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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; .

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Multiplicative Thinking






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Slide1

Multiplicative Thinking

cc-by-

sa 3.0 unported unless otherwise noted

Ted Coe, Ph.D

Director, Mathematics

Achieve, Inc.

2/5/2015Slide2

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 Slide3

Too much math never killed anyone.Slide4

The PlotSlide5

Teaching and Learning Mathematics

Ways of doing Ways of thinkingHabits of thinkingSlide6

The FootSlide7

From http://www.healthreform.gov/reports/hiddencosts/index.html (6/3/2011)Slide8

The BroomsticksSlide9

The RED broomstick is three feet long

The YELLOW broomstick is four feet long

The GREEN broomstick is six feet long

The BroomsticksSlide10

Source

: http://tedcoe.com/math/wp-content/uploads/2013/10/broomsticks-for-nctm.docSlide11

Source

: http://tedcoe.com/math/wp-content/uploads/2013/10/broomsticks-for-nctm.docSlide12

12

Source: http://tedcoe.com/math/wp-content/uploads/2013/10/broomsticks-for-nctm.docSlide13
Slide14

14

Source:Slide15

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.24Slide16

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.

From the CCSS: Grade 4

16

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.29Slide17

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

.

From the CCSS: Grade 4

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.29Slide18

5.NF.5aInterpret 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.From the CCSS: Grade 5Source: 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.36Slide19

http://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdfSlide20

“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.49Slide21

Learning Trajectories

Source

: Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction.

Daro

, et al., 2011

.Slide22

Learning Trajectories

Source

: Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction.

Daro

, et al., 2011. Slide23

What is “it”?

Is the perimeter a measurement?

…or is “it” something we

can

measure?

PerimeterSlide24

Perimeter

Is perimeter a one-dimensional, two-dimensional, or three-dimensional thing?

Does this room have a perimeter?Slide25
Slide26
Slide27

From the AZ STD's (2008)

Perimeter

: the sum of all lengths of a polygon.

DiscussSlide28

Wolframalpha.com

4/18/2013:Slide29

What do we mean when we talk about “measurement”?

MeasurementSlide30

“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”?

MeasurementSlide31

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.

MeasurementSlide32

From Fractions and Multiplicative Reasoning, Thompson and

Saldanha

, 2003. (

pdf

p. 22) Slide33

Create your own…

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. Slide34

What is a circle?Slide35
Slide36

Draw a circle

with a diameter equal to your international standard unit band lengthSlide37

What is circumference?Slide38

From the AZ STD's (2008)

the total distance around a closed curve like a circle

CircumferenceSlide39

So.... how do we measure circumference?

CircumferenceSlide40
Slide41

The circumference is three and a bit times as large as the diameter.

http://tedcoe.com/math/circumferenceSlide42

The circumference is about how many times as large as the diameter?

The diameter is about how many times as large as the circumference?Slide43

Tennis BallsSlide44

Circumference

If I double the RADIUS of a circle what happens to the circumference?Slide45

How many Rotations?Slide46
Slide47

What is an angle?

AnglesSlide48

Using objects at your table measure the

angle

You may

not

use degrees.

You must focus on the attribute you are measuring.

AnglesSlide49

What attribute are we measuring when we measure angles

?

Think about: What is one degree?

AnglesSlide50

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

Grade 4 CCSS: 4.MD.5Slide51

Source:Slide52

http://

tedcoe.com/math/radius-unwrapper-2-0Slide53

What is the length of “d”? You may choose the unit.Slide54

What is the measure of the angle? You may choose the unit.Slide55

Indiana (1896)

House Bill 296, Section 2: “…that the ratio of the diameter and circumference is as five-fourths 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

Slide56

Illustration:

 Slide57

Define: AreaSlide58

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 GlossarySlide59

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.21Slide60
Slide61
Slide62
Slide63
Slide64
Slide65

What about the kite?Slide66
Slide67
Slide68

http://

geogebratube.org/student/m279 (cc-by-sa)

http://geogebratube.org/material/show/id/279

Slide69

Geometric FractionsSlide70

If =

. What is 1?

How can you use this to show that

 

Check for Synthesis:

70

Source:Slide71
Slide72

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.42Slide73

Assume:

Angles that look like right angles

are

right angles

Lengths that look to be the same as AB can be verified using a compassSlide74

Find the dimensions of the rectangle

Find the area of the rectangle

Find a rectangle somewhere in the room

similar

to the shaded triangleSlide75
Slide76

Or not…

http://goldenratiomyth.weebly.com/Slide77

When I say two figures are similar I mean…

When I say two figures are similar I mean…

Hint: We haven’t defined “proportional” so you cannot use it.Slide78

What is a scale factor?

Teaching Geometry According to

the Common

Core

Standards

, H

.

Wu Revised

: April 15, 2012

. Grade 7 notes, p.49:Slide79
Slide80
Slide81

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.”Slide82

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.17Slide83

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.46Slide84

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.46Slide85

From 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

From the ProgressionsSlide86

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.56Slide87

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.pdfSlide88

CCSS: HS Geometry (p.74)Slide89
Slide90

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.77Slide91

Assume

http://tedcoe.com/math/geometry/similar-trianglesSlide92

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)Slide93

CCSS: HS Geometry (p.74)

CCSS: HS Geometry (p.74)Slide94

From http://www.healthreform.gov/reports/hiddencosts/index.html (6/3/2011)Slide95

What does it mean to say something is “out of proportion”?Slide96

“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)Slide97

On the Statue of Liberty the distance from heel to top of head is 33.86m

How

wide is her mouth?Slide98

http://www.nps.gov/stli/historyculture/statue-statistics.htmSlide99

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.77Slide100

Source

: http://tedcoe.com/math/geometry/pythagorean-and-similar-triangles Slide101

Note: Points A,B, and C are the centers of the indicated circular arcs.

Find all lengths and areas.Slide102

Volume

What is “it”?Slide103
Slide104
Slide105

http://

tedcoe.com/math/cavalieri Slide106

Connection to AlgebraSlide107

http://tedcoe.com/math/algebra/constant-rate

http://tedcoe.com/math/algebra/constant-rateSlide108
Slide109

http://tedcoe.com/math/algebra/constant-rate

http://tedcoe.com/math/algebra/constant-rateSlide110

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.54Slide111

From the progressions documents

Source:

http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_functions_2013_07_02.pdf p.5Slide112
Slide113

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.70Slide114

Source:

You have an investment account that grows from $60 to $103.68 over three years.Slide115

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

A tangent:Slide116

Cut this into 408 pieces

Copy one piece 577 times

It will never be good enough.Slide117

Hippasus, however, was not lauded for his efforts:  

http://

en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012Slide118

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. 11/2/2012Slide119

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. 11/2/2012Slide120

…except

Hippasus

Too much math never killed anyone.Slide121

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. 11/2/2012Slide122

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.jpgSlide123

…except

Hippasus…and Archimedes

Too much math never killed anyone.Slide124

Teaching and Learning Mathematics

Ways of doing Ways of thinkingHabits of thinkingSlide125

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 CCSSSlide126

Habits?

How did I do?Slide127

Creative

Commons

http://creativecommons.org

Slide128

Contact

Ted Coetcoe@achieve.orgtedcoe.com

@drtedcoeAchieve1400 16th St NW, Suite 510Washington, DC. 20036202-641-3146