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
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Let's Break It Down

Unit 3: Equivalent Expressions
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".

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Math, Expressions (Algebra), distributive property, Mental Math
  54 minutes
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