Multiplicative Thinking Ted Coe, Ph.D Director, Mathematics Achieve, Inc. cc-by-sa 3.0 unported unless otherwise noted

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.

08/13/09 08/13/09 The Plot 4

Teaching and Learning Mathematics Ways of doing Ways of thinking Habits of thinking

The Foot

From http://www. healthreform. gov/reports/hiddencosts/index

08/13/09 8 08/13/09 The Broomsticks 8

The Broomsticks 08/13/09 The RED broomstick is three feet long The YELLOW broomstick is four feet long The GREEN broomstick is six feet long 9

Source: http://tedcoe

Source: http://tedcoe

Source: http://tedcoe

08/13/09 1313 08/13/09 13

Source:

From the CCSS: Grade 3 Source: CCSS Math Standards, Grade 3, p. 24 (screen capture)

From the CCSS: Grade 4 4.OA.1, 4.OA.2 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. 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: CCSS Grade 4

From the CCSS: Grade 4 4.OA.1, 4.OA.2 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. 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: CCSS Grade 4

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: CCSS Grade 5

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.” Source: Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction. Daro, et al., 2011. p.49

Learning Trajectories From Learning Trajectories in Mathematics (2011) Daro, et al. CPRE.

Learning Trajectories From Learning Trajectories in Mathematics (2011) Daro, et al. CPRE.

Perimeter What is “it”? Is the perimeter a measurement? 08/13/09 2323 08/13/09 08/13/09 Perimeter What is “it”? Is the perimeter a measurement? …or is “it” something we can measure? 23 23

2424 08/13/09 08/13/09 Perimeter Is perimeter a one-dimensional, two- dimensional, or three-dimensional thing? Does this room have a perimeter? 24

08/13/09 2525 08/13/09 08/13/09 25 25

08/13/09 2626 08/13/09 08/13/09 26 26

Perimeter: the sum of all lengths of a polygon. 08/13/09 2727 08/13/09 08/13/09 From the AZ STD's (2008) Perimeter: the sum of all lengths of a polygon. Discuss 27 27

Wolframalpha.com 4/18/2013:

Measurement What do we mean when we talk about “measurement”? 08/13/09 2929 08/13/09 Measurement What do we mean when we talk about “measurement”?

3030 08/13/09 08/13/09 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”? 30

Measurement Determine the attribute you want to measure 3131 08/13/09 08/13/09 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. 31

From Fractions and Multiplicative Reasoning, Thompson and Saldanha, 2003. (pdf p. 22)

Create your own… With a rubber band. International standard unit of length. With a rubber band. Use it to measure something. Use it to measure the length of someone else’s band. Use their band to measure yours.

What is a circle?

What is circumference?

Circumference From the AZ STD's (2008) 3737 08/13/09 Circumference From the AZ STD's (2008) the total distance around a closed curve like a circle 37

3838 08/13/09 Circumference So.... how do we measure circumference? 38

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? 08/13/09 08/13/09 The circumference is about how many times as large as the diameter? The diameter is about how many times as large as the circumference? 41 41

Tennis Balls

Circumference If I double the RADIUS of a circle what happens to the circumference?

How many Rotations?

4646 08/13/09 Angles What is an angle? 46

Angles Using objects at your table measure the angle 4747 08/13/09 Angles Using objects at your table measure the angle You may not use degrees. You must focus on the attribute you are measuring. 47

Angles What attribute are we measuring when we measure angles? 4848 08/13/09 Angles What attribute are we measuring when we measure angles? Think about: What is one degree? 48

CCSS, Grade 4, p.31

Source:

http://tedcoe.com/math/radius-unwrapper-2-0

What is the length of “d”? You may choose the unit. 5252 What is the length of “d”? You may choose the unit.

What is the measure of the angle? You may choose the unit. 5353 What is the measure of the angle? You may choose the unit.

Define: Area 54

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 55

Area: Grade 3 CCSS

57

What about the kite?

Area of whole square is 4r^2 08/13/09 08/13/09 Area of whole square is 4r^2 Area of red square is 2r^2 Area of circle is… 65 65

Are of whole square is 4r^2 08/13/09 08/13/09 Are of whole square is 4r^2 Area of red square is 2r^2 Area of circle is… 66 66

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 ? Source:

Geometric Fractions

Angles that look like right angles are right angles 08/13/09 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 72

Find the dimensions of the rectangle Find the area of the rectangle 08/13/09 Find the dimensions of the rectangle Find the area of the rectangle Find a rectangle somewhere in the room similar to the shaded triangle 73

Or not… http://goldenratiomyth.weebly.com/

When I say two figures are similar I mean… Hint: We haven’t defined “proportional” so you cannot use it.

Using only your meaning of similarity (and no formulas) explain how to find x. (Assume the figures are similar)

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.”

From http://www. healthreform. gov/reports/hiddencosts/index

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)

CCSS: Geometry (G-SRT.6, p. 77) Source: CCSS High School Geometry (screen capture)

Assume http://tedcoe.com/math/geometry/similar-triangles

On the Statue of Liberty the distance from heel to top of head is 33 On the Statue of Liberty the distance from heel to top of head is 33.86m How wide is her mouth? 3 feet

8686 http://www.nps.gov/stli/historyculture/statue-statistics.htm 08/13/09 http://www.nps.gov/stli/historyculture/statue-statistics.htm 86

Source:

Source: http://tedcoe

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.56)

CCSS: HS Geometry (p.74)

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

What is a scale factor? Got to here on Saturday 2. Teaching Geometry According to the Common Core Standards, H. Wu Revised: April 15, 2012. Grade 7 notes, p.49:

CCSS: HS Geometry (p.74)

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

Source:

You have an investment account that grows from $60 to $103 You have an investment account that grows from $60 to $103.68 over three years. Source:

An informal approach…

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

Cut this into 408 pieces Copy one piece 577 times It will never be good enough.

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:Domenico-Fetti_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

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 appropriately tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Mathematical Practices from the CCSS Habits of Thinking?

How did I do? Habits?

Creative Commons http://creativecommons.org

www.achieve.org/equip EQuIP

EQuIP Quality Review: Process & Dimensions The EQuIP quality review process is a collegial process that centers on the use of criteria-based rubrics for mathematics. The criteria are organized into four dimensions: The Four Dimensions Alignment to the depth of the CCSS Key shifts in the CCSS Instructional supports Assessment

The EQuIP Rubric for Mathematics

Contact Ted Coe tcoe@achieve.org tedcoe.com @drtedcoe Achieve 1400 16th St NW, Suite 510 Washington, DC. 20036 202-641-3146