As I started to think about our next unit in math, factors/prime/composite numbers, I knew I wanted to dig in to the concept of multiplication using an array model to see what my students remember from last year. I took one of my favorite lessons from recent years and tweaked it–and I needed it to be “sub friendly” as I was going to be out of my classroom for a training. Here’s how things all turned out!

The Task:

Students should be told that they are working at a factory that

packages square chocolates. They have

been asked to help the packaging department figure out all the different sized

boxes they need to package different amounts of chocolate. Because it is easiest to stack and ship

rectangular boxes, chocolates can only be arranged into rectangular (arrays)

boxes.

packages square chocolates. They have

been asked to help the packaging department figure out all the different sized

boxes they need to package different amounts of chocolate. Because it is easiest to stack and ship

rectangular boxes, chocolates can only be arranged into rectangular (arrays)

boxes.

The goal is for students to understand that certain numbers have MANY different ways for the chocolates to be arranged while others have only one way. I wanted to make sure students started in a very concrete way, so we actually wanted to BUILD the arrays. Here are the steps I recommend.

1. Model the concept of the array with students. Make sure they are clear that an array is rectangular. I cannot lie. I love using a certain type of square, foil

wrapped chocolates to show the concept of arrays. This is particularly powerful motivation—as

the students can each eat part of the modeled array when they finish the lesson

for the day! Consider a “medium sized” array to

start—perhaps 12 candies. This allows

them to see arrays of different shapes and sizes.

wrapped chocolates to show the concept of arrays. This is particularly powerful motivation—as

the students can each eat part of the modeled array when they finish the lesson

for the day! Consider a “medium sized” array to

start—perhaps 12 candies. This allows

them to see arrays of different shapes and sizes.

2. Show the students how arrays can face in different directions but

are really worth the same amount. I also show the students how to use manipulatives to model this. You can use 1 inch tiles or even cut paper tiles–brown paper tiles can look an awful lot like chocolate if you use your imagination!

are really worth the same amount. I also show the students how to use manipulatives to model this. You can use 1 inch tiles or even cut paper tiles–brown paper tiles can look an awful lot like chocolate if you use your imagination!

3. Model how students can sketch the arrays they build on the left

side of their work space and write the coordinating number sentences on the right. Tell them to skip the “prime” and “composite”

part for now. You decide if you want them

to write BOTH numbers sentences (1 x 3= 3 AND 3 x 1 = 3) or if you only want

them to write one of them.

side of their work space and write the coordinating number sentences on the right. Tell them to skip the “prime” and “composite”

part for now. You decide if you want them

to write BOTH numbers sentences (1 x 3= 3 AND 3 x 1 = 3) or if you only want

them to write one of them.

4. After you have done the first few number combinations with the class, you can send them off to work more independently. I ask students to come and show me their work after they reach 12

chocolates so I can catch misconceptions.

I do ask students to use the tiles or paper squares to show their

modeling before drawing up to the point where I “release” them. Some students will need to continue to model

throughout the activity, but others have a strong sense of multiplication and

area and can derive all the possible combinations without a pictorial

representation.

chocolates so I can catch misconceptions.

I do ask students to use the tiles or paper squares to show their

modeling before drawing up to the point where I “release” them. Some students will need to continue to model

throughout the activity, but others have a strong sense of multiplication and

area and can derive all the possible combinations without a pictorial

representation.

NOTE: Some students struggle

to find some of the “middle” arrays.

They can see the long, skinny ones but struggle with the others. One prompt I offer is “Can you build a box

that has 2 rows? How about 3?” and see if they can move from there.

to find some of the “middle” arrays.

They can see the long, skinny ones but struggle with the others. One prompt I offer is “Can you build a box

that has 2 rows? How about 3?” and see if they can move from there.

5. On day 2 of this investigation, revisit the array you made on day

one with the wrapped chocolates.

Introduce the term “factor” as you rebuild (using candies, tiles, paper

squares, or sketches) all the possible arrays you can for that number. I show the students how to identify all the

numbers that are “factors”. For example,

for the number 12, students should have found:

one with the wrapped chocolates.

Introduce the term “factor” as you rebuild (using candies, tiles, paper

squares, or sketches) all the possible arrays you can for that number. I show the students how to identify all the

numbers that are “factors”. For example,

for the number 12, students should have found:

1 x 12 = 12, 2 x 6 = 12, 3 x 4 = 12, 4 x 3 = 12, 6 x 2 = 12, 12 x 1

= 12

= 12

6. I then show how I find ALL the different numbers that can be

multiplied to make 12 and I circle them on the list. I have marked them in red to show this. Students quickly see that the “turn around

fact” is simply the factors doubled. In

this example, twelve has SIX factors.

multiplied to make 12 and I circle them on the list. I have marked them in red to show this. Students quickly see that the “turn around

fact” is simply the factors doubled. In

this example, twelve has SIX factors.

1 x 12 = 12, 2 x 6 = 12, 3 x 4 = 12, 4 x 3 = 12, 6

x 2 = 12, 12 x 1 = 12

x 2 = 12, 12 x 1 = 12

7. The final step of this lesson sequence is to define prime and

composite numbers. One of the easiest

definitions for students this age to understand is the following:

composite numbers. One of the easiest

definitions for students this age to understand is the following:

●A prime number is a whole number GREATER THAN ONE which has exactly

two factors—”one” and itself.

two factors—”one” and itself.

●A composite number is a whole number GREATER THAN ONE which can be

divided by itself, one, and at least one other number.

divided by itself, one, and at least one other number.

After I started writing everything down and getting everything organized, I thought it might be something that others could use as well so I turned it into my latest “Scoop” resource! I have been getting great feedback that these lessons have been super helpful–the photos and flexibility with different options make them useful for all sorts of situations. Interested? Here is the link!