1. Unit 7: Proportional Reasoning Algebraic and Geometric Thinking

2. Numeration Quantity/ Magnitude Base Ten Equality Form of a Number Proportional Reasoning Algebraic and Geometric Thinking The Components of Number Sense © 2007 Cain/Doggett/Faulkner/Hale/NCDPI Language

3. Proportional Reasoning Defining the Concept Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application

9. Diagnosis Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application

12. Where the research meets the road Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application

17. Classroom Application: Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application

24. Problems that Encourage Proportional Sense Adapted from Esther Billings Mathematical Teaching in the Middle School NCTM

38. Let’s use visuals to help us think this through…

39. Prototype for lesson construction Touchable visual Discussion: Makes sense Of concept 1 2 Learn to Record these ideas V. Faulkner and DPI Task Force adapted from Griffin Symbols Simply record keeping! Mathematical Structure Discussion of the concrete Quantity Concrete display of concept

45. Now, back to the earth

46. Direct Variation, Linear Relationships and a connection Between Geometry and Algebra! Y = mX + 0

47. $ 225 200 175 150 125 100 75 50 25 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Lawns Mowed You have a lawn mower and your parents will pay for maintenance and gas if you make use of your summer by starting a lawn mowing business. You will make $25 for each lawn you mow. This means that, if I tell you how many lawns you mow you can figure out how much you earned OR if I tell you how much your earned, you can tell me how many lawns you mowed. Each point has TWO pieces of information! Plot the following points with Your small group: You mowed 0 lawns (0, ___) You mowed 1 lawn (1, ___) You mowed 2 lawns (2, ___) You mowed 3 lawns (3, ___) You earned $100 dollars (___, 100) You earned $200 dollars (___, 200) Look at the point that I plotted. Did I plot it correctly? How do you know?

48. $ 225 200 175 150 125 100 75 50 25 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Lawns Mowed I have a lawn mower and will mow Lawns this summer for$25.00 a lawn Money earned = $25 * Lawns mowed Y = mX

49. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 DIAMETER Pi is a proportion and the Diameter Varies Directly with the Circumference. This is just like mowing the lawns! Do you see that these two parts of the circle Vary Directly and therefore, when you plot them they form a Line? y = mx Circumference = ~3(Diameter) CIRCUMFERENCE 30 27 24 21 18 15 12 9 6 3 0 SLOPE ~ 3/1 3 1

50. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 RADIUS What is the relationship between The RADIUS and the Circumference? Circumference = ~6(Radius) y = mx CIRCUMFERENCE 30 27 24 21 18 15 12 9 6 3 0 Slope ~ 6/1

51. Circumference/6 = Radius 30 5

52. Circumference/6 = Radius 50 ?

53. Circumference/6 = Radius 100 ?

54. Circumference/2  = Radius 100 + 100 ?

55. Circumference/6 = Radius 50 + 100 ?

56. Circumference/6 = Radius 1,000,000 + 100 ?

57. 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 RADIUS CIRCUMFERENCE 250 225 200 175 150 125 100 75 50 25 0 Plot these points: Radius of 5 (5, ~___) Radius of 10 (10, ~ ___) Circumference of 72 (~___, 72) On the Graph we have plotted (~16, 1 00) and what other point? (____, _____). Look at the two points that are plotted. Using these two points answer the following: For every 100 change in Circumference, about what will be the change in Radius? What is the slope of this linear relationship? Δ X = Δ Y = 100 (~16, 100)

58. 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 RADIUS CIRCUMFERENCE 250 225 200 175 150 125 100 75 50 25 0 Plot these points: Radius of 5 (5, ~ ___) Radius of 10 (10, ~ ___) Circumference of 72 (~___, 72) On the Graph we have plotted (~16, 100) and what other point? (____, _____). Look at the two points that are plotted. Using these two points answer the following: For every 100 gain in Circumference, about what will be the gain in Radius? What is the slope of this linear relationship? Δ X Δ Y (~16, 100) (32, 200) = ~16 = 100 100 ~16 = ~6

60. r = C 2 

61. 100 2 

62. r = C 2  + 100 2 

64. There is a linear relationship between C and r. Whenever C is increased by 100 cm, r will be increased by 100/(2 π )cm.

65. No matter what size our sphere is (initial radius), if the circumference is increased by 100 centimeters, the distance added from the center of the sphere will always be 100/6 or about 16 centimeters!

66. Pi and the earth Pi is a ratio Pi is a constant 1 2 V. Faulkner and DPI Task Force adapted from Griffin Symbols Record keeping: Generalization Mathematical Structure Sense-making of the concrete Quantity Concrete display of concept c/2*Pi + 100/ 2* Pi

67. Numeration Quantity/ Magnitude Base Ten Equality Form of a Number Proportional Reasoning Algebraic and Geometric Thinking Components of Number Sense © 2007 Cain/Doggett/Faulkner/Hale/NCDPI Language

68. Houston , we have another problem… Tip o’ the nib to Cindy Shackelton

70. Algebraic and Geometric Thinking

71. Numeration Quantity/ Magnitude Base Ten Equality Form of a Number Proportional Reasoning Algebraic and Geometric Thinking The Components of Number Sense © 2007 Cain/Doggett/Faulkner/Hale/NCDPI Language

72. Algebraic Thinking Defining the Concept Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application

79. Fundamental Algebraic Ideas Algebra as abstract arithmetic Algebra as the language of mathematics Algebra as a tool to study functions and mathematical modeling Dr. Shelley Kriegler, UCLA, Teacher Handbook , Part 1 Mathematical Thinking Tools Problem solving skills Representation skills Reasoning skills COMPONENTS OF ALGEBRAIC THINKING

80. Diagnosis Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application

85. Where the Research Meets the Road Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application

90. Classroom Application: Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application

99. No Correlation…

100. Prototype for Lesson Construction Touchable Visual Discussion: Makes Sense of Concept 1 2 Symbols: Simply record keeping! Mathematical Structure Discussion of the concrete Quantity: Concrete display of concept Learn to Record these Ideas V. Faulkner and DPI Task Force adapted from Griffin

101. Even Algebra can fit this mold! (av. cost)(items) + gas = grocery $ mx + b = y 1 2 Symbols Simply record keeping! Mathematical Structure Discussion of the concrete Quantity: Concrete display of concept Faulkner adapting Leinwald, Griffin How do we organize data to make predictions and decisions? Why is the slope steeper for Fancy Food then Puggly Wuggly? What about gas cost? Is there a point at which the same items would cost the same at both stores?, etc.

103. Unit 1 Concept Development Dollar Deals

104. Dollar Deals Learning Objectives Dollar Deals 1 2 Symbols Simply record keeping! Math Structure Discussion of the concrete Quantity: Concrete display of concept 1 2 3 3 2 1 x f(x) y x=y 1x = y 1x = y+0 Ratio; Forms of same info; 1:1; Identity function

106. Unit 2 Dollar Deals vs. Puggly Wuggly

107. Puggly Wuggly vs. Dollar Deals Learning Objectives 1 2 Symbols Simply record keeping! Math Structure: Discussion of the concrete Quantity: Concrete display of concept 1 2 3 3 2 1 x f(x) y ? Coefficient and slope How does cost affect slope? Why does PW have a steeper slope? How do you model PW? 2x=y

109. Unit 3 Puggly Wuggly vs. Fancy Food

110. Puggly Wuggly vs. Fancy Foods Learning Objectives 1 2 Symbols Simply record keeping! Math Structure: Discussion of the concrete Quantity: Concrete display of concept 1 2 3 3 2 1 x f(x) y ? Coefficient and slope How does cost affect slope? What does it mean that the slope lines meet at 0? How do you model FF and PW? 2.5x=y 4x = y

113. Other Forms of the Number Relationship! 20 12.5 5 16 10 4 12 7.5 3 8 5 2 4 2.5 1 Y=4x FF Y=2.5x PW X (Items)

114. Unit 4 Feeding the DumDee

115. Feeding the DumDee Learning Objectives 1 2 Symbols Simply record keeping! Verbal: Mathematical Structures Quantity: Concrete display of concept 1 2 3 3 2 1 x f(x) y Why do the lines intersect? When is it cheaper to shop At fancy foods? What is a Y intercept and what is a constant? 2.5x=y 4x = y

116. Feeding the DumDee What Does it Look Like? Communicating with “cell phones” and marking their points on the Chart as they go through that point in the script.

119. Mathematical Models The Showdown: PW + Gas vs. FF on foot + +

120. Fancy Food vs. Puggly Wuggly + Gas! $24.00 $24.00 6 $21.50 $20.00 5 $19.00 $16.00 4 $16.50 $12.00 3 $14.00 $8.00 2 $11.50 $4.00 1 Y=2.5x + 9 PW Y=4x FF X Items

121. Feeding the DumDee--Jenn V. Jermaine Learning Objectives 1 2 Symbols Simply record keeping! Verbal: Discussion of the concrete Quantity: Concrete display of concept 1 2 3 3 2 1 x f(x) y Why don’t the lines intersect? Same slope, different Y-intercept—Will there ever be the same cost for the same number of items What is a Y= intercept and what is a constant? 2.5x=y 4x = y

124. Mathematical Models + +

125. + +

126. Jenn and Jermaine’s Shopping Trip $10.00 $9.50 4 $8.00 $7.50 3 $6.00 $5.50 2 $4.00 $3.50 1 $2.00 $1.50 0 Y=2x + 2 Jenn Y=2x + 1.5 Jermaine X Items

127. Y = mx +b

128. r = C + 100 2  2  r = 1 C + 100 2  2 

129. Y = mx +b r = 1 C + 100 2  2 

130. Numeration Quantity/ Magnitude Base Ten Equality Form of a Number Proportional Reasoning Algebraic and Geometric Thinking The Components of Number Sense © 2007 Cain/Doggett/Faulkner/Hale/NCDPI Language

132. Geometric Thinking

133. Numeration Quantity/ Magnitude Base Ten Equality Form of a Number Proportional Reasoning Algebraic and Geometric Thinking The Components of Number Sense © 2007 Cain/Doggett/Faulkner/Hale/NCDPI Language

134. Geometric Thinking Defining the Concept Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application

136. Diagnosis Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application

138. Where the research meets the road Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application

140. Classroom Application: Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application

141. Foundation Level Training Understanding the Relationship between Area and Perimeter Using Griffin’s Model Mayer’s Problem Solving Model Components of Number Sense

143. Prototype for Lesson Construction Area (Squared Measure) vs. Perimeter (Linear Measure) 1 2 Symbols Record Keeping! Math Structure: Discussion of the Concrete Quantity: Concrete Display of Concept A=L x W=10 P= 2L + 2W=14 V. Faulkner and DPI Task Force adapted from Griffin

159. Challenge 1 What other perimeter could be shown that has the same area? Area = _____ P = ____

160. Challenge 2 A = ______ P = _____ Rearrange the area to show a diagram with a different perimeter .

161. Challenge 3 Add a square so that the area is 6 and the perimeter remains 12. A = ____ P = ____ Can you find other arrangements with an area of 6 and a perimeter of 12?

167. How Can We Organize the Data? A = 9 sq. units P= 20 units 1 9 A = 16 sq. units P= 20units 2 8 A = 21 sq. units P= 20units 3 7 A = 24 sq. units P= 20units 4 6 A = 25 sq. units P= 20 units 5 5 A = 24 sq. units P= 20 units 6 4 A = 21 sq. units P= 20 units 7 3 A = 16 sq. units P= 20 units 8 2 A = 9 sq. units P= 20 units 9 1 A = L x W P= 2L + 2 W Width Length