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Diana Cheng & Rachel Jones (Towson University, Maryland)
With thanks to Dicky Ng (North Carolina State University) &
Einav Aizikovitsh (Beit Berl College, Israel)
Divvying Up the Practice Field:
Student Solutions
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• Engaging in a task for which solution is unknown•No given path for solution•High – level cognitive demand tasks•Allow multiple solution methods• Require students to make connections among
and between mathematical concepts
Reference: Stein, Grover & Henningsen (2006)
Problem solving
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• “Mathematical Problem Solving” graduate course •Middle & secondary
school teachers• 3 to 22 years’ teaching
experience• 3 to 9 graduate
mathematics courses taken previously•Masters degree in
Mathematics Education candidates
Summer 2013 course students
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Square Field Problem
A team uses half of a rectangular-shaped grass field on which to practice. The coach divided the resulting square field into four parts as shown in the figure below. The coach let the goalies practice on the small square area. He then divided the rest of the team into two smaller groups, Group 1 and Group 2, and assigned them their practice areas as shown in the figure below (not drawn to scale).
• Which group, Group 1 or Group 2, had more space on which to practice? Solve the problem in two different ways.
• Explain and justify your solutions, and explain how your two solution methods are different from each other.
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Non-generalizable Student Solution: Trisection with Numeric Values
Group 1: 2 squares & 2 triangles
2*(1/9 + 1/9) = 4/9 units2
Group 2: Trapezoid
½ * (1 + 1/3) * (2/3) = 4/9 units2
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Non-generalizable Student Solution: Trisection with Variables
Group 1: (4/9) x2 units2
Group 2: Trapezoid
(4/9) x2 units2
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Generalizable Student Solution: Trapezoid Area Formula
Group 1: 2 Trapezoids
Area = ½ d(b+1) + ½ e(b+1)Distributive property ½ (d+e)(b+1)
Substitute (d + e) = (1 – b)Area = ½ (1-b)(b+1) units2
Group 2: Trapezoid
Area = ½ (1-b)(b+1) units2
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e
d
c
b
a
F
E
C
I H
G
D
BA
Generalizable Student Solution: Triangle Area Formula Extend AC and IF to point E
Group 1: [EIH-EFG] + [ABE-CDE]
[½ e(a+b) – ½ b(b+c)] + [½ e(c+d) – ½ c(b+c)] Substitute e = a + b + c + d
Group 2: [AEI-CEF] = ½ e2 – ½ (b+c)(b+c)
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Construct TS to be d away from left side of square
Group 1: triangles JKL & PQR, rectangles KXWL & PVUQ
Group 2: triangles JML & PNR, rectangles LPST & TSNM
Generalizable Student Solution: Geometric
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Transformational solution method
Group 1
Group 2
Goalie
a+b
c
c
a+b
B
C
I H
D
A
K
L
• Translate goalie box up b units• Group 2’s new trapezoid has same height & bases as original
trapezoida+b
c
b
c
a
B
F
C
I H
G
D
A
K
L
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Find the area of right triangles & special quadrilaterals… (6.G.A.1)
Solve real-world & mathematical problems involving area of 2D objects composed of triangles & quadrilaterals. (7.G.B.6)
Understand congruence & similarity using physical models or geometry software; Describe the effect of dilations, translations, rotations, & reflections on 2D figures using coordinates. (8.G.A.3)
Apply geometric concepts in modeling situations
Common Core State Standards – Mathematics Grades 6, 7, 8, High School