From Grade 5 to High School
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- Johnathan Randall
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1 From Grade 5 to High School the arc of an investigation Al Cuoco Education Development Center in collaboration with Alicia Chiasson Newbury (MA) Elementary School (Special Thanks to Ken Levasseur, UMASS Lowell) Curtis Center, 2015
2 Outline 1 Introduction Mathematical practice The context 2 The problem: Grade 5 6 A Simulation and Some Representations Some Conjectures 3 The same problem: Later on An Algebraic Approach A Geometric Approach 4 Parting Thoughts
3 Mathematical practice Outline 1 Introduction Mathematical practice The context 2 The problem: Grade 5 6 A Simulation and Some Representations Some Conjectures 3 The same problem: Later on An Algebraic Approach A Geometric Approach 4 Parting Thoughts
4 Mathematical practice Featured mathematical habits Look for and express regularity in repeated reasoning. Abstract general principles from a set of special cases.
5 Mathematical practice Featured mathematical habits Look for and express regularity in repeated reasoning. Abstract general principles from a set of special cases. Look for and make use of structure. Reason from the form of a calculation rather than its value.
6 Mathematical practice Featured mathematical habits Look for and express regularity in repeated reasoning. Abstract general principles from a set of special cases. Look for and make use of structure. Reason from the form of a calculation rather than its value. Attend to precision. Shoehorn nascent insights into precise language.
7 Mathematical practice Featured mathematical habits Look for and express regularity in repeated reasoning. Abstract general principles from a set of special cases. Look for and make use of structure. Reason from the form of a calculation rather than its value. Attend to precision. Shoehorn nascent insights into precise language. Use appropriate tools strategically. Use technology to build insight and to visualize thought experiments.
8 Mathematical practice Featured mathematical habits Look for and express regularity in repeated reasoning. Abstract general principles from a set of special cases. Look for and make use of structure. Reason from the form of a calculation rather than its value. Attend to precision. Shoehorn nascent insights into precise language. Use appropriate tools strategically. Use technology to build insight and to visualize thought experiments. Model with mathematics Use formal algebra as a modeling tool.
9 The context Outline 1 Introduction Mathematical practice The context 2 The problem: Grade 5 6 A Simulation and Some Representations Some Conjectures 3 The same problem: Later on An Algebraic Approach A Geometric Approach 4 Parting Thoughts
10 The context Framing question When rolling 2 or more dice, are certain sums more likely to come up than others?
11 A Simulation and Some Representations Outline 1 Introduction Mathematical practice The context 2 The problem: Grade 5 6 A Simulation and Some Representations Some Conjectures 3 The same problem: Later on An Algebraic Approach A Geometric Approach 4 Parting Thoughts
12 A Simulation and Some Representations The Investigation began with a simulation: 1) How many combinations are there when tossing two dice? 2) What are the possible sums when tossing two dice? 3) Which sums do you think are most likely to come up? Toss two dice 20 times and record the outcomes: Outcome Tally Total Make a bar graph of your outcomes. Why are certain sums more likely to come up than others? Why is 7 often called a lucky number?
13 Some Conjectures Outline 1 Introduction Mathematical practice The context 2 The problem: Grade 5 6 A Simulation and Some Representations Some Conjectures 3 The same problem: Later on An Algebraic Approach A Geometric Approach 4 Parting Thoughts
14 Some Conjectures Students realized that certain sums were more likely than others. c o m b i n a ti o n s They made a table to determine all possible sums. They noticed a bell curved pattern in the number of sums with the most likely sum (7) in the middle : :1 5:2 6: :1 4:2 4:3 5:3 6: :3 3:2 3:3 3:4 4:4 5:4 6: :1 2:2 2:3 2:4 2:5 3:5 4:5 5:5 6: :1 1:2 1:3 1:4 1:5 1:6 2:6 3:6 4:6 5:6 6: Sums
15 Some Conjectures Others found sums using this table: The sum of 7 appears most often. dice
16 Some Conjectures These are all the probabilities with three dice. C O B I N A T I O N S :3:1 6:4:1 25 6:2:2 6:3:2 25 6:2:1 6:1:3 6:2:3 6:5:1 6:1:2 5:4:1 6:1:4 6:4:2 5:3:1 5:3:2 5:5:1 6:3:3 21 5:2:2 5:2:3 5:4:2 6:2:4 21 6:1:1 5:1:3 5:1:4 5:3:2 6:1:5 6:6:1 5:2:1 4:4:1 4:5:1 5:2:4 5:6:1 6:5:2 5:1:2 4:3:2 4:4:2 5:1:5 5:5:2 6:4:3 4:3:1 4:2:3 4:3:3 4:6:1 5:4:3 6:3:4 4:2:2 4:1:4 4:2:4 4:5:2 2:3:4 6:2:5 15 4:1:3 3:5:1 4:1:5 4:4:3 5:2:5 6:1:6 15 5:1:1 3:4:1 3:4:2 3:6:1 4:3:4 5:1:6 5:6:2 6:6:2 4:2:1 3:3:2 3:3:3 3:5:2 4:2:5 4:6:2 5:5:3 6:5:3 4:1:2 3:2:3 3:2:4 3:4:3 4:1:6 4:5:3 5:4:4 6:4:4 3:3:1 3:1:4 3:1:5 3:3:4 3:6:2 4:4:4 5:3:5 6:3:5 10 3:2:2 2:5:1 2:6:1 3:2:5 3:5:3 4:3:5 4:2:6 6:2:6 10 4:1:1 3:1:3 2:4:2 2:5:2 3:1:6 3:4:4 4:2:6 4:6:3 5:6:3 6:6:3 3:2:1 2:4:1 2:3:3 2:4:3 2:6:2 3:3:5 3:6:3 4:5:4 5:5:4 6:5:4 3:1:2 2:3:2 2:2:4 2:3:4 2:5:3 3:2:6 3:5:4 4:4:5 5:4:5 6:4:5 6 2:3:1 2:2:3 2:1:5 2:2:5 2:4:4 2:6:3 3:4:5 4:3:6 5:3:6 6:3:6 6 3:1:1 2:2:2 2:1:5 1:6:1 2:1:6 2:3:5 2:5:4 3:3:6 3:6:4 4:6:4 5:6:4 6:6:4 2:2:1 2:1:3 1:5:1 1:5:2 1:6:2 2:2:6 2:4:5 2:6:4 3:5:5 4:5:5 5:5:5 6:5:5 3 2:1:2 1:4:1 1:4:2 1:4:3 1:5:3 1:6:3 2:3:6 2:5:5 3:4:6 4:4:6 5:4:6 6:4:6 3 2:1:1 1:3:1 1:3:2 1:3:3 1:3:4 1:4:4 1:5:4 1:6:4 2:4:6 2:6:5 3:6:5 5:3:6 5:6:6 6:6:5 1 1:2:1 1:2:1 1:2:3 1:2:4 1:2:5 1:3:5 1:4:5 1:5:5 1:6:5 2:5:6 3:5:6 4:5:6 5:6:5 6:5:6 1 1:1:1 1:1:2 1:1:3 1:1:4 1:1:5 1:1:6 1:2:6 1:3:6 1:4:6 1:5:6 1:6:6 2:6:6 3:6:6 4:6:6 5:6:6 6:6: SUMS
17 Some Conjectures Sums 1,, Sums 2,,
18 Some Conjectures Students made this table to take a closer look: # of dice # combos Lowest Highest Lucky # roll roll 2 6 squared = cubed = & to the power of 4 = 1, to the power of 5 = 7, & to the power of 6 = 46,
19 Some Conjectures Students found that the middle number (median) was the lucky number. Lucky # (median) 7 10 & & , ,4 +3 Notice the pattern between medians on the table to the left!
20 Some Conjectures Students determined these rules: # dice # combo Lowest roll n 6 to the power n Highest roll Lucky # n 6 n Median When looking at the medians students discovered that: Every other median is a multiple of 7. Each median is 3 more then the previous median. Even number of dice have one median Odd number of dice have two medians The median number is also the mean.
21 An Algebraic Approach Outline 1 Introduction Mathematical practice The context 2 The problem: Grade 5 6 A Simulation and Some Representations Some Conjectures 3 The same problem: Later on An Algebraic Approach A Geometric Approach 4 Parting Thoughts
22 An Algebraic Approach Structure in Expressions The table that showed that 7 is the most likely sum: dice also tells you how the other sums are distributed.
23 An Algebraic Approach Structure in Expressions This distribution of all sums can be read off from the table by collecting the like occurrences of each integer: dice Sum Number of occurrences
24 An Algebraic Approach Structure in Expressions What does this have to do with algebra?
25 An Algebraic Approach Structure in Expressions What does this have to do with algebra? Look at the expansion box for (x 2 + 3x+2)(2x 2 + 5x+1) Poly 1 Poly 2 x 2 3x 2 2x 2 2x 4 6x 3 4x 2 5x 5x 3 15x 2 10x 1 x 2 3x 2
26 An Algebraic Approach Structure in Expressions The product can can be read off from the table by collecting the like terms:
27 An Algebraic Approach Structure in Expressions and hence Term x 4 x 3 x 2 x 1 Number of occurrences (x 2 + 3x+2)(2x 2 + 5x+1) = 2x x x x+2
28 An Algebraic Approach Structure in Expressions There a polynomial calculation whose expansion box has the same structure as the distribution of sums on two dice.
29 An Algebraic Approach Structure in Expressions There a polynomial calculation whose expansion box has the same structure as the distribution of sums on two dice. Consider: (x+x 2 + x 3 + x 4 + x 5 + x 6 ) 2
30 An Algebraic Approach Structure in Expressions There a polynomial calculation whose expansion box has the same structure as the distribution of sums on two dice. Consider: (x+x 2 + x 3 + x 4 + x 5 + x 6 ) 2 (x+x 2 + x 3 + x 4 + x 5 + x 6 )(x+x 2 + x 3 + x 4 + x 5 + x 6 )
31 An Algebraic Approach Structure in Expressions There a polynomial calculation whose expansion box has the same structure as the distribution of sums on two dice. Consider: (x+x 2 + x 3 + x 4 + x 5 + x 6 ) 2 (x+x 2 + x 3 + x 4 + x 5 + x 6 )(x+x 2 + x 3 + x 4 + x 5 + x 6 ) 1 What is the coefficient of x 4 in the expansion?
32 An Algebraic Approach Structure in Expressions There a polynomial calculation whose expansion box has the same structure as the distribution of sums on two dice. Consider: (x+x 2 + x 3 + x 4 + x 5 + x 6 ) 2 (x+x 2 + x 3 + x 4 + x 5 + x 6 )(x+x 2 + x 3 + x 4 + x 5 + x 6 ) 1 What is the coefficient of x 4 in the expansion? 2 What is the coefficient of x 9 in the expansion?
33 An Algebraic Approach Structure in Expressions Die 1 Die
34 An Algebraic Approach Structure in Expressions Die 1 Die poly 1 poly 2 x 1 x 2 x 3 x 4 x 5 x 6 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 6 x 7 x 8 x 9 x 10 x 11 x 12 Scratchpad
35 An Algebraic Approach Structure in Expressions Take it Further Scratchpad What is the distribution of sums when 3 dice are thrown?
36 An Algebraic Approach Structure in Expressions Take it Further Scratchpad What is the distribution of sums when 3 dice are thrown? How many possibilities are there when 3 dice are thrown?
37 An Algebraic Approach Structure in Expressions Take it Further Scratchpad What is the distribution of sums when 3 dice are thrown? How many possibilities are there when 3 dice are thrown? Are there more even or odd sums when k dice are thrown?
38 An Algebraic Approach Structure in Expressions Take it Further Scratchpad What is the distribution of sums when 3 dice are thrown? How many possibilities are there when 3 dice are thrown? Are there more even or odd sums when k dice are thrown? What is the distribution of sums for two dice labeled {1, 2, 2, 4, 4, 6} and {4, 4, 4, 3, 3, 3}
39 An Algebraic Approach Structure in Expressions Take it Further Scratchpad What is the distribution of sums when 3 dice are thrown? How many possibilities are there when 3 dice are thrown? Are there more even or odd sums when k dice are thrown? What is the distribution of sums for two dice labeled {1, 2, 2, 4, 4, 6} and {4, 4, 4, 3, 3, 3} Is it possible to label the faces of two dice with positive integers different from the standard labeling so that the distribution of the sums is the same?
40 An Algebraic Approach Weird Dice: An algebraic strategy Is it possible to label the faces of two dice with positive integers, different from the standard labeling, so that the distribution of the sums is the same? Essence of the approach: Factor x+x 2 + x 3 + x 4 + x 5 + x 6 : x+x 2 + x 3 + x 4 + x 5 + x 6 = x(x+1)(x 2 + x+1)(x 2 x+1) So (x+x 2 +x 3 +x 4 +x 5 +x 6 ) 2 = x 2 (x+1) 2 (x 2 +x+1) 2 (x 2 x+1) 2 Now just rearrange these prime factors into two sets such that The product of the factors in each set has positive coefficients The product of the factors in each set, when evaluated at 1, yields 6 From there...
41 An Algebraic Approach Weird Dice: An algebraic strategy Is it possible to label the faces of two dice with positive integers, different from the standard labeling, so that the distribution of the sums is the same? Essence of the approach: Factor x+x 2 + x 3 + x 4 + x 5 + x 6 : x+x 2 + x 3 + x 4 + x 5 + x 6 = x(x+1)(x 2 + x+1)(x 2 x+1) So (x+x 2 +x 3 +x 4 +x 5 +x 6 ) 2 = x 2 (x+1) 2 (x 2 +x+1) 2 (x 2 x+1) 2 Now just rearrange these prime factors into two sets such that The product of the factors in each set has positive coefficients The product of the factors in each set, when evaluated at 1, yields 6 From there... It s a nice thing to think about on the drive home.
42 An Algebraic Approach From CME Precalculus Minds in Action episode 23 Tony and Derman look at Exercise 9 from Lesson 7.3. Tony So, it s a wheel with 1, 2, 3, and 10 fish on it. And we want to know the average from one spin. 9. A local market has a prize wheel. Lucky customers can spin the wheel to win free fish. On one spin, it is possible to win 1 fish, 2 fish, 3 fish, or 10 fish. a. What is the average number of fish the market can expect to give away, per spin? b. Three customers spin the wheel. What is the most likely total number of fish that they win? How likely is this? 10. Take It Further Ten customers spin the Wheel of Fish from Exercise 9. Find the probability that the total number of fish they win is even. Derman I ll just add and divide. The sum is c16. So the average is 4. Tony Sounds good. Now what about two spins? Derman I think it s going to be 8. Two times four. Two spins, 4 fish each? Tony I m not sure it works that way. Let s just write out the sample space. There are only 16 ways it can go. Derman A table it is n Tony Cool. I m going to add all these numbers then divide by 16. Since the sample space is small, I won t bother making a frequency table. The sum is 128. So, the expected value for two spins is 128 over hey, you were right! It is 8. Derman It was bound to happen sometime. If I m right, for three spins the expected value should be 12. The table s going to be a mess! Tony Well, we should use a polynomial power instead. The outcomes are 1, 2, 3, and 10, so the polynomial for one spin should be (x 1 1 x 2 1 x 3 1 x 10 ) Derman Isn t x 1 just x? Tony I like writing x 1, it makes it more clear where it came from. I d do that for x 0 instead of writing 1. Scratchpad Derman Fair enough. And we raise that to the third power, since it s three spins. Expand that and... Tony Let s write that out.
43 An Algebraic Approach A Curiousity :3:1 6:4:1 25 6:2:2 6:3:2 25 6:2:1 6:1:3 6:2:3 6:5:1 6:1:2 5:4:1 6:1:4 6:4:2 5:3:1 5:3:2 5:5:1 6:3:3 21 5:2:2 5:2:3 5:4:2 6:2:4 21 6:1:1 5:1:3 5:1:4 5:3:2 6:1:5 6:6:1 5:2:1 4:4:1 4:5:1 5:2:4 5:6:1 6:5:2 5:1:2 4:3:2 4:4:2 5:1:5 5:5:2 6:4:3 4:3:1 4:2:3 4:3:3 4:6:1 5:4:3 6:3:4 4:2:2 4:1:4 4:2:4 4:5:2 2:3:4 6:2:5 15 4:1:3 3:5:1 4:1:5 4:4:3 5:2:5 6:1:6 15 5:1:1 3:4:1 3:4:2 3:6:1 4:3:4 5:1:6 5:6:2 6:6:2 4:2:1 3:3:2 3:3:3 3:5:2 4:2:5 4:6:2 5:5:3 6:5:3 4:1:2 3:2:3 3:2:4 3:4:3 4:1:6 4:5:3 5:4:4 6:4:4 3:3:1 3:1:4 3:1:5 3:3:4 3:6:2 4:4:4 5:3:5 6:3:5 10 3:2:2 2:5:1 2:6:1 3:2:5 3:5:3 4:3:5 4:2:6 6:2:6 10 4:1:1 3:1:3 2:4:2 2:5:2 3:1:6 3:4:4 4:2:6 4:6:3 5:6:3 6:6:3 3:2:1 2:4:1 2:3:3 2:4:3 2:6:2 3:3:5 3:6:3 4:5:4 5:5:4 6:5:4 3:1:2 2:3:2 2:2:4 2:3:4 2:5:3 3:2:6 3:5:4 4:4:5 5:4:5 6:4:5 6 2:3:1 2:2:3 2:1:5 2:2:5 2:4:4 2:6:3 3:4:5 4:3:6 5:3:6 6:3:6 6 3:1:1 2:2:2 2:1:5 1:6:1 2:1:6 2:3:5 2:5:4 3:3:6 3:6:4 4:6:4 5:6:4 6:6:4 2:2:1 2:1:3 1:5:1 1:5:2 1:6:2 2:2:6 2:4:5 2:6:4 3:5:5 4:5:5 5:5:5 6:5:5 3 2:1:2 1:4:1 1:4:2 1:4:3 1:5:3 1:6:3 2:3:6 2:5:5 3:4:6 4:4:6 5:4:6 6:4:6 3 2:1:1 1:3:1 1:3:2 1:3:3 1:3:4 1:4:4 1:5:4 1:6:4 2:4:6 2:6:5 3:6:5 5:3:6 5:6:6 6:6:5 1 1:2:1 1:2:1 1:2:3 1:2:4 1:2:5 1:3:5 1:4:5 1:5:5 1:6:5 2:5:6 3:5:6 4:5:6 5:6:5 6:5:6 1 1:1:1 1:1:2 1:1:3 1:1:4 1:1:5 1:1:6 1:2:6 1:3:6 1:4:6 1:5:6 1:6:6 2:6:6 3:6:6 4:6:6 5:6:6 6:6:
44 An Algebraic Approach A Curiousity (x+x 2 + x 3 + x 4 + x 5 + x 6 ) 3 = x x x x x x x x x x x x x 6 + 6x 5 + 3x 4 + x 3
45 An Algebraic Approach A Curiousity (x+x 2 + x 3 + x 4 + x 5 + x 6 ) 3 = x x x x x x x x x x x x x 6 + 6x 5 + 3x 4 + x 3 Why triangular numbers for a while, then not, then they come back?
46 A Geometric Approach Outline 1 Introduction Mathematical practice The context 2 The problem: Grade 5 6 A Simulation and Some Representations Some Conjectures 3 The same problem: Later on An Algebraic Approach A Geometric Approach 4 Parting Thoughts
47 A Geometric Approach Building on the tables
48 A Geometric Approach Building on the tables Imagine that these are stacked decks. And look at a particular sum, say, 9
49 A Geometric Approach Building on the tables
50 A Geometric Approach Building on the tables Stacked up, the decks would be slices of a cube, and 9s would lie on a plane that passes through the cube.
51 A Geometric Approach Out[29]=
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67 A Geometric Approach For more on uses of formal algebra For more on formal algebra from our course Developing Mathematical Practice see: For more about Developing Mathematical Practice, see:
68 Some Conclusions
69 Some Conclusions Mathematics is a low-threshold, high-ceiling enterprise.
70 Some Conclusions Mathematics is a low-threshold, high-ceiling enterprise. Sophisticated mathematical thinking shows up early.
71 Some Conclusions Mathematics is a low-threshold, high-ceiling enterprise. Sophisticated mathematical thinking shows up early. Ideas often emerge without the machinery needed to express them precisely.
72 Some Conclusions Mathematics is a low-threshold, high-ceiling enterprise. Sophisticated mathematical thinking shows up early. Ideas often emerge without the machinery needed to express them precisely. Curricular themes can create coherent arcs across the grades.
73 Thanks Alicia Chiasson Al Cuoco Slides:
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