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!
