### week1best2014web

```The Arizona Mathematics
Partnership:
Week 1 Geometry
Ted Coe, June 2014
cc-by-sa 3.0 unported unless otherwise noted
Teaching and Learning
Mathematics
Ways of doing
Ways of thinking
Habits of thinking
THE Rules of Engagement
Speak meaningfully — what you say should carry meaning to others;
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 others — allow them the opportunity to
think, reflect and construct. When assisting your peers, pose questions to help
them construct meaning rather than show them how to get the answer.
Marilyn Carlson, Arizona State University, Project Pathways
http://hub.mspnet.org/media/data/Classroom_Rules_of_Engagement-Carlson.pdf?media_000000007898.pdf
The Plot...
The Foot
Define
Square
Triangle
Angle
A square is a rectangle.
True or False?
Rectangle
Square
From the AZ Glossary, 2008:
But that isn’t the only possibility…
Some are based on symmetry
An Example of CCSS 5.G…
From Illustrativemathematics.org
From Illustrativemathematics.org
From Illustrativemathematics.org
Wolframalpha.com (6/11/2013)
From the Progression Documents
http://ime.math.arizona.edu/progressions/
http://ime.math.arizona.edu/progressions/
Act I Scene II
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
The Willis tower (formerly the Sears tower) is 1730 feet high.
The Burj Khalifa (formerly Burj Dubai) is 2717 feet high.
The Burj is ______________ times as
large as the Willis tower.
The Willis tower is _____________times as
large as the Burj
The Burj is _____________ percent the
size of the Willis tower.
The Willis tower is _____________ percent
the size of the Burj.
Source: CCSS Math Standards, Grade 3, p. 24 (screen capture)
30
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
31
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
32
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
33
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.
34
“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
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
Progressions
Progressions:
http://ime.math.arizona.edu/progressions/
http://commoncoretools.files.wordpress.com/2012/07/ccss_progression_gm_k5
_2012_07_21.pdf p.16.
4/18/2013:
Wolframalpha.com
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”?
Van de Walle, J. (2001) Elementary School Mathematics : Elementary & Middle School Mathematics Teaching
Developmentally.
Measurement
•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)
http://pat-thompson.net/PDFversions/2004FracsMultRsng.pdf
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.
Which is a circle?
What is circumference?
From the AZ STD's (2008)
Circumference
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
Tennis Balls
The circumference is about how many times
as large as the diameter?
The diameter is about how many times as large
as the circumference?
Circumference
If I double the RADIUS of a circle what
happens to the circumference?
How many Rotations?
•What is an angle?
Angles
•Using objects at your table measure the angle
Angles
•What attribute are we measuring when we measure
angles?
Angles