Rachel Jones (Towson University, Maryland)

rjones39@students.towson.edu

With thanks toDr. Diana Cheng (Towson University)

Dr. Dicky Ng (North Carolina State University) & Dr. Einav Aizikovitsh-Udi (Beit Berl College, Israel)

Presented at the 2014 Joint Mathematics Meetings (Baltimore, MD) & 2015 Undergraduate Research Conference (Towson University, MD)

• 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)

• “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

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.

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

Group 1:(4/9) x2 units2

Group 2: Trapezoid

(4/9) x2 units2

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

Extend AC and IF to point EGroup 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 + dGroup 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 & PVUQGroup 2:

triangles JML & PNR, rectangles LPST & TSNM

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

trapezoid

a+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