Mathematical Models Tools for ReTeaching

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Mathematical ModelsTools for Re-Teaching

• Logan Toone
• Davis School District Curriculum Department

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Re-Teaching

• What do we do when a kid doesnt understand

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Re-Teaching
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Re-Teaching

• Effective re-teaching is not merely teaching
  again.

Insanity Doing the same thing over and over
again and expecting different results. --Albert
Einstein (1879-1955
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One Way to Teach Differently

• Mathematical Models

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What is a Mathematical Model?

• Could mean one of 2 things.
• A model for a real-life situation using
  mathematical language
• Example population growth modeled by
  exponential functions

Remember
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What is a Mathematical Model?

• Not all real-life situations can be modeled
  mathematically

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What is a Mathematical Model?

• The other definition of mathematical model
• A concrete representation used to give meaning
  and clarity to a mathematical concept.
• Well look at two examples of models.
• A model for multiplying integers
• The bar model for problem solving

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Multiplying IntegersA Model for Teaching

• Developed by
• Logan Toone
• and
• Patty Norman

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Common Models for Multiplication

• Algebraic Model - Multiplication as repeated
  addition of sets
• Example

3 x 5
3 x 5 5 5 5 15
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Common Models for Multiplication

• Geometric Model - Multiplication as the area of a
  rectangle
• Example

3 x 5
3 x 5 15
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A Comprehensive Model ?

• These two models for multiplication work
  primarily with whole numbers, which is great
• Is there a model that includes negative integers
  as well?

but
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A Model for All Integers

• We can map the possible combinations of positive
  and negative integers on a quadrant grid.

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A Model for All Integers

• We continue the model of multiplication as
  repeated addition of sets.

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A Model for All Integers
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In Other Words
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Convention for this Model

• The first number tells us how many sets.
• The second number tells us how many elements in
  each set.
• The commutative property would also allow us to
  work with five sets of three, but we chose this
  convention for the model.

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Quadrant 1Positive x Positive
Start with zero sets
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Positives
Negatives
3 x 5 add 3 sets of positive 5
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Quadrant 2Negative x Positive
Start with zero equivalent (3 sets of negative 5
and 3 sets of positive 5)
-15
Positives
Negatives
(-3) x 5 subtract 3 sets of positive 5
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Quadrant 3Negative x Negative
Start with zero equivalent (3 sets of negative 5
and 3 sets of positive 5)
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Positives
Negatives
(-3) x (-5) subtract 3 sets of negative 5
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Quadrant 4Positive x Negative
Start with zero sets
-15
Positives
Negatives
3 x (-5) add 3 sets of negative 5
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Summary
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Bar ModelsA Model for Teaching

• Logan Toone
• Davis School District Curriculum Department
• Examples adapted from
• Bisk, R. (2008). Problem Solving with Model
  Drawing. wwwfac.wrocester.edu/smip

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What is a Bar Model?

• In a bar model, the bars are representative of
  numbers (known or unknown) with the length of the
  bar corresponding to the magnitude of the number.

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What is a Bar Model?

• The bars can be divided into units which may be
  known or unknown.

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3x
5
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5x
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Simple Application

• A worker earned three times as much this week as
  she did last week. If she earned a total of 200
  in the two weeks, how much more did she earn this
  week than last week?

200
???
4 units 200 1 unit 50 Difference 2 units
100
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Simple Application

• The sum of two numbers is 30. The smaller number
  is two-thirds of the larger number. What are the
  two numbers?

30
5 units 30 1 unit 6 Larger number 3 units
18 Smaller number 2 units 12
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Simple Application

• One of your stock certificates lost 2/5 of its
  total value in the last two weeks. If it
  originally was worth 60.00, how much is it worth
  now?

60
???
5 units 60 1 unit 12 Current value 3 units
36
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More Challenging Application

• A box of Cheez-Its was divided among Paul, Rita,
  and Anna in the ratio of 357. If Rita got 40
  Cheez-Its, how many more Cheez-Its did Anna get
  than Paul?

60
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More Challenging Application

• In a stadium, one fourth of the seats are in the
  top tier. One third of the seats are in the
  center tier, and there are 11,250 seats on the
  floor. How many total seats are there in the
  stadium

11,250
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More Challenging Application

• You spent 60 of your paycheck during the first
  week of the month. You spent 1/3 of the
  remainder during the second week. If you spent
  220.00 in the two weeks, how much of the
  paycheck was left.

110
???
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Reminder

• As with all other models, bar models are not
  intended to be a comprehensive strategy for
  problem solving. (They dont work smoothly for
  all problems).
• Rather, they are a way of helping students
  visualize what is happening so they can begin to
  move from concrete ideas to abstract thought and
  mathematical representation.