Using a dot array, students will calculate the products of one- and two-digit factors. They will: 
- break apart arrays (representing the distributive property of multiplication over addition) to compute a two-digit factor multiplied by a one-digit factor. 
- find multiple ways to divide an array in order to calculate the product of two factors.

- How can mathematics help to quantify, compare, depict, and model numbers? 
- How may patterns be used to describe mathematical relationships?

- Equivalence: The relationship between expressions which have an equal value. 
- Multiple: The product of a given number and a whole number.

- Base-ten grid paper (M-4-2-2_Base-Ten Grid) 
- Array Pictures sheet (M-4-2-2_Array Pictures) 
- Array Practice worksheet and KEY (M-4-2-2_Array Practice and KEY)

- Use the Array Practice worksheet and KEY (M-4-2-2_Array Practice and KEY) to assess the student's understanding. 
- Use the think-pair-share exercise in the Routine section of the Extension to assess student knowledge. 
- Observe students solve multiplication problems with arrays to assess their understanding of the content.

Scaffolding, Active Engagement, Modeling, and Explicit Instruction 
W: Introduce students to arrays. Allowing students to find the total number of objects in the arrays without providing any instructions will lead them to identify methods to split up the assignment. 
H: Visual arrays can help students grasp problems and write numerical sentences. 
E: Explain how to solve an array problem by dividing it into rectangles based on place value (e.g., tens by tens, tens by ones). 
R: Have students solve array problems from the problem set. 
E: To determine if students understand the relationship between arrays and number sentences, have them answer the same problems by drawing the arrays on grid paper, decomposing them, and labeling each part. 
T: Adjust the lesson based on the strategies given in the Extension section. 
O: The lesson aims to teach students how to create and break down arrays for multiplication. In doing so, students develop a graphical model of distributing multiplication across addition and learn about the concept of partial products. At this stage, students are learning a number of ways to solve multiplication problems. By discussing each strategy, students get a greater knowledge of the multiplication process, especially when it applies to multidigit numbers. 

"Today, we are going to talk about arrays. An array is a rectangular grouping of things in equal rows and columns. Let's look at few examples."

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

"How could you determine the number of objects shown in each array?" (repeated addition, multiplying rows by the number in each row, counting, grouping, and skipping counting)

"Believe it or not, we probably see arrays on a daily basis. For the next few minutes, you can find some arrays right here in our classroom (or hallway)." Allow students a few minutes to walk around the classroom or hallway to find real-life arrays. Make a class list of the arrays they found. (Examples include crayons in a box, chalk in a box, seats in a row, and a monthly wall calendar.)

Provide extra settings that aid reasoning about arrays: 

There are 3 dozen eggs. How many eggs are in three dozen? 
There are five twelve-packs of soda. How many Coke cans are in five 12-packs?

Instruct students to draw a picture of the problem. Students can use photos, dots, stars, or any other shape to represent the objects in their images. 

"How many different number sentences can we write to represent this problem?" Students should create equivalent equations to those they practiced in Lesson 1. For example, a student may write (5 × 6) + (5 × 6) = 5 × 12. Note that the sentence can alternatively be expressed as (6 × 5) + (6 × 5) = 12 × 5 using the commutative principle. (10 × 3) + (2 × 3) = 12 × 3. Look for ways to highlight the qualities in the students' solutions.

Present the following problem:

Four girls saw a strip of strange stickers for $0.45 at the school store. Each girl had a dime and two pennies leftover from buying lunch. The girls combined their coins on the store's counter. Did they have enough to buy the strip of stickers? 
Encourage students to describe how they would address this challenge. Try to get them to focus on the concept of grouping like coins together. For example, combining 4 dimes gives $0.40 (4 × 10), but adding all pennies gives 8 pennies (4 × 2). 

Encourage students to illustrate the grouping of 10s and singles using pictures (four rows of 10 dots each and four rows of two dots each) presented right next to the previous quantity. Use the base-ten grid paper (M-4-2-2_Base-Ten Grid) to color the quantity (4 × 10) + (4 × 2). Ask students to label the components of the rectangular grid. As demonstrated below, label the width "4" and the length of the rectangle "10 + 2" respectively.



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

Ask probing questions regarding number relationships and properties (such as the distributive property). For example, "What if each of the 4 girls has two dimes and two pennies. What would we need to change in our rectangle array, and why? How does this answer relate to $0.48?" An example would look like this:



The setting of the problem encourages students to think about 10s, thus the problem is decomposed in a way that emphasizes place value. 

Present the following problem:

Four children saw a used kite in the window of a thrift store. Each child held a quarter and three pence. The kite's price was posted at $1.15. Can the children combine their coins to purchase the kite? Explain. 
This problem context encourages students to add all of the quarters together to form $1.00 and the pennies together to make 12 cents.

Outline the problem's rectangular array using base-ten grid paper. $0.28 × 4. (four rows, 28 squares) The base-ten grid paper highlights the groups of 10 in 28 with the 8 added on. If necessary, reinforce understanding with extra related multiplication problems. 

Give students problem sets (such as the ones below) to help them practice more. Encourage students to apply whatever strategies they have learnt so far in the unit to solve these problems. Ask students to explain the relationship between the problems in each set.

Students should draw rectangular arrays on base-ten grid paper that correspond to the problem sets listed above. Students should identify the rectangular array and draw connections to the related problems. Students should be asked to explain one of the drawings or expressions to demonstrate their understanding of multiplication by decomposition using arrays. Remind students to focus on 10s in these problems.

Extension:

Routine: As a warm-up for the following day's lecture, write a one-digit by two-digit multiplication problem on the whiteboard. Students should consider using an array to solve the problem, then draw the array and solve it. (To create the array, they can use grid paper, draw freehand, draw dots, or write Xs. Engage students in a think-pair-share activity to share their solutions with a partner and quickly discuss methods. Then ask the students to explain how they arrived at the solution. 

Expansion: Students who have rapidly or easily grasped the concept might consider how to solve a two-digit by two-digit multiplication problem with arrays. Make them test their theory. Students could write word problems to solve and then swap them with a partner. This will give you more practice and make it more difficult to solve a problem by converting a story into symbols. 

Technology Connection: For students who haven't mastered the concept yet, visit http://www.printable-math-worksheets.com/multiplication-array.html to practice using arrays for multiplication without the difficulty of two-digit numbers. This website provides practice with multiplication arrays through one-digit by one-digit problems, allowing students to learn the concept before going on to two-digit multiplication.