identical meaning. For example, ratios are
often used to make “part-part” comparisons but fractions are not.
6.RP.2
A unit rate expresses a ratio as part-to-one, comparing a quantity in terms of one unit of another quantity. Common unit rates are cost per
item or distance per time.
Students are able to name the amount of either quantity in terms of the other quantity. Students will begin to notice that related unit rates
(i.e. miles / hour and hours / mile) are reciprocals as in the second example below. At this level, students should use reasoning to find these
unit rates instead of an algorithm or rule.
In 6th grade, students are not expected to work with unit rates expressed as complex fractions. Both the numerator and denominator of the
original ratio will be whole numbers.
Example 1:
There are 2 cookies for 3 students. What is the amount of cookie each student would receive? (i.e. the unit rate)
Solution: This can be modeled as shown below to show that there is
1 1
2 3
2
3
2
2
of a cookie for 1 student, so the unit rate is : 1.
3
3


Example 2:
On a bicycle Jack can travel 20 miles in 4 hours. What are the unit rates in this situation, (the distance Jack can travel in 1 hour and the
amount of time required to travel 1 mile)?
Solution: Jack can travel 5 miles in 1 hour written as
the relationship between 20 miles and 4 hours.
5 mi
1 hr
and it takes
1
5
of a hour to travel each mile written as
1
hr
5
1 mi
. Students can represent
6.RP.3 Ratios and rates can be used in ratio tables and graphs to solve problems. Previously, students have used additive reasoning in
tables to solve problems. To begin the shift to proportional reasoning, students need to begin using multiplicative reasoning. Scaling up or
down with multiplication maintains the equivalence. To aid in the development of proportional reasoning the cross-product algorithm is not
expected at this level. When working with ratio tables and graphs, whole number measurements are the expectation for this standard.
Example 1:
At Books Unlimited, 3 paperback books cost $18. What would 7 books cost? How many books could be purchased with $54.
Solution: To find the price of 1 book, divide $18 by 3. One book costs $6. To find the price of 7 books, multiply $6 (the cost of one book
times 7 to get $42. To find the number of books that can be purchased with $54, multiply $6 times 9 to get $54 and then multiply 1 book
times 9 to get 9 books. Students use ratios, unit rates and multiplicative reasoning to solve problems in various contexts, including
measurement, prices, and geometry. Notice in the table below, a multiplicative relationship exists between the numbers both horizontally
(times 6) and vertically (ie. 1 • 7 = 7; 6 • 7 = 42). Red numbers indicate solutions.
Number
of Books
(n)
1
3
7
9
Cost
(C)
6
18
42
54
Students use tables to compare ratios. Another bookstore offers paperback books at the prices below. Which bookstore has the best buy?
Explain your answer.
Number
of Books
(n)
Cost
(C)
4
20
8
40
To help understand the multiplicative relationship between the number of books and cost, students write equations to express the cost of
any number of books. Writing equations is foundational for work in 7 th grade. For example, the equation for the first table would be C = 6n,
while the equation for the second bookstore is C = 5n.
The numbers in the table can be expressed as ordered pairs (number of books, cost) and plotted on a coordinate plane.
Students are able to plot ratios as ordered pairs. For example, a graph of Books Unlimited would be:
Books Unlimited
35
30
Cost
25
20
15
10
5
0
0
1
2
3
4
5
6
Number of Books
Example 2:
Ratios can also be used in problem solving by thinking about the total amount for each ratio unit.
The ratio of cups of orange juice concentrate to cups of water in punch is 1: 3. If James made 32 cups of punch, how many cups of orange
did he need?
Solution: Students recognize that the total ratio would produce 4 cups of punch. To get 32 cups, the ratio would need to be duplicated 8
times, resulting in 8 cups of orange juice concentrate.
Example 3:
Using the information in the table, find the number of yards in 24 feet.
Feet
Yards
3
1
6
2
9
3
15
5
24
?
Solution:
There are several strategies that students could use to determine the solution to this problem:
o Add quantities from the table to total 24 feet (9 feet and 15 feet); therefore the number of yards in 24 feet must be 8 yards (3 yards
and 5 yards).
o Use multiplication to find 24 feet: 1) 3 feet x 8 = 24 feet; therefore 1 yard x 8 = 8 yards, or 2) 6 feet x 4 = 24 feet; therefore 2 yards
x 4 = 8 yards.
Example 4:
Compare the number of black circles to white circles. If the ratio remains the same, how many black circles will there be if there are 60 white
circles?
Black
White
4
3
40
30
20
15
60
45
?
60
Solution:
There are several strategies that students could use to determine the solution to this problem
o Add quantities from the table to total 60 white circles (15 + 45). Use the corresponding numbers to determine the number of black
circles (20 + 60) to get 80 black circles.
o Use multiplication to find 60 white circles (one possibility 30 x 2). Use the corresponding numbers and operations to determine the
number of black circles (40 x 2) to get 80 black circles.
Students recognize the use of ratios, unit rate and multiplication in solving problems, which could allow for the use of fractions and decimals.
Example 1:
In trail mix, the ratio of cups of peanuts to cups of chocolate candies is 3 to 2. How many cups of chocolate candies would be needed for 9
cups of peanuts?
Peanut
s
Chocola
te
3
2
Solution:
One possible solution is for students to find the number of cups of chocolate candies for 1 cup of peanuts by dividing both sides of the table
2
cup of chocolate for each cup of peanuts. To find the amount of chocolate needed for 9 cups of peanuts, students multiply
3
2
the unit rate by nine (9 • ), giving 6 cups of chocolate.
3
by 3, giving

Example 2:
If steak costs $2.25 per pound, how much does 0.8 pounds of steak cost? Explain how you determined your answer.

Solution:
The unit rate is $2.25 per pound so multiply $2.25 x 0.8 to get $1.80 per 0.8 lb of steak.
This is the students’ first introduction to percents. Percentages are a rate per 100. Models, such as percent bars or 10 x 10 grids should be
used to model percents.
Students use ratios to identify percents.
Example 1:
What percent is 12 out of 25?
Solution: One possible solution method is to set up a ratio table:
Multiply 25 by 4 to get 100. Multiplying 12 by 4 will give 48, meaning
that 12 out of 25 is equivalent to 48 out of 100 or 48%.
Part
Whole
12
?
25
100
Students use percentages to find the part when given the percent, by recognizing that the whole is being divided into 100 parts and then
taking a part of them (the percent).
Example 2:
What is 40% of 30?
Solution: There are several methods to solve this problem. One possible solution using rates is to use a 10 x 10 grid to represent the whole
amount (or 30). If the 30 is divided into 100 parts, the rate for one block is 0.3. Forty percent would be 40 of the blocks, or 40 x 0.3, which
equals 12.
See the web link below for more information.
http://illuminations.nctm.org/LessonDetail.aspx?id=L249
Students also determine the whole amount, given a part and the percent.
(Solution: 20)
Example 3:
If 30% of the students in Mrs. Rutherford’s class like chocolate ice cream, then how many students are in Mrs. Rutherford’s class if 6 like
chocolate ice cream?
Example 4:
A credit card company charges 17% interest fee on any charges not paid at the end of the month. Make a ratio table to show how much the
interest would be for several amounts. If the bill totals $450 for this month, how much interest would you have to be paid on the balance?
Solution:
Charge $1
$50
$100 $200 $450
s
Interest $0.17 $8.5 $17
$34
?
0
One possible solution is to multiply 1 by 450 to get 450 and then multiply 0.17 by 450 to get $76.50
A ratio can be used to compare measures of two different types, such as inches per foot, milliliters per liter and centimeters per inch.
Students recognize that a conversion factor is a fraction equal to 1 since the numerator and denominator describe the same quantity. For
example, 12 inches is a conversion factor since the numerator and
1 foot
denominator equal the same amount. Since the ratio is equivalent to 1, the identity property of multiplication allows an amount to be
multiplied by the ratio. Also, the value of the ratio can also be expressed as
1 foot allowing for the conversion ratios to be expressed in a format so that units will “cancel”.
12 inches
Students use ratios as conversion factors and the identity property for multiplication to convert ratio units.
Example 1:
How many centimeters are in 7 feet, given that 1inch ≈ 2.54 cm.
Solution:
7 feet x 12 inches x 2.54 cm = 7 feet x 12 inches x 2.54 cm
1 foot
1 inch
1 foot
1 inch
= 7 x 12 x 2.54 cm = 213.36 cm
Note: Conversion factors will be given. Conversions can occur both between and across the metric and English systems. Estimates are
not expected.
Focus Standards for Mathematical Practice
MP.1 Make sense of problems and persevere in solving them. Students make sense of and solve real world and mathematical ratio,
rate, and percent problems using representations, such as tape diagrams, ratio tables, the coordinate plane, and/or double number line
diagrams. They identify and explain the correspondences between the verbal descriptions and their representations and articulate how the
representation depicts the relationship of the quantities in the problem. Problems include ratio problems involving the comparison of three
quantities, multistep changing ratio problems, using a given ratio to find associated ratios, and constant rate problems including two or more
people or machines working together.
MP.2 Reason abstractly and quantitatively. Students solve problems by analyzing and comparing ratios and unit rates given in tables,
equations, and graphs. Students decontextualize a given constant speed situation, representing symbolically the quantities involved with the
formula, distance = rate x time.
MP.5 Use appropriate tools strategically. Students become proficient using a variety of representations that are useful in reasoning with
rate and ratio problems such as tape diagrams, double line diagrams, ratio tables, a coordinate plane and equations. They then use
judgment in selecting appropriate tools as they solve ratio and rate problems.
MP.6 Attend to precision. Students define and distinguish between ratio, the value of a ratio, a unit rate, a rate unit, and a rate. Students
use precise language and symbols to describe ratios and rates. Students learn and apply the precise definition of percent.
MP.7 Look for and make use of structure. Students recognize the structure of equivalent ratios in solving word problems using tape
diagrams. Students identifying the structure of a ratio table and use it to find missing values in the table. Students make use of the structure
of division and ratios to model 5 miles/2 hours as a quantity 2.5 mph.
Skills and Concepts
Prerequisite Skills/Concepts: Students should already know how to…

4.OA.2 Multiply or divide to solve word problems involving
multiplicative comparison, e.g., by using drawings and equations
with a symbol for the unknown number to represent the problem,
distinguishing multiplicative comparison from additive comparison.
Advanced Skills/Concepts: Some students may be able to…
Skills: Students will be able to …
 Use ratio language to describe a ratio relationship between two
quantities. (6.RP.1)
 Represent a ratio relationship between two quantities using
Apply and extend previous understandings of multiplication and
division to multiply and divide fractions.
 5.NF.3 Interpret a fraction as division of the numerator by the
denominator (a/b = a ÷ b). Solve word problems involving division of
whole numbers leading to answers in the form of fractions or mixed
numbers, e.g., by using visual fraction models or equations to
represent the problem. For example, interpret 3/4 as the result of
dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that
when 3 wholes are shared equally among 4 people each person has
a share of size 3/4. If 9 people want to share a 50-pound sack of
rice equally by weight, how many pounds of rice should each person
get? Between what two whole numbers does your answer lie?

Convert like measurement units within a given measurement
system.
 5.MD.1 Convert among different-sized standard measurement units
within a given measurement system (e.g., convert 5 cm to 0.05 m),
and use these conversions in solving multi-step, real world
problems.

Graph points on the coordinate plane to solve real-world and
mathematical problems.
 5.G.1 Use a pair of perpendicular number lines, called axes, to
define a coordinate system, with the intersection of the lines (the
origin) arranged to coincide with the 0 on each line and a given point
in the plane located by using an ordered pair of numbers, called its
coordinates. Understand that the first number indicates how far to
travel from the origin in the direction of one axis, and the second
number indicates how far to travel in the direction of the second
axis, with the convention that the names of the two axes and the
coordinates correspond (e.g., x-axis and x-coordinate, y-axis and ycoordinate).





manipulatives and/or pictures, symbols and real-life situations.
(a to b, a:b, or a/b) (6.RP.1)
Represent unit rate associated with ratios using visuals, charts,
symbols, real-life situations and rate language. (6.RP.2)
Use ratio and rate reasoning to solve real-world and
mathematical problems. (6.RP.3)
Make and interpret tables of equivalent ratios. (6.RP.3)
Plot pairs of values of the quantities being compared on the
coordinate plane. (6.RP.3)
Use multiple representations such as tape diagrams, double
number line diagrams, or equations to solve rate and ratio
problems. (6.RP.3)
Solve unit rate problems (including unit pricing and constant
speed). (6.RP.3)
Solve percent problems, including finding a percent of a
quantity as a rate per 100 and finding the whole, given the part
and the percent. (6.RP.3).

5.G.2 Represent real world and mathematical problems by graphing
points in the first quadrant of the coordinate plane, and interpret
coordinate values of points in the context of the situation.
Academic Vocabulary












Ratio Rate
Unit Rate
Value of a Ratio
Equivalent Ratios
Percent (Percent of a quantity is a rate per 100.)
Associated Ratios (e.g., if a popular shade of purple is made by mixing 2 cups of blue paint for every 3 cups of red paint, not only can we
say that the ratio of blue paint to red paint in the mixture is 2:3, but we can discuss associated ratios such as the ratio of cups of red paint
to cups of blue paint, the ratio of cups of blue paint to total cups of purple paint, the ratio of cups of red paint to total cups of purple paint,
etc.)
Double Number Line
Ratio Table
Convert
Tape Diagram
Coordinate Plane
Equation
Unit Resources