Cumberland County

1
Cumberland County

• Thinking Maps Follow Up
• In Mathematics
• Hope Mills Middle School
• January 28,2008
• Janie B. MacIntyre

2
Agenda

• Building Communities of Learners
• Cheat Sheets for Test Prep
• Additional Curricular Samples
• Thinking Maps Throughout a unit in Geometry
• Connections to EOG Released Items
• EOG Triggers and Tricks
• QA

3
Getting Ready Take a sheet of colored,legal-
length paper

• Fold in halfhamburger bun-style
• Re-open the paper, revealing the 4 sections
  (front and back)
• Number in the top left hand corner of
• each section 1 3
• 2
  4
• 
  front back

4
In Section 1, please construct and use a Circle
Map to answer the following questions.
Has the frequency of TMs use changed since our
1st session?
Have you been able to share your TMs?
Crates?
What TMs have you used?
Reflection Observation
How were they used within your instruction?
How would you rate your current comfort level?
Successes?.Barriers?
What impact on student learning have you
detected?
5
Take a closer look

• Place your Thinking Maps samples on the tables
• Get ideas from others for use with your students
• Record the ideas on your tree map

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Thinking Maps Ideas in Math
Section 2 Ideas from Other Teachers

• Goal Map Type Map title Comment

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In Section 1, please add the Frame of Reference
to your Circle Map and respond to
What insights have you gained from others?
What are the effects of this kind of dialogue?
Has the frequency of TMs use changed since our
1st session?
Have you been able to share your TMs?
Crates?
What TMs have you used?
Reflection Observation
How were they used within your instruction?
How would you rate your current comfort level?
SuccessesBarriers?
What impact student learning have you detected?
How can this interaction be continued?
How do you plan to use what you have learned?
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Section 3Classroom Applications

• Reflect on individual progress.
• Share related work samples.
• Gain insights from others.
• Reflect on the interpersonal benefits for the
  whole groups long term goals.

Make notes about using this strategy to develop a
community of learners in your class.
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Take out
"Cheat Sheets" for Test Prep
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What are the Cheat Sheets?

• Originally used for EOG Review
• Not all inclusiveadd or delete as needed
• Subsequent use was on-going
• developed prior to unit assessment
• included key aspects students needed
• cheat sheet wall
• Lets take a closer look

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Students love the idea of a cheat sheet!!!
R.F. When you encounter a problem involving ___
think about or use ____.
Continue to develop as an on-going project.
Include it as a cornerstone in the students EOG
review packet.
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Section 4Use Cheat Sheets
In your table groups discuss different ways you
can use Cheat Sheets with your
students. Record each idea on a post-it
note. Place on chart paper for whole group
sharing of ideas.
Include the whole groups ideas in section 4.
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Have you examined the additional curricular
samples of math Thinking Maps?
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FOCUS ON
Maps throughout a unit in Geometry
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Classify Essential Concepts with a Tree Map
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Color-coding greatly aids student
understanding as each special angle type is
taught.
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Concepts develop logically and sequentially to
aid in student understanding, and to support them
in the learning of new material to follow.
Triangles are classified in two ways.
Triangle classification by congruent sides.
Triangle classification by angle types.
Facts to aid in triangle classification.
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Visual models of abstract concepts
provide students a concrete tool for learning.
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Have students use the Multi-flow Map to determine
what makes pairs of triangles congruent.
ASA
?
Congruent Triangles
CPCTC
?
?
Students would include the specifics from a given
pair of triangles.
24
Bridge Maps help students visualize the related
congruent parts of triangles

• AB AC BC
• XY XZ YZ

A B C
x y z
RF _______?__________
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Students can use a Bridge Map to solve for
similar figures. It will enable them to set up
the proportion appropriately.
Students will then replace the last as with an
cross-multiply, simplify.
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Have students begin with circle map to access
prior knowledge.
How would you define it?
What does it look like?
The Pythagorean Theorem
When is it used?
How is it used?
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Some students may require more specific task
analysis.
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Use a tree map to have students Identify
patterns, record findings, extend multiples.
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Any questions so far?

• Todays power point slides will be saved on
• your computer.

31
Getting Ready Take a sheet of colored,legal-
length paper

• Fold in halfhamburger bun-style
• Re-open the paper, revealing the 4 sections
  (front and back)
• Number in the top left hand corner of
• each section 5 7
• 6
  8
• 
  front back

32
Section 5Connections to EOG items
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Some Ways to Solve Math Problems

• Pencil Paper
• Calculator
• Chart
• Sketch
• Guess Check
• Equation

• Pattern
• Proportion
• Estimate
• Measure
• Formula
• Working Backwards

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Having a clear picture or understanding of the
problem is critical in arriving at its solution.
The analysis is done as students prepare to
solve math problems.
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Please read the following problem, then

• select the best answer choice.

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• Solve this problem
• Two flagpoles stand outside the NC Department of
• Transportation Building. The one with the US
  flag stands
• 200 feet tall. The flagpole with the NC flag
  stands 170
• feet tall. The two flagpoles are seventy-five
  feet apart.
• What is the distance from the top of the US
  flagpole to
• the top of the NC flagpole?
• 75 ft.
• 30 ft.
• Sq. root of 6525
• Approx. 213 ft.

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Would the best solution include a sketch?

• Would it makes a difference in how one would
  analyze and solve this problem.

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• Solve this problem
• Two flagpoles stand outside the NC Department of
• Transportation Building. The one with the US
  flag stands
• 200 feet tall. The flagpole with the NC flag
  stands 170
• feet tall. The two flagpoles are seventy-five
  feet apart.
• What is the distance from the top of the US
  flagpole to
• the top of the NC flagpole?
• 75 ft.
• 30 ft.
• Sq. root of 6525
• Approx. 213 ft.

75
200
170
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Students can be taught how to be more analytical
as they approach and solve the problems they
face.
One method uses
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Questions to Aid in Item Analysis

• What are you given?
• What do you see?
• What are you asked to do?
• What kind of thinking is needed?
• Which Thinking Map would be helpful?
• What is the best method to use?
• How would you use that method to solve this
  problem?
• How do the answer foils try to trick you?

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• Solve this problem
• Two flagpoles stand outside the NC Department of
• Transportation Building. The one with the US
  flag stands
• 200 feet tall. The flagpole with the NC flag
  stands 170
• feet tall. The two flagpoles are seventy-five
  feet apart.
• What is the distance from the top of the US
  flagpole to
• the top of the NC flagpole?
• 75 ft.
• 30 ft.
• Sq. root of 6525
• Approx. 213 ft.

Make a sketch Right Triangle length of the
hypotenuse Pythagorean Theorem Sequencing Flow
Map a. distance b/w flagpoles
b. difference in height of flagpoles
c. answer d. incorrect use of
measurements w/the
Pythagorean Theorem
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Activity Released EOG Item Analysis

• Each of you will be given the released items for
  one goal for your grade level.
• Analyze each release item by using key questions.
  
• Neatly write your responses with the item
• Choose 1 item to share.
• (Copy problem possible answers onto paper.
  Indicate grade level and goal on paper. Include
  answers to item analysis questions and be
  prepared to share with whole group.)
• Turn in your completed goal (Dont let me forget
  to do this.)
• Receive a copy the collective results for all
  goals in your grade level.
• Put an asterisk beside the number of any
  misleading or tricky items. Put a P beside any
  item that can be solved using a proportion.

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As you analyze the released test items answer the
following questions.
What do you see? What are you given? What are you
asked to do? What type of thinking is
needed? Which Thinking Maps will be
appropriate? What is the best method to use? How
would you use that method to solve the
problem? What is the correct answer? Why? How do
you know? How are the foils trying to trick
your students? Special Note
or underlining key terms may be very
helpful.
Highlighting
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Its Time to Model Our EOG Item Analysis
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Section 5Classroom Applications EOG Item
Analysis

• Instructional Process all year
• EOG Review
• Modeling for students by the process of
• I doWe doYou do
• then, they will use this approach
  automatically during EOG Math Test sessions.

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EOGTriggers Tricks

• The reoccurring words and phrases in math that
  signal how we should answer or solve a given
  problem can be called triggers.
• Some of the areas that may be indicated by
  triggers are
• Operations Estimation
• Proportions Pythagorean
  Theorem
• Units of Measure Equations

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Section 6 Proportion Triggers

• Look at the EOG items you marked with a P for
  Proportionand underline the triggers.
• Include them on your Proportion Triggers circle
  map.

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Write the Proportional TriggersYou Detect on the
Chart Paper Circle Map for Your Grade Level

• Cut the corresponding proportion-triggering
  problems answers from the
• CLEAN SET of ITEMS.
• Tape the problems in the circle maps
• Frame of Reference

How would your students benefit from this type of
item exposure?
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Develop Circle Maps withTriggers Problems for

• Other areas like
• Estimating
• Using the Pythagorean Theorem
• Percentages
• Equations
• Others???

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Proportion Triggers
Make sure your students are fully aware of the
multiple opportunities to solve problems by using
proportions.
Any time two different quantities are set up as
equals. Similar figures At this rate
If..then. Enlargements and reductions
Recipe conversions Can be performed in ___,
how long would it take to How many
should.if Any ratio comparisons involving
change or variation Amount for x dayshow much
for ___days
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Use Bridge Maps to set up Solve for
Proportional Problems
A 6-foot man is standing beside a tree. He is
casting a shadow that is 4 feet long. If the
tree is casting a shadow that is 20 feet long,
how tall is the tree? ___Man___ ____6____
___4___ Tree x
20
6(20) 4(x)
120 4x
30 x
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Never Fail Percent Proportionis of
100

• Flow Map How to use
• proportion
• Write percent proportion
• Connect the terms within the problem.
• Substitute values.
• Cross Multiply
• Divide by Coefficient

• Examples
• 15 is what percent of 90
• 15 is what percent of 90
• 15 x
• 90 100
• 1500 90X
• 16.6 X
• What is 7 of 165
• 12 is 15 of what number

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Key Operation Terms
Addition
Subtraction
Multiplication
Division
Sum Total In all Altogether Plus Increase Deposit
Gain More than Greater than Taller than Higher
than
Difference Minus Decrease Reduce Less Least Withdr
awal Less than Fewer than Smaller than Shorter
than Lower than
Product Of Multiply Per Doubled Tripled Times Twic
e
Quotient Same Size Divided by Equal
amounts Shared equally Same size
Items should be written from right to left!
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Estimation Triggers

• What are the triggers for estimation?

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Estimation Techniques
Calculator Active
Calculator Inactive
Whole Numbers
Decimals
Fractions
Other areas?
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Caution Your Students !

• Exact answers are often included as
• possible answer choices on
• estimation problems!
• Make sure your students understand that an exact
  answer will be an INCORRECT response to an
  estimation problem.

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Removing the Tricks!!!

• Okayso what are they?

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Be careful with pattern extensions.
Be careful with Units of Measure in answer
choices.
Be careful with operations in answer choices.
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Tricks with Patterns
Whats your answer?

• A set of numbers is shown below. Should its
  pattern
• Continue, what would be the eighth member of the
  set?
• 2,4,8,16
• 32
• 8
• 128
• d. 256

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• A set of numbers is shown below. Should its
  pattern
• Continue, what would be the eighth member of the
  set?
• 2, 4, 8, 16,32, 64, 128,256
• 32
• 8
• 128
• d. 256

Our brains are seekers of patterns and and quite
naturally want to select the very next member of
the set. Without fail, the next member of the
set is the very first answer choice. Have
students underline as if for placeholders to
the required position. Otherwise, and
even though the pattern was determined
and extended, the answer would be wrong!
Why was answer b given?
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Notice how simultaneous Flow Maps were used by
this student to arrive at the correct response.
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Tricks Within Answer Selections

• Students typically anticipate the straightforward
  question or problem
• Students need to become more aware of the
  likelihood of misleading answer choices that can
  be exacerbated by the need to work as quickly.
• Modeling can be very helpful in increasing
  student awareness.

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Tricks are often used in answer choices
involving units and degree of measure

• How much carpeting would be required for a
  rectangular
• bedroom measuring 10 x 14 feet?
• 48 ft²
• 140 ft³
• 96 ft
• 140 ft²

Make students aware of this trick!
Might time be a factor in student answer
selection?
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• If 45 were written in decimal
• notation, which of the following
• number statements would be true?
• 44444
• 5x5x5x5
• 4x4x4x4x4
• 4x5

What would your students choose?
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Question Triggers
Use clear and concise techniques to help
students effectively comprehend
question triggers. This should aid them with
time management, and mathematical focus
on essential information.
Focus on questions containing except, not,
but.
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Measurement with Overlays
What is the area of the shaded region?
4
30
18
6
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Measurement with Overlays
Use multiple transparencies, or pieces of
construction paper to demonstrate the two shapes,
independently. Once students can visualize the
simplicity of the problem, they will be able to
find the difference of the two respective area
dimensions.
30
4
18
6
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Measurement with Overlays

30
18
4
6
30
6
18
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Area 684 sq.
-240 sq Shaded Area 444 sq
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Tricky Wording of Questions

• What is?
• A standard six sided random number generator.

• What is this question about?
• Two clocks were on a wall. One clock was
  larger than the other. The minute hand on the
  smaller clock was 6 inches. The minute hand on
  the larger clock was 10 inches. In the course of
  1 hour, how much greater a distance did the
  minute hand of the larger clock travel than the
  minute hand of the smaller clock.

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Why not call a die, just that!
Mmm.. Did anyone here author any EOG test ?s

• Is this question about timedistance or the
  difference in the circumference of two circles?

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Other Tricks

• In your table group, what other tricks might
  students face on math EOG tests?
• Be prepared to share

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Now.turn in the sets ofreleased items.

• Thank you.

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Section 7Potential EOG Tricks

• Items needing a sketch
• Pattern extensions
• Units or powers of answer choices
• Misleading questions
• Misleading operation symbols
• Misleading placement of decimals (scientific
  notation)
• Confusing questions

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Map Making

• Create the Thinking Maps you will need for the
  next unit of instruction you will be teaching.
• We will share with the whole group.

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Section 8Thinking Maps for Unit on _______

• Include a list of the maps you have constructed
  for instructional use with your students
• Concept Type of Map Title of Map

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Thinking Maps Labels

• The master sheet of Thinking Maps labels can be
  used to print actual peel and stick labels.
• Use them in your lesson plans
• Teacher edition text
• Other uses????

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Any Questions?
Janie MacIntyre 812 Nichole Lane Rocky Mount,
N.C. 27803 jmacin0722_at_aol.com 252-903-7274
(cell)