As third graders develop their understanding of multiplication and division, they find that there are a number of specific contexts surrounding “equal groups”. One specific context is through the arrangement of rows and columns. Students need an understanding of the language of arrays as well as time to explore rows and columns with hands-on materials.
The 5 examples listed are certainly not an exhaustive list of tools and lessons that could be used to teach this skill.
They are rather a sample progression that includes all types of array word problems.
Which step represents your students’ current level of understanding?
Introduce the concept of arrays through pictures or photographs that show arrays. Show these pictures to your students and discuss what they see. Use the words “rows” and “columns” as you describe the arrays and ask your students to use these new words as well!
Laminating your pictures makes it easy to circle rows or columns so that you can also related arrays to the idea of “equal groups” which is likely a well known multiplication and division representation to your students.
Given a number of rows and the number of objects in a row, could your students find the total?
Start this exploration by providing your students with array grids of varying sizes. Ask your students how many rows they see in a given array. How many squares are in that row? Relate the representation to a multiplication equation for your students.
Lay a counter, such as a mini eraser or centimeter cube, in each square of the grid and count to find the total. If your students have been introduced to the concept of skip counting you might also skip count by rows or columns to find the product.
Begin the lesson by providing each student with a pile of 12 counting bears. Ask your students if they can explore and arrange the bears into equal rows. After your students have had some time to explore ask them to share the different ways in which they were able to arrange the bears.
After your students have had an opportunity to explore on their own, provide your students with varying amounts of counting bears and ask them to divide them into a specific number of rows. Relate your students findings to a division equation.
As your students arrange their counting bears, you can also ask your students to draw an array drawing that matches the scenario. This will help when you are ready to strip concrete materials away at a later date!
Provide your students with a context to explore! Tell your students that you will pretend that they are working at an art gallery. Provide your students with a pile of 10 square tiles and tell them that they are going to pretend that each tile is a painting. Tell your students that you would like them to put the tiles into rows of 2 paintings and you’re wondering how many rows that would create.
As your students explore with hands-on materials, record their findings both as an array drawing and division equation.
Provide your students with a variety of hands on materials. Prompt students with an array word problem with a missing product, missing number of rows or missing row size.
Allow your students to explore with hands on materials and record their thinking as an array drawing and equation.
If your students struggle at this point when array problems are mixed, first look at the type of problems they are struggling with. Were they successful with these problems in isolation? If not, they may need more time for hands-on exploration. If your students were successful in isolation, they may need to spend more time discussing word problems and noticing what piece of information is missing.