Content Deepening
th
7 Grade Math
September 23, 2013
Jeanne Simpson
AMSTI Math Specialist
2
Welcome
•Name
•School
•Classes you teach
•What are you hoping to learn
today?
3
4
He who dares to teach
must never cease to
learn.
John Cotton Dana
acos2010.wikispaces.com
• Electronic version of handouts
• Links to web resources
6
Five Fundamental Areas Required for
Successful Implementation of CCSS
Instruction
Content
Collaboration
Intervention
Assessment
7
How do we teach?
Instruction
Content
• Deep conceptual understanding
• Collaborative lesson design
• Standards for Mathematical Practice
• Fewer standards with greater depth
• Understanding, focus, and coherence
• Common and high-demand tasks
Collaboration
Intervention
Assessment
• Common required response to
intervention framework response
• Differentiated, targeted, and intensive
response to student needs
• Student equity, access, and support
• PLC teaching-assessing-learning cycle
• In-class formative assessment
processes
• Common assessment instruments as
formative learning opportunities
8
Standards for Mathematical
Practice
SMP1 - Make sense of problems and persevere in solving
them
SMP2 - Reason abstractly and quantitatively
SMP3 - Construct viable arguments and critique the
reasoning of others
SMP4 - Model with mathematics
SMP5 - Use appropriate tools strategically
SMP6 - Attend to precision
SMP7 - Look for and make use of structure
SMP8 - Look for and express regularity in repeated
reasoning
9
What Are The Practice Standards?
• Capture the processes and proficiencies
that we want our students to possess
• Not just the knowledge and skills but how
our students use the knowledge and skills
• Describe habits of mind of the
mathematically proficient student
• Carry across all grade levels, K-12
10
Standards of Mathematical Practice
•√ I already do this.
• ! This sounds exciting!
• ? I have questions.
11
High-Leverage Strategies
•
•
•
•
•
•
•
•
Problem solving
Demanding tasks
Student understanding
Discussion of alternative strategies
Extensive mathematics discussion
Effective questioning
Student conjectures
Multiple representations
12
What are we teaching?
Instruction
Content
• Deep conceptual understanding
• Collaborative lesson design
• Standards for Mathematical Practice
• Fewer standards with greater depth
• Understanding, focus, and coherence
• Common and high-demand tasks
Collaboration
Intervention
Assessment
• Common required response to
intervention framework response
• Differentiated, targeted, and intensive
response to student needs
• Student equity, access, and support
• PLC teaching-assessing-learning cycle
• In-class formative assessment
processes
• Common assessment instruments as
formative learning opportunities
13
PICS
• Share your section and record what you hear
• Describe one connection you notice
• How could you use this with students
Critical Focus Areas
Ratios and Proportional
Reasoning
Applying to problems
Graphing and slope
Number Systems,
Expressions and
Equations
Standards 4-10
Standards 1-3
Geometry
Statistics
Scale drawings,
constructions, area, surface
area, and volume
Drawing inferences
about populations
based on samples
Standards 11-16
Standards 17-20
Probability – Standards 21-24
Recommend Emphases
from PARCC Model Content Framework for Mathematics
16
7th Grade Domains
1.
2.
3.
4.
5.
Ratios and Proportional Reasoning
The Number System
Expressions and Equations
Geometry
Statistics and Probability
Analysis Tool
7.SP3 Informally assess the degree of visual overlap of two numerical
data distributions with similar variabilities, measuring the difference
between centers by expressing it as a multiple of a measure of variability.
Content
Standard
Cluster
Draw
informal
comparative
inferences
about two
populations.
Which
What’s New
Standards in
or
the Cluster
Challenging in
Are Familiar?
These
Standards?
Which
Standards in
the Cluster
Need
Unpacking or
Emphasizing?
How Is This
Cluster
Connected to
the Other 6-8
Domains and
Mathematical
Practice?
18
Unpacking the Standards
19
Unpacking the Standards
“To increase student achievement
by ensuring educators
understand specifically what
the new standards mean a
student must know,
understand, and be able to
do. (Unpacking) may also be
used to facilitate discussion
among teachers and curriculum
staff and to encourage
coherence…(Unpacking), along
with on-going professional
development is one of many
resources used to understand and
teach the CCSS.”
-North Carolina Dept of Public
Instruction
Step 1: Target a standard
Step 2: Chunk the Main
Categories
Step 3: Identify all standard
components
Step 4: Identify the
Developmental
Progression
Step 5: Identify Key
Vocabulary
Step 6: Add Clarifying
Information
20
Why are we Unpacking
Standards?
 To understand what the standards are asking
students to know, understand, and be able to do
 To make time for professional discussion about
the standards
 To build upon and use common terminology
when discussing the implementation of the
standards
Unpacking is standards is not a substitute
document for the Common Core Standards, it is
a record of the conversation of those who are
involved in the process of digging into the
standards.
21
Step 1 – Target a Standard
• What standard(s) do you need to
explore further?
• Find a group of 2-4 teachers who
will explore that topic with you.
2.G.3
Partition circles and
rectangles into two,
three, or four equal
shares
The final product….
Partition
Partition circles and
rectangles into two
equal shares, using
the word halves,
half of
2.G.3
Describe
Partition circles and
rectangles into three
equal shares, using
the word thirds,
third of
2.G.3
circle
Partition a
shape into
fourths in
different ways
rectangle
Equal
shares
Pattern
Blocks
Recognize
Partition circles and
rectangles into four
equal shares, using
the word fourths,
fourth of
2.G.3
Describe the whole
as two halves, three
thirds, four fourths
2.G.3
Needed for 3.G.2
Builds on 1.G.3
partition
22
Halves
Half of
Thirds
Third of
Fraction
Bars/Circles
Fourths
Fourth of
whole
2/2 = one
whole
Identical
whole
Recognize that equal
shares of identical
wholes need not have
the same shape
2.G.3
23
Step 2: Chunk the Main Categories
Example
2.G.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.
1. All Standard(s) in
the cluster(s)
2. Identify Key Verbs
Partition
2.G.3
Partition
circles and
rectangles
into two,
three, or four
equal shares
Describe
Recognize
24
lt blue
Step 3: Identify all
standard components
Components from CCSS:
Analyze nouns and verbs
What do students need to do?
Include bullets, examples, footnotes,
etc.
Take standard apart according to the
verbs to separate skills within the
standard
What do the students need to know?
25
Example
2.G.3
Partition circles
and rectangles
into two, three,
or four equal
shares
Partition
Partition circles
and rectangles
into two equal
shares, using
the word halves,
half of
2.G.3
Partition circles
and rectangles
into three equal
shares, using
the word thirds,
third of
2.G.3
Describe
Partition circles
and rectangles
into four equal
shares, using
the word
fourths, fourth of
2.G.3
Recognize
Describe the
whole as two
halves, three
thirds, four
fourths
2.G.3
Recognize that
equal shares of
identical wholes
need not have
the same shape
2.G.3
26
Step 4: Identify the Developmental
Progression
Questions to consider when looking at the
developmental progression of the standards…
• How would you utilize these chunks (blue) for
scaffolding toward mastery of the entire standard?
• Where would you start when teaching this standard?
• What is the chunk that demonstrates the highest level
of thinking?
27
Vertical Alignment
Using the progression document(s)
from Ohio Department of Education and CCSS Writing Team:
Look to the grade level(s) below to see if
the standard is introduced.
Look to the grade level(s) above to see if
the standard is continued.
Code each standard on the poster with:
 builds on
 introduced
 needed for
 or mastered
and the grade level to which the standard aligns.
28
2.G.3
Partition circles
and rectangles
into two, three,
or four equal
shares
Partition
Partition circles
and rectangles
into two equal
shares, using
the word halves,
half of
2.G.3
Partition circles
and rectangles
into three equal
shares, using
the word thirds,
third of
2.G.3
Introduced?
Mastered?
Needed for?
Builds on?
Describe
Partition circles
and rectangles
into four equal
shares, using
the word
fourths, fourth of
2.G.3
Recognize
Describe the
whole as two
halves, three
thirds, four
fourths
2.G.3
Builds on 1.G.3
Needed for 3.G.2
Recognize that
equal shares of
identical wholes
need not have
the same shape
2.G.3
29
Step 5: Identify Key Vocabulary
Identify content vocabulary
directly from the standard.
Identify additional vocabulary
students will need to know to meet
the standard.
green
2.G.3
Partition circles and
rectangles into two,
three, or four equal
shares
Partition
Partition
circles
Partition
circles
and
and rectangles
rectangles
into two
equal
using
into shares,
two equal
the wordusing
halves,
shares,
the
half
of
word halves,
2.G.3
half of
2.G.3
Describe
Partition circles and
rectangles into three
equal shares, using
the word thirds,
third of
2.G.3
partition
circle
rectangle
Equal
shares
Recognize
Partition circles and
rectangles into four
equal shares, using
the word fourths,
fourth of
2.G.3
Builds on 1.G.3
Halves
Half of
Thirds
Third of
30
Describe the whole
as two halves, three
thirds, four fourths
2.G.3
Needed for 3.G.2
Fourths
Fourth of
whole
Identical
whole
Recognize that equal
shares of identical
wholes need not have
the same shape
2.G.3
31
Step 6: Add Clarifying Information
Kid-friendly language to add clarity
Clarifying pictures, words, or phrases
Definitions, examples
Symbols, formulas, pictures, etc.
CAUTION: do not replace important
vocabulary that is included in the standard.
yellow
2.G.3
Partition circles and
rectangles into two,
three, or four equal
shares
Partition
Partition
circles
and
Partition
circles
rectangles
into
two
and rectangles
equal shares, using
into two equal
the word halves,
shares,
using
half
of the
word2.G.3
halves,
half of
2.G.3
Describe
circle
Partition a
shape into
fourths in
different ways
rectangle
Equal
shares
Pattern
Blocks
Recognize
Partition circles and
rectangles into four
equal shares, using
the word fourths,
fourth of
2.G.3
Partition circles and
rectangles into three
equal shares, using
the word thirds,
third of
2.G.3
Builds on 1.G.3
partition
32
Halves
Half of
Thirds
Third of
Fraction
Bars/Circles
Describe the whole
as two halves, three
thirds, four fourths
2.G.3
Needed for 3.G.2
Fourths
Fourth of
whole
2/2 = one
whole
Identical
whole
Recognize that equal
shares of identical
wholes need not have
the same shape
2.G.3
33
Transfer Unwrapping to Chart
2.G.3
Partition circles and
rectangles into two,
three, or four equal
shares
Main Idea of Standard
Key Verbs
Partition
34
Describe
Partition circles and
Partition
circles
and
Partition
circles
Partition
circles
and
rectangles
into to
four

Take
standard
apart
according to the
verbs
rectangles
into
two
and
rectangles
rectangles into three
equal shares, using
equal
shares, using
separate
the standard.
equal shares,
using
into
two equal skills within
the word fourths,
the word halves,
the
word
thirds,
fourth of
shares,
using
the
 Use
all components
of
standard.
half
of
third
of
2.G.3
halves,
2.G.3
 word
Put
in a logical sequence
2.G.3
Recognize
Describe the whole
as two halves, three
thirds, four fourths
2.G.3
half of
2.G.3
Builds on 1.G.3
Vertical alignment
partition
rectangle
Vocabulary
circle
Partition a
shape into
fourths in
different ways
Equal
shares
Pattern
Blocks
Clarifying information,
Halves
Half of
Thirds
Third of
Fraction
Bars/Circles
student-friendly
Needed for 3.G.2
Fourths
Fourth of
whole
2/2 = one
whole
Identical
whole
Recognize that equal
shares of identical
wholes need not have
the same shape
2.G.3
35
The Number System
36
The standards stress not only procedural skill but
also conceptual understanding, to make sure
students are learning and absorbing the critical
information they need to succeed at higher levels
– rather than the current practices by which
many students learn enough to get by on the
next test, but forget it shortly thereafter, only to
review again the following year.
(CCSSI, 2012)
37
Mathematics consists of pieces that make sense; they
are not just independent manipulation/skills to be
practiced and memorized – as perceived by many
students.
These individual pieces progress through different
grades (in organized structures we called “flows”) and
can/should be unified together into a coherent whole.
Jason Zimba, Bill McCallum
38
Mathematical Fluency
39
Fluency
• The word fluent is used in the Standards to
mean fast and accurate. Fluency in each grade
involves a mixture of just knowing some answers
from patterns (e.g., “adding 0 yields the same
number”), and knowing some answers from the
use of strategies.
• Progressions for the Common Core State Standards in
Mathematics
40
Fluency
• Fluent in the standards means “fast and
accurate.” It might also help to think of fluency
as meaning more or less the same as when
someone is said to be fluent in a foreign
language. To be fluent is to flow; fluent isn’t
halting, stumbling, or reversing oneself.
Jason Zimba
41
Fluency Expectations
Grade
Required Fluency
K
Add/subtract within 5
1
Add/subtract within 10
2
Add/subtract within 20
Add/subtract within 100 (pencil and paper)
3
Multiply/divide within 100
Add/subtract within 1000
4
Add/subtract within 1,000,000
5
Multi-digit multiplication
6
Multi-digit division (6.NS.2)
Multi-digit decimal operations (6.NS.3)
7
Solve px + q = r, p(x + q) = r
Integers
• Most of the numbers you have worked
with in math class this year have been
greater than or equal to zero. However,
numbers less than zero can provide
important information.
• Where have you seen numbers less than
zero outside of school?
44
7NS - Understanding
• 7.NS.1b – Understand p + q as the number located a
distance |q| from p, in the positive or negative direction
depending on whether q is positive or negative.
• 7.NS.1c – Understand subtraction of rational numbers as
adding the additive inverse, p – q = p + (–q).
• 7.NS.2a – Understand that multiplication is extended
from fractions to rational numbers by requiring that
operations continue to satisfy the properties of
operations, particularly the distributive property, leading
to products such as (–1)(–1) = 1 and the rules for
multiplying signed numbers.
• 7.NS.2b – Understand that integers can be divided,
provided that the divisor is not zero, and every quotient
of integers (with nonzero divisor) is a rational number.
Progressions Documents
•
•
•
•
•
•
K–6 Geometry
6-8 Statistics and Probability
6–7 Ratios and Proportional Relationships
6–8 Expressions and Equations
6-8 Number Systems
3-5 Number and Operations: Fractions
• These are the documents currently available. They are working on
documents for the other domains (Functions, Geometry 7-8).
http://ime.math.arizona.edu/progressions/
46
Jigsaw
• Read your assigned section
• Chart paper
▫ Summarize what you read.
▫ How can this document help you in your
classroom?
• Be prepared to share
47
Adding Integers
48
Opposites
49
Subtracting Integers
50
Subtracting Integers
51
Fractions
• Difficulty with learning fractions is pervasive
and is an obstacle to further progress in
mathematics and other domains dependent on
mathematics, including algebra. It has also been
linked to difficulties in adulthood, such as failure
to understand medication regimens.
National Mathematics Panel Report, 2008
52
Fractions
• “Students who are asked to practice the
algorithm over and over…stop thinking. They
sacrifice the relationships in order to treat the
numbers simply as digits.”
Imm, Fosnot, Uittenbogaard (2012)
53
Unit Fractions
54
Fraction Multiplication in Grade 5
55
Fraction Multiplication in Grade 5
56
Fraction Multiplication in Grade 5
57
Fraction Multiplication in Grade 5
58
5th Grade Division
59
5th Grade Division
60
5th Grade Division Problems
• How much chocolate will each person get if 3
people share ½ pound equally?
61
5th Grade Division Problems
• How many 1/3 cup servings are in 2/3 cups of
raisins?
62
Fraction Division in Grade 6
• 6.NS.1 – Interpret and compute quotients of
fractions, and solve word problems involving
division of fractions, e.g., by using visual fraction
models and equations to represent the problem.
• Examples:
▫ Create a story context…
▫ Use a visual fraction model to show the quotient…
▫ Explain division using its relationship with
multiplication
▫ Sample problems
63
6th Grade Division
64
6th Grade Division
65
Ratios and Proportional
Relationships
66
Kim and Bob ran equally fast around a track.
Kim started first. When she had run 9 laps, Bob
had run 3 laps.
When Bob had run 15 laps, how many laps had
Kim run?
Explain your reasoning.
67
Solving Proportions
Solve
Kanold, p. 94
• If two pounds of beans cost $5,
how much will 15 pounds of
beans cost?
• The traditional method of
creating and solving
proportions by using crossmultiplication is deemphasized (in fact it is not
mentioned in the CCSS)
because it obscures the
proportional relationship
between quantities in a given
problem situation.
68
Ratios and Proportional Relationships
Progression, pages 6-7
• Although it is traditional to
move students quickly to
solving proportions by setting
up an equation, the Standards
do not require this method in
Grade 6. There are a number
of strategies for solving
problems that involve ratios.
As students become familiar
with relationships among
equivalent ratios, their
strategies become increasingly
abbreviated and efficient.
69
6.RP.3 Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by
reasoning about tables of equivalent ratios, tape
diagrams, double number line diagrams, or
equations.
a.
b.
c.
d.
Make tables of equivalent ratios relating quantities with
whole-number measurements, find missing values in the
tables, and plot the pairs of values on the coordinate plane.
Use tables to compare ratios.
Solve unit rate problems including those involving unit
pricing and constant speed.
Find a percent of a quantity as a rate per 100; solve
problems involving finding the whole, given a part and the
percent.
Use ratio reasoning to convert measurement units;
manipulate and transform units appropriately when
multiplying or dividing quantities.
70
1. The ratio of free throws that Omar made to the
ones he missed at practice yesterday was 7:3. If
he attempted 90 free throw at practice, how
many free throws did Omar make?
made 7
9
9
9
9
9
9
9
Attempted
90
missed 3
9
9
9
90 ÷ 10 = 9
9 x 7 = 63
Omar made 63 free throws.
71
2. At FDR High School, the ratio of seniors who
attend college to those who do not is 5:2. If 98
seniors do not attend college, how many do?
72
3. At Mesa Park High School, the ratio of
students who have driver’s licenses to those
who don’t is 8:3. If 144 students have driver’s
licenses, how many students are enrolled at
Mesa Park High School?
73
4. Of the black and blue pens that Mrs. White has
in a drawer in her desk, 18 are black. The ratio
of black pens to blue pens is 2:3. When Mrs.
White removes 3 blue pens, what is the new
ratio of black pens to blue pens?
74
Expressions and Equations
75
Understanding
• 7.EE.2 – Understand that rewriting an
expression in different forms in a problem
context can shed light on the problem, and how
the quantities in it are related.
76
Fluency Expectations
Grade
Required Fluency
K
Add/subtract within 5
1
Add/subtract within 10
2
Add/subtract within 20
Add/subtract within 100 (pencil and paper)
3
Multiply/divide within 100
Add/subtract within 1000
4
Add/subtract within 1,000,000
5
Multi-digit multiplication
6
Multi-digit division (6.NS.2)
Multi-digit decimal operations (6.NS.3)
7
Solve px + q = r, p(x + q) = r
77
Fluency Expectations
Grade
Required Fluency
K
Add/subtract within 5
1
Add/subtract within 10
2
Add/subtract within 20
Add/subtract within 100 (pencil and paper)
3
Multiply/divide within 100
Add/subtract within 1000
4
Add/subtract within 1,000,000
5
Multi-digit multiplication
6
Multi-digit division (6.NS.2)
Multi-digit decimal operations (6.NS.3)
7
Solve px + q = r, p(x + q) = r
The Mystery Bags Game
Solving Equations
The Mystery Bags Game
There once was a
king who loved to
watch his many
bags of gold.
• But it can get very
boring watching gold
all day, so he had the
court jester make up
games for him to
pass the time. The
game the king loves
best is the Mystery
Bags game.
• First, the jester
takes one or more
empty bags and
fills each bag with
the same amount
of gold. These
bags are called the
“mystery bags.”
• Next, the jester digs into
his collection of lead
weights. He takes out his
pan balance and places
some combination of
mystery bags and lead
weights on the two pans
so that the sides balance.
• The game is to figure out
the weight of each
mystery bag.
Your Task
• The game sound rather easy, but it can get
very difficult for the king. See if you can win
the mystery bags game in the various
situations described on your worksheet by
figuring out how much gold there is in each
mystery bag.
• Explain how you know you are correct. You
may want to draw diagrams to show what’s
going on.
• Write an equation to
represent this situation
Use your methods from Problem 3.2 to find the number
of gold coins in each pouch
Write down a similar method using the equation that
represents this situation
Question 1
• There are 3 mystery bags on one side of the
balance and 51 ounces of lead weights on the other
side.
51 oz.
2
• There are 1 mystery bag and 42 ounces of
weights on one side, and 100 ounces of
weights on the other side.
3
• There are 8 mystery bags and 10 ounces of
weights on one side, and 90 ounces of weights
on the other side.
4
• There are 3 mystery bags and 29 ounces of
weights on one side, and 4 mystery bags on
the other side.
5
• There are 11 mystery bags and 65 ounces of
weights on one side, and 4 mystery bags and
100 ounces of weights on the other side.
6
• There are 6 mystery bags and 13 ounces of
weights on one side, and 6 mystery bags and
14 ounces of weights on the other side. (The
jester could get in a lot of trouble for this
one!)
7
• There are 15 mystery bags and 7 ounces of
weights on both sides. (At first the king
thought this one was easy, but then he found
it to be incredibly hard.)
8
• The king wants to be able to win easily all of
the time, without calling you in. Therefore,
your final task in this assignment is to describe
in words a procedure by which the king can
find out how much is in a mystery bag in any
situation.
103
104
105
106
• 7.EE.4 – Use variables to represent quantities in
real-world or mathematical problems, and
construct simple equations and inequalities to
solve problems by reasoning about the
quantities.
107
Mathematics Assessment Project
• Tools for formative and summative assessment that make
knowledge and reasoning visible, and help teachers to guide
students in how to improve, and monitor their progress. These tools
comprise:
• Classroom Challenges: lessons for formative assessment, some
focused on developing math concepts, others on non-routine
problem solving.
• Professional Development Modules: to help teachers with the
new pedagogical challenges that formative assessment presents.
• Summative Assessment Task Collection: to illustrate the
range of performance goals required by CCSSM.
• Prototype Summative Tests: designed to help teachers and
students monitor their progress, these tests provide a model for
examinations that may replace or complement current US tests.
http://map.mathshell.org/
Writing Algebraic Expressions
Area of rectangle = _ _ _ _ _ _ _ _ _ _ _ _
Projector Resources
Steps to Solving Equations
P-108
Writing Algebraic Expressions
Perimeter of rectangle = _ _ _ _ _ _ _ _ _ _ _ _
Projector Resources
Steps to Solving Equations
P-109
Writing Algebraic Expressions
Which two expressions are equivalent?
Projector Resources
Steps to Solving Equations
P-110
Which Equations Describe The Story?
A pencil costs $2 less than a
notebook.
Let x represent the cost of
notebook.
A pen costs 3 times as much
as a pencil.
The pen costs $9
A:
3x - 6 = 9
B:
x-6 = 9
C:
3x - 2 = 9
D:
3(x - 2) = 9
Which of the four equations
opposite describe this story?
Projector Resources
Steps to Solving Equations
P-111
112
Geometry
113
The van Hiele
Theory of Geometric Thought
114
115
Statistics and Probability
Resources
116
Understanding
• 7.SP.1 – Understand that statistics can be used
to gain information about a population by
examining a sample of the population;
generalizations about a population from a
sample are valid only if the sample is
representative of that population. Understand
that random sampling tends to produce
representative samples and support valid
inferences.
At a nearby school, teachers decided to get rid of
pizza Fridays. After a survey of all teachers,
counselors, and administrators, it was
overwhelmingly decided that pizza would be
replaced with broccoli with ranch dip.
After surveying 83 students in 3
classes, 70% responded that girls
should be allowed to go to lunch
two minutes early every day and
boys will go at the regular time.
 Do you think this is an accurate statistic?
 Who do you think the sample population
was?
 Each group will need to assign the following
roles:
 Facilitator – keeps group on task and ensures
equal participation
 Materials Manager – collects and returns
materials
 Recorder – writes group answer on chart paper
 Reporter – presents group answer to the class
 Discuss and complete the handout as a group. Begin
with the multiple choice questions.
 Choose one biased survey to present to the class on
chart paper. Include the following in your
presentation:
 Original survey
 Why you think it is biased
 How you would correct it
Question
Population
Sample group
 In a poll of Mrs. Simpson’s math class, 67%
of the students say that math is their
favorite academic subject. The editor of the
school paper is in the class, and he wants to
write an article for the paper saying that
math is the most popular subject at the
school. Explain why this is not a valid
conclusion, and suggest a way to gather
better data to determine what subject is
most popular.
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Illustrative Mathematics
• Illustrative Mathematics provides guidance to
states, assessment consortia, testing companies,
and curriculum developers by illustrating the
range and types of mathematical work that
students experience in a faithful implementation
of the Common Core State Standards, and by
publishing other tools that support
implementation of the standards.
http://www.illustrativemathematics.org/
What Do You Expect?
Compound Probability
How is probability used in real
life?
Event
 Theoretical probability
 Experimental probability
 Outcome

What’s in the Bucket?
What’s in the Bucket?
Without looking in or
emptying the bucket,
how could we
determine the fraction
of blocks that are red,
yellow, or blue?
Problem 1.1





How many blocks drawn by your class
were blue?
How many were yellow?
How many were red?
Which color block do you think there are
the greatest of in the bucket?
Which color block do you think there are
the least number of?
Problem 1.1

Based on our experimental
data, predict the fraction of
blocks in the bucket that
are blue, that are yellow,
and that are red.
Problem 1.1

How do the fraction of
blocks that are blue, yellow,
and red compare to the
fractions of blue, yellow,
and red drawn during the
experiment?
Match / No-Match Rules



Spin the spinner twice for each turn.
If both spins land on the same color, you
have made a MATCH.
Player A scores 1 point.
If the two spins land on different colors,
you have made a NO-MATCH.
Player B scores 2 points.
Experimental Probability of
Match
number of turns that are matches
total number of turns
Experimental Probability of
No-Match
number of turns that are no-matches
total number of turns
What are the possible
outcomes in this game?
Color on 1st spin – Color on 2nd spin
Are all outcomes equally likely?
Theoretical Probability of
No-Match
number of outcomes that are no- matches
total number of turns
Theoretical Probability of
Match
number of outcomes that are matches
total number of turns
A.
B.
Compare the experimental and
theoretical probabilities for match and
for no-match.
Is Match/No-Match a fair game?
If you think the game is fair, explain
why. If you think it is not fair, explain
how the rules could be changed to
make it fair.
Making Purple
RED
BROWN
YELLOW
GREEN
BLUE
ORANGE
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Contact Information
Jeanne Simpson
UAHuntsville AMSTI
jeanne.simpson@uah.edu
acos2010@wikispaces.com
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Feedback
Praise
Question
Polish