commutative property of multiplication

by C. Elkins, OK Math and Reading Lady

Yes, you can even use ten frames to teach multiplication concepts! Here are my mini ten-frames with dot cards from 1 – 10: Click HERE to get a free copy. These are helpful to use, especially if you don’t have enough tens/ones blocks . . . or you prefer manipulatives that are slightly easier to manage. These provide a strong connection to place value, and the commutiative and distributive properties.

I recommend two sets of the cards (1-9) per student. Each set has multiple copies of the same number. They can be laminated, cut, and placed in a baggie for ease in handing out and storage.

Multiplication Examples:

Single digits (basic facts):
- For the problem 3 x 6, the ten frame is really helpful for the student to see 3 x 6 is almost like 3 x 5 with one more group of 3 added on (by being familiar with the fact that the top row on a ten frame is 5).
- Because of the commutative property, I know these two facts will have the same answer. But which of these below do you think might be “easier” to solve? Students don’t often know they have a choice in how they can use the numbers to their advantage!
  
  6 x 3 (six groups of 3)
  
  3 x 6 (three groups of 6)
Double digit x 1 digit:
- Use of these also provides a strong connection of place value and multiplication. Notice how students can see the breakdown on the 4 x 12 problem (4 groups of 12 = 4 x 10 plus 4 x 2). Great introduction to the distributive property of multiplication!
- Here is where application of the commutative property also comes in handy. Which of the methods below would you rather use to solve: count by 4’s or count by 12’s? Again, show students how to use their strengths to decide which way to think about solving the problem.
- Even though the number of total pieces might seem to be a little overwhelming, it definitely is worth the effort for a few lessons so students get a visual picture of the magnitude of the products.
  
  12 x 4 (12 groups of 4)
  
  4 x 12 (4 groups of 12): or 4 x 10 plus 4 x 2
Here are other ways to model multiplication problems with manipulatives like base ten rods or base ten disks.

Notice: 4 groups of 30 fand 4 groups of 2

4 x 32 = (4 x 30) + (4 x 2)

After students get practice using these manipulatives (concrete), then proceed to pictorial models to draw them as simply as possible. This will give them a good foundation to apply to the abstract (numbers only) problems. I always pitch for the CPA progression whenever possible!!!

I will pause a while for the summer and just post once a month until school starts up again. Take care, everyone! But please don’t be shy. Post your comments, ask your questions, etc.

by C. Elkins, OK Math and Reading Lady

In the last post, I shared my thoughts about multiplication strategies using the repeated addition strategy. This time I will focus on using arrays. Do you have some arrays in your classroom? Look for them with bookshelves, cubbies, windows, rows of desks, floor or ceiling tiles, bricks, pocket charts, etc. Students need to know arrays are everywhere! It is also very helpful for students to build arrays with objects as well as draw them. This assists students with moving from concrete to pictorial representations — then the abstract (numbers only) can be conceptualized and visualized more easily. Some good materials for arrays:

cubes
tiles
circular disks
flat stones
pinto beans (dry)
grid or graph paper
bingo stamper (to stamp arrays inside grids)
mini stickers
candy (Skittles, M&Ms, jellybeans)

Array Basics:

Arrays form rectangular shapes.
Arrays are arranged in horizontal rows and vertical columns. This vocabulary is very important!
The number of objects in each row (and column) in an array are equal.
Arrays can be formed by objects, pictures, or numbers.
Arrays can be described using numbers: If there are 4 rows and 3 columns, it is a 4 by 3 array.
The number of rows and number in each row are the factors. The product is the total.
When an array is rotated, this shows the commutative property.

4 x 6

6 x 4

Ways to incorporate arrays into story problems:

Desks in a class (5 rows, 4 desks in each row)
Chairs in a classroom or auditorium (10 rows of chairs, 8 chairs in each row)
Plants in a garden (6 rows of corn, 8 corn plants in each row)
Boxes in a warehouse (7 stacks, 5 boxes in each stack)
Pancakes (3 stacks, 5 pancakes in each stack)
Cars in a parking lot (4 rows, 5 cars in each row)
Bottles of water in a crate (3 rows, 8 bottles in each row)
Donuts or cupcakes in a box (how many rows? how many in each row)

Activities to encourage concrete and pictorial construction of arrays:

Start off using manilla grid paper you probably have available with the construction paper supply at your school. This will help students keep their rows and columns even. Pose a problem and allow students to use manipulatives you have available to construct the array. If you say, “Build an array for this multiplication problem: 3 x 5,” do they know the 3 refers to # of rows and the 5 refers to the number in each row? Starting off using this graph paper may help when students freehand their own array (to help keep rows and columns lined up and not going astray).
Turn the paper after building the above array to see the commutative property. Now the picture shows 5 x 3 (5 rows with 3 in each row). The product is still 15.
Use the manilla grid paper along with bingo dobbers to create the array. The grids can also be completed with mini stickers (I get them all the time in junk mail) or drawings.
When using pictures of arrays, direct your students to always label 2 sides of the array (the rows and columns). Try to label different sides of the array so it’s not always presented in the same format.
Find the product: The whole point of using an array as a multiplication strategy is to visualize the rows and columns to help calculate the product. If students create rows and columns and then just count the objects one-by-one, then this does not accomplish the objective. Show students how to skip count using the # of objects in the rows or columns. Believe me, students don’t always know to do this without a hint from the teacher. Or better yet, before actually telling them to do this, ask students this question: “How did you get the total number of objects?” When you pose this question, you are honoring their strategy while secretly performing an informal assessment. Then when the student who skip counted to find the total shares their strategy, you give them the credit: “That is an efficient and fast way to count the objects, thank you for sharing! I’d be interested to see if more of you would try that with the next problem.” Plus now students have 2 strategies.
Use the distributive property to find the product: Let’s suppose the array was 6 x 7. Maybe your students are trying to count by 6’s or 7’s to be more efficient – but the problem is that counting by 6’s or 7’s is difficult for most students. Break up (decompose) the array into smaller sections in which the student can use their multiplication skills. Decomposing into rows or columns of 2’s and 5’s would be a good place to start. This is the distributive property in action – and now the students have 3 strategies for using an array!! This is a great way to use known facts to help with those being learned.Here is a link to Math Coach’s Corner (image credited above) and a great array resource: Multiplication arrays activities from TPT $5.50. Here is my FREE guided teaching activity to help students decompose an array into 2 smaller rectangles. Click HERE for the free blank template.
Use the online geoboard I described in a previous post to create arrays using geobands. Click here for the link: Online geoboard Click here for the previous post: Geometry websites (blog post)
Try these freebies: Free array activities from k-5mathteachingresources.com. Here’s a sample.
Play this game I call “Block-It.” This is a competitive partner game in which students must create arrays on grid paper. Click here for a FREE copy of the directions: Block-It Game Directions
Relate use of arrays when learning strategies for division and area.