##
* *Reflection: Developing a Conceptual Understanding
Let's Break It Down - Section 2: Number Talk

Because so many of my students learned (or failed to learn) their math facts by rote memorization they lack the flexibiltiy and the ability to problem solve and persevere by using what they know to figure out what they don't. They do not understand the relationships between numbers or the meaning of operations. I front load this number talk be first reviewing the definition of multiplication. I tell them that before we start out it may be really useful to remember something they learned when they were first learning to multiply back in elementary school. On the board I write 5x15 (which comes straight from the problem in their warm up) and remind them that in elementary school they may have learned that this multiplication means 5, 15 times or that it means "15 sets of 5" and that it will be really useful to keep that in mind when we start doing big multiplication problems in our heads. This can be really intimidating to students who are below grade level or who have a history of struggling in math, because they know they don't have a solid foundation in their math facts. I admit that I don't know what 15 sets of 5 is, but I do know what 10 sets of 5 is and I write 5x10 next to 5x15. At this point I hope that students think "hey, I know what that is too, maybe I'm okay". I ask what it is and they say 50, which I write below 5x10. Then I say "But that's not 15 sets of 5, that's only 10 sets, hmm" (some kids will start to get the idea here) "how many more sets of 5 do we need?" When they say 5, I say "Oh, good, I know what 5 sets of 5 is too" and I write 5x5 next to 5x10. Again I want kids to think "I know that one too, I can do this". Then I ask "if 10 sets of 5 is 50 and 5 sets of 5 is 25, what is 15 x 5?" They see that it is 75. I tell them we found this by breaking up the 5(15) into manageable parts 5(10+5) and doing the multiplication in "friendlier chunks". I tell them I call this strategy "Front end multiplication". I write the traditional method for multiplying by stacking the numbers and starting to multiply from the back of the number, carrying, etc. Then I show how instead of starting with 5x5, then 5x1 (or actually 10), we just did it in reverse by multiplying 5x10 then 5x5. I tell them that Farmer's John and Fred really help us to show why this math works and I demonstrate with the area models that they have become familiar with.

Later, when we did 7x12, students also suggested splitting it into other easier-to-do-in-your-head problems like 7x6 and then doubling.

*Using definitions*

*Developing a Conceptual Understanding: Using definitions*

# Let's Break It Down

Lesson 3 of 23

## Objective: 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".

*54 minutes*

#### Warm Up

*10 min*

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.

#### Resources

*expand content*

#### Number Talk

*35 min*

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.

*expand content*

#### White Boards & Homework

*9 min*

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.

*expand content*

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- UNIT 1: Order of operations & Number properties
- UNIT 2: Writing expressions
- UNIT 3: Equivalent Expressions
- UNIT 4: Operations with Integers
- UNIT 5: Writing and comparing ratios
- UNIT 6: Proportionality on a graph
- UNIT 7: Percent proportions
- UNIT 8: Exploring Rational Numbers
- UNIT 9: Exploring Surface Area
- UNIT 10: Exploring Area & Perimeter

- LESSON 1: Farmer John and Farmer Fred Day 1 of 2
- LESSON 2: Farmer John and Farmer Fred Day 2 of 2
- LESSON 3: Let's Break It Down
- LESSON 4: Halloween Candy to Zombies
- LESSON 5: Extending Farmer Frank's Field with the Distributive Property
- LESSON 6: Who's Right?
- LESSON 7: To Change or Not to Change
- LESSON 8: Let's Simplify Matters
- LESSON 9: Clarifying Our Terms
- LESSON 10: Breaking Down Barriers
- LESSON 11: Number System Assessment
- LESSON 12: Garden Design
- LESSON 13: Ducks in a Row!
- LESSON 14: The Power of Factors
- LESSON 15: Forgetful Farmer Frank
- LESSON 16: Common Factor the Great!
- LESSON 17: Naughty Zombies
- LESSON 18: Reducing Fields
- LESSON 19: Common Factor the Great Defeats the Candy Zombies!
- LESSON 20: The Story of 1 (Part 1)
- LESSON 21: The Story of 1 (Part 2)
- LESSON 22: Simple Powers
- LESSON 23: Equivalent expression assessment