Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
5E Lesson Plan Math
Grade Level: 6
Lesson Title: Unit 5
Proportional Reasoning with Ratios and
Rates
THE TEACHING PROCESS
Subject Area: Math
Lesson Length: 15 days
Lesson Overview
This unit bundles student expectations that address representing and solving problems
with ratios and rates, including those involving percents and converting units within a
measurement system using proportions and unit rates. According to the Texas
Education Agency, mathematical process standards including application, a problemsolving model, tools and techniques, communication, representations, relationships,
and justifications should be integrated (when applicable) with content knowledge and
skills so that students are prepared to use mathematics in everyday life, society, and
the workplace.
During this unit, students are formally introduced to proportional reasoning with the
building blocks of ratios, rates, and proportions. Students examine and distinguish
between ratios and rates as they give examples of ratios as multiplicative comparisons
of two quantities describing the same attribute and examples of rates as the
comparison by division of two quantities having different attributes. Students extend
previous work with representing percents using concrete models and fractions.
Additionally, students are introduced to generating equivalent forms of fractions,
decimals, and percents using ratios, including problems that involve money. Students
solve and represent problem situations involving ratios and rates with scale factors,
tables, graphs, and proportions. Students also represent real-world problems involving
ratios and rates, including unit rates, while converting units within a measurement
system. These representations allow students to develop a sense of covariation when
using proportional reasoning to solve problems, which means they are able to
determine and analyze how related quantities change together. Students use both
qualitative and quantitative reasoning to make both predictions and comparisons in
problem situations involving ratios and rates. Students revisit solving real-world
problems to find the whole given a part and the percent, to find the part given the whole
and the percent, and to find the percent given the part and the whole, including the use
of concrete and pictorial models. Methods for solving real-world problem situations
involving percents, such as the use of proportions or scale factors between ratios, are
included within this unit. Extensive and deliberate development of proportional
reasoning skills is foundational for all future mathematics coursework, much of which
concentrates on the concept of proportionality.
Unit Objectives:
Students will…
 Be formally introduced to proportional reasoning with the building blocks of
ratios, rates, and proportions
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Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
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examine and distinguish between ratios and rates as they give examples of
ratios as multiplicative comparisons of two quantities describing the same
attribute and examples of rates as the comparison by division of two quantities
having different attributes
extend previous work with representing percents using concrete models and
fractions
students are introduced to generating equivalent forms of fractions, decimals,
and percents using ratios, including problems that involve money
solve and represent problem situations involving ratios and rates with scale
factors, tables, graphs, and proportions
represent real-world problems involving ratios and rates, including unit rates,
while converting units within a measurement system
develop a sense of covariation when using proportional reasoning to solve
problems, which means they are able to determine and analyze how related
quantities change together
use both qualitative and quantitative reasoning to make both predictions and
comparisons in problem situations involving ratios and rates
Students revisit solving real-world problems to find the whole given a part and
the percent, to find the part given the whole and the percent, and to find the
percent given the part and the whole, including the use of concrete and pictorial
models
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Standards addressed:
TEKS:
 6.1A Apply mathematics to problems arising in everyday life, society, and the
workplace.
 6.1B Use a problem-solving model that incorporates analyzing given
information, formulating a plan or strategy, determining a solution, justifying the
solution, and evaluating the problem-solving process and the reasonableness of
the solution.
 6.1C Select tools, including real objects, manipulatives, paper and pencil, and
technology as appropriate, and techniques, including mental math, estimation,
and number sense as appropriate, to solve problems.
 6.1D Communicate mathematical ideas, reasoning, and their implications using
multiple representations, including symbols, diagrams, graphs, and language as
appropriate.
 6.1E Create and use representations to organize, record, and communicate
mathematical ideas.
 6.1F Analyze mathematical relationships to connect and communicate
mathematical ideas.
 6.1G Display, explain, and justify mathematical ideas and arguments using
precise mathematical language in written or oral communication.
 6.4B Apply qualitative and quantitative reasoning to solve prediction and
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Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
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comparison of real-world problems involving ratios and rates.
6.4C Give examples of ratios as multiplicative comparisons of two quantities
describing the same attribute.
6.4D Give examples of rates as the comparison by division of two quantities
having different attributes, including rates as quotients.
6.4E Represent ratios and percents with concrete models, fractions, and
decimals.
6.4G Generate equivalent forms of fractions, decimals, and percents using realworld problems, including problems that involve money.
6.4H Convert units within a measurement system, including the use of
proportions and unit rates.
6.5A Represent mathematical and real-world problems involving ratios and rates
using scale factors, tables, graphs, and proportions.
6.5B Solve real-world problems to find the whole given a part and the percent, to
find the part given the whole and the percent, and to find the percent given the
part and the whole, including the use of concrete and pictorial models.
ELPS:
 ELPS.c.1A use prior knowledge and experiences to understand meanings in
English
 ELPS c.1E internalize new basic and academic language by using and reusing it
in meaningful ways in speaking and writing activities that build concept and
language attainment
 ELPS.c.2D monitor understanding of spoken language during classroom
instruction and interactions and seek clarification as needed
 ELPS.c.2E use visual, contextual, and linguistic support to enhance and confirm
understanding of increasingly complex and elaborated spoken language
 ELPS.c.3D speak using grade-level content area vocabulary in context to
internalize new English words and build academic language proficiency
 ELPS.c.3H narrate, describe, and explain with increasing specificity and detail
as more English is acquired
 ELPS c.4F use visual and contextual support and support from peers and
teachers to read grade-appropriate content area text, enhance and confirm
understanding, and develop vocabulary, grasp of language structures, and
background knowledge needed to comprehend increasingly challenging
language
 ELPS.c.5B write using newly acquired basic vocabulary and content-based
grade-level vocabulary
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Misconceptions:
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Some students may generate an “equivalent” ratio by exchanging the numbers in
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Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
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a ratio without their appropriate labels rather than interpreting the ratio as a
comparison that must maintain the same relationship. (e.g., 2 girls:3 boys is not
equivalent to 3 girls:2 boys)
Some students may think that the order of the terms in a ratio or proportion is not
important.
Some students may think that generating an equivalent ratio is different from
generating an equivalent fraction.
Some students may think that all ratios are fractions, rather than understanding
that a ratio may represent a part-to-part or part-to-whole relationship.
Some students may think that rates are not related to ratios.
Some students may think ratios and rates may not be represented on a graph
rather than realizing all ratios and rates can be viewed as ordered pairs.
Some students may think that a unit rate must have a denominator of one rather
than understanding that a unit rate is a ratio between two different units where
one of the terms is one.
Some students may only use additive thinking rather than multiplicative thinking
when solving proportions.
Underdeveloped Concepts:
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Some students may forget to multiply or divide both of the terms in a ratio by the
same number to find an equivalent ratio.
Some students may think the value of 43% of 35 is the same value of 43% of 45
because the percents are the same rather than considering that the wholes of 35
and 45 are different, so 43% of each value will be different.
Some students may not realize which operation is easier to use when converting
between number forms.
Some students may have difficulty recognizing the part and the whole in problem
situations.
Some students may believe every fraction relates to a different rational number
instead of realizing equivalent fractions relate to the same relative amount.
Some students may try to convert a fraction to a decimal by placing the
denominator in the dividend, rather than the numerator.
Some students may think that
is equivalent 0.78.
Vocabulary:
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Comparison by division of two quantities– a proportional comparison in which
one quantity can be described as a ratio of the other
Multiplicative comparison of two quantities – a proportional comparison in
which one quantity can be described as a multiple of the other
Percent – a part of a whole expressed in hundredths
Positive rational numbers – the set of numbers that can be expressed as a
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Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
fraction
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, where a and b are whole numbers and b≠ 0, which includes the
subsets of whole numbers and counting (natural) numbers (e.g., 0, 2,
etc.)
Qualitative – a broad subjective description (e.g., The speed of car A is slower
than the speed of car B.)
Quantitative – a narrowed objective description associated with a quantity (e.g.,
The ratio of blue cars to red cars is 6:3; therefore, there are twice as many blue
cars as red cars.)
Rate – a multiplicative comparison of two different quantities where the
measuring unit is different for each quantity
Ratio– a multiplicative comparison of two quantities
Scale factor – the common multiplicative ratio between pairs of related data
which may be represented as a unit rate
Strip diagram – a linear model used to illustrate number relationships
Unit rate – a ratio between two different units where one of the terms is 1
Related Vocabulary:
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10 by 10 grid
Compare
Decimal
Decimal notation
Denominator
Equivalent
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Fraction
Fraction circle
Fraction notation
Improper fraction
Mixed number
Number line
List of Materials:
 Pancake Activity handout (Day 1)
 3 bowls (Day 1)
 Pancake mix (Day 1)
 Water (Day 1)
 10 x 10 grids (Day 2)
 Ratios in Recipes as Fractions, Decimals and Percents handout (Day 3)
 Determining Unit Rates handout (Day 4)
 Qualitative and Quantitative Reasoning handout (Day 5)
 Proportional Reasoning Handout (Day 6)
 Setting Up and Solving Proportions handout (Day 7)
 Coordinate Grids 4 on sheet handout (Day 8)
 Tables and Graphs with Proportional Relationships handout (Day 8)
 6th Grade Math Reference Materials (Day 9)
 K-W-L chart (Day 9)
 Measurement Conversions Tables handout (Day 9)
 Are You Scaly? Handout (Day 10)
 Set of Four Cards (Day 11)
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Nume
Prope
Propo
Propo
Ratio
Unit fr
Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
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What is Missing? Handout (Day 11)
Percent Proportions handout (Day 12)
Copies of 2 different grocery store ads (HEB and Brookshire Bros.) (Days 13 and
14)
Unit Rates Center handout. (Days 13 and 14)
http://www.youtube.com/watch?v=IhtgKHYZti0 (Days 13 and 14)
http://www.youtube.com/watch?v=XKCZn5MLKvk (Days 13 and 14)
Laptop or chrome book station (Days 13 and 14)
Webcam or video camera (Days 13 and 14)
Blank paper for drawing (Days 13 and 14)
Ratios and Rates Center activity handout (Days 13 and 14)
INSTRUCTIONAL SEQUENCE
Phase ____Engage Day 1___
Pancake Activity
Materials List
 Pancake Activity handout (Day 1)
 3 bowls (Day 1)
 Pancake mix (Day 1)
 Water (Day 1)
Activity: Pancake Activity
The teacher will lead the students in an introduction to the vocabulary of rates and
ratios while conducting an activity comparing the ratio of pancake mix to water in
different recipes for pancakes (Pancake Activity handout). The students will be
investigating the different pancake recipes to compare which has the best water to mix
ratio to make pancakes. They will be using their prior knowledge of comparing
quantities and any baking experience they may have to determine why Recipe C has
the best water to mix ratio. Then, the teacher will lead the students in answering
questions relating ratios to other recipes, such as cookies and tacos.
What’s the teacher doing?
What are the students doing?
The teacher is mixing the mix and water
together following the pancake recipes A, B
and C. The teacher is also leading the
students in discussion of what is happening
to the mix.
 What makes a thinner batter?
Possible response: more water would
make the batter thinner
 What makes a thicker batter?
The students are assisting the teacher
in mixing the pancake recipes. They
are investigating what is happening with
the mix to water ratio. They are using
any prior knowledge of baking and
number comparisons to determine the
best recipe and generating questions
and answers about proportional
reasoning.
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Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
Possible response: a higher ratio of
mix to water would make the batter
thicker
 Why is Recipe C the best recipe?
Possible response: Recipe C is the
best recipe because the ratio of water
and mix is appropriate to make it not
too thin or thick
 How would the recipe change for a
double batch?
Possible response: All of the
ingredients would be doubled
 How would the recipe change for a
half batch?
Possible response: All of the
ingredients would be divided by 2 or
cut in half
 How would the recipe change for a
triple batch?
Possible response: All of the original
ingredients would need to be
multiplied by 3
 Can you apply this to other
recipes?
Possible response: To double any
recipe, you could multiply the
ingredients by 2, to half you could
divide all of the ingredients by 2 and to
triple, you could multiply all of the
ingredients by 3.
Refer to the Pancake Activity handout for
more questions, as well.
Phase ____Explore/Explain (Day 2)______
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Creating Recipes and Ratios
including Models
Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
Materials List
 10 x 10 grids (Day 2)
Activity:
Involve the students in a whole class discussion to develop 4 recipes. Within those 4
recipes, you want to choose two main ingredients to compare. For example, the
students could discuss the ingredients necessary to make a chocolate cake. They
might want to focus on the ratio of sugar to eggs. If it takes 2 cups of sugar to every 3
eggs, then be sure to use ALL verbal representations of this ratio:
2 to 3, 2 per 3, 2 parts to 3 parts, 2 for every 3, 2 out of every 3, 2:3, etc.
Break the students into groups of 3. In their groups, have the students draw two
pictoral models for each ratio (2 ingredients specified) in each of the 4 recipes. The
students may choose actual objects, fraction circles, strip diagrams, 10 x 10 grids or
number lines to represent each ratio.
Examples are listed below:
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Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
When students are finished creating their pictoral models, the teacher may pose the
following questions:
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How can ratios or percents be representing numerically with fractions and
decimals?
Possible response: A ratio can be written as
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𝑎
𝑏
comparing two values by division
Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
and then, a fraction can be converted into a decimal by solving 𝑎 ÷ 𝑏
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What types of models can be used to represent ratios and percents?
Possible response: fraction circles, fraction strips, number lines and 10x10 grids
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Which recipe would be sweeter?
Possible response: The recipe with the higher ratio of sugar would be sweeter
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Which recipe would have a thinner or thicker batter?
Possible response: The recipe with the higher ratio of liquid ingredients would be
thinner and the recipe with the higher ratio of dry ingredients would be thicker.
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How could you adjust the recipe to make a sweeter cake?
Possible response: Making the recipe have a higher sugar ratio would make it
sweeter
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How could you adjust the recipe to make the cookies more moist?
Possible response: You could make the recipe have a higher ratio of liquid
ingredients, such as vegetable oil or eggs, to make the cookies more moist
What’s the teacher doing?
Leading student discussion of pictoral
models, leading the students in ratio
comparisons of recipes and creating ratios
from recipes. The teacher is monitoring the
groups’ progress with their pictoral models
and posing questions that begin to
demonstrate qualitative reasoning to
compare.
What are the students doing?
The students are participating in
developing the recipes and listing the 2
ingredients for each recipe as a ratio.
The students are also developing the
pictoral models and beginning to
answer questions of qualitative
reasoning to compare the ratios.
Phase ____Explore/Explain (Day 3)______
Expressing ratios as a fraction,
decimal and percent
Materials List
 Ratios in Recipes as Fractions, Decimals and Percents handout (Day 3)
Activity:
The students will work in groups of 2 on the Ratios in Recipes as Fractions, Decimals
and Percents handout. They will use the recipe given to generate ratios within the
recipe and then convert them into fractions, decimals and percents. The teacher will be
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Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
monitoring their progress in their groups and prodding investigation through asking the
following questions:
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How can an equivalent fraction be generated when given a decimal or
percent?
Possible response: An equivalent fraction can be generated through using the
67
place value system. For example, 0.67 is equivalent to 100 because it is read as
sixty-seven hundredths. A percent can be converted into a fraction by placing it
over 100 and simplifying.
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How can an equivalent decimal be generated given a fraction or percent?
Possible response: A fraction can be converted into a decimal by dividing the
numerator by the denominator. A decimal can be converted into a percent by
multiplying it by 100 or moving the decimal point two places to the right.
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How can an equivalent percent be generated given a decimal or fraction?
Possible response: A decimal can be converted into a percent by dividing it by
100 or moving the decimal point two places to the right. A fraction can be
converted into a percent by making a common denominator of 100 or converting
it into a decimal first and then into a percent.
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What relationships exist between equivalent fractions, decimals, and
percents?
Possible response: Any fraction, decimal or percent can be converted into all 3
equal forms of ratios.
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How are equivalent forms of fractions, decimals, and percents related to
money?
Fractions, decimals and percents are related to money in that it takes 100
pennies to make one dollar so money is also a ratio out of 100
After completing the handout in their groups, the students will journal their responses to
the teacher’s questions (above) in their interactive journals
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Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
What’s the teacher doing?
The teacher is monitoring progress in the
groups on the ratios in recipes handout and
asking the groups the questions concerning
how they are generating each of the
subsequent numbers.
Phase ___Explore/Explain (Day 4)_______
What are the students doing?
The students are working in groups to
connect the development of the
definition of ratios to converting
fractions, decimals and percents. They
will also be discussing in their groups
the questions posed by the teacher and
recording their answers in their
interactive journal.
Unit Rates
Materials List
 Determining Unit Rates handout (Day 4)
Activity:
The teacher introduces the concept of unit rates by asking the students:
 Which is a better deal: buying 5 pieces of gum for $1.55 or buying 3 pieces
of gum for $0.99?
Possible response: The 5 pieces of gum is the better deal because each piece
costs $0.31 and the price of each piece in the 3-piece pack is $0.33
The teacher will describe to the students that the terms per, for or in are used to
describe unit rates. Present the problem: The cost of a 12 oz. bottle of shampoo
is $2.88, what is the cost per ounce of the shampoo?
Possible response: The price of the bottle $2.88 can be divided by the number of
ounces, 12, to find the unit rate.
Through class discussion, elicit division or two equivalent ratios (proportion when they
reach this term) can be used to solve the problem.
Have the students work in groups of 2 on the Finding Unit Rates activity.
The teacher will monitor student progress and ask the students the following questions
that the students can discuss in their pairs and then journal in their interactive journals.
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How do you determine a unit rate?
Possible response: The unit rate can be determined by dividing the two
quantities to find the amount “per” something else
How does a unit rate differ from ratios that we have seen thus far?
Possible response: Ratios always have a denominator of 1, something “per”
something else
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Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
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Create 2 real-world examples of unit rates, such as, the number of students
per teacher in the school.
Possible response: There are 320 students in our school and 16 teachers.
Therefore, there are 20 students per teacher
What’s the teacher doing?
What are the students doing?
The teacher is leading the students in
developing the concept of unit rates and then
monitoring their discovery of unit rates in their
group project. The teacher will then lead the
students in discussion of the listed questions
to be discussed in their groups and then
answered in their interactive journals.
The students are developing a definition
of unit rates and then working in pairs to
solve the unit rates on the handout.
The students are also journaling the
answers into their interactive journals.
Phase ___Explore/Explain (Day 5)_______
Unit Rates and Qualitative and
Quantitative Reasoning
Materials List
 Qualitative and Quantitative Reasoning handout (Day 5)
Activity:
The terms Qualitative and Quantitative reasoning are very new to our students, so the teacher
would introduce the terms and develop definitions with the students through the following
examples:
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Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
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Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
After this discussion and introduction, the students would work in groups of 3 on the qualitative
and quantitative reasoning handout.
While they are working, the teacher could walk around monitoring the groups’ progress and
elicit group discussion of the following questions to be answered in their interactive journals.
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Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
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What is the difference between qualitative and quantitative reasoning?
Possible response: Qualitative reasoning is a broad subjective description and
quantitative reasoning is a narrow objective description with a specified numerical
difference.
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How can ratios and rates be described qualitatively?
Possible response: Qualitative descriptions are broad, such as, this recipe is sweeter or
this car drove faster.
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How can qualitative reasoning be used to make predictions and comparisons in
problems involving ratios and rates?
Possible response: Use broad descriptions, such as, this car drives slower and compare
it to another amount to make predictions.
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How can ratios and rates be described quantitatively?
Possible response: A quantitative description is specific and is numerically based, such
as, Student A earns three times as much as Student B.
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How can quantitative reasoning be used to make predictions and comparisons in
problems involving ratios and rates?
Possible response: With quantitative reasoning, a unit rate can be found and then used
to make predictions and comparisons.
What’s the teacher doing?
What are the students doing?
The teacher is introducing the topic of
quantitative and qualitative reasoning to
the students and developing the
concept. Then, the teacher is
monitoring students’ progress with the
handout and checking for understanding
through questioning.
The students are defining qualitative and
quantitative reasoning. Then, they are using
this newly acquired knowledge to make
predictions and reason the comparisons on
the handout and answering the teacher’s
questions in their interactive journal.
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Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
Phase __Explore/Explain (Day 6)__
Setting up and Solving Proportions
Materials List
 Proportional Reasoning Handout (Day 6)
Activity:
The teacher develops the idea of a proportion being two equal ratios by giving the
students a recipe example:
Pancake Mix
4 cups of mix
? cups of water
Water
3 cups of mix
9 cups of water
The students discuss how they can find this unknown quantity and how to set up the
problem as two equivalent ratios labelling the two units (mix and water).
The students will then be broken into groups of 3 to apply this process into setting up
and solving proportional reasoning real-world problems on the Proportional
Reasoning handout. The teacher will be prompting group discussion with the students
by asking
 What is a proportion?
Possible response: A proportion is two equivalent ratios that can be used to
determine an unknown quantity
 How do you solve for the unknown quantity?
Possible response: To solve for an unknown quantity, set up a proportion and
then use multiplication and division to solve for the unknown quantity.
 How do you determine the order to set the proportion as two equivalent
ratios?
Possible response: You must read the two items being compared, such as, dogs
and cats and set up a word ratio to ensure that the proportion is also set up
correctly.
 Is there more than one correct way to set up a proportion?
Possible response: Yes, for instance, if the word ratio is unicorns to ponies, then
it can be set up as ponies to unicorns in a different order and still solved to find
the same answer.
The students will then write a problem involving two different units that can be set up as
a proportion and solved to find a missing quantity. The students will then pass their
question to another group member to set up the proportion and solve it to find the
unknown quantity.
What’s the teacher doing?
The teacher is leading the students to
define proportion and developing the
process to correctly setting up and
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What are the students doing?
The students are developing the definition of
proportions, developing strategies for setting
up and solving proportions from word
Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
solving the proportion. Then, the
teacher is posing questions to the
groups and overseeing their written
proportional relationship questions and
group members’ answers.
problems and then creating a word problem
of their own and solving a problem written by
a group member.
Phase __Explore/Explain (Day 7)___
Setting up and Solving Proportions
Materials List
 Setting Up and Solving Proportions handout (Day 7)
Activity:
The teacher will break the students into pairs and give them a Setting Up and Solving
Proportions handout. In their pairs, the students will identify the two items being
compared and set up two equivalent ratios to solve for the unknown quantity.
Then, the students will journal their responses to the teacher-led questions:
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How are the two ratios related in a proportion?
Possible response: Two ratios in a proportion are equivalent
If the first ratio is comparing dogs to cats, can the second ratio compare
cats to dogs, respectively, and be equivalent?
Possible response: No, the ratios must be set up in their respective order to be
equivalent; however, when solving for an unknown quantity in a proportion, the
word ratio may be set up either way
How do you determine which number to multiply the numerator and
denominator by when solving the problem?
Possible answer: Look for the relationship between the two ratios and then
multiply or divide by the same quantity
Create a real-world problem that can be solved by setting up and solving a
proportion.
Possible response: If it takes two watermelons to feed 15 people, then how many
watermelons are needed to feed 60 people? (8 watermelons)
What’s the teacher doing?
What are the students doing?
The teacher is monitoring student
progress by checking to see that they
The students are working in pairs to set up
and solve proportional relationship examples.
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Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
are setting the proportions up correctly
using the units in the same order, for
example, both ratios would need to be
set up in the order teachers to students
to solve for the unknown quantity. The
teacher is also prompting the students
with questions to check their
understanding of proportional
relationships.
The students should be discussing how they
are setting up their proportions and the order
of the units, for instance, teachers to
students, when setting it up and then,
discussing the process of solving the
proportion. They may discuss the teacherled questions in their pairs and then journal
the answers in their own words.
Phase ____Explore/Explain (Day 8)__
Representing Ratios and Proportions with
Tables and Graphs
Materials List
 Coordinate Grids 4 on sheet handout (Day 8)
 Tables and Graphs with Proportional Relationships handout (Day 8)
Activity:
The teacher can introduce the lesson to the students by using speed as an example.
The students can even choose the speed to use (150 miles in 3 hours). Then, the
teacher and students can develop a table to represent this information, while discussing
the labels for the table (time and distance) and discussing how to determine the unit
rate (using prior knowledge).
An example table might look like the one below.
Time
1
2
3
4
5
(hours)
Distance
50
150
(miles)
Once the students find the unit rate, then they can discuss how to use this and the
amount of time listed as a proportional relationship to find the missing distance in miles.
Another way to display proportional relationships is with a coordinate grid.
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Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
A situation for this grid could be: Grant walks 5 miles every 2 hours. How many miles
does he walk in 4 hours? 6 hours? 8 hours? 1 hour?
After discussing this example, have the students discuss in groups of 3 the following
questions and journal their responses into their interactive journals.




How can you use ratios and rates to represent the relationship between two
quantities using tables and graphs?
Possible response: Have the students use the proportional relationship between
the two ratios to find the missing values in a table and create coordinate points
from the unit rate and equivalent ratios.
How do you complete a table that has a proportional relationship between
quantities?
Possible response: Set up your ratios as equivalent and solve the proportion to
find the missing value
How can tables be used to solve problems involving ratios? Rates?
Possible response: The ratios in the table can be used to determine the unit rate
and then set up proportions to solve for the missing values.
How can graphs be used to solve problems involving ratios? Rates?
Possible response: The coordinate points can be written as a ratio and then set
up and solve proportions for the missing values. The points can also be divided
to find the unit rates.
Then, have the students work in groups of 3 on the Tables and Graphs with
Proportional Relationships handout and the Coordinate Grids 4 on sheet.
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Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
What’s the teacher doing?
What are the students doing?
The teacher will lead the students in a
discussion of how proportional
relationships can be displayed on a
table or grid and ask the students
questions concerning proportional
relationships in tables and graphs and
then have the students create tables
and graphs in groups of 3.
The students are discussing how unit rates
and proportional relationships can be used to
create tables and graphs. The students will
then complete tables and graphs to represent
these relationships.
Phase:__Explore/Explain (Day 9)___
Measurement Unit Conversions
Materials List
 6th Grade Math Reference Materials (Day 9)
 K-W-L chart (Day 9)
 Measurement Conversions Tables handout (Day 9)
Activity:
The students were introduced to unit measurement conversions on the previous day’s
handout of Tables and Graphs in Proportional Relationships. Today, give each
student a copy of the 6th Grade Math Reference Materials. When introducing the
lesson on measurement conversions, pull the student’s prior knowledge through a K-WL chart because many times they have been previously introduced to this concept and
may be able to relate the pneumonic device of “KHDMDCM”.
Place the students into pairs. Have them draw tables to find the unit rate of the
conversion and then fill in the table for the missing values.
Examples include:
Yards
Inches
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1
2
72
3
Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
Meters
1000
2000
5000
Kilometers
5
The students can work in pairs to complete the tables on the Measurement
Conversions Charts handout.
While the students are completing the tables, the teacher can ask the students:
 Is there a proportional relationship between these conversions?
Possible response: Yes, unit conversions can be set up and solved as
proportions
 Can you solve unit conversions through setting up a proportion?
Possible response: Yes, they are proportional relationships
 The conversion of gallons to cups is not on your 6th Grade Math Reference
Materials chart, how will you determine the unit rate for how many cups are
in a gallon?
Possible response: One option is to use Gallon Guy or convert from gallons to
quarts and then to pints and then to cups
 How is converting metric measurements different from converting
customary measurements?
Possible response: The metric system is all base ten where customary has
different numbers in each unit.
The students can discuss these questions in their groups and then answer them in
their interactive journals using their own vocabulary.
What’s the teacher doing?
What are the students doing?
The teacher is posing questions to have
the students pull from their prior
knowledge of conversions and
proportions the proportional relationship
of unit conversions. Also, the teacher is
leading the students in questioning
strategies to develop that proportional
relationship.
Phase:_Explore/Explain (Day 10)____
The students are completing tables of
measurement conversions to develop the
proportional relationship that is involved in
measurement conversions. They are also
discussing in their groups the teacher’s
questions and answering them in their
interactive journals.
Measurement Conversions and Scale
Drawings
Materials List
 Are You Scaly? Handout (Day 10)
Activity:
The teacher will break the students into groups of 3 and the students will begin the
day’s lesson with discussing the following questions and journaling their responses.
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Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6

What is the process for converting units within the same measurement
system?
Possible response: Units can be converted through setting up and solving a
proportion.

How can proportions and unit rates be used to convert units of measure?
Possible response: Proportions can be set up with a word ratio and then the unit
rate as one equivalent ratio to find the unknown quantity.
The students will then create 3 problems each over measurement conversions and then
hand their paper to another group member who will answer their questions and then
they can return them to check for understanding.
The teacher can then show the students a scale drawing of their classroom and pose
questions to the class on how it can also relate to proportionality
 Are the scale drawing and the actual room proportional?
Possible response: Yes, they should be proportional based on the proportional
relationship of the scale
 If given the dimensions of the scale drawing and the scale itself, could you
determine the measurements of the actual room? How?
Possible response: The scale can be set up as one ratio and then, the scale
drawing measurement can be used to find the missing actual room dimensions.
 If given the dimensions of the actual room and the scale used to create the
drawing, could you determine the measurements of the scale drawing?
Possible response: The scale can be set up as one ratio and then, the actual
room measurement can be used to find the scale drawing dimensions.
Then, have the students work on the Are You Scaly? Handout in their groups of 3.
What’s the teacher doing?
The teacher is facilitating discussion on
unit conversions, introducing discussion
on scale drawings and introducing the
students to the steps needed to solve
scale drawing problems.
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What are the students doing?
The students are discussing how unit
conversions are proportional relationships
and then discussing how scale drawings are
also proportional relationships and
developing a strategy to solve these
problems.
Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
Phase ___Explore/Explain (Day 11)__
Percent Proportions
Materials List
 Set of Four Cards (Day 11)
 What is Missing? Handout (Day 11)
Activity:
The teacher will lead the students in a discussion of how to solve percent proportion
problems and even give the students a few examples from the What is Missing?
Handout. The teacher will also give the students each a set of four cards (handout)
each labelled Part, Whole, % and 100. The students will discuss how to set them up to
solve percent proportions. Questions that the teacher might pose while leading the
discussion could be:
 What is always out of 100?
Possible response: Percent is a ratio out of 100
 When writing ratios, is the part out of the whole or the whole out of the
part?
Possible response: Part is out of whole
 Are the two ratios equivalent?
Possible response: Yes, they are proportional
The students should set them up as the proportion
𝑃𝑎𝑟𝑡
%
=
𝑊ℎ𝑜𝑙𝑒 100
The students can then work in pairs using their set of four cards to determine how to set
up each problem and answer the question, what is missing? On the handout.
The teacher would ask the following questions to lead into tomorrow’s lesson on setting
up and solving percent proportions.
 What additional steps would need to be done to solve these problems?
Possible response: The proportion needs to be solved
 How would you solve the problem when missing the part?
%
Possible response: Use the ratio 100and create an equivalent ratio out of the
whole
 How would you solve the problem when missing the whole?
%
Possible response: Use the ratio 100and create an equivalent ratio part out of the
missing quantity
 How would you solve the problem when missing the percent?
𝑃𝑎𝑟𝑡
Possible response: Use the ratio 𝑊ℎ𝑜𝑙𝑒 and the equivalent ratio out of 100 to find
the missing quantity
The students could discuss these questions in their groups and answer them in their
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Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
interactive journals.
What’s the teacher doing?
What are the students doing?
The teacher is using the four card sets
to develop the percent proportion of
𝑃𝑎𝑟𝑡
%
=
𝑊ℎ𝑜𝑙𝑒 100
and then facilitating discussion as they
set up the percent proportions and then
asking them questions about what will
be needed to solve the percent
proportions for tomorrow’s lesson.
The students are developing the proportion
𝑃𝑎𝑟𝑡
%
=
𝑊ℎ𝑜𝑙𝑒 100
To solve percent proportions and then setting
up problems in their groups and answering
questions to prepare themselves for
tomorrow’s lesson.
Phase__Explore/Explain (Day 12)___
Percent Proportions
Materials List
 Percent Proportions handout (Day 12)
Activity:
The students apply their prior knowledge of percents and proportional reasoning to
solve real-world problems where they are asked to find the whole when given a part and
the percent. They are also asked to find the part when given the percent and the whole
and find the percent when given the part and whole.
The teacher will group the students in groups of 3 to collaboratively work on the
Percent Proportions handout.
While they are working on this handout in their groups, the teacher will monitor their
progress and prompt the students with the following questions that they can discuss in
their own words in their groups and use to create 2 percent proportion word problems of
their own.

How can concrete and pictorial models be used to determine the whole
when given a part and the percent?
Possible response: They can be used to shade the percent and then find the
equivalent ratio of part out of whole

How can proportions or scale factors between ratios be used to determine
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Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
the whole when given a part and the percent?
%
Possible response: Use the ratio 100and create an equivalent ratio out of the
whole

How can concrete and pictorial models be used to determine the part when
given the whole and the percent?
Possible response: You can represent the percent and the part out of the whole

How can proportions or scale factors between ratios be used to determine
the part when given the whole and the percent?
%
Possible response: Use the ratio 100and create an equivalent ratio part out of the
missing quantity

How can concrete and pictorial models be used to determine the percent
when given the part and the whole?
Possible response:

How can proportions or scale factors between ratios be used to determine
the percent when given the part and the whole?
𝑃𝑎𝑟𝑡
Possible response: Use the ratio 𝑊ℎ𝑜𝑙𝑒 and the equivalent ratio out of 100 to find
the missing quantity
What’s the teacher doing?
The teacher is monitoring the students’
progress ensuring that the students are
correctly setting up and solving the
percent proportions to find the unknown
quantity.
Phase__Elaborate (Days 13 and 14)_
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What are the students doing?
The students are working collaboratively to
set up and solve the percent proportion
problems to find the missing part, whole or
percent. They will also discuss the teacher’s
questions in their groups and create their
own problem that can be solved using a
percent proportion.
Proportional Relationships Centers
Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
Materials List
 Copies of 2 different grocery store ads (HEB and Brookshire Bros.) (Days 13 and
14)
 Unit Rates Center handout. (Days 13 and 14)
 http://www.youtube.com/watch?v=IhtgKHYZti0 (Days 13 and 14)
 http://www.youtube.com/watch?v=XKCZn5MLKvk (Days 13 and 14)
 Laptop or chrome book station (Days 13 and 14)
 Webcam or video camera (Days 13 and 14)
 Blank paper for drawing (Days 13 and 14)
 Ratios and Rates Center activity handout (Days 13 and 14)
Activity:
The classroom will be broken into 5 centers:
 Unit Rates and Qualitative and Quantitative Reasoning
 Percent Proportions
 Ratios and Fraction, Decimal and Percent Relationships with Recipes
 Unit Conversions
 Tables and Scale Drawings
The students will be allowed to move about the room working at each center with a
partner. The teacher should split them into the different centers so that no one center is
crowded with too many students.
The Unit Rates and Qualitative and Quantitative Reasoning station should have at least
two sets of two different grocery store ads (HEB and Brookshire Bros.). They should
also have a Unit Rates Center handout.
At the Percent Proportions Center, each student will create 3 percent proportion word
problems. One problem should be missing the part, one missing the whole and the
third missing the percent. When both partners have completed this task, they will switch
papers and set up and solve each problem created by their partner.
At the Unit Conversions station, the students can watch the following videos (one or
both depending on time) at a laptop or chromebook station or even iPad station
depending on availability.
http://www.youtube.com/watch?v=IhtgKHYZti0
http://www.youtube.com/watch?v=XKCZn5MLKvk
Then, the students will work with their partner to create their own paper slide video
demonstrating how to use proportions to solve unit conversions (metric and customary
example).
At the Scale Drawings and tables station, the pairs will design together a scale drawing
of a car. They will use the scale of 2 cm = 5 inches to then create a larger car model.
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Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
The students will create a table using the scale and dimensions of the car.
For example, this could be the start of a table:
Scale
Length
Cm
2
24
Inches
5
120
Width
Hood Length
At the Recipes and Ratios station, the students will work on the Recipes and Ratios
handout to list five ingredients for a recipe and then write these ratios as fractions,
decimals and percents. The students will finally double the ingredients to make twice
as many servings.
What’s the teacher doing?
What are the students doing?
The teacher is facilitating instruction by
monitoring the students’ progress
through the centers and assessing their
knowledge through observation and the
products that they create at each
station.
The students are working with partners
moving from center to center completing
each activity. They will be asking any
questions that they may have at any station
and reflecting in their interactive journal at
the end of Day 14.
Phase: _Evaluate (Day 15)_________
Unit 5 Assessment
Activity:
Assess student understanding of related concepts and processes by using the
Performance Assessment (s) aligned to this lesson.
Performance Assessment:
Unit 5 PA 1
Analyze the problem situation(s) described below. Organize and record your work for
each of the following tasks. Using precise mathematical language, justify and explain
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Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
each solution process.
1) Max and Pedro both clean pools. Each charges a flat fee to clean a pool. Max cleans
12 pools in 8 hours and earns $270. Pedro cleans 9 pools in 7.5 hours and earns
$195.75.
a) Use qualitative reasoning to compare and predict the amount of time it takes each
boy to clean 7 pools.
b) Use quantitative reasoning to compare and predict the amount of time it takes each
boy to clean 7 pools.
c) Identify a ratio that compares the number of pools Max cleans to the number of pools
Pedro cleans and describe the multiplicative relationship between the quantities.
d) Identify the rate each boy charges per pool and describe the relationship between the
quantities and division.
Unit 5 PA 2
Analyze the problem situation(s) described below. Organize and record your work for
each of the following tasks. Using precise mathematical language, justify and explain
each solution process.
1) Nallely wants to landscape her rectangular backyard that measures 765 square feet
and has a length of 30 feet. She is not certain of the landscape design, so she created a
scale drawing of the backyard to plan the layout of her design. The scale drawing of her
garden had a length of 20 inches.
a) Determine the scale factor that was used to create the scale drawing and create a
replica of the drawing.
b) Nallely would like to landscape 535.5 square feet of her backyard. Determine the
percent of her backyard that she would like landscaped and represent this percent
using concrete models, fractions, and decimals.
c) On Nallely’s scale drawing, she drew a flowerbed with the dimensions of 3.5 inches
by
inches. Determine the actual dimensions of the flowerbed with the ratio used to
create the scale drawing.
d) Nallely decided to purchase a decorative border to outline her garden that sells for
$7.99 per yard. Use a table, graph, or proportion to determine the number of yards of
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Unit 5 Proportional Reasoning with Ratios and Rates
Grade 6
border Nallely needs to purchase to go around her garden.
What’s the teacher doing?
What are the students doing?
Monitoring student progress and using
the assessment data to determine any
areas of misconception that may need
reteaching.
Working diligently on the assessment and, if
needed, after the assessment, journaling any
questions to be asked the next day over
material that they had questions concerning.
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