1. Visualizing Middle and High School Mathematics with Color Tiles
ATOMIC 2015 Fall Conference
Jennifer Silverman

2. Presenter Jennifer Silverman Independent Math Consultant, Inventor
Visit jensilvermath.com & proradian.net

3. 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.

4. 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)

5. 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.

6. Support productive struggle
“What do you notice?”
“What do you wonder?”

7. Practice the Practices
Color tiles are
manipulatives
students can
use to build
understanding
at every grade
level!

8. 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.

9. Kindergarten - Example
Use two colors to fill each ten-frame. Write a number sentence.
Open-ended, flexible (start with one addend for example), leads to counting-on

10. Grade 1 - Content Standard
CCSS.Math.Content.1.MD
Represent and interpret data.

11. Grade 1 - Example
Make a bar graph of the number of differentbugs in the picture.

12. Grade 1 - Example

13. 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.

14. Grade 2 - Example

15. 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.

16. Grade 3 - Example Find each missing side length.
Write a number sentence for each.

17. 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.

18. 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.

19. Grade 5 - Content Standard
CCSS.Math.Content.5.NF
Use equivalent fractions as a strategy to add and subtract fractions.

20. 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?

21. Grade 5 - Example + =

22. Grade 6 - Content Standard
CCSS.Math.Content.6.NS.B
Compute fluently with multi-digit numbers and find common factors and multiples.

23. 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.

24. Grade 6 - Example What rectangle could contain them both?
The LCM is the area of this new
rectangle.
LCM is 4 x 6 = 24

25. Grade 7 - Content Standard
CCSS.Math.Content.7.RP.A.2
Recognize and represent proportional relationships between quantities.

26. 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
Green 4
4/2
Yellow 3 6
6/3
Blue 8
8/4

27. 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).

28. Grade 8 - Example 1

29. Grade 8 - Example 2

30. Number - HS Standard CCSS.Math.Content.HSA-SSE.A.2
Use the structure of an expression to identify ways to rewrite it.

31. 2(R + 2B) +(3Y + 2G) + 3R - 2(B + G) - Y
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

32. Number - Example
2R + 4B + 3Y + 2G + 3R - 2B - 2G -Y
5R + 2B +2

33. 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.

34. Make as many rectangles as you can whose perimeter is 12.
Algebra - Example
What is the greatest rectangular area that can be enclosed by 12 meters of fencing?
Make as many rectangles as you can whose perimeter is 12.

35. Algebra - Example Make a table of area as a function of height.
Color
Height
Area
Red 1 5
Yellow 2 8
Green 3 9
Blue 4
Make a graph of area
as a function of height.

36. Find an equation for area as a function of height.
Algebra - Example
Find an equation for area as a function of height.

37. 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.

38. Start at 3 blocks and add 2 blocks for the next stage.
Functions - Example
Build 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.

39. Functions - Example
The domain of a sequence is not all real numbers, so the graph should be points (discrete), not a connected line (continuous).
Number of squares
Height of the plant
Number of the term
Number of days

40. Geometry - HS Standard CCSS.Math.Content.HSG-CO
Experiment with transformations in the plane

41. 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).

42. 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.

43. 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

44. 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!

45. Resources Principles to Standards Executive Summary
Math Forum: I Notice, I Wonder Intro
Rhode Island Monument photo:
Crumb Cake photo:
Graph with sliders made on
Transformations applet and most diagrams made with free software at