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## But Why?

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**But Why?**Barney Ricca AMTRA – 2012**If we weren’t math teachers…**Ours is not to reason why; Just invert and multiply!**But we are math teachers**• Why does this work? • And, how can we help our students learn about this? • Because “invert & multiply” really only prevents students from succeeding in high school & college math. • We’ll look at a few things that might help us…**Let’s Divide!**• Some problems: 5 ÷ 1 10 ÷ 2 20 ÷ 4 40 ÷ 8 • How can you use one of these to solve another one of them? Why? • Which number is “the whole” in each of these? • How can you interpret each problem?**So what?**• Some more problems: 40 ÷ 8 20 ÷ 4 10 ÷ 2 5 ÷ 1 2½ ÷ ½ • Can you use something (other than the rule) to solve the last one of these? How? Why?**More problems**• How about this one? 81 ÷ 9 27 ÷ 3 9 ÷ 1 3 ÷ 1/3 • Hmm… • Or maybe this one: 81 ÷ 18 27 ÷ 6 9 ÷ 2 3 ÷ 2/3 • Hmm…**In the Classroom**• Your students will need more time than just this • And more discussion • But for us, what does this tell us we are “really” doing when we invert & multiply? • One re-interpretation of division: 40 ÷ 8 can mean “How many 8s are needed to make 40?”**But, wait! There’s more**• Divide 30 gold pieces among 5 pirates • One for you, one for you, one for you, one for you, one for me…. • a.k.a. “partitive” • How many pirates can we pay from 30 gold pieces, if each is owed 5? • 5, 10, 15, 20, 25, 30 • a.k.a. “quotative” • Both of these look like 30 ÷ 5, though**Let’s explore with colored tiles**• The problem: 24 ÷ 4. • Look at one way • Hmm…12 ÷ 2 can be gotten from this…just cover up two columns, but we’ve done exactly the same distributing! • What about 6 ÷ 1? • What about 3 ÷ ½? (Hint: You have scissors!)**There’s still more: Let’s Multiply!**• 3/5 x 4: • Let’s watch the not-YouTube not-video**So what?**• “I knew that” • “And I didn’t need to go through all that work” • Ah…but we (and our students!) do need to go through all that work in order to learn it! • In the same way that the problem strings at the beginning were helpful in showing us what to do, the thing we already know might help us here. • What is “the whole” in 3/5 x 4? Why?**Let’s Divide!**• 3/5 ÷ 4: • Another not-YouTube not-video**Yeah, but what about fractions?**• Well, you asked for it: 2/3 ÷ 3/4? (I.e., how many groups of ¾ can we get from 2/3?) • Still another non-YouTube non-video**Hey, kids:**• Don’t try this at home! • Mere presentation of this won’t do significantly better than what we’re already doing. • This is just one way to re-present the problem so that students might get to play around with it**The “big ideas”**• Partitive vs. Quotative • Fair sharing vs. skip counting • Equivalence • What is “the whole”? • Playing around • Not to be underestimated! • Note that we sort of did division by adding! Kamii found that this is how kids naturally like to divide.