This presentation shows possible solutions to a middle school level problem involving algebra and geometry.

1. Rachel Jones
(Towson University, Maryland)
rjones39@students.towson.edu
With thanks to
Dr. 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)

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. e
d
c
b
a
F
E
C
I H
G
D
BA
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)

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. 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

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