ATOMIC 2015 Fall ConferenceJennifer Silverman

Visualizing Middle and High School Mathematics with

Color Tiles

Presenter

Jennifer Silverman

Independent Math Consultant, Inventor

Visit jensilvermath.com & proradian.net

Email jensilvermath@gmail.com

Twitter @jensilvermath

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.

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)

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.

“What do you notice?”

Support productive struggle

“What do you wonder?”

Practice the Practices

Color tiles aremanipulativesstudents canuse to buildunderstandingat every gradelevel!

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.

Kindergarten - ExampleUse two colors to fill each ten-frame. Write a number sentence.

Grade 1 - Content Standard

CCSS.Math.Content.1.MD

Represent and interpret data.

Grade 1 - Example

Make a bar graph of the number of differentbugs in the picture.

Grade 1 - Example

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.

Grade 2 - Example

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.

Grade 3 - Example

Find each missing side length.Write a number sentence for each.

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.

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.

Grade 5 - Content Standard

CCSS.Math.Content.5.NF

Use equivalent fractions as a strategy to

add and subtract fractions.

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?

+

Grade 5 - Example

+ =

Grade 6 - Content Standard

CCSS.Math.Content.6.NS.B

Compute fluently with multi-digit numbers

and find common factors and multiples.

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.

Grade 6 - Example

What rectangle could contain them both?

The LCM is the area of this newrectangle.

LCM is 4 x 6 = 24

Grade 7 - Content Standard

CCSS.Math.Content.7.RP.A.2

Recognize and represent proportional

relationships between quantities.

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

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).

Grade 8 - Example 1

Grade 8 - Example 2

Number - HS Standard

CCSS.Math.Content.HSA-SSE.A.2

Use the structure of an expression to

identify ways to rewrite it.

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

Number - Example

5R + 2B +2

2R + 4B + 3Y + 2G + 3R - 2B - 2G -Y

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.

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?

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

Algebra - ExampleFind an equation for area as a function of height.

https://www.desmos.com/calculator/b1racffu2r

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.

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.

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

Geometry - HS Standard

CCSS.Math.Content.HSG-CO

Experiment with transformations in the

plane

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).

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.

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

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!

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