READING MATH. Valerie Faulkner NC State University Elementary Education-Math Methods SAS Hi-Five Math Summit Summer 2013

Transcription

1 READING MATH Valerie Faulkner NC State University Elementary Education-Math Methods SAS Hi-Five Math Summit Summer 2013 Contact: Valeriefaulknermathclub.com

2 What does the expression/equation say? Pretend it is a sentence in a novel Pretend it is a line in an article Connect it to a picture and a story: A visual image Let it speak to you like a character in a context...

3 NOTICE: CHARACTER DEVELOPMENT? Is the story you are telling a procedure without CONTEXT? Do this to that? All PLOT with no CHARACTER? Let s DEVELOP the CHARACTERS first before we follow them around. How do CONNECT to the BIG PICTURE Math Story?

4 Anna Karenina: All PLOT and no CONTEXT Marry her off to a bureaucrat Have her fall in love with a young soldier Throw her under a train.

5 Anna Karenina: Really about Marriage... Anna s brother cheats and his marriage remains intact Society Anna cheats and it ruins her life Levin and Kitty have an ideal marriage in the country

6 What story does this expression tell? 4 1 A procedure or a character?

7 4 1 4 in 1 dimension A linear 4

8 What story does this expression tell? 4 2 An action without a character?

9 height 16 base 16 = = 4 4 The BASE The Square Root So, the exponent literally means the dimensions you go in using that base.we live in a three dimensional world, so for exponents to the 1st, 2nd, and 3rd power, we can model this and understand it visually! SAS - How I access 4 squared changes how I access so many other connected math topics: So many Connections! Exponents, Geometry, Measurement, Pythagorean Theorem, Square Roots, Distance Formula, Trigonometry, etc. So what does this mean about Algebra? How is our tendency to teach procedurally working? How is it affecting our outcomes and our student thinking? How many different forms of the number 16 do you see here? Think about the power for students to understand algebra if they are firmly entrenched in forms of a number. Take your particularity concrete kid---they can now understand this concept using quantity and magnitude We can emphasis for them the equality of the different forms of this number 16 and if the student really needs concrete support to understand the concept they count the squares to attach to the concept. We are already hooking into our geometric thinking at a very early age! As we talk about this---we are making stronger implementation choices that helps students make stronger connections within the mathematics. In addition, we are differentiating our instruction with various strategies at different places where students can get it.

10 What story does this expression tell? 4 3

11 What story does this expression tell? 4 4

12 What story does this expression tell? a 2

13 a 2 a 2 a The Square Root (or the Base of the square) So, the exponent literally means the dimensions you go in using that base.we live in a three dimensional world, so for exponents to the 1st, 2nd, and 3rd power, we can model this and understand it visually! SAS - How I access 4 squared changes how I access so many other connected math topics: So many Connections! Exponents, Geometry, Measurement, Pythagorean Theorem, Square Roots, Distance Formula, Trigonometry, etc. So what does this mean about Algebra? How is our tendency to teach procedurally working? How is it affecting our outcomes and our student thinking? How many different forms of the number 16 do you see here? Think about the power for students to understand algebra if they are firmly entrenched in forms of a number. Take your particularity concrete kid---they can now understand this concept using quantity and magnitude We can emphasis for them the equality of the different forms of this number 16 and if the student really needs concrete support to understand the concept they count the squares to attach to the concept. We are already hooking into our geometric thinking at a very early age! As we talk about this---we are making stronger implementation choices that helps students make stronger connections within the mathematics. In addition, we are differentiating our instruction with various strategies at different places where students can get it.

14 What story does this equation tell? a 2 + b 2 = c 2

15 Pythagorean Theorem: c 2 a 2 c a b b 2

16 Conclusion Plot Climax Rising Action Setting Tone Theme or Moral Character Development Denouement Supporting Detail Supporting Detail Supporting Detail

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18 Just as in literature... Where else do you see this theme or structure? Distance formula Quadratic formula Fermat s Theorem

19 Fermat s Theorem a 3 + b 3 c 3 + whole number root whole number root whole number root

20 Are students reading? Are they reading Dimensions? Link to Pi valeriefaulknermathclub.com

21 2 r r 2

22 What story does this expression tell? 2 r

23 2 r 2 r d

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31 day activities? Huge Circle on the football field: How many band members around? How many band members across? Do you have to measure the circumference and the diameter to know the answer to both?

32 d 2 r why in terms of r?

33 The radius is a more helpful character... r 2

34 r 2 ~3 squared rs

35 What story does this equation tell? A = r 2

36 area ~ area area area ~3 squared rs

37 area = r 2 r 2 r 2 r 2 * squared rs

38 Adjust your task analysis... If a student uses the wrong formula for a circumference or circle area problem, the remediation is not getting them to memorize which formula is which, the remediation is...

39 Adjust your task analysis... Distance Formula What s the real issue if a student is not doing well with the distance formula? Hint: It is not more repetitions memorizing the formula at the symbolic level (It is not the plot it is the characters )

40 Height Base Derr, so the distance formula is just the Pythagorean Theorem. Why didn t you tell me that? So I have to Create my own right triangle so that I can use the Pythagorean theorem. And then the annoying x1 - x2 is just a way to get the length of its base and y1 - y2 is how I can get the length of its height? That s so dumb. And then I just get the squares, and add them and then get the square root to get the linear measure just like the Pythagorean thereom. Duh, that s stupid.

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42 What story does this equation tell? x = y

43 What story does this equation tell? 3x + 7 = 0

44 Students dealing with this equation because they have been taught procedurally... 3x + 7 =

45 Students metacognition when they understand the story... 3x + 7 = 0 Well 7 must be the opposite of 3x so... 3x = -7

46 What story does this equation tell? 3x * 7 = 0

47 procedural instruction yields... 3x * 7 = 0 7 7

48 procedural instruction yields... 3x = 0 3 3

49 procedural instruction yields... x = 0

50 Which demonstrates a superficial relationship to the story!

51 Metacognition if I understand (and have been explicitly taught to think about) the story... 3x * 7 = 0 either 3 or x or 7 must be zero, so 3 = 0, x = 0 or, 7 = 0 well, derr, only x could be 0 in this case.

52 What type of literature is this? (x + 3)

53 Mystery Number - Part we know, part we don t (x + 3)

54 (x + 2) (x + 3) square rectangle with x sides rectangle 3 by x rectangle 2 by x 2 by 3 rectangle that s 6 (square units)

55 Mystery Novel If I understand the Theme of *0... 2x 2-11x - 21 = 0 (2x + 3) (x - 7) = 0

56 (2x + 3) (x - 7) = 0

57 Discuss with neighbors What thought process do you want your students to have when dealing with 2x 2-11x - 21 = 0?

58 Discuss with neighbors Now, how does that extend to this: 2x 2-11x = 21?

59 If I understand the power of *0...

60 Can your students picture factoring a difference of squares? (or do they just have a rule memorized?)

61 Expression x 2-9 (x + 3) (x -3)

62 (x + 3) (x - 3) square rectangle with x sides minus a rectangle 3 by x rectangle 3 by x minus a square rectangle 3x3

63 (x - 3) (x + 3)

64 So the area of the overall shape will be the area of the first product minus the area of the last product. x 2 9

65 What if I knew that x 2-9 = 0... Does that (HUGE) clue help me solve my mystery?

66 (x + 3) (x - 3) = 0 So...when y equals 0,... I need x to equal either 3 or - 3.

67 So the area of the overall shape will still be the area of the first product minus the area of the last product. x 2 9

68 (-3,0) (3, 0)

69 (x + 3) (x - 3) = 0 MAKE SURE to consider non-examples so that this is truly understood. What if I said, x = 4. What would happen?

70 What if I want to build a garden with a patio surrounding it. I like the look of squares. What are my options for the area of my patio. I want to try and figure out the costs... x 2 9

71 X 2-9sq ft of garden = Y The story is... I am not sure how much patio I can afford!

72 (-3,0) (3,0) (If Patio is 3x3, 0 garden space)

73 (4, 7)

74 4x4 patio - 9sqft garden = 7 sqft of patio I want more patio! (4, 7)

75 In patio and garden terms, what story might this expression tell? r 2 - r 2

76 r2 - r2

77 Why did we choose the shapes we chose? r 2 - r 2

78 What does the expression/equation say? Pretend it is a sentence in a novel Pretend it is a line in an article Connect it to a picture and a story: A visual image Let it speak to you like a character in a context...

79 Questions? READING MATH Valerie Faulkner NC State University Elementary Education-Math Methods SAS Hi-Five Math Summit Summer 2013 Contact: Valeriefaulknermathclub.com