MASTA Proof and Reasoning 9-12 Module
Algebraic and Symbolic Reasoning
Handout - Page 1 of 13
1. What are the three categories of common algebra errors as described by
Matz?
MASTA Proof and Reasoning 9-12 Module
Algebraic and Symbolic Reasoning
Handout - Page 2 of 13
2. Categorize the following errors according to Matz’s framework.
(i)
AX  BY
 A B
X Y

1 1
3
 2  2  6x 2
(ii) x x
x
x  1  3  6x 2

(x  a)(x  b)  c
(iii)
(x  a)  c or (x  b)  c
x  c  a or x  c  b
(iv)
A
A A
 
BC B C


(v) If

x 1 5
 , then x  2 and x  4.
x4 6
MASTA Proof and Reasoning 9-12 Module
Algebraic and Symbolic Reasoning
3. Find the error in the argument below.
xx
x  11  x  0
(1 x) 1  (x 1)  1
(1 x) 1  (x 1)  1
1 x 1  x  1 1
04
How would you describe this type of error?

Handout - Page 3 of 13
MASTA Proof and Reasoning 9-12 Module
Algebraic and Symbolic Reasoning
Handout - Page 4 of 13
4. Analyze the argument below.
xx
(x) 2  (x) 2
(x  5) 2  5 2  (x) 2
x 2  10x  25  25  x 2
x 2  10x  x 2
10x  0
x 0

Why does the last statement contradict the first?
What is the error?
How would you describe this type of error?
MASTA Proof and Reasoning 9-12 Module
Algebraic and Symbolic Reasoning
Handout - Page 5 of 13
5. Challenge problem (if time permits):
Find the error in the argument below.
24  24
16  40  36  60
16  40  25  36  60  25
4 2  2  4  5  52  62  2  6  5  52
(4  5) 2  (6  5) 2
4 5 65
46
6. Reasoningwith Magic Squares
Place the digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 in the square below so that the sum
of each digits along the each row, column and diagonal is the same?
MASTA Proof and Reasoning 9-12 Module
Algebraic and Symbolic Reasoning
Handout - Page 6 of 13
7. More Magic Squares
Do the entries in a magic square have to be 1, 2, …, n2 ?
What other sequences could you use?
Can the set of numbers {1, 3, 5, 7, 8, 10, 12, 14, 16} be used to form a 3 X 3
magic square? Or {0, 2, 3, 4, 5, 7, 8, 9, 11}?
In a 3X3 square, the center square is always ___________ of the magic sum.
What is the general form of a 3X3 magic square if a is the value of the center?
MASTA Proof and Reasoning 9-12 Module
Algebraic and Symbolic Reasoning
Mathematics as sensemaking (Schoenfeld, 1991):
A. establishing subgoals
B. working backwards
C. focusing on key points
D. exploiting extreme cases
E. exploiting symmetry
F. working forwards
G. using systematic generating procedures
8. What sort of mathematics did you use?
Handout - Page 7 of 13
MASTA Proof and Reasoning 9-12 Module
Algebraic and Symbolic Reasoning
Handout - Page 8 of 13
What types of mathematical reasoning did you use? (Think about last
session on Mathematical Reasoning)
9. Prior to manipulating symbols, students must use reasoning in determining
appropriate representations to _______________________________________________
___________________________________________________________________________________.
Many ideas in algebra can be expressed with multiple representations
i.e. Rule of 3 + 1
10. Reasoning is often used in ________________________________________________
________________________________________________________________________________.
MASTA Proof and Reasoning 9-12 Module
Algebraic and Symbolic Reasoning
Handout - Page 9 of 13
11. The graph and table below show how values of two functions change.
x
y
…
…
-3
1
-2
0
-1
-1
0
-2
1
-3
2
-4
3
-5
Choose functions from below that have a property in common with one or both of
the functions above. Explain your point of view and find as many viewpoints as you
can.
2
y x
3
1
y
x

Viewpoints:

y  x
y  2x  1
y  x2




1
y  x 2
2

y  x2
…
…
MASTA Proof and Reasoning 9-12 Module
Algebraic and Symbolic Reasoning
Handout - Page 10 of 13
12. Student Professor Problem
Using the letter S to represent the number of students (at a university) and
the letter P to represent the number of professors, write an equation that
summarizes the following sentence “There are six times as many students as
professors at this university”.
13. If a student answered 6S = P, how might you go about helping this student?
What areas do you think the student needs help with?
14. Clement (1982) suggested that the two essential competencies for solving
the problem are:
Recognizing that the letters represented quantities (notion of
variable)
Creating a “hypothetical operation” to make the two quantities (such
as the number of students and professors) equal.
MASTA Proof and Reasoning 9-12 Module
Algebraic and Symbolic Reasoning
15. Let’s prove the following:
1 + 3 + 5 + … + (2n-1) = n2
Proof that proves:
Handout - Page 11 of 13
MASTA Proof and Reasoning 9-12 Module
Algebraic and Symbolic Reasoning
Proof that explains:
1 + 3 + 5 + … + (2n-1) = n2
Why does this diagram justify the equation?
What must you assume is true for the diagram?
Handout - Page 12 of 13
MASTA Proof and Reasoning 9-12 Module
Algebraic and Symbolic Reasoning
Handout - Page 13 of 13
16. What equation is represented by the shapes below? Look closely at the
symbols as well.
17. Baseline Assessment
18. Interview Protocol