Area with Multiplying and Adding

Explicit Instruction/Teacher modeling

• Model how to draw a rectangle on graph paper and portion it off so that it is a large rectangle that is 30-by-20. Make sure students are in agreement with the length and width measurement, and then cut off a piece of the rectangle so there are two rectangles, one that's 10-by-9 and the other that is 20-by-11.
• Tape the 10-by-9 rectangle on the board and label the sides and color in the boxes with a marker. Cut the 20-by-11 rectangle so that it becomes two separate rectangles where one measures 15-by-5 and the other measures 5-by-6 and have different colors. Tape both of those rectangles on the board and label the sides.
• Calculate the area of each of the rectangles using the area equation and counting the squares of one for reinforcement. Solicit help from students and make sure to use key vocabulary as you think aloud the process to find the areas of all three of the rectangles.
• "Reattach" each rectangle one at a time so that it becomes a 30-by-20 rectangle with different colored areas. Retape them on the board together and label the colored lengths and width. Recalculate the area of the rectangle and show that it would be the same if they added up the individual areas of the rectangles. Write the equation that represents the joined together rectangles (i.e., (10 + 15 + 5) x (9 + 5 + 6) = 600).

Guided Practice

• Choose students one by one to recall the steps you made to decompose the rectangle and add the areas again. Allow students to assist their peers and offer suggested answers that the presenter will repeat after you accept the suggestion as correct.
• Distribute one sheet of graph paper and copy paper to partnerships and ask students to cut out a rectangle that would be large enough for them to decompose twice. Then, have them write down "Rectangle #1" and, next to it, write the area equation that corresponds to the rectangle (e.g., 34 x 10 = 340). Have students follow that same process for the second and third rectangles they'll create in partnerships, but have students label them "Rectangle A," "Rectangle B," and "Rectangle C."
• Model asking questions about the rectangles as you're decomposing them. For example, "Why do you think that the area, when added together, equals the original rectangle?" or "Would the area be the same if I cut the rectangle vertically rather than horizontally?"

Group work time

• Tell students they'll now practice composing partitioned rectangles in an information gap card game that requires them to find their group members by matching three cards together. Explain to students they cannot show anyone their cards and they need to ask their potential partner questions to see if they are a match. For example, they can ask "What is the length and width of your rectangle?" or "Do you have a rectangle or the area on the card?"
• Distribute the Info Gap Cards: Area as Additive (one card per student) and review the instructions. Have students find their partners by asking questions to match three cards together. Have them use the back of their copy paper as scratch paper for their calculations.
• Remind students that the areas of two rectangles will add together to equal the area listed on the third card. Once they find their partners, have them create the appropriate equation when they merge the two rectangles together and input the lengths and widths into the area equation (e.g., (10 + 15 + 5) x (9 + 5 + 6) = 600).
• Have students find their partners and explain how they know they got the right answer. Then, ask students to ask questions about their area cards and questions they still have about adding areas and multiplication. (For example, "I wonder if I can create irregular figures with the rectangles and the area will still be the same," or "Why do you think my rectangle has equal sides and my partner has unequal sides?")
• Provide the following sentence frames to guide student questions or statements, and ask students to provide more sentence frames if they can:
  • "I was wondering why ____?"
  • "How come ____?"
  • "Why do you think ____?"
• Ask students to share some of their ponderings and see if students can answer some of the questions.