Multiplication Concepts Part 2: Arrays

by C. Elkins, OK Math and Reading Lady

Last week I posted 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:

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

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?
  • 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 a few posts back 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.

In a future post I will show some ways to use manipulatives and pictures arrays for double digit multiplication problems. Stay tuned!!

Division Basics Part 1: Arrays and Area Model

by OK Math and Reading Lady

Division seems to be the hot topic with classes I have been visiting lately, so I thought I’d focus on that for now. Let’s look at some of the basics.  Students as young as first grade actually start thinking about division when working on fraction standards such as:  Determine fair share — equal parts. Most students have had practical experience with dividing sets of objects in their real life to share with friends, classmates, or family (cookies, pizza, crayons, money, pieces of paper). So now our job as teachers is to relate this real-life experience with the division algorithm.

Much like multiplication, there are different aspects of division children should get familiar with:

  • Arrays 
  • Equal Groups
  • Repeated Subtraction
  • Number lines
  • Skip counting

In this post, I will break down the benefits and uses for arrays (and the related area model) to help children understand division (and how it is related to multiplication). There’s a fun FREE game (Block-It) at the end of the post.

What is an array?  An array is a rectangular model made up of rows and columns.  When an array is constructed, the factors are represented by the number of rows and columns.  So, do your students know the difference in a row and column?  (Rows go horizontally, while columns are vertical.)  These are important math terms students should be using.

  • Give students experience constructing arrays with manipulative objects (tiles, chips, cubes, etc.):
    • You can be specific, such as: “Build an array using a total of 12 tiles. Put them in 3 rows.  How many columns did you create?” In this scenario, there is only 1 way to show this array. Students would be modeling 12 ÷ 3 = 4. Twelve is the dividend (the total amount you started with). The # of rows is the divisor (how it was divided).  The quotient is the result (in this case the # of columns).
    • You can also be a little more open ended such as:  “Build an array using 12 tiles. Is there more than one way to do this?” If students are given the opportunity to explore, they hopefully find arrays such as 3 x 4; 4 x 3; 2 x 6; 6 x 2; 1 x 12; or 12 x 1. Students would be modeling 12 ÷ 4; 12 ÷ 2; 12 ÷1, etc.
  • Give students experience drawing arrays:
    • You can be specific or open-ended (as above).
    • Children can free-hand draw or use grid paper.  If using grid paper, then these can be cut out and displayed as “Different ways to divide 12.”
  • Give students experience using pre-drawn arrays:
    • Students should label the sides of the array with numbers.
    • Use the numbers shown to determine the fact family.  Example:  3 x 4 = 12; 4 x 3 = 12; 12 ÷ 3 = 4; and 12 ÷ 4 = 3
  • After the array is made, ask questions or explore more such as:
    • How many 3’s are in 12? (count the columns)
    • How many 4’s are in 12? (count the rows)
    • Circle the rows and / or columns to see the groups more easily.
    • Help children make up story problems to match the array:  “I have 12 desks that I need to arrange in 3 rows. How many desks will be in each row?” or “I need to put 12 books equally onto 3 shelves. How many books will go on each shelf?

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Multiplication Strategies Part 2: Decomposing and distributive property to learn facts

by C. Elkins, OK Math and Reading Lady

In part 2, I will show you some ways to help students decompose a multiplication problem into 2 (or more) easier multiplication problems. Most students know problems with factors of 2, 5, and 10. The decomposing will allow students to use what they know to work on the unknown / unmemorized fact.

I frequently see students struggle with solving an unknown multiplication problem. Often they choose skip counting, but if they miss just one number in the sequence, the answer comes out wrong. I also see them use their multiplication chart, but this doesn’t do much to help them apply number sense. Other times I see students draw circles with dots inside, but this is time consuming and it often becomes just a counting practice. This method is using the distributive property. Students can break apart one of the factors into “friendly” addends. I usually advise making one of the addends a 2, 5, or 10 since those are usually easier to compute or are already memorized. Here are some examples:

 

 

I have also attached a class activity sheet in which students cut out grids, glue them on the worksheet and then decompose them. Get it free here: Distributive property teaching chart  Another resource for teachers is my multiplication strategies guide which shows some ways to break down each factor’s family. Get it here free: Multiplication fact strategies chart  Finally, here is a link to a TPT source with a freebie for using the distributive property with arrays: Distrib. Property of Multip. freebie by Tonya’s Treats for Teachers

Have a great Spring Break for Oklahoma teachers!! I will be back in 2 weeks with more multiplication strategies.