• Use the Array Practice worksheet and Array Practice KEY (M-4-2-2_Array Practice.docx and M-4-2-2_Array Practice KEY.docx) to find the degree of student understanding.
• Use the think-pair-share activity in the Routine section of the Extension to help establish the level of student knowledge.
• Observe students solving multiplication problems using arrays to evaluate student comprehension of the materials.

“Today, we are going to talk about arrays. An array is defined as a rectangular arrangement of objects in equal rows or columns. Let’s look at some examples.”

Show students the Array Pictures (M-4-2-2_Array Pictures.docx). Have them make observations about how the arrays are arranged.

“How could you determine the number of objects shown in each array?” (repeated addition, multiplying rows by the number in each row, counting, grouping together, skip counting)

“Believe it or not, we probably see arrays every day. For the next few minutes, you will find some arrays right here in our classroom (or hallway).” Give students a few minutes to walk around the classroom or hallway and find real-life arrays. Construct a class list of arrays they found. (Examples might be crayons in a box, chalk in a box, chairs in a row, monthly wall calendar, etc.)

Provide additional contexts that facilitate thinking about arrays:

• There are three dozen eggs. How many eggs is three dozen?
• There are five 12-packs of soda. How many cans of soda are in five 12-packs?

Ask students to draw a picture of the problem. Students can use pictures, dots, stars, or any shape to represent the objects in their pictures.

“How many different number sentences can we write to represent this problem?” Here students should generate equivalent equations like those they practiced in Lesson 1. For example, a student might write (5 × 6) + (5 × 6) = 5 × 12. Point out that the sentence also can be written as (6 × 5) + (6 × 5) = 12 × 5 by the commutative property. (10 × 3) + (2 × 3) = 12 × 3. Look for opportunities to point out the properties in the students’ solutions.

Present the following problem:

• Four girls saw a strip of unusual stickers for $0.45 at the school store. Each girl had a dime and two pennies left over from buying lunch. The girls combined their coins on the counter of the store. Did they have enough to buy the strip of stickers?

Encourage students to explain how they would approach this problem. Try to get them to focus on the idea of putting like coins together. For instance, putting the 4 dimes together gives $0.40 (4 × 10), and putting all the pennies together gives 8 pennies (4 × 2). Encourage students to represent the grouping of the 10s and singles using pictures (four rows of 10 dots each and four rows of two dots each) displayed right next to the previous amount. Use the base-ten grid paper (M-4-2-2_Base-Ten Grid.docx) to color in the amount (4 × 10) + (4 × 2). Ask students to label the parts of the rectangular grid. Use “4” to label the width and “10 + 2” to label the length of the rectangle, as shown here.

“Can someone explain what the first section represents?” (4 groups of 10, 4 girls each had a dime.) “And the second section?” (4 groups of 2, 4 girls each had 2 cents.)

Probe with questions that encourage thinking about number relationships and properties (such as the distributive property). For example, “What if the four girls each had two dimes and two pennies? What would we have to change in our rectangular array and why? How is this answer related to $0.48?” One example might look like this:

Notice that the context of the problem encourages students to think about 10s, decomposing the problem in a way that emphasizes place value.

Present the following problem:

• Four children saw a used kite in the window of a thrift shop. Each child had a quarter and three pennies. The price marked on the kite was $1.15. Can the children combine their coins to buy the kite? Explain.

This problem context encourages the children to combine all the quarters together to make $1.00 and combine the pennies together to make 12 cents.

Use base-ten grid paper to outline the rectangular array that will represent the problem $0.28 × 4. (four rows of 28 squares) The base-ten grid paper points out the groups of 10 in 28 with the 8 added on. Use additional similar multiplication problems to reinforce understanding if necessary.

Provide problem sets (like the following) for students to gain additional practice. Encourage them to use any strategy they have learned so far in the unit to solve these problems. Have students explain the relationship among the problems in each set.

Have students draw rectangular arrays on the base-ten grid paper to match the problem sets above. Students should label the rectangular array, making the connection to the related problems. Students should be asked to explain one of the drawings or expressions to demonstrate an understanding of multiplication by decomposition using arrays. Remind students to focus on 10s in these problems.

Extension:

• Routine: As a warm-up for the next day’s lesson, post a one-digit by two-digit multiplication problem on the board. Have students think about using an array to solve the problem and then draw the array and solve the problem. (They may use grid paper, draw freehand, draw lines of dots, or write Xs to make the array). Engage students in a think-pair-share to share their solutions with a partner and quickly discuss methodology. Then call on students to explain how they arrived at the solution.
• Expansion: Students who have mastered the concept quickly or easily can think about how to solve a two-digit by two-digit multiplication problem using arrays. Have them test their theory. Students could write word problems to solve and exchange their problems with a partner. This will provide additional practice and the added complexity of translating a story into symbols to reach a solution.
• Technology Connection: For students who have not yet mastered the concept, go to http://www.printable-math-worksheets.com/multiplication-array.html for additional practice using arrays for multiplication without the complexity of two-digit numbers. This Web site offers practice with multiplication arrays using one-digit by one-digit problems, so students can master the concept before moving on to two-digit multiplication.