California Mathematics Council South Friday, November, 0 rd Annual Mathematics Conference Palm Springs, California Ratio & Proportion: from Models to Fluency by Jeff Simpson Mastery Learning Systems mastery@pacific.net masterylearningsystems.com blog: http://masterylearningsystems.wordpress.com Topics to be addressed will include some or all of the following: teaching perspectives reviewing fraction equivalents (halves); establish a sense of what is true, then do the math multiplying/dilating; dividing/reducing; comparing top and bottom; cross multiplying using partner pages and timing to develop aniety-free fluency reviewing the forms of division kinds of more using pictures to teach rates (derive generalizations and definitions from repeated observations) comparing amounts (numbers with different units) with rates; why are we comparing? implied repetition word problems: begin with simplified phrasing in plain English (no math jargon) read a little, think a little, draw a little (or use pre-eisting pictures) comparing amounts (numbers with the same units) with ratios; why are we comparing? the forms of ratio notation (pay attention to word order) implied totals (/ ); pre-teach the thinking required for word problems proportions (equal ratios); why are we comparing ratios? draw cartoon character, then enlarge incorrectly ratio reciprocals; fraction:fraction ratios head torso legs 0 0 0 Snowman Snowman Snowman Snowman Snowman
Teaching Perspectives: Teach R/P as a way of perceiving reality, not as a belief system. Avoid arbitrary statements. Teaching R/P to kids: R/P is the subject, and Kids are the subject; know them both! Educate is from Latin educare, to draw out. We are not pouring R/P into their heads; we are drawing out their natural ability to notice, and to think and perform mathematically. I taught it, but they didn t learn it often means you didn t teach it in ways that they could learn it. Teaching is a game of Are they thinking what I m thinking? Meet students where they are, not where they are supposed to be. Struggling students must be taught differently. One size of instruction does not fit all. In differentiating instruction, use shifting small groups for a few minutes at a time. MIND THE GAPs. Many students have concept and skill gaps with the four basic operations, and with fractions (equivalent fractions; multiplying and dividing; reducing and dilating). How can we fill these gaps by teaching R/P? It is more productive in the long run to directly address these underlying concepts and skills, rather than ignoring them. Provide pictures and stories to make math real, credible, and memorable. Remember what it s like to not know something. Move from concrete to abstract. If the students can repeat what you (or the book) told them, does it really mean they understand it? If the students understand what you (or the book) told them, does it mean they will remember it? Slow down, and you ll all get there faster. ASK, DON T TELL. Don t tell the answer to a problem. Ask a series of questions (starting with the easiest) that will cause the students to tell you the answer with certainty in their voice! Begin by engaging their amount sense in something that is simple and true; develop the sense of relative magnitude, then link that to a a math process that can be trusted to the standard procedural logic. Derive generalizations and definitions from repeated observations of physical phenomena (inductive reasoning precedes deductive reasoning). Instead of saying, Any questions? or Do you understand?, ask individual students questions; their responses will reveal what they understand. Continually look for evidence of the students thinking. Don t wait for students to ask questions. Ask them the questions they should be asking you but are afraid to ask, or don t know how to ask. Don t wait for students to ask for help. Choose a problem of appropriate difficulty and say, Show me how to do this problem, or Let s work this one together, or May I watch you do this problem? Use guided discovery to clarify concepts, and rapid reconstruction to develop fluent, concept-connected, contet-derived memory. GUIDED DISCOVERY: tell them what to do show them how to do it watch them and make sure they do it right ask them questions RAPID RECONSTRUCTION: develop memory ( remembering ) the same way that you learn your way around a new city not by memorizing, the way first-graders learn the Pledge to the Flag. Words that make students think: Read the problem to me. What do you think it means? What do they want to know? What does this number stand for? What does this word mean? Draw a picture of it. How else could we do it? Show me how you got that. Try again. Why? Is this true? How do you know? How did you know that so fast? Could it be (give a false answer, to make them think about what must be true).) Try it, and see if it works.
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Other Names for 00 J. Simpson One Half name: date: corrected by: Write all the equivalents of one half that are pictured on the Halves Chart (-): 0 Look at all the numerators that you just wrote. What do you notice? Look at all the denominators that you just wrote. What do you notice? Study the fractions that you wrote above, then cover them up. Without looking at the Halves Chart, write all the equivalents of one half that are pictured there, in order: 0 Do the math for boes,, and below. Then continue the pattern of multiplication for the other fractions. 0 a F -
A Half of a Half 00 J. Simpson A Third of a Half I got a whole pizza. Start here. which was of the original pizza. I cut it in half. I gave of my to my friend I kept half of it. I cut my half in half. I got a whole pizza. Start here. which was of the original pizza. I cut it in half. I gave of my to my friend I kept half of it. I cut my half in thirds. F -0 of
Multiplying Fractions 00 J. Simpson name: date due: corrected by: of is the same as of is the same as multiply the numerator times the numerator multiply the denominator times the denominator Multiply the fractions as shown above, and write the products (the answers). a F -
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Fraction Timer Critical This page uses the multiplication facts that are hardest to remember plus four others. Use Carmen's Denominator Chart (- ff.) to picture the problems. Practice in different orders until you can say the answers in minute. name: date: corrected by: Progress Chart: :00 :00 :0 :00 : :0 : :00 00 J. Simpson F - of
A B Comparing 00 J. Simpson name: date due: Let s compare A to B. Which has more? B, of course. How much more? Most people would say, One more, using addition ( + ) as a way of comparing. But some would say, B has two times as much using multiplication ( ) to make the comparison. Both ways of comparing are useful. Look at this pattern: Check the way of comparing that best describes this pattern: + + + + Look at this pattern: Check the way of comparing that best describes this pattern: + + + + Look at this pattern: Check the way of comparing that best describes this pattern: In this chapter on ratio, proportion, and scale, we will use multiplication and division as our main way of comparing. Use multiplication ( times ) to compare the figures below. The first eample is done for you. A B C D E F 0 G H I J K L PRR -
Comparing with Multiplication 00 J. Simpson name: date due: Use multiplication ( times ) to compare the figures below. The first eample is done for you. A B C D E F G H I J Use multiplication to compare the top amount to the bottom amount. Then write what you see as a fraction. The first eample is done for you. A C E G I B D F H J Is the bottom number always two times as much as the top number? Is the top number always a half of the bottom number? A Do A and I contain the same amount of gray? I B Do B and J contain the same amount of gray? We can see that one half and five tenths are equal fractions. 0 J Here are five ways to use multiplication or division to compare fractions, and show that they are equal. 0 0 0 0 0 0 0 PRR - 0 is half of 0
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Thinking About Stuff # 00 J. Simpson Use The Coin Chart (-), The Dollars Per Hour Chart (-), CookieMultiSorter (-), Make-a-Team Chart (-), Dozens of One-Dollar Bills (-), or The Number Line (-0) to picture the problems.. Two dogs went into the kitchen, where someone put four dimes and si pennies on the table. How much money is that? Also, how many coins are on the table?. Amadi has two sisters and one brother. He also has five dimes and a quarter. How many cents is that? How many coins is that?. Becca has 0. But what she wants to buy costs. She is going to meet five friends at the mall at :00. How many pennies and how many dimes does she need? How many coins is that?. The boss paid Carlo ten dollars every time he worked an hour. He worked four hours. And he watched television for two hours every day. How much money did he earn?. Donna was paid eleven dollars for every hour she worked. She was fourteen years old. She worked five hours, and was given a three-dollar bonus! How much was she paid?. Evan agreed to work for ten dollars per hour. There were si other workers. On payday, he got 0. How many hours did he work?. Friede made cookies for herself and some friends. She worked in the kitchen for hours. How many friends did she invite so that everyone had cookies?. Greg went to a soccer tournament with other players on buses. The referee reminded them that there are eleven players on a team. How many teams were there?. Hannah s allowance is 0 a week and she saved her allowance for weeks. She bought a present for her grandmother, who is, for. How much money does Hannah have left? 0. Izzy has a dog. Her mother gave her cookies to share equally between her and two of her friends. How many cookies will be left over for her dog?. Jose went to summer camp to learn to play soccer. He sat on the bus each day for one and a half hours with other guys. How many teams were there and how many had to stand on the sidelines and watch?. Katherine has. She needs to go on the ski trip to Lake Tahoe. She has a brother and a sister who are not going on the trip this year. How much does she need to earn so she can go on the trip? a MD -
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