options: Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.
problem_activity: Activity
This activity continues work done in the previous lesson that connected various representations of proportional relationships including images, equations, and descriptions. In this activity students match diagrams, descriptions, and equations that represent a proportional relationship, involving variables
\(x\)
and
\(y\)
. Students then create their own diagram to represent an equation.
Identify students that create diagrams representing each of the unmatched equations, so they can share their work during the discussion.
Launch
Give students 7–10 minutes of quiet think time on the questions. Follow with whole-class discussion.
Representation: Internalize Comprehension.
Begin the activity with concrete or familiar contexts. For example, display a tape diagram with concrete values for
\(x\)
and
\(y\)
such as 20 and 15 respectively. Ask students “How much did 20 decrease by to get to 15?” Highlight the connections between the concrete and abstract tape diagrams. In the concrete tape diagram, 20 decreased by
\(\frac{1}{4}\)
to get 15. In the abstract tape diagram,
\(x\)
also decreased by
\(\frac{1}{4}\)
to get
\(y\)
.
Supports accessibility for: Conceptual processing; Memory
Student Facing
Match each diagram with a description and an equation.
Diagrams:
[IMG]
Descriptions:
An increase by
\(\frac14\)
An increase by
\(\frac13\)
An increase by
\(\frac23\)
A decrease by
\(\frac15\)
A decrease by
\(\frac14\)
Equations:
\(y=1.\overline{6}x\)
\(y=1.\overline{3}x\)
\(y=0.75x\)
\(y=0.4x\)
\(y=1.25x\)
Draw a diagram for one of the unmatched equations.
Student Response
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Activity Synthesis
Ask previously identified students to share their diagrams and explain how they represent the equation they chose. Then, ask students that did not share their diagrams to explain whether
\(x\)
(or
\(y\)
) is increased or decreased by some amount in the discussed equations.
Representing, Writing: MLR3 Clarify, Critique, Correct.
Present an incorrect diagram to represent one of the unused equations that reflects a possible misunderstanding from the class. For example, for the equation
\(y=0.75x\)
, draw a diagram where
\(y\)
is greater than
\(x\)
. Prompt students to identify the error, and then write a correct diagram to represent the equation. This will support students to understand the relationship between equations and diagrams.
Design Principle(s): Maximize meta-awareness


correct_option_index: 7