Transcript Slide 1

Coherence: Multiplicative
Reasoning Across the
Common Core/AZCCRS
AATM September 20, 2014
Too much math never killed anyone.
Teaching and Learning
Mathematics
Ways of doing
Ways of thinking
Habits of thinking
Ways of Doing?
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
Source: http://tedcoe.com/math/wp-content/uploads/2013/10/broomsticks-for-nctm.doc
Source: http://tedcoe.com/math/wp-content/uploads/2013/10/broomsticks-for-nctm.doc
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Source: http://tedcoe.com/math/wp-content/uploads/2013/10/broomsticks-for-nctm.doc
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Source: http://tedcoe.com/math/wp-content/uploads/2013/10/broomsticks-for-nctm.doc
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Source: http://tedcoe.com/math/wp-content/uploads/2013/10/broomsticks-for-nctm.doc
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Key Shifts in the AZCCRS
• Focus
• Coherence
• Rigor
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Ways of Thinking?
Learning Progressions in the
AZCCRS
From the CCSS: Grade 3
Source: CCSS Math Standards, Grade 3, p. 24 (screen capture)
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From the CCSS: Grade 3
3.OA.1:
Interpret products of whole numbers, e.g., interpret 5 × 7
as the total number of objects in 5 groups of 7 objects
each. For example, describe a context in which a total
number of objects can be expressed as 5 × 7.
Soucre: CCSS Grade 3. See: Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum,
Assessment, and Instruction. Daro, et al., 2011. pp.48-49
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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: CCSS Grade 4
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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: CCSS Grade 4
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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
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“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
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.
Measurement
Source: Fractions and Multiplicative Reasoning, Thompson and Saldanha, 2003. (pdf p. 22)
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
•What is an angle?
Angles
•What attribute are we measuring when we
measure angles?
Angles
CCSS, Grade 4, p.31
Source: CCSS Grade 4, 4.MD.5
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http://tedcoe.com/math/radius-unwrapper-2-0
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Check for Synthesis:
If
𝟐
𝟑
= . What is 1?
How can you use this to show that
Source:
𝟏
𝟐
𝟑
=
𝟑
𝟐
?
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Similar Figures
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Similar Figures
CCSS: Grade 7 (p.46)
Source: CCSS Grade 7, p.46
Source:
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Source: http://tedcoe.com/math/geometry/pythagorean-and-similar-triangles
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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 (8.EE.6, p.54)
Source: CCSS Grade 8
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You have an investment account that grows from $60
to $103.68 over three years.
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Assume
http://tedcoe.com/math/geometry/similar-triangles
CCSS: Geometry (G-SRT.6, p. 77)
Source: CCSS High School Geometry (screen capture)
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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
Copy one piece 577 times
Cut this into 408 pieces
It will never be good enough.
577
 1.4142156
408
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
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
Standards for Mathematical Practice
Eight Standards for Mathematical Practice
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the understanding of others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for and express regularity in repeated reasoning
Source: CCSS
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Materials?
Source: http://ime.math.arizona.edu/progressions/
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Ted Coe
Director, Mathematics
Achieve, Inc.
[email protected]
Twitter: @drtedcoe
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