![[½ e(a+b) – ½ b(b+c)] + [½ e(c+d) – ½ c(b+c)]
e
d
[AEI-CEF] = ½ e2 – ½ (b+c)(b+c)
c
b
a
Extend AC and IF to point E
Group 1:
[EIH-EFG] + [ABE-CDE]
Substitute e = a + b + c + d
Group 2:
A B
F
D
E
C
G
I H](https://image.slidesharecdn.com/divvyingupthepracticefieldstudentsolutions-141203122258-conversion-gate01/85/Divvying-up-the-practice-field-student-solutions-8-320.jpg)
Divvying up the practice field: student solutions
- 1. Diana Cheng & Rachel Jones (Towson University, Maryland) With thanks to Dicky Ng (North Carolina State University) & Einav Aizikovitsh (Beit Berl College, Israel)
- 2. • 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)
- 3. • “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
- 4. 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.
- 5. 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
- 6. Group 1: (4/9) x2 units2 Group 2: Trapezoid (4/9) x2 units2
- 7. 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
- 8. [½ e(a+b) – ½ b(b+c)] + [½ e(c+d) – ½ c(b+c)] e d [AEI-CEF] = ½ e2 – ½ (b+c)(b+c) c b a Extend AC and IF to point E Group 1: [EIH-EFG] + [ABE-CDE] Substitute e = a + b + c + d Group 2: A B F D E C G I H
- 9. 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
- 10. • Translate goalie box up b units • Group 2’s new trapezoid has same height & bases as original K Group 1 Group 2 Goalie a+b c c a+b B C D A I L H trapezoid a+b b c c a B C K F D G A I L H
- 11. 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