Designs with Nines: Patterns in Multiplication and Division (Day 1)
Lesson 5 of 6
Objective: Students will be able to create a design by using products of equations in which 9 is a factor, decompose designs into a new pattern by dividing all products by 3 and express their observations about the mathematical patterns they discover.
I ask the students to count by ones, and they laugh and do so. Then I ask them to count by twos and they smile and are eagerly successful. I do the same for fives and tens. I remind them that if they are met with a fact that they don't yet know by heart (that will come with time) then they can always break it down into familiar parts. There is almost never one right way to solve a problem in math, instead there are many alternative approaches.
Using pattern blocks (or whatever materials you have available) students work either independently or with a partner to create a design or series of designs to represent multiplication facts with a factor of 9.
I write the nine facts on the board (1 x 9, not 9 x 1) and next to them I put the examples of how to use the tens strategy.
1 x 9 = 9 (1 x 10) -1 = 9
2 x 9 = 18 (2 x 10) -2 = 18 and so on.
Students don't need to have the facts memorized to do this activity, in fact, one of the purposes of this activity is to gain familiarity with the facts through hands-on practice and referring to written examples of the facts. Another option would be to have a multiplication chart out with the nines highlighted.
The students can get so caught up in making their design that some of them forget what the overarching purpose is. Gently remind these creative children that when asked, they need to be able to explain which shape represents which math fact, and what the product of that fact is.
Students divide each group of shapes into 3. For this, they will record their answers on paper.
81 divided by 3 = 27
72 divided by 3 = 24
63 divided by 3 = 21
When they have completed the divided by 3 design, simply ask them to make observations and be prepared to share at least one with you, a peer, or the class. I also suggest that they write down one observation, in a complete sentence. If they make an inaccurate observation (example: I notice that the amount in each group goes down by 4 each time) then redirect through questioning.
In an organized fashion (perhaps moving by table group clockwise, for example) have the students move around the room and observe other people's designs.
Questions: How did they represent the 9 facts in a different way than you did?
Is there a fact that is especially easy for you to find? Why? (Prompt them to go beyond "1 x 9" because it's small).
For homework, I ask the students to collect objects that represent the products of the 9 facts. I have my students write this down and take it home to their parents, and there was some confusion in a few households because this wasn't a typical homework assignment.
For this reason, I made a parent note in English and Spanish explaining the assignment so that the parents can help out and have fun!
So yes, they are to bring one bag with 9 objects, one bag with 18 different objects, and so on up until 81. Some children worry that they have to buy things and would be unable to do so. Absolutely unnecessary! That is one of the reasons I made my model with grass and leaves from shrubs in my apartment complex parking lot. This needs to be an activity that's accessible to everyone.
That said, I am sure that in some home environments, students may go out and purchase some fun little items from crafts stores to complete this activity. I'm sure those will be interesting to look at!
Some items used by my students included:
Cereals (messy) such as Mini Wheats, Cheerio, Fruit Loops
M&Ms, other assorted candies of that type
Wide assortment of natural items (twigs, leaves, pebbles)
They use these found items to create a design, again with nines, about which they will write a very short math story. It will be a creative story that doesn't need to have any basis in reality, other than the accuracy of the math facts. I give my students 2 days to gather the items and most are successful.