In Topic A, students examine situations carefully to determine if they are describing a proportional relationship. Their analysis is applied to relationships given in tables, graphs, and verbal descriptions. At the end of the unit, students will use proportional and non-proportional relationships to solve real-world problems.

Content Standard(s):

Mathematics MA2019 (2019) Grade: 7

2. Represent a relationship between two quantities and determine whether the two quantities are related proportionally.

a. Use equivalent ratios displayed in a table or in a graph of the relationship in the coordinate plane to determine whether a relationship between two quantities is proportional.

b. Identify the constant of proportionality (unit rate) and express the proportional relationship using multiple representations including tables, graphs, equations, diagrams, and verbal descriptions.

c. Explain in context the meaning of a point (x,y) on the graph of a proportional relationship, with special attention to the points (0,0) and (1, r) where r is the unit rate.

Unpacked Content

Evidence Of Student Attainment:

Students:

Decide whether a relationship between two quantities is proportional.

Recognize that not all relationships are proportional.

Use equivalent ratios in a table or a coordinate graph to verify a proportional relationship.

Identify the constant of proportionality when a proportional relationship exists between two quantities.

Use a variety of models (tables, graphs, equations, diagrams and verbal descriptions) to demonstrate the constant of proportionality.

Explain the meaning of a point (x, y) in the context of a real-world problem.

Example, if a boy charges $6 per hour to mow lawns, this relationship can be graphed on the coordinate plane. The point (1,6) means that after 1 hour of working the boy makes $6, which shows the unit rate of $6 per hour.

Teacher Vocabulary:

Equivalent ratios

proportional

Coordinate plane

Ratio table

Unit rate

Constant of proportionality

Equation

ordered pair

Knowledge:

Students know:

(2a) how to explain whether a relationship is proportional.

(2b) that the constant of proportionality is the same as a unit rate. Students know:

where the constant of proportionality can be found in a table, graph, equation or diagram.

(2c) that the constant of proportionality or unit rate can be found on a graph of a proportional relationship where the input value or x-coordinate is 1.

Skills:

Students are able to:

(2a) determine if a proportional relationship exists when given a table of equivalent ratios or a graph of the relationship in the coordinate plane.

(2b) identify the constant of proportionality and express the proportional relationship using a variety of representations including tables, graphs, equations, diagrams, and verbal descriptions.

(2c) model a proportional relationship using coordinate graphing.

Explain the meaning of the point (1, r), where r is the unit rate or constant of proportionality.

Understanding:

Students understand that:

(2a) A proportional relationship requires equivalent ratios between quantities. Students understand how to decide whether two quantities are proportional.

(2b) The constant of proportionality is the unit rate. Students are able to identify the constant of proportionality for a proportional relationship and explain its meaning in a real-world context.
(2c) The context of a problem can help them interpret a point on a graph of a proportional relationship.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: M.7.2.1: Define proportions and proportional relationships.
M.7.2.2: Demonstrate how to write ratios as a fraction.
M.7.2.3: Define equivalent ratios and origin.
M.7.2.4: Locate the origin on a coordinate plane.
M.7.2.5 Show how to graph on Cartesian plane.
M.7.2.6: Determine if the graph is a straight line through the origin.
M.7.2.7: Use a table or graph to determine whether two quantities are proportional.
M.7.2.8: Define a constant and equations.
M.7.2.9: Create a table from a verbal description, diagram, or a graph.
M.7.2.10: Identify numeric patterns and finding the rule for that pattern.
M.7.2.11: Recall how to find unit rate.
M.7.2.12: Recall how to write equations to represent a proportional relationship.
M.7.2.13: Discuss the use of variables.
M.7.2.14: Define ordered pairs.
M.7.2.15: Show how to plot points on a Cartesian plane.
M.7.2.16: Locate the origin on the coordinate plane.

Prior Knowledge Skills:

Recall basic addition, subtraction, multiplication, and division facts.

Define ordered pair of numbers.

Define x-axis, y-axis, and zero on a coordinate.

Specify locations on the coordinate system.

Define ordered pair of numbers, quadrant one, coordinate plane, and plot points.

Label the horizontal axis (x).

Label the vertical axis (y).

Identify the x- and y- values in ordered pairs.

Model writing ordered pairs.

Define quantity, fraction, and ratio.

Reinterpret a fraction as a ratio. Example: Read 2/3 as 2 out of 3.

Write a ratio as a fraction.

Create a ratio or proportion from a given word problem, diagram, table, or equation.

Alabama Alternate Achievement Standards

AAS Standard: M.AAS.7.2 Use a ratio to model or describe a real-world relationship.

Mathematics MA2019 (2019) Grade: 7 Accelerated

2. Represent a relationship between two quantities and determine whether the two quantities are related proportionally.

a. Use equivalent ratios displayed in a table or in a graph of the relationship in the coordinate plane to determine whether a relationship between two quantities is proportional.

b. Identify the constant of proportionality (unit rate) and express the proportional relationship using multiple representations including tables, graphs, equations, diagrams, and verbal descriptions.

c. Explain in context the meaning of a point (x,y) on the graph of a proportional relationship, with special attention to the points (0,0) and (1, r) where r is the unit rate. [Grade 7, 2]

Unpacked Content

Evidence Of Student Attainment:

Students:

Decide whether a relationship between two quantities is proportional.

Recognize that not all relationships are proportional.

Use equivalent ratios in a table or a coordinate graph to demonstrate a proportional relationship.

Identify the constant of proportionality when a proportional relationship exists between two quantities.

Interpret a variety of models (tables, graphs, equations, diagrams and verbal descriptions) to identify the constant of proportionality.

Explain the meaning of a point (x, y) in the context of a real-world problem.

Example: If a boy charges $6 per hour to mow lawns, this relationship can be graphed on the coordinate plane. The point (1, 6) contains the unit rate or constant of proportionality, 6.

Teacher Vocabulary:

Equivalent ratios proportional

Coordinate plane

Ratio table

Unit rate

Constant of proportionality

Equation

Ordered pair

Knowledge:

Students know:

(2a) how to explain whether a relationship is proportional.

(2b) that the constant of proportionality is the same as a unit rate.

(2b) where the constant of proportionality can be found in a table, graph, equation or diagram.

(2c) that the constant of proportionality or unit rate can be found on a graph of a proportional relationship where the input value or x-coordinate is 1.

Skills:

Students are able to:

(2a) model a proportional relationship using a table of equivalent ratios.

Use a coordinate graph to decide whether a relationship is proportional by plotting ordered pairs and observing whether the graph is a straight line through the origin.

(2b) translate a written description of a proportional relationship into a table, graph, equation or diagram.

Read and interpret these to find the constant of proportionality.

(2c) model a proportional relationship using coordinate graphing.

Explain the meaning of the point (1, r), where r is the unit rate or constant of proportionality.

Understanding:

Students understand that:

(2a) a proportional relationship requires equivalent ratios between quantities. Students understand how to decide whether two quantities are proportional.

(2b) the constant of proportionality is the unit rate. Students are able to identify the constant of proportionality for a proportional relationship and explain its meaning in a real-world context.

(2c) the context of a problem can help them interpret a point on a graph of a proportional relationship.

Diverse Learning Needs:

Mathematics MA2019 (2019) Grade: 8

7. Determine whether a relationship between two variables is proportional or non-proportional.

Unpacked Content

Evidence Of Student Attainment:

Students:

Describe a given relationship as proportional or non-proportional when given in various contexts.

Teacher Vocabulary:

Ratio

Proportion

Proportional

Independent variable

Dependent variable

Knowledge:

Students know:

how to use rates and scale factors to find equivalent ratios.

What a unit rate is and how to find
it when needed.

Skills:

Students are able to:

Recognize whether ratios are in a proportional relationship using tables and verbal descriptions.

Understanding:

Students understand that:

a proportion is a relationship of equality between quantities.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: M.8.7.1: Define proportional, independent variable, dependent variable, unit rate.
M.8.7.2: Recall equivalent ratios and origin on a coordinate (Cartesian) plane.
M.8.7.3: Recall how to write a ratio of two quantities as a fraction.
M.8.7.4: Identify the unit rate of two quantities.
M.8.7.5: Recall that for a relationship to be proportional, both variables must start at zero.

Prior Knowledge Skills:

Define unit rate, proportion, and rate.

Create a ratio or proportion from a given word problem.

Calculate unit rate by using ratios or proportions.

Write a ratio as a fraction.

Define ratio, rate, proportion, percent, equivalent, input, output, ordered pairs, diagram, unit rate, and table.

Create a ratio or proportion from a given word problem, diagram, table, or equation.

Calculate unit rate or rate by using ratios or proportions with or without a calculator.

Restate real-world problems or mathematical problems.

Construct a graph from a set of ordered pairs given in the table of equivalent ratios.

Calculate missing input and/or output values in a table with or without a calculator.

Draw and label a table of equivalent ratios from given information.

Identify the parts of a table of equivalent ratios (input, output, etc.).

Compute the unit rate, unit price, and constant speed with or without a calculator.

Create a proportion or ratio from a given word problem.

Mathematics MA2019 (2019) Grade: 8

10. Compare proportional and non-proportional linear relationships represented in different ways (algebraically, graphically, numerically in tables, or by verbal descriptions) to solve real-world problems.

Unpacked Content

Evidence Of Student Attainment:

Students:

Analyze and explain the difference between a proportional and non-proportional linear relationship given in various contexts.

Use evidence gathered from data given by linear relationships to make sense of and solve real-world problems.

Teacher Vocabulary:

Proportional

Knowledge:

Students know:

the difference between proportional and non-proportional linear relationships.

What rate of change/slope represents as well as the meaning of initial value/y-intercepts when given in a variety of contexts.

Skills:

Students are able to:

qualitatively and quantitatively compare linear relationships in different ways when those relationships are presented within real-world problems.

Understanding:

Students understand that:

real-world linear relationships can be compared using any representation they choose. based on their understanding of proportions and functions.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: M.8.10.1: Define proportional and nonproportional.
M.8.10.2: Recall that for two relationships to be proportional they must have the same unit rate and pass through the origin on a coordinate plane.
M.8.10.3: Apply the rule of proportional relationship to real-world context.

Prior Knowledge Skills:

Define unit rate, proportion, and rate.

Calculate unit rate by using ratios or proportions.

Write a ratio as a fraction.

Define proportions and proportional relationships.