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

• 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

• “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

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.

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

Non-generalizable Student Solution: Trisection with Variables

Group 1: (4/9) x2 units2

Group 2: Trapezoid

(4/9) x2 units2

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

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)

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

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

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