Download Grade 7 Mathematics

Grade 6 Mathematics, Quarter 2, Unit 2.3
Understanding Ratios and Rates
Overview
Number of instructional days:
8
(1 day = 60 minutes)
Content to be learned
Mathematical practices to be integrated

Understand the concept of ratio.
Reason abstractly and quantitatively.

Use ratio language to describe a ratio
relationship between two quantities.

Make sense of quantities and their
relationships in problem situations.

Understand the concept of unit rate.

Attend to the meanings of quantities.

Write a ratio as a unit rate.


Use rate language in the context of a ratio
relationship.
Are able to flow between contextual and noncontextual situations during problem solving
and make meaning of numbers and symbols.
Attend to precision.

Communicate their understanding of
mathematics to others.

Use clear definitions and state the meaning of
the symbols they choose.

Specify units of measure and label correctly.

Strive for accuracy.
Essential questions

What is a ratio?

How would you describe a ratio relationship?

What is a rate?


How do write a ratio as a unit rate?
What are the similarities and differences
between ratios, rates, and unit rates?

How do you use a ratio to describe the
relationship between two quantities?
Warwick Public Schools, in collaboration with
the Charles A. Dana Center at the University of Texas at Austin
C-25
Grade 6 Mathematics, Quarter 2, Unit 2.3
Understanding Ratios and Rates (10 days)
Written Curriculum
Common Core State Standards for Mathematical Content
Ratios and Proportional Relationships
6.RP
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.”*
*Note: Expectations for unit rates in this grade are limited to non-complex fractions.
Common Core State Standards for Mathematical Practice
2
Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem situations.
They bring two complementary abilities to bear on problems involving quantitative relationships: the
ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the
representing symbols as if they have a life of their own, without necessarily attending to their referents—
and the ability to contextualize, to pause as needed during the manipulation process in order to probe into
the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent
representation of the problem at hand; considering the units involved; attending to the meaning of
quantities, not just how to compute them; and knowing and flexibly using different properties of
operations and objects.
5
Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem.
These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a
spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient
students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions
about when each of these tools might be helpful, recognizing both the insight to be gained and their
limitations. For example, mathematically proficient high school students analyze graphs of functions and
solutions generated using a graphing calculator. They detect possible errors by strategically using
estimation and other mathematical knowledge. When making mathematical models, they know that
technology can enable them to visualize the results of varying assumptions, explore consequences, and
compare predictions with data. Mathematically proficient students at various grade levels are able to
identify relevant external mathematical resources, such as digital content located on a website, and use
them to pose or solve problems. They are able to use technological tools to explore and deepen their
understanding of concepts.
Warwick Public Schools, in collaboration with
the Charles A. Dana Center at the University of Texas at Austin
C-26
Grade 6 Mathematics, Quarter 2, Unit 2.3
Understanding Ratios and Rates (10 days)
Clarifying the Standards
Prior Learning
In grade 5, students will analyze patterns and relationships by identifying apparent relationships between
corresponding terms. They interpret division of a whole number by a unit fraction, and compute. Students
also solve real-world problems involving division of unit fractions by non-zero whole numbers and
division of whole numbers by unit fractions.
Current Learning
Students use reasoning about multiplication and division to solve ratio and rate problems about quantities.
Students will connect their understanding of multiplication and division with ratios and rates by viewing
equivalent ratios and rates derived from a multiplication table and analyzing simple drawings. Students
expand the scope of problems for which they can use multiplication and division to solve problems, and
they connect ratios and fractions. Students will solve a variety of problems involving ratios and rates.
Future Learning
In grade 7, students will continue to compute unit rates associated with ratios of fractions, extending this
to ratios of lengths and areas. They will identify the constant of proportionality (unit rate) in tables,
graphs, equations, and diagrams. Students will analyze proportional relationships and use them to solve
real-world and mathematical problems. In grade 8, students will recognize a unit rate as the slope of a
line. They will also understand the connections between proportional relationships and linear equations.
Additional Findings
According to Curriculum Focal Points, students use simple reasoning about multiplication and division to
solve ratio and rate problems. According to the Atlas of Science Literacy, Volume 1, when something is
bigger than something else, we can characterize the relationship by how much bigger it is or how many
times bigger. The fact that a/b implies a special kind of comparison of a to b is critical. In this case, it is
related to unit rate.
Warwick Public Schools, in collaboration with
the Charles A. Dana Center at the University of Texas at Austin
C-27
Grade 6 Mathematics, Quarter 2, Unit 2.3
Warwick Public Schools, in collaboration with
the Charles A. Dana Center at the University of Texas at Austin
Understanding Ratios and Rates (10 days)
C-28