ccby sa 30 unported unless otherwise noted Ted Coe PhD Director Mathematics Achieve Inc 252015 The Rules of Engagement Speak meaningfully what you say should carry meaning ID: 230280
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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.docSlide13Slide14
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?Slide25Slide26Slide27
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?Slide35Slide36
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?
CircumferenceSlide40Slide41
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?Slide46Slide47
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.21Slide60Slide61Slide62Slide63Slide64Slide65
What about the kite?Slide66Slide67Slide68
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:Slide71Slide72
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 triangleSlide75Slide76
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:Slide79Slide80Slide81
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)Slide89Slide90
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”?Slide103Slide104Slide105
http://
tedcoe.com/math/cavalieri Slide106
Connection to AlgebraSlide107
http://tedcoe.com/math/algebra/constant-rate
http://tedcoe.com/math/algebra/constant-rateSlide108Slide109
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.5Slide112Slide113
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