standard_description: Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
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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