Friday, June 20, 2014

At Your Request - Multiplication/Division Flower Representations

I hope you're all enjoying the start of summer!  I just finished working with a great group of teachers who tried out a new idea -- flower representations for Goals 3, 4, and 5.

Goal 3 (x 2s)

Goal 4 (x 4s)

Goal 5 (x 8s)

Students can share copies of these flowers, and can work together to answer questions such as:  "What do you notice?" and "How can the patterns you're noticing help you remember your facts for this goal?".

After students have found many of the patterns, you may want to organize the numeric patterns in two rows so that students can see the distributive property in action.  For instance, with Goal 5, it could look like this:

Students will be able to notice the patterns, that 48 is 40 more than 8, 56 is 40 more than 16, and so on.  With your help, they can figure out that this pattern happens because of the distributive property, and because 5 x 8 makes 40.  Since 40 is a "friendly 10", the ones place repeats.  For instance:

8 x 1 =   8
+  8 x 5 = 40
8 x 6 = 48       or, more traditionally       8 x 6 = 8 x (1 + 5) = (8 x 1) + (8 x 5) = 8 + 40 = 48

8 x 2 = 16
+  8 x 5 = 40
8 x 7 = 56      or, more traditionally      8 x 7 = 8 x (2 + 5) = (8 x 2) + (8 x 5) = 16 + 40 = 56

The distributive property helps explain similar patterns for Goal 3, Goal 4, and Goal 7 as well.

2 x 4 =  8
+  2 x 5 = 10
2 x 9 = 18       or, more traditionally     2 x 9 = 2 x (4 + 5) = (2 x 4) + (2 x 5) = 8 + 10 = 18

This "friendly ten" pattern shows up in all of the even factor goals (e.g., x2, x4, x6, and x8).

And, since Goal 7 is part of the 3s, 6s, 9s sequence, I decided to change the image from flowers to apples.  It's also on the "flowers" file linked at the top of this post.

If you decide to use these with students, it would be great if you could share what you and your students learned.  I hope you all enjoy looking for patterns and making connections to algebraic thinking! 