Ratio Representations with Context
Using Colored Tiles
See Holt Course 2 Lesson 5-2 and Holt Course 3, Lesson 5-2
Objective/Goal: Use colored tiles to support understanding a composed unit and a unit ratio.
Benchmark(s) Emphasized
6.1.2.1 Identify and use ratios to compare quantities; understand that comparing quantities using ratios
is not the same as comparing quantities using subtraction.
6.1.2.4 Use reasoning about multiplication and division to solve ratio and rate problems.
6.1.2.2 Apply the relationship between ratios, equivalent fractions and percents to solve problems in
various contexts, including those involving mixtures and concentrations.
Misconceptions:
 Students may believe that 8:4 and 2:1 represent different ratios;
 Students may not understand that order matters in a ratio. For example, students may believe that 3:1
and 1:3 are the same ratios;
 Given the ratio 3 boys to every 7 girls, students may think there are exactly only 3 boys and 7 girls;
 Students see little difference between fractions and ratios, believing that all ratios express part-to-whole
relationships;
 Students may misinterpret or misrepresent ratios expressed in words. For example, students may
believe that the ratio of problems wrong to problems correct is 1:6 when a student gets one out of every
six problems wrong rather than 1:5;
 Students believe that adding or subtracting the same number to the numerator and denominator
6
produces an equivalent ratio (e.g., 48 is equivalent to 4+2
or10
).
8+2
 When scaling up by non-integer values, students revert to additive structures (e.g., When asked, "If it
takes 6 pizzas to feed 24 people, how many pizzas will it take to feed 36 people," students add 6 + 12
rather than multiply 6 x 1.5);
 Students do not understand unit rates as fractions (e.g., 25 students per bus means 25 𝑆𝑡𝑢𝑑𝑒𝑛𝑡𝑠
.
1 𝐵𝑢𝑠
Reference: scimathmn.org website
Materials:
 Colored tiles
 Ratio Representation Recording Document
 Ratio Mat
 Student Language Frames
 Highlighters or Markers
 Pencil
Progression:
1. Many activities using manipulatives (colored tiles) to build ratios and determine part-to-part ratios and
part-to-whole ratios.
2. Many activities using manipulatives to understand composed units and unit ratios.
3. Many activities using manipulatives to understand unit rate.
4. Many activities using manipulatives to understand slope.
Stages:
1. Read and discuss what a problem is asking.
2. Use the given information in the problem to find other combinations of the parts to determine the
composed unit.
3. Relate unit fractions to unit ratios. Build and draw a representation to model the arithmetic. Compare
solutions to understand the unit ratios.
1
Directions:
 Read the problem and discuss the given information
 Record the given information and the ratio in the first column of the recording document.
 Use colored tiles on the mat to represent the given information in the problem.
 Read Part A. Discuss what it is asking you to find.
 Build the Composed Unit on the mat and determine the new ratio.
 Use markers or highlighters to draw the initial ratio and the composed unit.
 Write the initial ratio and composed ratio next to the corresponding representation.
 Remove the composed unit from the mat.
 Read and discuss Part B.
 Keep the initial ratio on the mat and build the unit ratio on the mat. Determine the unit ratio.
 Use markers or highlighters to draw a representation of both the basic ratio and the unit ratio.
 Write the initial ratio and unit ratio next to the corresponding representation.
 Write the solution using words.
 Write the equivalent ratios.
Variation:
 Eliminate the manipulatives and mat when your students are proficient and only use visual
representations.
 When your students are proficient with finding unit rates where the denominator is smaller than the
numerator without manipulatives, then find unit rates where the denominator is larger than the
numerator. This will be difficult to make a visual representation so students need to be able to do the
mathematical calculations.
Purposefulness of Chosen Problems:
“There are two ways to form a ratio, both of which involve coordinating two quantities. One way is by
comparing two quantities multiplicatively. The second way is by joining or composing the two quantities in a
way that preserves a multiplicative relationship” (Van de Walle, J. A. (2012). Elementary and Middle School
Mathematics: Teaching Developmentally. New York, NY: Pearson, page 358).
A Ratio as a Composed Unit
“Composed unit refers to thinking of the ratio as one unit. For example, if kiwi are 4 for $1.00, then
you can think about other multiples that would also be true, like 8 for $2.00, 16 for $4.00, and so on.
(Each of these would be a unit composed of the original ratio.) This is iterating. It can also be
partitioned: 2 for $0.50 and 1 for $0.25. Any number of kiwis can be priced through using these
composed units.”(Van de Walle, J. A. (2012). Elementary and Middle School Mathematics:
Teaching Developmentally. New York, NY: Pearson, page 358). Another way to think about
composed units is to think of it as scaling the values up or scaling the values down.
A Ratio as a Unit Ratio
“A unit rate is a ratio that has a denominator of 1. A unit rate is also called a unit ratio. (They mean
the same thing)” ( www.mathvillage.info).
40 to 10, 45 and 3:8, $6 for a 24 case of soda are ratios, but they are not unit ratios.
4 to 1, 1.25
1 and 0.375: $0.25 for 1 can of soda, are unit ratios.
Any ratio can be converted into a unit ratio by dividing the numerator and the denominator by the
denominator.


2
Problems:
1. We are going to frost cupcakes and sell them at school. In a recipe for frosting, the
instructions state to frost one batch of cupcakes with aqua-colored frosting, you will need 9
drops of blue food coloring and 3 drops of green food coloring.
a. If we wanted to make two batches of cupcakes, how many drops of blue and green food
coloring would you need? Write the composed ratio.
b. If I only had one drop of green food coloring left, how many drops of blue food coloring
would I need?
2. Last Saturday I babysat Elsie for 2 hours and was paid $11.
a. I am going to babysit her again. If the time is tripled, how much will I expect to make?
Write the composed ratio.
b. How much will I make if I only babysat for one hour?
3. My friend and I run after school. I ran 6 city blocks and my friend ran 5 city blocks.
a. If we each ran the same distances for 3 days (I ran 6 city blocks and my friend ran 5 city
blocks). How many total city blocks did each of us run? Write the composed ratio.
b. For every city block my friend ran, how many city blocks did I run?
4. We are having company this evening. My mom asked me to make the lemonade. The recipe
uses 7 cans of water with 3 cans of lemonade concentrate.
a. I needed to make 1 12 batches of lemonade. How many cans of lemonade concentrate and
cans of water would I need? Write the composed ratio.
b. If you only had one can of lemonade concentrate, how much water should you mix with it?

3
Student Math Notebook Samples
Title written at the top of the page: "Ratio Representations"
Directions for Partner Work:
 Read the problem and discuss the given information
 Record the given information and the ratio in the first column of the recording document.
 Use colored tiles on the mat to represent the given information in the problem.
 Read Part A. Discuss what it is asking you to find.
 Build the Composed Unit on the mat and determine the new ratio.
 Use markers or highlighters to draw the initial ratio and the composed unit.
 Write the initial ratio and composed ratio next to the corresponding representation.
 Remove the composed unit from the mat.
 Read and discuss Part B.
 Keep the initial ratio on the mat and build the unit ratio on the mat. Determine the unit ratio.
 Use markers or highlighters to draw a representation of both the basic ratio and the unit ratio.
 Write the initial ratio and unit ratio next to the corresponding representation.
 Write the solution using words.
 Write the equivalent ratios.
Student Language Frames (what the students say to each other):
 The two parts we are comparing are ____________ to _____________.


(part)
(part)
To find the composed unit we need to iterate the multiplicative comparison ________ times. Our new
ratio is ____________.
The numerator tells us we need _________ parts in our visual representation. To find the unit ratio, we
partition both parts by our denominator of ______. For every ______ _____________________, we
need one _______________________.
(number)
(part)
(part)
4
Student Language Frames
Student Language Frames:
 The two parts we are comparing are ____________ to _____________.


(part)
(part)
To find the composed unit we need to iterate the multiplicative comparison ________ times. Our new
ratio is ____________.
The numerator tells us we need _________ parts in our visual representation. To find the unit ratio, we
partition both parts by our denominator of ______. For every ______ _____________________, we
need one _______________________.
(number)
(part)
(part)
Student Language Frames:
 The two parts we are comparing are ____________ to _____________.


(part)
(part)
To find the composed unit we need to iterate the multiplicative comparison ________ times. Our new
ratio is ____________.
The numerator tells us we need _________ parts in our visual representation. To find the unit ratio, we
partition both parts by our denominator of ______. For every ______ _____________________, we
need one _______________________.
(number)
(part)
(part)
Student Language Frames:
 The two parts we are comparing are ____________ to _____________.
(part)


(part)
To find the composed unit we need to iterate the multiplicative comparison ________ times. Our new
ratio is ____________.
The numerator tells us we need _________ parts in our visual representation. To find the unit ratio, we
partition both parts by our denominator of ______. For every ______ _____________________, we
need one _______________________.
(number)
(part)
(part)
Student Language Frames:
 The two parts we are comparing are ____________ to _____________.
(part)


(part)
To find the composed unit we need to iterate the multiplicative comparison ________ times. Our new
ratio is ____________.
The numerator tells us we need _________ parts in our visual representation. To find the unit ratio, we
partition both parts by our denominator of ______. For every ______ _____________________, we
need one _______________________.
(number)
(part)
(part)
5
Ratio Representations
Composed Unit
(Other Combinations)
B.
Unit Ratio
Secondary Ratio
Initial Ratio
Initial Ratio
A.
For every
______________________________________________,
Secondary Ratio
Ratio
Name _________________________ Date ________
I need one
______________________________________________.
Equivalent Ratios:
Secondary Ratio
Initial Ratio
Ratio Mat