Middle School Math
Concepts:
“Is this reasonable?”
Esther Kim
estherkim39@gmail.com
2011 API By Subgroup
Subgroup
API
% of school
population
All Students
736
596
100%
5%
963
895
730
2%
3%
84%
784
726
5%
69%
English Language
Learners
668
19%
Students with Disabilities
546
African American
Asian
Filipino
Hispanic
White
Socio-economically
disadvantaged
Percent of students scoring
proficient or advanced
Parent
Education
6th grade
math
7th grade
Pre-algebra
8th grade
General
Math
8th grade
Algebra 1
Not a high
school
graduate
8%
36%
0%
35%
High school
graduate
22%
49%
10%
34%
Some college
(includes AA
degree)
24%
51%
25%
34%
College
graduate
39%
55%
No students
available in
this category
44%
Over 50% or under 50%?
Thumbs up if you think over 50%
Thumbs down if you think under 50%
Over 50% or under 50%?
2–6+3–1
a) -3
c) 3
b) -2
d) 6
Over 50% or under 50%?
2–6+3–1
a) -3
c) 3
b) -2
d) 6
81.5% of the students
responded correctly.
2–6+3–1
a) -3
c) 3
b) -2
d) 6
Over 50% or under 50%?
45% of 61.4 is a number between
a) 3 and 30
b) 30 and 60
c) 60 and 240
d) 240 and 2800
Over 50% or under 50%?
45% of 61.4 is a number between
a) 3 and 30
b) 30 and 60
c) 60 and 240
d) 240 and 2800
Approximately 20% of the
students chose C or D.
45% of 61.4 is a number between
a) 3 and 30
b) 30 and 60
c) 60 and 240
d) 240 and 2800
(0.45)(61.4)=27.63
Over 50% or under 50%?
Which of the following is the largest?
a) 3/4
c) 12/13
b) 6/7
d) 17/25
Over 50% or under 50%?
Which of the following is the largest?
a) 3/4
c) 12/13
b) 6/7
d) 17/25
31% of the students chose
A.
Which of the following is the largest?
a) 3/4
c) 12/13
b) 6/7
d) 17/25
Over 50% or under 50%?
(5/6)(30)
a) 25
c) 150
b) 36
d) 156
70.8% of the students
responded correctly.
(5/6)(30)
a) 25
c) 150
b) 36
d) 156
35.8% of the students responded
correctly, and 32.5% of the students
chose A.
Order the following from least to greatest.
3/8, 1/3, 0.36, 0.39
a) 1/3, 0.36, 0.39, 3/8
3/8, 1/3
b) 0.36, 0.39,
c) 1/3, 0.36, 3/8, 0.39
0.39, 1/3
d) 0.36, 3/8,
What has worked?
Multiple Representations: One size does
not fit all
–
–
–
–
Pictures, Charts, Tables
Verbal
Algebraic
Numeric
Connecting concepts within lessons
Math 7
Subgroup
Black, not
Latino
Hispanic, not
White
Special
Education
% proficient
or advanced
(whole
school)
29%
% proficient
or advanced
(my
students)
60%
48%
64%
18%
36%
Need
Challenge
Possible
Solution
More opportunities Teachers did not
for students to
learn this way
engage conceptually
and build
automaticity
through relatable
experiences or
manipulatives
Teachers need
training, spaces, and
time to practice
Conceptual games
and apps
Forums for
professionals in
different sectors to
work together
Availability and
limitations of
expertise
Critical Concepts in Middle
School Mathematics
Mark Ellis
CSU Fullerton
mellis@fullerton.edu
*Note: If you have an iPhone or Andriod phone
with data access, please install the free Socrative
Student Clicker app. My room # is 51016
Once you’re in the “room” feel free to text in
your thoughts and questions.
Shifting Focus
• Traditional U.S.
– How can I teach my kids to get the
answer to this problem?
– Evaluate answers to determine proficiency.
• Common Core
– How can I use this problem to teach the
mathematics of this unit?
– Examine responses to uncover student thinking
and inform next steps pedagogically as all
students move toward big idea(s) of a unit.
Reasoning about Division
• Divide 365/4 (by hand) using three methods.
Reasoning about Division
Sense Making: Fraction
Division
1. What question might this expression
answer?
2. Find the quotient in a way that
makes sense.
3. Share your reasoning with your
neighbor(s) and/or send to Socrative
#51016
– What justifies your method?
– What prior knowledge is needed?
1 1
1 
2 3
Learning Trajectories
K -2
3-6
Rates, proportional
and linear
relationships
Equal
Partitioning
Unitizing in
base 10 and in
measurement
Number line in
Quantity and
measurement
Properties of
Operations
From Phil Daro, CMC-S
7 - 12
Rational number
Fractions
Rational
Expressions
Two “Big Ideas”
• Equivalence
• Proportionality
Misconception about
Equality
If you learned to interpret the equal
sign as an operation,
•3+8=
• 23 x 7 =
how would you make sense of these?
• 4 x 97 = 4 (90 + 7)
• 2x – 7 = x + 11
Reasoning about One-half
Euclid’s Algebra Challenge
• Model this geometrically, then with
algebraic symbols.
• If a straight line segment be cut at
random, the square on the whole is
equal to the squares on the segments
and twice the rectangle contained by the
segments.
(Euclid, Elements, II.4, 300 B.C.)
How Not to Learn
Proportional Reasoning
• What is not
developed
when
students
“learn” this
first?
• Why does
this
algorithm
work?
Reasoning Proportionally
•
John’ s mixture was 3 spoonfuls of sugar and 12
spoonfuls of lemon juice. Mary’ s mixture was 4
spoonfuls of sugar and 13 spoonfuls of lemon
juice. Whose lemonade is sweeter, John’ s or
Mary’ s? Or would they taste equally sweet?
•
Eva and Alex want to paint the door of their
garage. They mix 2 cans of white paint and 3
cans of black paint to get a particular shade of
gray. They then add one more can of each
color. Will the new shade of gray be lighter,
darker, or the same?
Role of Technology
Support and advance
mathematical sensemaking, reasoning,
problem solving, and
communication
http://www.skill-guru.com/gmat/wpcontent/uploads/2011/02/thinking-cap.gif
(NCTM Position Statement, 2011)
http://www.nctm.org/about/content.aspx?id=31734
Rhythm Wheel: Exploring
Multiplicative Reasoning
After 4 loops of this wheel:
1. How many individual sounds will
be played? How do you know
this?
2. How many times will the “open”
hand drum sound
be played?
How do you know?
3. How many times more will the
“open” hand drum sound be
played than the “slap” drum
sound? How do you know?
Rhythm Wheel (part 2)
1. How many loops would the 1st and 2nd
wheels each need to make so they stop
playing at the same time? Why?
2. If the 2nd wheel does 18 loops, how
many loops will the 1st wheel need to
make so it stops at the same time?
3. If the neck cowbell
sound gets
played 30 times, how many times will
the open hand drum sound
be
played? Prove it! (Assume the wheels
stop at the same time.) What if the
neck is played x times?
Proportionality and Linear
Functions
• If two quantities vary proportionally, that relationship
can be represented as a linear function.
• If two quantities vary proportionally,
– the ratio of corresponding terms is constant,
– the constant ratio can be expressed in lowest terms
(a composite unit) or as a unit amount, and
– the constant ratio is the slope of the related linear
function.
• When you graph the terms of equal ratios as ordered
pairs (first term, second term) and connect the points,
the graph is a straight line.
CCSS-Math: Content Domains and Conceptual Categories
K
1
2
3
4
5
6
7
8
HS
Counting
&
Cardinality
Number and Operations in Base Ten
Number and
OperationsFractions
Ratios and
Proportional
Relationships
Number &
Quantity
The Number System
Operations and Algebraic Thinking
Expressions and
Equations
Algebra
Functions
Geometry
Measurement and Data
Statistics and Probability
Two Categories of Tools
Content Exploration
Communication &
Collaboration
Resources
• Charles, R. (2005). Big Ideas and Understandings as the Foundation for
Elementary and Middle School Mathematics.
http://www.authenticeducation.org/bigideas/sample_units/math_samples/BigId
eas_NCSM_Spr05v7.pdf
• Learning Progressions/Trajectories in Mathematics
– http://ime.math.arizona.edu/progressions/
– http://www.turnonccmath.com/
– http://www.cpre.org/images/stories/cpre_pdfs/learning%20trajectories%20in%20math_ccii%2
0report.pdf
• Math Reasoning Inventory, https://mathreasoninginventory.com/
• Seigler, R. (2012). Knowledge of Fractions and Long Division Predicts LongTerm Math Success http://youtu.be/7YSj0mmjwBM and
http://www.psychologicalscience.org/index.php/news/releases/knowledge-offractions-and-long-division-predicts-long-term-math-success.html
• To Half or Not lesson,
http://www.pbs.org/teachers/mathline/lessonplans/pdf/esmp/half.pdf