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ATOMIC 2015 Fall ConferenceJennifer Silverman
Visualizing Middle and High School Mathematics with
Color Tiles
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Presenter
Jennifer Silverman
Independent Math Consultant, Inventor
Visit jensilvermath.com & proradian.net
Email [email protected]
Twitter @jensilvermath
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Teaching and Learning
An excellent mathematics program requires effective teaching that engages students in meaningful learning through individual and collaborative experiences that promote their ability to make sense of mathematical ideas and reason mathematically.
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Principles to Actions
The primary purpose of
Principles to Actions is to fill
the gap between the adoption
of rigorous standards and the
enactment of practices,
policies, programs, and
actions required for successful
implementation of those
standards. NCTM (2014)
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Mathematics Teaching Practices
• Establish mathematics goals to focus learning.
• Implement tasks that promote reasoning & problem solving.
• Use and connect mathematical representations.
• Facilitate meaningful mathematical discourse.
• Pose purposeful questions.
• Build procedural fluency from conceptual understanding.
• Support productive struggle in learning mathematics.
• Elicit and use evidence of student thinking.
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“What do you notice?”
Support productive struggle
“What do you wonder?”
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Practice the Practices
Color tiles aremanipulativesstudents canuse to buildunderstandingat every gradelevel!
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Kindergarten - Content Standard
CCSS.Math.Content.K.OA
Understand addition as putting together
and adding to, and understand
subtraction as taking apart and taking
from.
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Kindergarten - ExampleUse two colors to fill each ten-frame. Write a number sentence.
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Grade 1 - Content Standard
CCSS.Math.Content.1.MD
Represent and interpret data.
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Grade 1 - Example
Make a bar graph of the number of differentbugs in the picture.
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Grade 1 - Example
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Grade 2 - Content Standard
CCSS.Math.Content.2.G.A.3
Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.
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Grade 2 - Example
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Grade 3 - Content Standard
CCSS.Math.Content.3.MD
Geometric measurement: recognize
perimeter as an attribute of plane figures
and distinguish between linear and area
measures.
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Grade 3 - Example
Find each missing side length.Write a number sentence for each.
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Grade 4 - Content Standard
CCSS.Math.Content.4.MD.A.3
Apply the area and perimeter formulas for
rectangles in real world and mathematical
problems.
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Grade 4 - Example
The base of a monument has an area of 24 square feet and a perimeter of 20 feet. Find the length and width of the base.
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Grade 5 - Content Standard
CCSS.Math.Content.5.NF
Use equivalent fractions as a strategy to
add and subtract fractions.
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Grade 5 - Example
To make crumb cake, you need ⅔ of a cup of butter for the topping and ¾ of a cup of butter for the cake. How much butter do you need for the whole recipe?
+
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Grade 5 - Example
+ =
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Grade 6 - Content Standard
CCSS.Math.Content.6.NS.B
Compute fluently with multi-digit numbers
and find common factors and multiples.
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Grade 6 - Example
Find the GCF and LCM of 12 and 8.
GCF - make two rectangles with the greatest possible common dimension.
GCF is 4.
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Grade 6 - Example
What rectangle could contain them both?
The LCM is the area of this newrectangle.
LCM is 4 x 6 = 24
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Grade 7 - Content Standard
CCSS.Math.Content.7.RP.A.2
Recognize and represent proportional
relationships between quantities.
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Grade 7 - Example
Are the sides of these rectangles proportional? How do you know? How can you show it?
Color B H H/B k
Red (O) 1 2 2/1 2
Green 2 4 4/2 2
Yellow 3 6 6/3 2
Blue 4 8 8/4 2
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Grade 8 - Content Standard
CCSS.Math.Content.8.F.A.2
Compare properties of two functions each
represented in a different way
(algebraically, graphically, numerically in
tables, or by verbal descriptions).
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Grade 8 - Example 1
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Grade 8 - Example 2
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Number - HS Standard
CCSS.Math.Content.HSA-SSE.A.2
Use the structure of an expression to
identify ways to rewrite it.
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Number - Example
Simplify by combining like terms.2(R + 2B) +(3Y + 2G) + 3R - 2(B + G) - Y
2R + 4B + 3Y + 2G + 3R - 2B - 2G -Y
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Number - Example
5R + 2B +2
2R + 4B + 3Y + 2G + 3R - 2B - 2G -Y
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Algebra - HS Standard
CCSS.Math.Content.HSA-CED.A.2
Create equations in two or more variables
to represent relationships between
quantities; graph equations on coordinate
axes with labels and scales.
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Algebra - Example
Make as many rectangles as you can whose perimeter is 12.
What is the greatest rectangular area that can be enclosed by 12 meters of fencing?
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Algebra - ExampleMake a table of area as a function of height.
Make a graph of area as a function of height.
Color Height Area
Red 1 5
Yellow 2 8
Green 3 9
Blue 4 8
Red 5 5
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Algebra - ExampleFind an equation for area as a function of height.
https://www.desmos.com/calculator/b1racffu2r
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Functions - HS Standard
CCSS.Math.Content.HSF-IF.A.3
Recognize that sequences are functions,
sometimes defined recursively, whose
domain is a subset of the integers.
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Functions - ExampleBuild the next term in the sequence.
Describe each term in the sequence using the language of recursion. (Each term is defined by the term before it.)
Start at 3 blocks and add 2 blocks for the next stage.
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Functions - ExampleThe domain of a sequence is not all real numbers, so the graph should be points (discrete), not a connected line (continuous).
Number of the term
Num
ber
of
squa
res
Number of days
Hei
ght
of t
he p
lant
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Geometry - HS Standard
CCSS.Math.Content.HSG-CO
Experiment with transformations in the
plane
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Geometry - Example
Color tiles can be used to model rigid transformations (also called isometries).
Online applets can also be used (this one is free from GeoGebraTube.org).
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Prob/Stats - HS Standard
CCSS.Math.Content.HSS-CP.B.8
Apply the general Multiplication Rule in a
uniform probability model,
P(A and B) = P(A)P(B|A) = P(B)P(A|B),
and interpret the answer in terms of the
model.
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Prob/Stats - Example
What is the probability that you will pick a red and a yellow?
P(R) = 1/4
P(Y|R) = 1/3
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Prob/Stats - Example
What is the sample space? How many elements are in it?
P(R and Y) = 1/12 = P(R)P(Y|R) =(¼)(⅓) order matters!
P(R and Y) = 1/12 = P(Y)P(R|Y) =(¼)(⅓) order matters!
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Resources
Rhode Island Monument photo: http://www.nps.gov/ande/historyculture/rhode-island_monument.htm
Crumb Cake photo:http://tinaschic.com/2013/06/the-perfect-crumb-cake/
Graph with sliders made on www.desmos.com
Transformations applet and most diagrams made with free software at www.geogebra.org
Principles to Standards Executive Summaryhttps://www.nctm.org/uploadedFiles/Standards_and_Positions/PtAExecutiveSummary.pdfMath Forum: I Notice, I Wonder Introhttp://mathforum.org/pubs/notice_wonder_intro.pdf