Area Models and Multiplication
Students will be able to create area models and decompose the factor on one side of a rectangle.
- Display the rectangle at the top of the page from the worksheet An Introduction to Area. Ask students to brainstorm with partners about the area of the rectangle and what might represent that size in real life (e.g., piece of paper, picture book, etc.).
- Allow partners to share their answers, making sure to write the answers or equations on the board.
- Write the key terms on the board and ask students to use the terms in their explanations if possible.
Explicit Instruction/Teacher modeling(8 minutes)
- Define the area as the measurement of the square units inside a shape. Tell them this shape is a rectangle, so they can find the area by adding up all the squares, or multiplying the length by the width.
- Explain that the factors are the numbers they multiply to get the product, or the answers to the multiplication problem.
- Show students how to draw a rectangle and then find the area of the rectangle. Then, think aloud how to break apart the rectangle into two smaller rectangles so that you have two numbers that add up to create the length and one number for the width. Use those three numbers to find the area again and prove that it is a valid method for finding the area of the problem.
Guided Practice(22 minutes)
- Model rolling the dice twice and getting two factors. Then, draw the rectangle on the Graphing Paper worksheet. Use two different colors to emphasize the lengths of the sides.
- Write a multiplication problem with the two sides and solve the problem. Verify your answer by counting the boxes inside the rectangle, or creating a repeated addition problem. Ask students for input throughout the process and have them pretend to be the teacher and give you instructions.
- Pair students with mixed ability partners and give each partnership one graph paper, two dice, and two different colored markers. Write the following instructions on the board:
- Partner 1 rolls the dice twice to get the factors and calculate the product.
- Partner 1 draws an area model on the graph paper and label the sides.
- Partner 2 repeats steps 1–2.
- Continue drawing rectangles on the sheet until one player doesn't have enough room to draw a rectangle.
- Have partners find the areas for the rectangles together by using the strategy of their choice. Then, have them decompose one side of the rectangle. (For example, (3 + 4) x 5 creates a rectangle with a length of 3 + 4 (i.e., 7), and then a width of 5 for a total area of 35 squares.)
- Choose partners to share aloud their rectangles and how they found the area. Make sure to choose partners that used different strategies (i.e., repeated addition, multiplication, counting the squares, etc.).
Independent working time(10 minutes)
- Distribute copy paper.
- Draw a 23 x 9 rectangle on the board. Ask students to find the area of the rectangle. Then, ask them to write one to two sentences explaining their process.
- Allow partners to share their answers and adjust as necessary.
- Debrief with the whole class to make sure everyone has the correct answer.
- Give students a list, with examples and visuals, of different strategies for finding the area.
- Use the worksheet An Introduction to Area in small groups to support struggling learners. Allow them to color code as you reinforce their understanding of the factors and area involved in the rectangles.
- Allow students to use colored manipulatives to create their area models and then draw them on the graph paper.
- Have them decompose one side of their rectangles only once (i.e., there will be two numbers that add up to create the length while there is one number for the width).
- Challenge students to decompose one side of all their rectangles three or more times. Ask them to write the multiplication problem for each of the rectangles they create as they slice the larger rectangle into smaller rectangles.
- Allow students to create four-digit-by-two-digit products and to draw their rectangles on the back of the Graphing Paper worksheet.
- Have students decompose the length of the 23 x 9 rectangle from the independent practice two times on the same copy paper. That means there will be three numbers that add up to the length and one number for the width (e.g., (10 + 10 + 3) x 9).
- Allow students to share their answers with their partners and compare the values of the addends for the length without changing their answers. Then, ask volunteers to explain why, even though there are different addends for the length, the total area is still the same.
Review and closing(5 minutes)
- Tell students to turn and talk to their partners about why it's important to know the basis of area when solving multiplication or division problems that are four-digit-by-two-digit multiplication problems. For example, they will understand how the area of a multiplication problem increases or decreases when they divide it. Also, they can visualize the product and quotient.
- Explain that multiplication and division have an inverse relationship and that this strategy will be great when they have to divide large numbers in the future.