Transcript Ratio and Proportion Strategies

Algebraic and
Symbolic Reasoning
Today’s Agenda

Mathematical Reasoning: Review & Sharing

Analyze student errors in symbolic reasoning.

Look at reasoning with Magic Squares.

Investigate algebraic reasoning:
◦ Among different representations
◦ By using geometry

Discuss baseline assessment.
Reasoning Review

Pass around and look at samples of student work from
the Mathematical Reasoning baseline assessments and
student interviews.

Based on this work, share your condensed summaries
of your students’ reasoning abilities.
1.
2.
3.
4.
Which problems did your students perform well on?
Which did they struggle with?
What classroom activities did you use to foster reasoning?
Describe one situation where you saw students exhibit
growth in their reasoning abilities.
Symbolic Reasoning
Arithmetic with Symbols
Algebra can be described as a generalization of
arithmetic so that equations can be solved once for all
numbers. It allows us to make general statements
about a process without being tied to one specific
example.
The use of a symbol for a changeable value is a hard
conceptual leap for many students.
Algebraic Errors
Many, if not most, errors in algebra involve incorrect
manipulation of symbols.
When we see a student make these errors, we often
think, “This student has no number sense,” or “This
student can’t handle the abstract thinking required for
algebra.”
Example:
a  b 
a b
Algebraic Errors… are Reasonable?
a  b 
a b
This error results from the assumption that the square
root function is linear. That indicates a basic
misunderstanding of the square root funtion…

…but is understandable if you think about the hundreds,
or even thousands, of times students have used the
distributive property:
3(a  b)  3a  3b
(a  b)  a  b
Generalized Distribution
Matz’s Classification of High
School Algebra Errors
1. Errors generated by an incorrect choice of
an extrapolation technique
2. Errors reflecting an impoverished (but
correct) base knowledge
3. Errors in execution of a procedure.
Matz’s Classification of High
School Algebra Errors
1. Errors generated by an incorrect choice of
an extrapolation technique
Example:
a  b 
a b
Matz’s Classification of High
School Algebra Errors
2. Errors reflecting an impoverished (but
correct) base knowledge
Examples:
If 4 X  46, then X  6
4 5
1
If X  , then X 
3 3
3
Matz’s Classification of High
School Algebra Errors
3. Errors in execution of a procedure.
Example:
Solve forx :
5
5

4
2 x 2 x
5(2  x)  5(2  x)  4
Algebra Errors – Your Turn
Look at the common algebra errors on your
handout. With your group members, classify
them according to Matz’s framework.
Symbolic Reasoning
• Find the error in the argument below.
• How would you describe this type of error?
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
Symbolic Reasoning
• Why does the last statement contradict the first?
• What is the error?
• How would you describe this type of error?
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
Reasoning with Magic (Squares)
Reasoning with Magic Squares
Place the digits 1-9 in the box so that the sum along
any row, column, or diagonal is the same.
More Questions on Magic Squares
1.
2.
3.
4.
5.
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?
Reasoning with Magic Squares
Reasoning strategies (Schoenfeld, 1991):
A. Establish subgoals
• What can you say about the sum of any
diagonal, row or column?
• What is the important square to find first?
B. Working backwards
• Assume the sum of any column is S.
• Can you find the value of S?
C. Exploit extreme cases
• Try the large and small values around the
perimeter – what works?
Reasoning with Magic Squares
Reasoning strategies (Schoenfeld, 1991):
D. Exploit symmetry
• Balance large values with small ones on
either side of the center.
E. Work forward to try solutions.
F. Be systematic about trying cases.
Reasoning with Magic Squares

What sort of mathematics did you use?

What types of mathematical reasoning did
you use? (Think about last session)
Algebraic Reasoning
Algebraic Reasoning
Prior to manipulating symbols, students
must use reasoning in determining
appropriate representations to set up,
manipulate, and solve problems
 Many ideas in algebra can be expressed
with multiple representations

◦ i.e. Rule of 3 + 1

Reasoning is often used in shifting among
representations in algebra
Reasoning about Different Algebraic
Representations
The graph and table below show how values of two
functions change.
Choose functions from below that have a
property in common with one or both of the
functions above. Explain your point of
view. Find as many viewpoints as you can.
Student Professor Problem
Take a minute and answer the following
question:
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”.
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”.
•How many wrote 6S = P?
•Why is this incorrect? Discuss at your tables.
•Why might students answer this way? What
reasoning are they using?
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”.
Two possible explanations for incorrect answer of
6S = P (Schoenfeld, 1985:
1. A direct translation of the words into symbols. (Six
times students is professors)
2. Students may visualize the following “classroom”:
S
S
S
S
S
S
d
e
s
k
Professor
Student Professor Problem

If a student answered 6S = P, how might
you go about helping this student?
◦ Discuss at your tables.

What areas do you think the student
needs help with?
◦ Discuss at your tables.
Student Professor Problem

Clement (1982) suggested that the two
essential competencies for solving the
problem are:
1. Recognizing that the letters represented
quantities (notion of variable)
2. Creating a “hypothetical operation” to
make the two quantities (such as the
number of students and professors) equal.
Recall Day 1 Proofs

Let’s prove the following:
1 + 3 + 5 + … + (2n-1) = n2

Proof that proves (only):
◦ Use Mathematical Induction

Next: Proof that explains (and proves)…
Proof That Explains (and proves):
1 + 3 + 5 + … + (2n -1) = n2
The Geometry of Algebra
Geometric representations of algebraic
processes can be helpful
 For example, interpreting “n2” as
a square of dimensions, n x n
 Let’s try another…

Completing the Square
What equation is represented by the shapes below?
Baseline
Assessment
Baseline Assessment
1. Explain why the following argument results in a
false statement. Be sure to identify which step(s)
are untrue.
x x
x 2 x 2
4(x  2)  4(x  2)
4(x  2)  4(x  2 3) 3
4(x  2)  4(x  1) 3
4x  8  4x  4  3
8  1
9 0
Baseline Assessment
2. The graph and table below show how values of
two functions change.
Choose functions from below that have a
property in common with one or both of the
functions above. Explain your point of
view. Find as many viewpoints as you can.