Algebraic Thinking PowerPoint
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Mathematics is not a way of hanging numbers on things so that quantitative answers to ordinary questions can be obtained. It is a language that allows one to think about extraordinary questions ... getting the picture does not mean writing out the formula or crunching numbers, it means grasping the mathematical metaphor. (James Bullock, 1994.) GOALS FOR TODAY Explore how representations help students reason about mathematics. Who, why, when to pull a student from the general ed. (core) math track. 9:00 ACTIVITIES MULTIPLE REPRESENTATIONS BREAK ACTIVITIES WHAT WERE YOU THINKING? CONVERSATION ABOUT PULL-OUT 12:00 INTRODUCTIONS LUNCH MORE ACTIVITIES BREAK LESSON DISCUSSION ACTIVITY 3:15 WRAP– UP TRAFFIC JAM . . . . . . . There are seven stepping stones and six people. On the three left-hand stones, facing the center, stand three of the people. The other three people stand on the three right-hand stones, also facing the center. The center stone is not occupied. The challenge: exchanging places THE CHALLENGE *Everyone must move so that the people originally standing on the right-hand stepping stones are on the left-hand stones. *Those originally standing on the left-hand stepping stones are on the right-hand stones. *The center stone again unoccupied. THE RULES: 1. After each move, each person must be standing on a stepping stone. 2. If you start on the left, you may only move to the right. If you start on the right, you may only move to the left. 3. You may "jump" another person if there is an empty stone on the other side. You may not "jump" more than one person. 4. Only one person can move at a time. PROCESS IT... LOOK FOR A PATTERN What if there are only 2 people and 3 spaces? How many moves does it take for the two people to exchange positions? What if there are 4 people and 5 spaces? How many moves does it take for 4 people to exchange positions? What about 6? What about 8? What about 10? ..... Can you find a pattern for any number of people? WRITE IT ALGEBRAICALLY number of pairs number of people 1 2 3 4 5 6 2 4 6 8 10 12 minimum number of moves 3 8 15 ... ... ... ONE SOLUTION: number of pairs number of people 1 2 3 4 5 6 ... n 2 4 6 8 10 12 ... 2n min. number of moves 3 8 15 24 35 48 ... n^2 + 2(n) another view 1^2 + 2(1) = 3 2^2 + 2(2) = 8 3^2 + 2(3) = 15 4^2 + 2(4) = 24 5^2 + 2(5) = 35 6^2 + 2(6) = 48 ... n^2 + 2(n) = n(n + 2) Represent ¾ REPRESENT ¾ A pizza is cut into 8 equally sized pieces. How many pieces did Isaiah eat if he ate ¾ of the whole pizza? As a percent The Lesh Translation Model Realistic Pictorial Manipulatives Symbolic Language REPRESENTATION STANDARD Create and use representations to organize, record and communicate mathematical ideas; Select, apply, and translate among mathematical representations to solve problems; Use representations to model and interpret physical, social, and mathematical phenomena. “Principles and Standards for School Mathematics” National Council of Teachers of Mathematics - 2000 REPRESENTATION When students gain access to mathematical representations and the ideas they represent, they have a set of tools that significantly expand their capacity to think mathematically. WHY ARE REPRESENTATIONS IMPORTANT? Representations are tools that significantly expand students’ capacity to think mathematically. Representations support students’ understanding of mathematical ideas. Representations help students communicate understanding to themselves and to others. Representations facilitate connections among concepts. PAINTED TOWER Stage 1 Stage 2 Stage 3 Stage 4 PROCESS IT… Stage 1 Stage 2 Stage 3 1. How many faces would be painted in stage 5? How did you determine your answer? 2. How many faces would be painted in stage 16? How did you determine you answer? 3. Can you determine a rule for the number of faces that would be painted in any stage? Stage 4 PROGRESSION OF REASONING Empirical Pre-formal Formal “Focus in High School Mathematics – Reasoning and Sense Making” pp. 10 – 11 National Council of Teachers of Mathematics DEVELOPING REASONING HABITS ♣ Provide tasks that require students to figure things out for themselves. ♣ Give students time to analyze a problem intuitively, explore the problem further by using models, and then proceed to a more formal approach. ♣ Resist the urge to tell students how to solve a problem when they become frustrated; find other ways to support them as they think and work. DEVELOPING REASONING HABITS ♣ Ask students questions that will prompt their thinking—for example, Why does this work?” or “How do you know?” ♣ Provide adequate wait time after a question for students to formulate their own reasoning. ♣ Encourage students to ask probing questions of themselves and one another. ♣ DEVELOPING REASONING HABITS ♣ Establish a classroom climate in which students feel comfortable sharing their mathematical arguments and critiquing the arguments of others in a productive manner. “Focus in High School Mathematics – Reasoning and Sense Making” pp. 10 – 11 National Council of Teachers of Mathematics A teacher’s use of questioning plays a vital role in focusing learning on foundational mathematical ideas and promoting mathematical connections. Mr. Short is six paper clips in height. If he is measured in large buttons he is four large buttons in height. Mr. Tall is similar to Mr. Short but is six large buttons in height. Predict the height of Mr. Tall in paper clips. Explain. PROCESS IT… Reasoning Explanation Multiplicative Reasoning 1 He is nine paper clips tall. Each button is equal to one and a half paper clips. If he is six buttons tall you multiply six time one and a half to get nine paper clips. Multiplicative Reasoning 2 Mr. Tall is 1 ½ times as high as Mr. Short. Since Mr. Short is 6 clips high, Mr. Tall must be 6 * 1 ½ = 9 clips high. Multiplicative Reasoning using addition For every two buttons there are three paper clips. Mr. Tall is 2 buttons taller than Mr. Short so he must be 3 paper clips taller. 6 + 3 = 9 paper clips. Additive Reasoning 1 Mr. Tall is 8 paper clips high. Mr. Short is 4 large buttons high and 6 paper clips high. So the buttons are 2 less than the paper clips. Since Mr. Tall and Mr. Short are similar, and Mr. Tall is 6 buttons high, he must be 8 paper clips high. Additive Reasoning 2 Mr. Tall is two more buttons taller than Mr. Short so he will also be two more paper clips taller than Mr. Short resulting in 8 paper clips. Estimate Nine, I figured he would be a bit taller. Haphazard Since Mr. Tall is 2 more buttons than Mr. Short, I took the 6 paper clips and multiplied by 2 to get 12 paper clips. What Were They Thinking? Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω 2x9+7 Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω 1 + 3 + 5 + 7 + 5 + 3 +1 Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω 4x3+2x7-1 Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω 2 x 16 - 7 Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω 4x4+3x3 PROCESS IT… WHAT WERE THEY THINKING? Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω __________________ Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω __________________ Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω __________________ Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω __________________ Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω __________________ Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω __________________ Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω __________________ Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω __________________ Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω __________________ Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω 1. What are advantages/disadvantages to pulling a student from the general-ed (core) mathematics class? 2. When (if ever) is it appropriate to pull a student from the general-ed (core) mathematics class? How do you know? Philosophical Discussion MONSTER IN THE ROOM During the night, a monster entered this room and, while groping in the dark for the light switch, the monster left a large smudge that looked like a hand print on the wall. Not wishing to leave this mark for someone else to clean up, I began to remove its trace from the wall when suddenly I began to wonder how tall this monster could be. I made copies of the monster’s hand print. Can you help me find the approximate height of the monster? PROCESS IT… Stage 1 Stage 2 Stage 3 Swimming Pool Walkway Stage 4 PROCESS IT… Stage 1 Stage 2 Stage 3 1. How many blocks would there be in stage 5? How did you determine your answer? 2. How many blocks would there be in stage 20? How did you determine you answer? 3. Can you determine a rule for the number of blocks in any stage? 4. Which stage consists of 134 blocks? 5. Which stage consists of 108 blocks? 6. Explain why there cannot be a stage that contains 99 blocks? Stage 4 On Sunday (the first day), Grandma gave Lauren 3 apples. On Monday (the second day), Grandma again gave Lauren 3 apples but later in the day, Lauren’s horse ate one of the apples. On Tuesday (the third day), Grandma again gave Lauren 3 apples. On Wednesday (the fourth day), Grandma again gave Lauren 3 apples but later in the day, Lauren’s horse ate one of the apples. If this pattern of giving by Grandma and taking by Lauren’s horse continues, how many apples would Lauren have by the end of the 15th day? Represent the solution to this problem in a way that your students would. On Sunday (the first day), Grandma gave Lauren 3 apples. On Monday (the second day), Grandma again gave Lauren 3 apples but later in the day, Lauren’s horse ate one of the apples. On Tuesday (the third day), Grandma again gave Lauren 3 apples. On Wednesday (the fourth day), Grandma again gave Lauren 3 apples but later in the day, Lauren’s horse ate one of the apples. How would your students represent the solution if this pattern continued for ANY number of days? PROCESS IT… Day 1 Day 2 Day 3 Day 4 Tom Muchlinski Barb Scierka Julie Seldon
