Multiplicative
Thinking
cc-by-sa 3.0 unported unless
otherwise noted
Ted Coe, Ph.D
Director, Mathematics
Achieve, Inc.
2/5/2015
The Rules of Engagement
Speak meaningfully — what you say should carry meaning;
Exhibit intellectual integrity — base your conjectures on a logical
foundation; don’t pretend to understand when you don’t;
Strive to make sense — persist in making sense of problems and
your colleagues’ thinking.
Respect the learning process of your colleagues — allow them
the opportunity to think, reflect and construct. When assisting
your colleagues, pose questions to better understand their
constructed meanings. We ask that you refrain from simply
telling your colleagues how to do a particular task.
Marilyn Carlson, Arizona State University
Too much math never killed anyone.
The Plot
Teaching and Learning Mathematics
Ways of doing
Ways of thinking
Habits of thinking
The Foot
From http://www.healthreform.gov/reports/hiddencosts/index.html (6/3/2011)
The Broomsticks
The Broomsticks
The RED broomstick is three feet long
The YELLOW broomstick is four feet long
The GREEN broomstick is six feet long
From the CCSS: Grade 3
Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington,
DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. P.24
From the CCSS: Grade 4
4.OA.1, 4.OA.2
1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as
a statement that 35 is 5 times as many as 7 and 7 times as many as 5.
Represent verbal statements of multiplicative comparisons as multiplication
equations.
2. Multiply or divide to solve word problems involving multiplicative comparison,
e.g., by using drawings and equations with a symbol for the unknown number
to represent the problem, distinguishing multiplicative comparison from
additive comparison.
Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington,
DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. P.29
From the CCSS: Grade 4
4.OA.1, 4.OA.2
1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as
a statement that 35 is 5 times as many as 7 and 7 times as many as 5.
Represent verbal statements of multiplicative comparisons as multiplication
equations.
2. Multiply or divide to solve word problems involving multiplicative comparison,
e.g., by using drawings and equations with a symbol for the unknown number
to represent the problem, distinguishing multiplicative comparison from
additive comparison.
Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington,
DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. P.29
From the CCSS: Grade 5
5.NF.5a
Interpret multiplication as scaling (resizing), by:
Comparing the size of a product to the size of one factor on the
basis of the size of the other factor, without performing the
indicated multiplication.
Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington,
DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. P.36
http://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdf
“In Grades 6 and 7, rate, proportional relationships and linearity
build upon this scalar extension of multiplication. Students who
engage these concepts with the unextended version of
multiplication (a groups of b things) will have prior knowledge
that does not support the required mathematical coherences.”
Learning Trajectories
Learning Trajectories
Perimeter
What is “it”?
Is the perimeter a measurement?
…or is “it” something we can measure?
Perimeter
Is perimeter a one-dimensional, twodimensional, or three-dimensional thing?
Does this room have a perimeter?
From the AZ STD's (2008)
Perimeter: the sum of all
lengths of a polygon.
Discuss
Wolframalpha.com
4/18/2013:
Measurement
What do we mean when we
talk about “measurement”?
Measurement
•“Technically, a measurement is a number that
indicates a comparison between the attribute of
an object being measured and the same attribute
of a given unit of measure.”
–Van de Walle (2001)
•But what does he mean by “comparison”?
Measurement
How about this?
•Determine the attribute you want to
measure
•Find something else with the same
attribute. Use it as the measuring unit.
•Compare the two: multiplicatively.
From Fractions and Multiplicative Reasoning, Thompson and Saldanha, 2003. (pdf p. 22)
Create your own…
International standard unit of length. With a rubber band.
1. Use it to measure something.
2. Use it to measure the length of someone else’s band.
3. Use their band to measure yours.
What is a circle?
Draw a circle
with a diameter equal to your international standard unit band
length
What is circumference?
Circumference
From the AZ STD's (2008)
the total distance
around a closed curve
like a circle
Circumference
•So.... how do we measure
circumference?
The circumference is three and a bit times as large as the diameter.
http://tedcoe.com/math/circumference
 The circumference is about how many times as large as the diameter?
 The diameter is about how many times as large as the circumference?
Tennis Balls
Circumference
If I double the RADIUS of a circle what happens to the
circumference?
How many Rotations?
Angles
•What is an angle?
Angles
•Using objects at your table
measure the angle
•You may not use degrees.
•You must focus on the attribute
you are measuring.
Angles
•What attribute are we measuring
when we measure angles?
•Think about: What is one degree?
Grade 4 CCSS: 4.MD.5
Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington,
DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. P.31
http://tedcoe.com/math/radius-unwrapper-2-0
What is the length of “d”?
You may choose the unit.
What is the measure of the angle?
You may choose the unit.
Indiana (1896)
House Bill 296, Section 2:
“…that the ratio of the diameter and circumference is as fivefourths to four;”
What is the mathematical value they are proposing for Pi?
From http://www.agecon.purdue.edu/crd/Localgov/Second%20Level%20pages/indiana_pi_bill.htm
5
Illustration: 4 ÷
4
Define: Area
Area has been defined* as the following:
“a two dimensional space measured by the
number of non-overlapping unit squares or
parts of unit squares that can fit into the
space”
Discuss...

State of Arizona 2008 Standards Glossary
*
Area: Grade 3 CCSS
Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington,
DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. P.21
What about the kite?
http://geogebratube.org/material/show/id/279
http://geogebratube.org/student/m279 (cc-by-sa)
Geometric Fractions
Check for Synthesis:
If
𝟐
𝟑
= . What is 1?
How can you use this to show that
𝟏
𝟐
𝟑
=
𝟑
𝟐
?
Geometric Fractions
Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington,
DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. P.42
Assume:
• Angles that look like
right angles are right
angles
• Lengths that look to be
the same as AB can be
verified using a compass
•
•
•
Find the dimensions
of the rectangle
Find the area of the
rectangle
Find a rectangle
somewhere in the
room similar to the
shaded triangle
Or not…
http://goldenratiomyth.weebly.com/
When I say two
are are similar I
Whenfigures
I say two figures
mean…
similar I mean…
Hint: We haven’t defined “proportional”
so you cannot use it.
What is a
scale factor?
Teaching Geometry According to the Common Core Standards, H. Wu Revised: April 15,
2012. Grade 7 notes, p.49:
Working with similar figures
“Similar means same shape
different size.”
“All rectangles are the same
shape. They are all rectangles!”
“Therefore all rectangles are
similar.”
CCSS: Grade 2 (p.17)
CCSS: Grade 2 (p.17)
CCSS: Grade
2 (p.17)
Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington,
DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. P.17
CCSS: Grade 7 (p.46)
CCSS: Grade 7 (p.46)
Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington,
DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. P.46
CCSS: Grade 7 (p.46)
CCSS: Grade 7 (p.46)
Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington,
DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. P.46
From
Progressions
From
the the
Progressions
ime.math.arizona.edu/progressions
https://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf p.11
CCSS: Grade 8 (p.56)
CCSS: Grade
8 (p.56)
Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington,
DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. P.56
Teaching Geometry According in Grade 8 and High School According to the Common Core Standards, H. Wu Revised: October 16,
2013, p.45 http://math.berkeley.edu/~wu/CCSS-Geometry.pdf
CCSS: HS Geometry (p.74)
CCSS: Geometry (G-SRT.6, p. 77)
Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington,
DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. P.77
Assume
http://tedcoe.com/math/geometry/similar-triangles
CCSS: HS Geometry (p.74)
CCSS: HS
Geometry
(p.74)
Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC:
National Governors Association Center for Best Practices and the Council of Chief State School Officers. P.36
CCSS: HS Geometry (p.74)
CCSS: HS
Geometry
(p.74)
From http://www.healthreform.gov/reports/hiddencosts/index.html (6/3/2011)
What does it mean to say something is
“out of proportion”?
“A single proportion is a relationship between
two quantities such that if you increase the size
of one by a factor a, then the other’s measure
must increase by the same factor to maintain
the relationship”
Thompson, P. W., & Saldanha, L. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, G. Martin & D. Schifter
(Eds.), Research companion to the Principles and Standards for School Mathematics (pp. 95-114). Reston, VA: National
Council of Teachers of Mathematics.(p.18 of pdf)
• On the Statue of Liberty the
distance from heel to top of
head is 33.86m
• How wide is her mouth?
http://www.nps.gov/stli/historyculture/statue-statistics.htm
Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington,
DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. P.77
Find all lengths
and areas.
Note: Points A,B,
and C are the
centers of the
indicated circular
arcs.
Volume
What is “it”?
http://tedcoe.com/math/cavalieri
Connection to Algebra
http://tedcoe.com/math/algebra/constant-rate
http://tedcoe.com/math/algebra/constant-rate
http://tedcoe.com/math/algebra/constant-rate
http://tedcoe.com/math/algebra/constant-rate
CCSS: Grade
8 (p.54)
Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington,
DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. P.54
From the progressions documents
Source: http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_functions_2013_07_02.pdf p.5
Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington,
DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. P.70
You have an investment account that grows from $60 to $103.68
over three years.
A tangent:
The first proof of the existence of irrational numbers
is usually attributed to
a Pythagorean (possibly Hippasus of
Metapontum),who probably discovered them while
identifying sides of the pentagram.The then-current
Pythagorean method would have claimed that there
must be some sufficiently small, indivisible unit that
could fit evenly into one of these lengths as well as
the other. However, Hippasus, in the 5th century BC,
was able to deduce that there was in fact no
common unit of measure, and that the assertion of
such an existence was in fact a contradiction.
http://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012
577

408
Cut this into 408 pieces
Copy one piece 577 times
It will never be good enough.
577
 1.4142156
408
Hippasus, however, was not lauded for his
efforts:
http://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012
Hippasus, however, was not lauded for his
efforts: according to one legend, he made his
discovery while out at sea,
http://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012
Hippasus, however, was not lauded for his
efforts: according to one legend, he made his
discovery while out at sea, and was subsequently
thrown overboard by his fellow Pythagoreans
“…for having produced an element in the
universe which denied the…doctrine that all
phenomena in the universe can be reduced to
whole numbers and their ratios.”
http://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012
Too much math never killed anyone.
…except Hippasus
Archimedes died c. 212 BC …According to
the popular account given by Plutarch,
Archimedes was contemplating a
mathematical diagram when the city was
captured. A Roman soldier commanded
him to come and meet General Marcellus
but he declined, saying that he had to
finish working on the problem. The soldier
was enraged by this, and killed
Archimedes with his sword.
http://en.wikipedia.org/wiki/Archimedes. 11/2/2012
The last words attributed
to Archimedes are
"Do not disturb my circles"
http://en.wikipedia.org/wiki/Archimedes. 11/2/2012
Domenico-Fetti Archimedes 1620 http://en.wikipedia.org/wiki/Archimedes#mediaviewer/File:DomenicoFetti_Archimedes_1620.jpg
Too much math never killed anyone.
…except Hippasus
…and Archimedes
Teaching and Learning Mathematics
Ways of doing
Ways of thinking
Habits of thinking
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of
others.
Habits of
Thinking?
4. Model with mathematics.
5. Use appropriately tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
 Mathematical Practices from the CCSS
Habits?
How did I do?
Creative
Commons
http://creativecommons.org
Contact
Ted Coe
tcoe@achieve.org
tedcoe.com
@drtedcoe
Achieve
1400 16th St NW, Suite 510
Washington, DC. 20036
202-641-3146