Content Deepening
7th Grade Math
February 6, 2014
Jeanne Simpson
AMSTI Math Specialist
Welcome
Name
School
Classes
you teach
What are you hoping to
learn today?
2
He who dares to
teach must never
cease to learn.
John Cotton Dana
3
Goals for Today
Implementation
of the Standards of
Mathematical Practices in daily
lessons
Understanding of what the CCRS
expect students to learn blended
with how they expect students to
learn.
Student-engaged learning around
high-cognitive-demand tasks used in
every classroom.
Agenda
Surface
Area &
Volume
Statistics
Fractions
Probability
5
acos2010.wikispaces.com

Electronic version of handouts

Links to web resources
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
9
Standards for Mathematical
Practice
Mathematically proficient students will:
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
SMP Proficiency Matrix
Students:
Make sense of
problems
(I) Initial
Explain their thought processes in solving a
problem one way.
1b
Persevere in
solving them
Stay with a challenging problem for more than Try several approaches in finding a solution, and
one attempt.
only seek hints if stuck.
Struggle with various attempts over time, and learn
from previous solution attempts.
2
Reason
abstractly and
quantitatively
Construct viable
arguments
Critique the
reasoning of
others.
Model with
Mathematics
Reason with models or pictorial
representations to solve problems.
Are able to translate situations into symbols for
solving problems.
Explain their thinking for the solution they
found.
Understand and discuss other ideas and
approaches.
Explain their own thinking and thinking of others
with accurate vocabulary.
Explain other students’ solutions and identify
strengths and weaknesses of the solution.
Convert situations into symbols to appropriately
solve problems as well as convert symbols into
meaningful situations.
Justify and explain, with accurate language and
vocabulary, why their solution is correct.
Compare and contrast various solution strategies
and explain the reasoning of others.
Use models to represent and solve a problem,
and translate the solution to mathematical
symbols.
Use the appropriate tool to find a solution.
Use models and symbols to represent and solve a
problem, and accurately explain the solution
representation.
Select from a variety of tools the ones that can be
used to solve a problem, and explain their reasoning
for the selection.
Incorporate appropriate vocabulary and symbols
when communicating with others.
Use a variety of models, symbolic representations,
and technology tools to demonstrate a solution to a
problem.
Combine various tools, including technology,
explore and solve a problem as well as justify their
tool selection and problem solution.
Use appropriate symbols, vocabulary, and labeling
to effectively communicate and exchange ideas.
1a
3a
3b
4
5
6
Use appropriate
tools
strategically
Attend to
precision
Communicate their reasoning and solution to
others.
(IN) Intermediate
(A)
Advanced
Explain their thought processes in solving a problem Discuss, explain, and demonstrate solving a
and representing it in several ways.
problem with multiple representations and in
multiple ways.
7
Look for and
make use
of structure
Look for structure within mathematics to help Compose and decompose number situations and
them solve problems efficiently (such as 2 x 7 x 5 has relationships through observed patterns in order to
the same value as 2 x 5 x 7, so instead of multiplying 14 x 5, which is (2 x
simplify solutions.
7) x 5, the student can mentally calculate 10 x 7.
See complex and complicated mathematical
expressions as component parts.
8
Look for and
express
regularity in
repeated
reasoning
Look for obvious patterns, and use if/ then
reasoning strategies for obvious patterns.
Discover deep, underlying relationships, i.e.
uncover a model or equation that unifies the
various aspects of a problem such as
discovering an underlying function.
Find and explain subtle patterns.
SMP Instructional
Implementation Sequence
1.
Think-Pair-Share (1, 3)
2.
Showing thinking in classrooms (3, 6)
3.
Questioning and wait time (1, 3)
4.
Grouping and engaging problems (1, 2, 3, 4, 5, 8)
5.
Using questions and prompts with groups (4, 7)
6.
Allowing students to struggle (1, 4, 5, 6, 7, 8)
7.
Encouraging reasoning (2, 6, 7, 8)
SMP Proficiency Matrix
1a
Students:
Make sense of
problems
(I) Initial
Explain their thought processes in solving a
problem one way.
Pair-Share
(IN) Intermediate
(A)
Advanced
Explain their thought processes in solving a problem Discuss, explain, and demonstrate solving a
and representing it in several ways.
problem with multiple representations and in
multiple ways.
Questioning/Wait Time
Grouping/Engaging Problems
1b
Persevere in
solving them
Stay with a challenging problem for more than Try several approaches in finding a solution, and
Problems
one attempt.
only Grouping/Engaging
seek hints if stuck.
Struggle with various attempts over time, and learn
from previous solution attempts.
2
Reason
abstractly and
quantitatively
Reason with models or pictorial
representations to solve problems.
Grouping/Engaging
Problems
Explain their thinking for the solution they
found.
Showing Thinking
Are able to translate situations into symbols for
solving problems.
Convert situations into symbols to appropriately
solve problems as well as convert symbols into
meaningful situations.
Explain other students’ solutions and identify
strengths and weaknesses of the solution.
Questioning/Wait Time
Grouping/Engaging Problems
3a
Construct viable
arguments
3b
Critique the
reasoning of
others.
Understand and discuss other ideas and
approaches.
4
Model with
Mathematics
Use models to represent and solve a problem, Use models and symbols to represent and solve a
and translate the solution to mathematical
problem, and accurately explain the solution
symbols.
representation.
5
Use appropriate
tools
strategically
Use the appropriate tool to find a solution.
6
Attend to
precision
Communicate their reasoning and solution to
others.
7
Look for and
make use
of structure
Look for structure within mathematics to help Compose and decompose number situations and
them solve problems efficiently (such as 2 x 7 x 5 has relationships through observed patterns in order to
the same value as 2 x 5 x 7, so instead of multiplying 14 x 5, which is (2 x
simplify solutions.
7) x 5, the student can mentally calculate 10 x 7.
Look for and
express
regularity in
repeated
reasoning
Look for obvious patterns, and use if/ then
reasoning strategies for obvious patterns.
8
Pair-Share
Grouping/Engaging
Problems
Grouping/Engaging Problems
Showing Thinking
Questions/Prompts
for Groups
Grouping/Engaging
Problems
Explain their own thinking and thinking of others
with accurate vocabulary.
Questioning/Wait Time
Questioning/Wait Time
Questions/Prompts for
Groups
Select from a variety of tools the ones that can be
Showing Thinking
Encourage
Reasoning
Justify and explain, with accurate language and
Grouping/Engaging
vocabulary,
why their solution is Problems
correct.
Compare and contrast various solution strategies
and Grouping/Engaging
explain the reasoning of others.
Problems
Use a variety of models, symbolic representations,
and technology tools to demonstrate a solution to a
problem.
Showing Thinking
Combine various tools, including technology,
used to solve a problem, and explain their reasoning explore and solve a problem as well as justify their
Grouping/Engaging
Problems
for the
selection.
tool selection and problem solution.
Showing Thinking
Incorporate appropriate vocabulary and symbols
when communicating with others.
Allowing Struggle
Allowing Struggle
Find and explain subtle patterns.
Allowing Struggle
Use appropriate symbols, vocabulary, and labeling
to effectively communicate and exchange ideas.
Encourage
See complex and complicated mathematical
Reasoning
expressions as component parts.
Encourage
Discover deep,
underlying relationships, i.e.
Reasoning
uncover a model or equation that unifies the
various aspects of a problem such as
Encourage
discovering an underlying function.
Reasoning
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
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
Geometry
 Draw,
construct, and describe geometrical
figures and describe the relationships
between them.
 Solve
real-life and mathematical problems
involving angle measure, area, surface
area, and volume.
15
Surface Area and Volume
7.G.6
- Solve real-world and
mathematical problems
involving area, volume and
surface area of two- and threedimensional objects composed
of triangles, quadrilaterals,
polygons, cubes, and right
prisms.
16
7.G.6 - Solve real-world and mathematical
problems involving area, volume and surface
area of two- and three-dimensional objects
composed of triangles, quadrilaterals, polygons,
cubes, and right prisms.
17
Accelerated
8.G.9
– Know the
formulas for the volumes
of cones, cylinders, and
spheres and use them to
solve real-world and
mathematical problems.
18

According to Bill McCallum, 7th graders need to be
able to find the surface area of pyramids, but not
the volume.

“Know the formula” means:
 Having
an understanding of why the formula works
 Being
able to use the formula to solve a problem
without being told which formula to use

Surface area formulas are not the expectation.
Students should recognize that finding the area of
each facing and adding the areas will give the
surface area.

For volume, focus on the area of the base times the
height.

Pyramids are considered a type of cone for the 8th
grade standard.
Connected Mathematics
1.
2.
3.
4.
5.
Building Boxes
Designing
Rectangular Boxes
Prisms and Cylinders
Cones, Spheres, and
Pyramids
Scaling Boxes
Connected Mathematics








Understand volume as a measure of filling an object and
surface area as a measure of wrapping an object.
Design and use nets to visualize and calculate surface
area of prisms and cylinders.
Explore patterns among the volumes of cylinders, cones,
and spheres.
Develop strategies for finding the volumes of square
pyramids, prisms, cylinders, cones, and spheres directly
and by comparison with known volumes.
Understand that three-dimensional figures my have the
same volumes but quite different surface areas.
Understand how changes in one or more dimensions of a
rectangular prism or cylinder affects the prism’s volume
and surface area.
Extend students’ understanding of similarity and scale
factor to three-dimensional figures.
Use surface area and volume to solve a variety of realworld problems.
Surface Area and Volume
Building
a Box
Patch Tool
Surface Area and Volume
Changing
Surface Areas
Packing
to
Perfection
Changing Surface Area
Changing Surface Areas
Packing to Perfection
Is
there a relationship between
surface area and volume?
Can rectangular prisms with
different dimensions have the
same volume?
Do rectangular prisms with the
same volume have the same
surface area?
Packing to Perfection
Packing to Perfection
Statistics
Use
random sampling to draw
inferences about a population.
Draw informal comparative
inferences about two
populations.
31
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.
32
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.
Use random sampling to draw
inferences about a population.
 7.SP.2
– Use data from a random sample to
draw inferences about a population with an
unknown characteristic of interest. Generate
multiple samples (or simulated samples) of the
same size to gauge the variation in estimates or
predictions. For example, estimate the mean
word length in a book by randomly sampling
words from the book; predict the winner of a
school election based on randomly sampled
survey data. Gauge how far off the estimate or
prediction might be.
Counting Trees
The diagram shows some trees in a tree farm.






































 

 
 
 
 
 
 

 
 


 


 

 
 
 

 

 

 

 


 
 


 
 
 
 

 
 
 
 
 
 

 
 



  
  





  
 
  
  
 
 


 
  
 


  
 
 


  


  
  
  
 
  
  
 
 
  
  
 


 
  


  
  
  
  
  

  
  
  

  
  
 
 
 
 
 
  
 
  
  
 
  
  

 
  
  
 
  


  
 
 


  
 


  


 


 
 


  

 
 


  


 
  
  



    
  


  

 
     

  
 

   
 


   
    
    
  

 

   
 
 

  

  
  
 
     
  
 
 
  
   

     


 
 
  



 


  

   
   

 
 
     
    
     
   

  

 
 
  
  

   
    
     
     
 

   

   

   
   
  
  
   





















 

 

 
 
 
 
 
 

 
 
 
 

 
 


 
 

 
 



 
 
 
 
 
 
 
 















  
  

  


 

  

 








  

  

  


  
  
  
  
  
 
  

 
  
 
  
 
  
  
 
  

  
  


 
  

  
  
  
  
  


 
 
 
  
 
 

 
 

 
 
  
 


  




  
  
 


  





  
  
  
 
  

 
 


  


  
 
  
 


 
 
 


  
  
 




  
  
 
 
  
  






 
 

 



 

 
 

 

 
 
 





 


 

 
 
 
 
 
 


 
 
 
 

 
 
 
 
 
 








































 
 
 

 
 
 
 
 


 
 

 
 



 

 
 

 
 
 


































  

  
  
 
 
  

 
  
 


  



  
 
  
  
  
 
  

  
  
  
 
  
  
  
 

  
  
  
  
 
  





  
  
  
  
  
  
  


 

  

 


  


 
 


   
   
   
   



   
   

 
   
   
   
  
 
 



   
 

  
   
   
   
   
 
 

 

 
 

   
   
  

 
   
   

 
 

   
 
 
   
  
 
  
 
  
 
 




  
  
  




  
 
 
 
  
 
  
  
  


  
 


  
 
 
 
  
 
  
 


 
 
 
  
 
  
  
 


  
  

  
 
 




 
 
 
 
 


 

 
 
 
 
 
 


 
 


 

 

 
 
 
 







 

 
 
 
  
  



  
  
  
 
  
  

 
  






  
  
  
  
 
































  


 

  
  
  


 
 
 
 

  
  
  
  


  
 
 
  
  
 
 
 
  



 
 



  

 
  
  

  






  
  
  
  
  






























































 
 


 
 
 
 
 


 
 

 
 

 
 
 



 
 
 
 

 
 




 
 
 
The circles  show
old trees and the
triangles  show
young trees.

 






































Tom wants to know how many trees there are of each type, but says it
would take too long counting them all, one by one.
Projector Resources
Estimating: Counting Trees
P-42
The Tree Farm






































 

 
 
 
 
 
 

 
 


 


 

 
 
 

 

 

 

 


 
 


 
 
 
 

 
 
 
 
 
 

 
 



  
  





  
 
  
  
 
 


 
  
 


  
 
 


  


  
  
  
 
  
  
 
 
  
  
 


 
  


  
  
  
  
  

  
  
  

  
  
 
 
 
 
 
  
 
  
  
 
  
  

 
  
  
 
  


  
 
 


  
 


  


 


 
 


  

 
 


  


 
  
  



    
  

Projector Resources

  

 
     

  
 

   
 


   
    
    
  

 

   
 
 

  

  
  
 
     
  
 
 
  
   

     


 
 
  



 


  

   
   

 
 
     
    
     
   

  

 
 
  
  

   
    
     
     
 

   

   

   
   
  
  
   





















 

 

 
 
 
 
 
 

 
 
 
 

 
 


 
 

 
 



 
 
 
 
 
 
 
 











 


  
  

  


 

  








 

 
 
 
 



 
 

  


  
  
  
  
  
 
  

 
  
 
  
 
  
  
 
  

  
  


 
  

  
  
  
  
  


 
 
 
  
 
 
 
  
 


  




  
  
 


  





  
  
  
 
  

 
 


  


  
 
  
 


 
 
 


  
  
 




  
  
 
 
  
  






 
 

 



 

 
 

 

 
 
 





 


 

 
 
 
 
 
 


 
 
 
 

 
 
 
 
 
 








































 
 
 

 
 
 
 
 


 
 

 
 



 

 
 

 
 
 


































  

  
  
 
 
  

 
  
 


  



  
 
  
  
  
 
  

  
  
  
 
  
  
  
 

  
  
  
  
 
  





  
  
  
  
  
  
  


Estimating: Counting Trees
 

  

 


  


 
 


   
   
   
   



   
   

 
   
   
   
  
 
 



   
 

  
   
   
   
   
 
 

 

 
 

   
   
  

 
   
   

 
 

   
 
 
   
  
 
  
 
  
 
 




  
  
  




  
 
 
 
  
 
  
  
  


  
 


  
 
 
 
  
 
  
 


 
 
 
  
 
  
  
 


  
  

  
 
 




 
 
 
 
 


 

 
 
 
 
 
 


 
 


 

 

 
 
 
 







 

 
 
 
  
  



  
  
  
 
  
  

 
  






  
  
  
  
 
































  


 

  
  
  


 
 
 
 

  
  
  
  


  
 
 
  
  
 
 
 
  



 
 



  

 
  
  

  






  
  
  
  
  






























































 
 


 
 
 
 
 


 
 

 
 

 
 
 



 
 
 
 

 
 




 
 
 

 






































P-43
Collaborative Work: Joint Solution
1. Share your method with your partner(s) and your ideas for
improving your individual solution.
2. Together in your group, agree on the best method for completing
the problem.
3. Produce a poster, showing a joint solution to the problem.
4. Make sure that everyone in the group can explain the reasons for
your chosen method, and describe any assumptions you have
made.
Projector Resources
Estimating: Counting Trees
P-44
Analyzing Sample Responses to Discuss
1. Does the approach make mathematical sense?
2. What assumptions has the student made?
3. How could the solution be improved?
4. What questions could you ask the student, to
help you understand their work?
Projector Resources
Estimating: Counting Trees
P-45
Sample Responses to Discuss: Laura
Projector Resources
Estimating: Counting Trees
P-46
Sample Responses to Discuss: Wayne
Projector Resources
Estimating: Counting Trees
P-47
Sample Responses to Discuss: Amber
Projector Resources
Estimating: Counting Trees
P-48
Draw informal comparative
inferences about two populations.

7.SP.3 – Informally assess the degree of visual
overlap of two numerical data distributions with
similar variabilities, measuring the difference
between the centers by expressing it as a multiple
of a measure of variability. For example, the mean
height of players on the basketball team is 10 cm
greater than the mean height of players on the
soccer team, about twice the variability (mean
absolute deviation) on either team; on a dot plot,
the separation between the two distributions of
heights is noticeable.
49
How MAD are You?
(Mean Absolute Deviation)
 Fist
to Five…How much do you know about
Mean Absolute Deviation?
0
= No Knowledge
5
= Master Knowledge
Create a distribution of nine
data points on your number
line that would yield a
mean of 5.
Card Sort
 Which
data set seems to differ the least
from the mean?
 Which
data set seems to differ the most
from the mean?
 Put
all of the data sets in order from
“Differs Least” from the mean to “Differs
Most” from the mean.
The mean in each set equals 5.
Find the distance (deviation) of each point from the
mean.
Use the absolute value of each
distance.
3
3
3
2
1
1
3
4
Find the mean of the absolute deviations.
6
 How
could we arrange the nine points
in our data to decrease the MAD?
 How could we arrange the nine points
in our data to increase the MAD?
 How MAD are you?
Draw informal comparative
inferences about two populations.
 7.SP.4
– Use measures of center and
measures of variability for numerical data
from random samples to draw informal
comparative inferences about two
populations. For example, decide whether
the words in a chapter of a seventh-grade
science book are generally longer than the
words in a chapter of a fourth-grade
science book.
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/
56
Offensive Linemen
X Marks the Spot
Shakespeare vs. Rowling
The Number System
Apply
and extend previous
understandings of operations
with fractions to add,
subtract, multiply, and
divide rational numbers.
60
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
61
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
62
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)
63
Unit Fractions
64
Fraction Multiplication in Grade 5
65
Fraction Multiplication in Grade 5
66
Fraction Multiplication in Grade 5
67
Fraction Multiplication in Grade 5
68
5th Grade Division
69
5th Grade Division
70
5th Grade Division Problems
How
much chocolate will
each person get if 3 people
share ½ pound equally?
71
5th Grade Division Problems
How
many 1/3 cup servings
are in 2/3 cups of raisins?
72
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
73
6th Grade Division
74
6th Grade Division
75
Probability
Investigate chance processes and develop,
use, and evaluate probability models.
 7.SP.5
– Understand that the probability of
a chance event is a number between 0and
1 that expresses the likelihood of the event
occurring. Larger numbers indicate greater
likelihood. A probability near 0 indicates an
unlikely event, a probability around 1/2
indicates an event that is neither unlikely
nor likely, and a probability near 1
indicates a likely event.
Two bags of jellybeans
I have two bags. Both contain red and yellow
jellybeans.
There are more red jellybeans in bag A than in
bag B.
If I choose one jellybean from each bag I am
more likely to choose a red one from bag A
than from bag B.
Beta Version
© 2011 MARS University of Nottingham
Projector resources:
78
True, False or Unsure?
• Take turns to select a card and decide whether it is a true or false
statement.
• Convince your partner of your decision.
• It is important that you both understand the reasons for the decision.
If you don’t agree with your partner, explain why. You are both
responsible for each other’s learning.
• If you are both happy with the decision, glue the card onto the paper.
Next to the card, write reasons to support your decision.
• Put to one side any cards you are unsure about.
Beta Version
© 2011 MARS University of Nottingham
Projector resources:
79
Investigate chance processes and develop,
use, and evaluate probability models.

7.SP.6 – Approximate the probability of a chance event by collecting data on
the chance process that produces it and observing its long-run relative
frequency, and predict the approximate relative frequency given the
probability. For example, when rolling a number cube 600 times, predict
that a 3 or 6 would be rolled roughly 200 times, but probably not exactly
200 times.

7.SP.7 – Develop a probability model and use it to find probabilities of
events. Compare probabilities from a model to observed frequencies; if the
agreement is not good, explain possible sources of the discrepancy.

7.a – Develop a uniform probability model by assigning equal probability to all
outcomes, and use the model to determine probabilities of events. For example, if
a student is selected at random from a class, find the probability that Jane will
be selected and the probability that a girl will be selected.

7.b – Develop a probability model (which may not be uniform) by observing
frequencies in data generated from a chance process. For example, find the
approximate probability that a spinning penny will land heads up or that a tossed
paper cup will land open-end down. Do the outcomes for the spinning penny
appear to be equally likely based on the observed frequencies?
Investigate chance processes and develop,
use, and evaluate probability models.

7.SP.8- Find probabilities of compound events using organized lists, tables,
tree diagrams, and simulation.

8.a – Understand that, just as with simple events, the probability of
a compound event is the fraction of outcomes in the sample space for
which the compound event occurs.

8.b – Represent sample spaces for compound events using method
such as organized lists, tables and tree diagrams. For an event
described in everyday language (e.g., “rolling double sixes”), identify
the outcomes in the sample space which compose the event.

8.c – Design and use a simulation to generate frequencies for
compound events. For example, use random digits as a simulation tool
to approximate the answer to the question: If 40% of donors have type
A blood, what is the probability that it will take at least 4 donors to
find one with type A blood?
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
Expected Value
Compound Probability
Design and use a simulation to generate frequencies for compound events. [7-SP8c]
Investigation 3
Expected Value
On April 14, 1993, during halftime of a basketball game between the
Chicago Bulls and the Miami Heat, Don Calhoun won $1 million by
making a basket from the free-throw line at the opposite end of the
court. Don was chosen at random to attempt the basket as part of a
promotional contest. A Sports Illustrated article explained:
The odds against one randomly chosen person given
one shot from the opposite foul line and making it are
considered astronomical. Scottie Pippen admitted
that after practice one day he and Michael Jordan
tried to hit the shot but couldn’t. (Source: Bessone,
Lisa, “Sports People: Don Calhoun.” Sports
Illustrated, April 26, 1993, vol 48.)
What Do You Expect Inv. 3
101
Will he hit it or miss it?
▪ In the section finals, William’s
basketball team is 1 point
behind with 6 seconds left.
William has just been fouled,
and he is in a one-and-one freethrow situation. This means that
William will try one free throw.
If he makes it, he tries a second
free throw. If he misses the first
time, he does not get a second
try. William’s free-throw
average is 60%.
One-and-One Free Throws
▪ What are the possible scores a
player can make in a one-and-one
free-throw situation?
▪ How can each score be made?
▪ How would you design an experiment to
analyze this situation?
Will he hit it or miss it?
▪ In the section finals, William’s
basketball team is 1 point
behind with 6 seconds left.
William has just been fouled,
and he is in a one-and-one freethrow situation. This means that
William will try one free throw.
If he makes it, he tries a second
free throw. If he misses the first
time, he does not get a second
try. William’s free-throw
average is 60%.
One-and-One Free Throws
▪ What are the possible scores a
player can make in a one-and-one
free-throw situation?
▪ How can each score be made?
▪ How would you design an experiment to
analyze this situation?
Which is a 60% shooter more likely to
score?
–0 points
–1 point
–2 points
▪ Use the spinner to simulate William’s oneand-one situation 20 times. Record the result
of each trial.
▪ Make an area model for this
situation using a 10 x 10 grid.
▪ What is the theoretical
probability that William will
score 0 points?
▪ That he will score 1 point?
▪ That he will score 2 points?
▪ Compare these with the
experimental probabilities we
found.
▪ Suppose William’s freethrow average is 70%.
How does this affect the
outcome?
▪ Suppose William’s freethrow average is 70%.
How does this affect the
outcome?
▪ How does this diagram
help you predict how
many times William will
score 2 points in 100 oneand-one situations? In 200
one-and-one situations?
Investigation 3.2 Finding Expected Value
In Problem 3.1, you looked at the probabilities
of different outcomes of Nishi’s one-and-one
free-throw situation. You might have been
surprised about which outcome is most likely.
In this Problem, you will look at the number of
points Nishi can expect to make each time
she is in a one-and-one free-throw situation.
What Do You Expect Inv. 3
111
Standards of Mathematical Practice
Standard 1:
Make sense of problems and persevere in
solving them.
Standard 2: Reason abstractly and quantitatively.
Standard 3: Construct viable arguments and critique
the reasoning of others.
Standard 4: Model with mathematics.
Standard 5: Use appropriate tools strategically.
Standard 6: Attend to precision.
Standard 7: Look for and make use of structure.
Standard 8: Look for and express regularity in repeated
reasoning.
An effective mathematical
task is needed to challenge
and engage students
intellectually.
Comparing Two
Mathematical Tasks
Solve Two Tasks:
• Martha’s Carpeting Task
• The Fencing Task
Comparing Two
Mathematical Tasks
How are Martha’s Carpeting Task
and the Fencing Task the same
and how are they different?
Similarities and Differences
Similarities
Differences
• Both are “area”
problems
• The amount of thinking
and reasoning required
• Both require prior
knowledge of area
• The number of ways the
problem can be solved
• Way in which the area
formula is used
• The need to generalize
• The range of ways to
enter the problem
Comparing Two
Mathematical Tasks
Do the differences between the
Fencing Task and Martha’s
Carpeting Task matter?
Why or Why not?
Criteria for low cognitive demand tasks
• Recall
• Memorization
• Low on Bloom’s Taxonomy
Criteria for high cognitive demand tasks
• Requires generalizations
• Requires creativity
• Requires multiple representations
• Requires explanations (must be “worth
explaining”)
Patterns of Set up, Implementation,
and Student Learning
Task Set Up
Task Implementation
Student Learning
A
.
High
High
High
B.
Low
Low
Low
C
.
High
Low
Moderate
Stein & Lane, 2012
•
•
•
•
•
•
Factors Associated with the Maintenance and
Decline of High-Level Cognitive Demands
Routinizing problematic aspects of the task
Shifting the emphasis from meaning, concepts, or
understanding to the correctness or completeness of the
answer
Providing insufficient time to wrestle with the demanding
aspects of the task or so much time that students drift into
off-task behavior
Engaging in high-level cognitive activities is prevented due
to classroom management problems
Selecting a task that is inappropriate for a given group of
students
Failing to hold students accountable for high-level products
or processes
(Stein, Grover & Henningsen, 2012)
Factors Associated with the Maintenance and
Decline of High-Level Cognitive Demands
• Scaffolding of student thinking and reasoning
• Providing a means by which students can monitor their own
progress
• Modeling of high-level performance by teacher or capable
students
• Pressing for justifications, explanations, and/or meaning
through questioning, comments, and/or feedback
• Selecting tasks that build on students’ prior knowledge
• Drawing frequent conceptual connections
• Providing sufficient time to explore
(Stein, Grover & Henningsen, 2012)
“Not all tasks are created equal, and
different tasks will provoke different
levels and kinds of student
thinking.”
(Stein, Smith, Henningsen, & Silver, 2011)
“The level and kind of thinking in
which students engage determines
what they will learn.”
(Hiebert et al., 2011)
Contact Information
Jeanne Simpson
UAHuntsville AMSTI
jeanne.simpson@uah.edu
acos2010@wikispaces.com
134
Feedback
Praise
Question
Polish
135