options: Reason with shapes and their attributes.

Experiment with transformations in the plane.

Prove theorems involving similarity.

Make geometric constructions.

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

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.

Analyze, compare, create, and compose shapes.

Understand congruence in terms of rigid motions.

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

Understand and apply the Pythagorean Theorem.

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

Find arc lengths and areas of sectors of circles.

Apply geometric concepts in modeling situations.

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

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

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

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

Use coordinates to prove simple geometric theorems algebraically.

Apply trigonometry to general triangles.

Prove geometric theorems.

Explain volume formulas and use them to solve problems.

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

Understand similarity in terms of similarity transformations.

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

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

Understand and apply theorems about circles.
problem_activity: Narrative
The purpose of this activity is for students to finish partitioning a rectangle into equal-size squares. This work will prepare students for partitioning rectangles on their own in later lessons. In the synthesis, students are invited to use what they know about the structure of arrays to anticipate how many equal-size squares will fill the rectangle without drawing any lines (MP8).
MLR8 Discussion Supports.
Synthesis: At the appropriate time, give students 2–3 minutes to make sure that everyone in their group can explain how they know how many squares will fill the rectangle. Invite groups to rehearse what they will say when they share with the whole class.
Advances: Speaking, Conversing, Representing
Required Materials
Materials to Gather
Colored pencils or crayons
Rulers
Launch
Give each student at least 2 different colors of colored pencils or crayons and a ruler.
Activity
“In the first activity, you used tiles to make a rectangle and shaded in squares to represent it. In this activity, you will start with a rectangle and decompose it into small equal-size squares by drawing lines with a ruler. For each rectangle, there will be some squares already outlined. Draw lines to continue making equal-size squares inside each rectangle.”
“After making the squares, use colored pencils or crayons to show patterns in the rows or columns.”
10 minutes: independent work time
Student Facing
Draw lines so that the rectangle is completely filled with equal-size squares.
[IMG]
Color the rows different colors.
How many rows of equal-size squares are there?
How many squares are in each row?
Write an equation to represent the sum of the squares in each row.
Draw lines so that the rectangle is completely filled with equal-size squares.
[IMG]
Color the columns different colors.
How many columns of equal-size squares are there?
How many squares are in each column?
Write an equation that represents the sum of squares in each column.
Draw lines so that the rectangle is completely filled with equal-size squares.
[IMG]
How many columns of equal-size squares are there? How many squares are in each column?
How many rows of equal-size squares are there? How many squares are in each row?
Write 2 equations to represent the number of equal-size squares in the rectangle.
Student Response
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Advancing Student Thinking
If students create squares one by one or create squares that are significantly different in size, consider pairing them with a peer who uses a ruler or looks for ways to draw longer lines across rows and columns. Ask students to compare their techniques and try each other’s techniques to complete one row or column of squares.
Activity Synthesis
Display the rectangle that has 4 squares in the first row, but is missing squares in each column.
“How could you find the total number of squares that would fill this rectangle without drawing the lines?” (I can tell there will be 4 columns based on the top row. I can tell there will be 5 rows based on the first column, even though it is missing 1 square.)


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