Let's Break It Down

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SWBAT use the distributive property to multiply mentally.

Big Idea

Students will understand the equivalence of a(b+c) and ab+ac using an area model and by understanding multiplication as "adding sets of numbers".

Warm Up

10 minutes

This warm up references the last two lessons (Farmer John and Farmer Fred Days 1 & 2 of 2) because it will help students relate the work we will be doing today with distributive property to the area models. In the warm up two students, Alissa and Juan, present two possible ways for finding the area of combined pumpkin fields. Alissa suggests that you must multiply to find each smaller area then add them together (5x10+5x5) while Juan says you must multiply the length and width of the larger combined field 5(10+5). I remind them to decide together in their math family as this may generate some dissagreement (MP3). Some students may decide one way works and not bother to check the other way. Almost always, however, someone in the group will decide the other works or that both works and they go through the process of convincing each other. If they are raising different claims and presenting evidence to support that claim, or providing counterclaims, this is a good indication that the practice of argumentation is taking place. Once they decide that both methods are correct, I ask them to spend 2 silent minutes making sense of this and writing down why they think both methods are equal. Then I have them share in their math family groups. This may be very hard for them to articulate and they may rely on area models for now. I tell them that our number talk will help them better explain or make sense of it.

Number Talk

35 minutes

I have done several number talks earlier in this unit (Let's talk addition!, What were they thinking?, and Delightful decimals). I am using them in this unit to teach the number properties in such a way that all the strategies come from the students and in such a way that they internalize their importance and relevence. In addition, number talks are really useful at highlighting multiple methods and making students gain more mathematical flexibility, better number sense, and confidence. Students have been taught silent signals to show me that they are either working on a strategy or already have a solution.

To introduce this number talk I refer back to the number talks mentioned earlier in which they came up with the properties of addition themselves (and I refer back to the poster Properties of Addition we put together describing them). I tell them that today I expect them to come up with some strategies for doing multiplication mentally and that I expect we will see some important properties of multiplication as well.

I start with 5x30. When students explain their strategy they usually say they ignored the zero for a minute and they mutliplied 5x3 then added the zero back on. I remind them that they're not really "adding" the zero, just tacking it on to the end and I model how they broke up the 30 into 3 x 10 and instead of doing 5x(3x10) they did (5x3)x10. We do a few more like 6x70 or 3x400, etc. modeling each on the board. I ask them which addition property this reminds them and refer to our poster, making sure they identify the similarities. 

I write 9x12 on the board and ask them how we could split this problem into more easily manageable problems to get 12 sets of 9 more easily or if we can't remember it. If they have trouble getting started I would say "I don't know what 12 sets of 9 is, but I do know what 10 sets of 9 is...and I prompt students for the answer, then I draw the arrow from the 9 to the 2 and ask how many more sets we need in order to make 12.  When they say 2 I write 9(10+2). 9x2 is 18 then 90+18 = 108. Other students might suggest 9x6 and 9x6, etc.

Next we try 7x12. They use their silent signals to let me know whether they are still working (index finger) or have a solution (thumbs up). These signals are kept down low so they are not distracting or intimidating. When I see many thumbs up I ask for solutions. I write any solutions up (right or wrong) before asking for how they did it. Then I ask a student how they broke up the 12 sets. They all probably did 7x10 and 7x2. Now I show them that they could break it up in other ways too. If I found 6 sets of 7 (7x6) how many more sets would I need?

I have them work on a problem like 4x26. After we have a solution I ask them how they broke up the 26 and what "friendlier chunks" of multiplication they did. Most probably did 4(20+6), but someone may have done 4(10+10+6). If they don't come up with any other ways I would ask how else we could break it into "friendly chunks". They may come up with 4(13x2), which is the associative property and I write their name next to it off to the side of the board and tell them I thought someone might come up with that and that we will talk about this property later. Other students may point out that it is not particularly friendly. 

White Boards & Homework

9 minutes

White boards are a way for me to check for understanding and provide individual feedback to make sure kids are understanding and can do their homework. I tell them I am not going to ask them for the final answer so I don't want them actually doing any multiplication. I am going to break up the difficult multiplication for them and they are going to show me the "friendlier chunks" of multiplication they would do.

I start by writing a problem like 7x13 and ask how I could break up the 13 sets...I take all their suggestions and I write

7(10+3) and ask them to write on their white board the "chunks" of multiplication. I should see 7(10)+7(3). If students forget the addition sign I will ask them to remember that they will be added together and to show that (I may refer them to another students' board). Some students may put 70+21, in which case I will show that class what it would look like after the multiplication, but I really just need to see the chunks. If they put 91, I tell them they are showing the solution, not the strategy.

I would continue with problems like 4(30+ 6), 6(5+4), (20+3)5, 3(5+5+5), 9(2+x). Each time I circulate around a different section of class providing scaffolding as needed.

Then I ask them to show how they could use this strategy/property to help them "chunk" 4(231). I circulate and may need to suggest that they try "front end" multiplication. If several students seem to be struggling I may just hold up one person's board who has it and ask if this would work. Then I would give them another example to work on.

With the remainder of time I allow them to work together on their homework.