Multiplication and Rectangles

Est. Class Sessions: 3

Developing the Lesson

Part 1. Exploring Factors Using Tiles

Give each student 25 square-inch tiles of the same color. Different colors might confuse students when they write number sentences to match the tiles. For example, if a student sees 5 yellow tiles and 1 blue tile on a 3 × 2 array, he or she may write the number sentence 5 + 1 = 6. This sentence would describe the colors instead of the arrangement of the tiles into 3 equal groups of 2.

Find Factors of 6. Distribute 25 square-inch tiles to each student. See TIMS Tip. Direct students' attention to the Multiplication and Rectangles page in the Student Guide. Use this page as an introduction to the lesson as you read the vignettes together. Remind students that rows are horizontal and refer to the tiles going across. Use a display set of square-inch tiles to recreate the 4 × 2 and 1 × 8 rectangles made on the Student Guide page. Have students recreate the same rectangles at their desks.

Make a configuration that is not a rectangle with the 6 tiles and ask:

Is this a rectangle? Why or why not? (Possible response: A rectangle has 4 sides and 4 square corners. The opposite sides are the same length.)

Then ask:

Arrange six tiles into rectangles in as many ways as you can.

Students should work individually and then compare their rectangles with those of a partner. Students should find four ways to arrange the six tiles, counting the arrangement of two rows and three columns, for example, as different from the arrangement of three rows and two columns. See Figure 2.

Show students a display of the rectangle with 2 rows and 3 tiles in each row.

Ask:

How many rows do you see in this rectangle? (2 rows and 3 columns)
How many tiles are in each row? (3)
Write a number sentence that matches this rectangle. (2 × 3 = 6)
Show me where the tiles are for each number in the sentence.

If students suggest number sentences using repeated addition (e.g., 2 + 2 + 2), link this with multiplication by showing how both number sentences match the rectangle. Emphasize that in this activity, students will write multiplication sentences.

Review the term factor with students. Two and three are factors of six because when they are multiplied together the answer (product) is 6.

Ask students to remove the Square-Inch Grid Paper pages from the Student Activity Book. As you display the page, tell students you are going to show them how to make a record of the rectangles they build with tiles.

Ask:

Each square on the grid paper will represent one tile. How many tiles did I use to make this rectangle? (6)
How many squares should I shade in on the grid to show my rectangle? (6)

Show students how to shade in 2 rows of 3 squares to record the rectangle. Discourage students from drawing the outline of the rectangle. Focus instead on shading in 6 squares to represent the area of the rectangle. Have students shade a 2 × 3 rectangle on the Square-Inch Grid Paper page and write the multiplication number sentence 2 × 3 near it.

Ask student volunteers to use the display set of square-inch tiles to share the 3 other rectangles they made with 6 tiles: a 3 × 2, a 1 × 6, and a 6 × 1. As each one is shared, demonstrate how to draw the rectangle on grid paper by shading in squares. Then have students draw the rectangle on grid paper and write a multiplication sentence near it. Display both a 3 × 2 and a 1 × 6 rectangle.

Ask:

What are some factors of 6? In other words, which numbers multiplied together give an answer of 6? (1 and 6, 2 and 3)
How is a 3 × 2 rectangle similar to a 1 × 6 rectangle? (They both use 6 tiles.)
How are they different? (One has 3 rows of 2 tiles and the other has 1 row of 6 tiles.)
Since they both use 6 tiles, can I use the same multiplication sentence to describe them? Why or why not? (No. Even though the product is the same, the number sentence does not match the arrangement of the tiles. See TIMS Tip.)

Tell students they are going to be using more tiles to build larger rectangles. The rectangles will be too large to fit on to the Square-Inch Grid Paper, so new grid paper is needed. Display the Centimeter Grid Paper Master.

Ask:

How is this grid different from the grid you used to draw 6-tile rectangles? (There are more squares. The squares are smaller.)

If students are having difficulty recording their rectangles on grid paper, have them focus on coloring in each square rather than trying to draw an outline of the rectangle. When students try to draw outlines they are concentrating on counting lines and thinking about perimeter rather than the area needed to represent their rectangle.

Connecting a number sentence with a picture may not be apparent to some students. Emphasize that number sentences must match the picture. Give students examples of number sentences that do not match a picture. For example, for a 5 × 4 array, students may write 2 × 10 = 20. Even though the product is the same, this number sentence does not match the picture. Point out that students should count the number of rows and columns in the rectangle.

Explain that you will still use one square on the grid to show each tile in your rectangle. Demonstrate how to shade in the 2 × 3 rectangle on Centimeter Grid Paper. Ask volunteers to demonstrate how to shade in other rectangles.

Find Factors of 12 and 18. Direct students' attention to the Exploring Factors Using Tiles pages of the Student Activity Book. Point out the centimeter grids on which students will draw their rectangles. Assign Questions 1–2 to student pairs. Students will arrange 12 and 18 square-inch tiles into rectangles in as many ways as they can, shade them on grid paper, and write a multiplication sentences near each one. Figure 3 shows the six rectangles that can be made with 12 tiles.

After they have finished, ask a pair of students to share their work with the class. Use discussion prompts similar to those in the Sample Dialog. Repeat a similar discussion for the six rectangles that can be made with 18 tiles.

Making Rectangles with 12 Tiles. Rosa and Josh drew four rectangles on a transparency and labeled them with number sentences:
3 × 4 = 12, 4 × 3 = 12, 2 × 6 = 12, 6 × 2 = 12

Teacher: Rosa and Josh, tell us about your rectangles and your number sentences. How do your number sentences match your rectangles? [Rosa and Josh look confused.] Tell us about the rows and columns and how they match your number sentences.

Rosa: This one has 3 rows and 4 going down. We wrote 3 × 4 = 12. This one has 4 rows and 3 this way and that's 12, too.

Teacher: The tiles going up and down vertically are in columns. Tell us about the other rectangles.

Josh: This one has 2 rows and 6 columns and this one is the same except it has 6 rows and 2 columns. The number sentences are 6 × 2 and 2 × 6. They all equal 12.

Teacher: [Hands are up in the classroom.] Rosa and Josh, there seem to be some questions. Call on someone to ask a question.

Rosa: Suzanne, do you have a question?

Suzanne: We found two more rectangles. Can I show you? [Suzanne comes to the display and adds rectangles for 1 × 12 and 12 × 1 and writes the corresponding number sentences.]

Teacher: Thank you, Suzanne. Look at all the rectangles and the number sentences. What numbers are factors of 12? What numbers multiplied together make 12?

Frank: 2 and 6 and 3 and 4.

Suzanne: Don't forget mine: 12 and 1.

Student Guide: 201

Answers