Developing a Lens of Structure to Build Mathematical Meaning
Casey Hawthorne
San Diego State University
caseyhawthorne@yahoo.com
CCSS.MATH.PRACTICE.MP1
CCSS.MATH.PRACTICE.MP2
CCSS.MATH.PRACTICE.MP3
CCSS.MATH.PRACTICE.MP4
CCSS.MATH.PRACTICE.MP5
CCSS.MATH.PRACTICE MP6
CCSS.MATH.PRACTICE.MP7
CCSS.MATH.PRACTICE.MP8
Beth Rackliffe
La Mesa Spring Valley SD
beth780@gmail.com
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically
Attend to precision
Look for and make use of structure.
Look for and express regularity in repeated reasoning
Mathematically proficient students look closely to discern a pattern or structure. Young
students, for example, might notice that three and seven more is the same amount as seven
and three more, or they may sort a collection of shapes according to how many sides the
shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in
preparation for learning about the distributive property. In the expression x2 + 9x + 14, older
students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an
existing line in a geometric figure and can use the strategy of drawing an auxiliary line for
solving problems. They also can step back for an overview and shift perspective. They can see
complicated things, such as some algebraic expressions, as single objects or as being
composed of several objects. For example, they can see 5 – 3(x -y)2 as 5 minus a positive
number times a square and use that to realize that its value cannot be more than 5 for any real
numbers x and y.
Components of Structural Sense
a) Seeing a mathematical expression as an entity, not as a process
b) Recognizing connections among different mathematical forms and ideas-identifying an
expression as a familiar structure
c) Dividing an entity into a variety of sub-structures according to context
d) Stopping to look at an expression to choose an appropriate manipulation that makes best
use of the structure, instead of automatically applying a set procedure.
Cover Up Method
Solve the following for x in two different ways
50
11 −
=6
𝑥−2
Seeing Structure in Notation
What do you see in the number 23?
What are the different ways one might see the number of tens in 347? 4 tens? 34 tens? 34.7 tens?
87 + 35
Algorithm
Base 10 structure
Number line structure
Compensation
Number Line Structure
Compensation
Structure
Algorithm
8
7
+3
+5
Base 10 Structure
(80 + 30 )+ (7 + 5)
3
87 90
90 + 40 = 130
130 – 8 = 122
22
10
100
122
Number Strings
a) 6 x 25
b) 16 x 25
c) 16 x 35
Algebra Strings
Knowing that x = 16 is a solution to
2x + 5 = 37, what are the solutions to
a) 2(x – 3) + 5 = 37
b) 2x2 + 5 = 37
My Number/Algebra String Problem
Relational Thinking
Solve the following for ⎕
a) 37 + 54 = ⎕ + 55
b) 98 + 48 – 48 = ⎕
c) 98 + 74 + 2 = ⎕
Jacobs, V. R., Franke, M. L., Carpenter, T. P., Levi, L., & Battey, D. (2007). Professional
development focused on children's algebraic reasoning in elementary school. Journal
for research in mathematics education, 258-288.
Equation
1
1
1
(1 − 𝑛+2) − (1 − 𝑛+2) = 132
1
1
1
1 − 𝑛+2 − 1 + 𝑛+2 = 72
1
𝑥
1
𝑥
(4 − 𝑥−1) − 𝑥 = 6 + (4 − 𝑥−1)
1
4
𝑥
1
𝑥
− 𝑥−1 − 𝑥 = 6 + 4 − 𝑥−1
Advanced
Intermediate
23% (3/13)
12.5% (2/16)
0% (0/16)
0% (0/15)
25% (4/16)
11.8% (2/17)
18.8% (3/16)
6.3% (1/16)
Hoch, M., & Dreyfus, T. (2004). Structure sense in high school algebra: The effect of
brackets. In Proceedings of the 28th Conference of the International Group for the
Psychology of Mathematics Education (Vol. 3, pp. 49-56).
My Relational Thinking Problem
Attending to structure allows you to see connections
Simplification
Are any of the following equal to 1? Why or why not?
𝑡+3
a) 𝑡+2 = 1
b)
𝑡+3
𝑡−3
=1
c)
𝑡−3
3−𝑡
=1
There is no simplest form.
Each form has different meanings and should be appreciated as such. Think of three forms of
quadratic
Factoring
a) (2x + 3)2 – 12(2x + 3) + 36
d) Prove that
𝑘(𝑘+1)(𝑘+2)
3
b) (2x – 3)(x2 – 4)
+ (𝑘 + 1)(𝑘 + 2) =
c) (x + 3)2 – (x – 3)2
(𝑘+1)(𝑘+2)(𝑘+3)
3
Solving
6
a) Solve the following for x
𝑥−6
𝑥
9
− 𝑥−6 = 9−𝑥
a) 3x + 6y = 4
b) 4y – 3(x – 2) = 7
2x – 3y = -5
2x – 4 = 5y
2 2
c) Solve the following for 2xy
12xy – 4x y = 34xy – 13
d) Solve the following for v
v√𝑢 = 1 + 2𝑣√1 + 𝑢
2
e) Knowing the solutions to 6x – 2x = 17x – 13 are x = 1 and -6.5,
what are the solutions to 12x2 – 4x4 = 34x2 – 26?
b) Solve the following using substitution
Chunking:
How can we compare/relate the different compounding interest formulas? While the all look
very different, each of these expressions have the same structure.
P(1 + r)t
𝑃(1 + 𝑟⁄𝑛)𝑛𝑡
Pert
P(1 + r)t
𝑃((1 + 𝑟⁄𝑛)𝑛 )𝑡
P(𝑒 𝑟 )𝑡
Imagine a rate of 5% compounded daily, monthly and continuously
12 𝑡
P(1 + .05)t
𝑃 ((1 + . 05⁄12) )
P(𝑒 .05 )𝑡
P(1.05)t
P(1.05116)t
P(1.05127)t
Graphing
What number makes sense to plug in, in order to see the y-value very easily?
y = -2x + 4
when we plug in x = 0, then y = 4, giving us the y-intercept of (0, 4)
y = 3(x – 2) + 1
when we plug in x = 2, then y = 4, giving us the point (2, 1)
From this perspective, the slope intercept and point-slope forms of linear equations have very
similar structure.
Flexibility
What other results can be deduced from the following product?
23 x 35 = 805
Connecting Algebra with Representation
Consider the algebraic expressions below:
(n+2)2 – 4
n2+4n
a. Use the figures below to illustrate why the expressions are equivalent
b. Use an algebraic method to verify the same result.
c. Can you show that n2+4n = (n+4)n using the diagram?
Add-Subtract-Multiply-Divide
a) 3476 + 2456 – 1476
b) 99 – 17
c) 99 × 17
d) 165/15
e) 6 × 43 + 6 × 7
f)50 x 35 x 2 ÷ 5
g) 125 × 13 × 80
3 1 4
h)  
7 9 7
5
i) 13
10
13
j) [ 1 (3  2 )  5  4 ( 5  2)]( 6  18 )
3
3
7 2
3
Mental Percentages
a) 15% of 60
d) 40% of 35
g) 96% of 200
b) 25% of 320
e) 90% of 500
h) 96% of 150
c) 4% of 400
f) 120% of 50
i) 96% of 75
Sequences
a) 1  2  3  4  5  6  7  8  9
2 3
4 5
6
7 8
9 10

c)
b) (1 +
1
1
1
1
)(1 + )(1 + )…(1 +
)
2
3
10
1
1
1

(telescoping series)
n n2

n 1
Integration
a) ∫ 𝑥√𝑥 2 + 1𝑑𝑥
c) Given that
i.

3
1

3
1
b) ∫ 𝑥 √𝑥 + 1𝑑𝑥
g ( x )dx = 10, deduce the value of
1
g ( x)dx;
2
ii.

3
1
( g ( x)  4)dx.
Quadratics
a) Can you see the vertex in the quadratic formula?
−𝑏±√𝑏2 −4𝑎𝑐
2𝑎
b) Can you see in the quadratic formula that the sum of roots is –b/a, or product of roots is c/a
(Vieta’s formula)
c) Can you see the factors forming a rectangle with integer sides? a square? Neither?
x2 + 2x + 1 = (x + 2)(x + 1)
x2 + 2x + 1 = (x + 1)2
x2 + x + 1
Gurl, T., Artzt, A., Sultan, A., & Curcio, F. (2014). Implementing the Common Core State
Standards through Mathematical Problem Solving: High School. Reston, VA:
National Council of Teachers of Mathematics.
Functional decomposition-Domain and Range
25
a) 𝑔(𝑥) = 8√1 − 𝑥 2
25
Domain: 𝑥 2 cannot be bigger than 1, or the radical will be irrational. So x must be greater than
5 or less than -5.
25
Range: The biggest number that 1 − 𝑥 2 can produce is 1. So the range will be less than or
equal to 8. The smallest value will be 0 which occurs when
25
𝑥2
gets closer to 1.
b) 𝑓(𝑥) = √𝑥 − 1 + 3
Domain: We know what’s inside a square root must be positive or zero. What is the largest
number that x can be so that x – 1 is positive? 1
Range: we first have to identify that the square root only produces positive numbers (0 to
infinity). If we increase the possibilities by 3, we get 3 to infinity.
c) ℎ(𝑥) = √𝑙𝑜𝑔3 𝑥
Domain: We can only take the square root of positive numbers. So the output of the log can
only be 0 to infinity. When will log’s be positive? When the argument is 1 or larger.
Range: Logs will produce all positive numbers and zero. So the range greater than or equal
to zero.