options: Understand congruence in terms of rigid motions.

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

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

Find arc lengths and areas of sectors of circles.

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

Prove theorems involving similarity.

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

Prove geometric theorems.

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

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

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

Define trigonometric ratios and solve problems involving right triangles.

Analyze, compare, create, and compose shapes.

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

Explain volume formulas and use them to solve problems.

Apply trigonometry to general triangles.

Apply geometric concepts in modeling situations.

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

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

Understand and apply the Pythagorean Theorem.

Use coordinates to prove simple geometric theorems algebraically.

Make geometric constructions.

Experiment with transformations in the plane.

Understand similarity in terms of similarity transformations.

Understand and apply theorems about circles.

Reason with shapes and their attributes.

Draw, construct, and describe geometrical figures and describe the relationships between them.
problem_activity: Optional activity
This activity gives students a chance to apply what they know about scale factors, lengths, and angles and create scaled copies without the support of a grid. Students work in groups of 3 to complete a jigsaw puzzle, each group member scaling 2 non-adjacent pieces of a 6-piece puzzle with a scale factor of
\(\frac12\)
. The group then assembles the scaled pieces and examines the accuracy of their scaled puzzle. Consider having students use a color in place of the cross hatching.
[IMG]
[IMG]
As students work, notice how they measure distances and whether they consider angles. Depending on how students determine scaled distances, they may not need to transfer angles. Look out for students who measure only the lengths of drawn segments rather than distances, e.g., between the corner of a square and where a segment begins. Suggest that they consider other measurements that might help them locate the beginning and end of a segment.
You will need the Scaling a Puzzle blackline master for this activity.
Teacher Notes for IM 6–8 Math Accelerated
If time allows, allow students to start this activity in class and then finish as part of the practice problem set.
Launch
Arrange students in groups of 3. Give pre-cut puzzle squares 1 and 5 to one student in the group, squares 2 and 6 to a second student, and squares 3 and 4 to the third. After students have answered the first question, give each student 2 blank squares cut from the second section of the blackline master, whose sides are half of the side length of the puzzle squares. Provide access to geometry toolkits.
Engagement: Develop Effort and Persistence.
Connect a new concept to one with which students have experienced success. Before students begin, ask them to predict which tools, from their geometry toolkits, they anticipate they will need, and to describe how they might use them. During the activity, make sure that students are using their rulers and protractors correctly.
Supports accessibility for: Visual-spatial processing; Conceptual processing; Fine-motor skills
Student Facing
Your teacher will give you 2 pieces of a 6-piece puzzle.
If you drew scaled copies of your puzzle pieces using a scale factor of
\(\frac12\)
, would they be larger or smaller than the original pieces? How do you know?
Create a scaled copy of each puzzle piece on a blank square, with a scale factor of
\(\frac12\)
.
When everyone in your group is finished, put all 6 of the original puzzle pieces together like this:
[IMG]
Next, put all 6 of your scaled copies together. Compare your scaled puzzle with the original puzzle. Which parts seem to be scaled correctly and which seem off? What might have caused those parts to be off?
Revise any of the scaled copies that may have been drawn incorrectly.
If you were to lose one of the pieces of the original puzzle, but still had the scaled copy, how could you recreate the lost piece?
Student Response
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for free access to Student Response.
Anticipated Misconceptions
Students may incorporate the scale factor when scaling line segments but neglect to do so when scaling distances between two points not connected by a segment. Remind them that all distances are scaled by the same factor.
Students may not remember to verify that the angles in their copies must remain the same as the original. Ask them to notice the angles and recall what happens to angles when a figure is a scaled copy.
Activity Synthesis
Much of the conversations about creating accurate scaled copies will have taken place among partners, but consider coming together as a class to reflect on the different ways students worked. Ask questions such as:
How is this task more challenging than creating scaled copies of polygons on a grid?
Besides distances or lengths, what helped you create an accurate copy?
How did you know or decide which distances to measure?
Before your drawings were assembled, how did you check if they were correct?
Student responses to these questions may differ: for example, for piece 6, the two lines can be drawn by measuring distances on the border of the puzzle piece; the angles work out correctly automatically. For piece 2, however, to get the three lines that meet in a point in the middle of the piece just right, students can either measure angles, or extend those line segments until they meet the border of the piece (and then measure distances).


correct_option_index: 26