Lesson 17 of 23
Objective: SWBAT distribute multiplication over subtraction and use it as a strategy to solve math mentally.
Students begin the Warm up Number Talk halfsies when they enter. It consists of two tables with multiplication problems they need to complete and then look for a pattern in the results. I tell them that the two patterns they are looking for will help do both simpler multiplication problems when they forget their facts as well as much more complex multiplication problems with ease.
The first table contains numbers being multiplied by 10 and then 5. The pattern I want them to notice is that the product of 5 is half of the product of 10. This is going to prepare us to use the distributive property to multiply by 15 and 16, etc. The second table contains a doubling pattern, numbers first multiplied by 2, then 4.
Once we have gone over the patterns I show them how they can use the patterns to help remember facts they forgot or do more complex problems. If you forget your 5s you can just cut your 10s in half and now we can find our 15s just by adding. We try the pattern with some other numbers to complete the table. In the other table if we can double our 2s to get our 4s what will get if we double our 4s? Can we double again? What multiplication did we just do? (our 16s...wow!). We try doubling with some other numbers to complete the Warm up Number Talk halfsies completed.
This is really important for my students, because they have such trouble remembering their multiplication facts and it causes them anxiety and frustration as well as resulting in incorrect work. This helps them see the relationships of the numbers, the operations, and plugs them back into their intuition.
In Number Talks I give students a problem to do mentally. They use silent signals to indicate to me whether they have a solution or are still working on a strategy, or have found more than one strategy. I always ask for and model multiple strategies and write students names on the board next to their strategy to give them a sense of ownership and also to give others a way to refer to their work. I have used the number talks as a way for students to intuitively "discover" the number properties in earlier lessons (Let's Talk Addition!, Delightful Decimals, Let's Break it Down) and this one will help them extend what they have already learned and are able to do with distributive property to addition of more than two terms as well as to subtraction.
I start with 14 x 15. After the warm up activity I expect them to do 14 x 10, know that 14 times 5 is half of it, and add them together, which I model on the board using an area model for the distributive property. I also model all other strategies they describe.
Next I do 14 x 16 and expect some of them to use the answer to the last one to help. The area model for this strategy shows distributive property in 3 parts: 14(10+5+1).
I may continue with 14 x 17, 11 x 18.
Next they do 8 x 20 and I model the ways they do this. They may do 8(10+10) for which I model the distributive property, but they will likely do (8 x 2) x 10 for which I model the associative property. I really am only having them do this problem so they can use it on the next one.
Now I ask them to do 8 x 19. I expect many of them to say they used the answer to 8 x 20 and subtracted 8 from it. I draw a rectangle to model the 8 x 20 and remind them of a previous number talk (Let's Talk Addition!) when we "added over" and then "backed it up". For example 19 + 45 = 20 + 45 -1. I tell them they are "multiplying over" and "backing it up" and I shade the area that they are removing from the large rectangle. Some students may subtract one from the 8 x 20, but I remind them we are working with "sets" of 8. I can also use the area model and ask what the area is of the section being removed.
I continue a couple of similar problems (14 x 19, 12 x 18, 24 x 19) until I see more people with "thumbs up" indicating a solution.
With no introduction I ask them to simplify 5(2x - 3).
We have had enough experience with distributive property at this point that I expect they will automatically distribute the multiplication. Some students may put up 10x + 15 in which case I would ask if they noticed the subtraction. When I go over this first one with them I will show them the area model so they can see the relationship to the number talk we just did.
After doing a couple more [ 2(3x - 1) and 3(x - 4) ] I ask them how this is similar to what we know about and have already done with distributive property. They may say that the multiplication is still distributed to all terms inside the parentheses. Many will refer to a story I told when first introducing distributive property (Halloween Candy to Zombies) about distributing Halloween candy to zombies. The story has helped them remember that all the "zombies" that come into my doorway (parentheses) get candy. Someone usually points out that the second group of zombies are getting their candy taken away.
If they were successful with the ones we have done so far I will continue with more complex problems. If some are having trouble I will put up two problems [ 4(2x -1) and 4(2x + 3y - 1) ]and let them choose their level of challenge.
More complex problems may include something along the lines of:
6(4e + 2c - 3) 5(4m - 3w - 2) 3(4x - 3) + 2x 5(2x - 5) - 2x
I am careful not to do problems like 3(3 - 4x) because when they try to put it into standard order they will not know how to represent the negative term.
Some students may still be having trouble simplifying multiple step expressions like the last two, so I include them here and may do more like them as a way to help them review how order of operations plays into simplifying expressions. If they seem to be getting it, I let them spend a couple of final minutes on homework.