options: Understand and apply theorems about circles.

Make geometric constructions.

Define trigonometric ratios and solve problems involving right triangles.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Understand similarity in terms of similarity transformations.

Analyze, compare, create, and compose shapes.

Prove theorems involving similarity.

Solve real-world and mathematical problems involving area, surface area, and volume.

Graph points on the coordinate plane to solve real-world and mathematical problems.

Translate between the geometric description and the equation for a conic section.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Classify two-dimensional figures into categories based on their properties.

Draw, construct, and describe geometrical figures and describe the relationships between them.

Understand congruence in terms of rigid motions.

Understand congruence and similarity using physical models, transparencies, or geometry software.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Understand and apply the Pythagorean Theorem.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Reason with shapes and their attributes.

Use coordinates to prove simple geometric theorems algebraically.

Explain volume formulas and use them to solve problems.

Experiment with transformations in the plane.

Apply trigonometry to general triangles.

Visualize relationships between two-dimensional and three-dimensional objects.

Find arc lengths and areas of sectors of circles.

Prove geometric theorems.

Apply geometric concepts in modeling situations.
problem_activity: Activity
Students have already looked at measurement conversions that can be represented by proportional relationships, both in grade 6 and in previous lessons in this unit. This task introduces a measurement conversion that is not associated with a proportional relationship. The discussion should start to move students from determining whether a relationship is proportional by examining a table, to making the determination from the equation.
Note that some students may think of the two scales on a thermometer like a double number line diagram, leading them to believe that the relationship between degrees Celsius and degrees Fahrenheit is proportional. If not mentioned by students, the teacher should point out that when a double number line is used to represent a set of equivalent ratios, the tick marks for 0 on each line need to be aligned.
Launch
Arrange students in groups of 2. Provide access to calculators, if desired. Give students 5 minutes of quiet work time followed by students discussing responses with a partner, followed by whole-class discussion.
Representation: Internalize Comprehension.
Activate or supply background knowledge. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Student Facing
The other day you worked with converting meters, centimeters, and millimeters. Here are some more unit conversions.
Use the equation
\(F =\frac95 C + 32\)
, where
\(F\)
represents degrees Fahrenheit and
\(C\)
represents degrees Celsius, to complete the table.
| temperature                       \((^\circ\text{C})\) | temperature                       \((^\circ\text{F})\) |
|--------------------------------------------------------|--------------------------------------------------------|
|                           20                           |                                                        |
|                           4                            |                                                        |
|                          175                           |                                                        |
Use the equation
\(c = 2.54n\)
, where
\(c\)
represents the length in centimeters and
\(n\)
represents the length in inches, to complete the table.
|             length (in)             | length (cm) |
|-------------------------------------|-------------|
|                 10                  |             |
|                  8                  |             |
| 3                       \(\frac12\) |             |
Are these proportional relationships? Explain why or why not.
Student Response
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Anticipated Misconceptions
Some students may struggle with the fraction
\(\frac95\)
in the temperature conversion. Teachers can prompt them to convert the fraction to its decimal form, 1.8, before trying to evaluate the equation for the values in the table.
Activity Synthesis
After discussing the work students did, ask, “What do you notice about the forms of the equations for each relationship?” After soliciting students’ observations, point out that the proportional relationship is of the form
\(y = kx\)
, while the nonproportional relationship is not.
Speaking: MLR8 Discussion Supports.
To provide a support students in producing statements when they compare and contrast equations with proportional and nonproportional relationships. Provide sentence frames such as: “I noticed that _____,” “What makes _____ different from the others is _____,” or “The patterns show _____.” Invite students to use these sentence frames when students comprehend orally the patterns they notice with proportional equations.
Design Principle(s): Support sense-making


correct_option_index: 10