Download Grade 5 - Example

Visualizing Middle
and High School
Mathematics with
Color Tiles
ATOMIC 2015 Fall Conference
Jennifer Silverman
Presenter
Jennifer Silverman
Independent Math Consultant, Inventor
Visit jensilvermath.com & proradian.net
Email [email protected]
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.
Support productive struggle
“What do you notice?”
“What do you wonder?”
Practice the Practices
Color tiles are
manipulatives
students can
use to build
understanding
at every grade
level!
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 - Example
Use 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
different
bugs 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 new
rectangle.
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
2R + 4B + 3Y + 2G + 3R - 2B - 2G -Y
5R + 2B +2
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
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.
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
8
Red
5
5
Make a graph of area
as a function of height.
Algebra - Example
Find 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 - 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.
Functions - Example
Height of the plant
Number of squares
The 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
Number of days
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
Principles to Standards Executive Summary
https://www.nctm.org/uploadedFiles/Standards_and_Positions/PtAExecutiveSummary.pdf
Math Forum: I Notice, I Wonder Intro
http://mathforum.org/pubs/notice_wonder_intro.pdf
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