Designs with Nines: Patterns in Multiplication and Division (Day 2)

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Objective

Students will increase familiarity with their "9 facts" by creating designs in which 9 is a factor or the dividend. They will count out the individual items or groups that make the product or dividend. Their entire design will represent facts from 1 x 9 through 9 x 9.

Big Idea

Repetition helps students build fluency with multiplication and division. The activities that contain the opportunities to practice their facts can be meaningful, creative, and open-ended.

Opener

12 minutes

I remind students of the designs they created on Day One, with pattern blocks, in Google Drawing, or on paper.

We review the objective: To create designs in which patterns related to 9 are visible and can be pointed out by the artist.

I activate their thinking by writing the basic facts on the board, and prompt my students to give the answers as these facts should be coming more readily to mind at this point (even though they haven't been filling out endless fact page worksheets!).

The facts we will work on are 1 x 9 = 9 through 9 x 9 = 81.  I leave out 9 x 10 because it is an easy fact for children, and 90 is a lot of objects to gather!

Then I write up the turnaround division facts, in which 9 is the dividend.  9 divided by 1 = 9 through 81 divided by 9 = 9.

Students have a brief peer talk, which I support using these questions. I also monitor their brief exchanges with one another.

What patterns do you see in the multiplication examples?

What patterns do you see in the division examples?

What patterns do the multiplication and division examples have in common?

How are the multiplication and division examples different?

Is there a particular set of facts that you especially like?  Why?

Next, I use more rigorous questions to challenge students to use what they know to analyze the 9 facts.

**  Can all 9 facts be broken down into threes?  Why or why not?

*** Can all 9 facts be broken down into sixes?  Why or why not?

**** Is there a way to make the multiplication pattern of  3 x 9 = (3 x 10) - 3 work for division?

Activity

45 minutes

In our prior lesson, students were prompted  to collect groups of objects from home to do today's activity. This direction to be given at least two days in advance, unless you plan to supply objects to students who don't bring in the groups the first day after the assignment.

I've deliberately chosen objects easily found by students so that everyone should be able to create these groups, because there is no cost.  I accepted grass, leaves, pebbles, tiny twigs, leaves, flower petals, Q-tips, cotton balls, business cards, envelopes, hair pins, and so on.

Assess the size of the objects students brought and make sure that their chosen work space is large enough for them to display their objects without much stacking, which makes it very difficult for anyone to see and internalize the pattern because so much of it is obscured.

As students are building their designs, ask questions such as:

Where is  __________ (example:  4 x 9)?

What is ________ (4 x 9)?

What division problem could be represented by _________________?

Is there a way you could move objects to represent _______ as a division problem?

This activity can easily go over the allotted 45 minutes, and as long as students are talking about numbers, products, dividends, factors, divisors, decomposing and other math patterns they see, I think it is time well spent.  Children who are earlier finishers, and I only had a few, may join in and help another student with their design.

One way I've found to support students is to suggest to students they make up stories to help guide their thinking when they create their math designs from their collection, using the listed facts (on the board).

If possible, leave students' math art out for for the writing in math activity. If space is at a premium, as it is in my room, photographing their designs prior to disassembly works well, and perhaps even better because when they move to the writing activity they don't get caught up in recreating parts of their design.

Wrap-up

5 minutes

It's important to tell students the purpose of their work, so I make sure to circle back to the purpose at the end of the lesson. In this case, practice with basic facts leads to fluency, comfort with the relationship between multiplication and division, and adds to ideas and strategies they already have in place to decompose parts of multiplication and perhaps division problems.

First, I have students share their designs with a peer who wasn't working near them.  I suggest that they can ask each other where certain facts can be found within the larger design.

To complete our wrap up, I bring the together and ask them to share what the purpose of this activity was.