The Fundamental Arithmetic Operations: , , x,

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The Fundamental Arithmetic Operations , -, x,

• Joe Hill

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Addition

• Always involves joining two sets
• Prerequisite skills
• Counting
• Place value
• Concepts to be learned
• Counting on
• Basic addition number facts
• Commutative (order) property
• Associative (grouping) property
• Algorithm for adding large numbers

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Example

• Aaron has 7 tennis balls. John gives him 4 more.
  How many does Aaron have now?

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Example

• An elementary school has 347 boys and 385 girls.
  How many students in all are there in this school?

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1
3 4 7 3 8 5
2
3
7
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Addition Dos and Donts

• Do teach with manipulatives like unifix cubes and
  base ten blocks
• Do present the concept in a problem context (not
  just fact practice)
• Dont use the term carry - use regroup
  instead
• Do encourage students to develop their own mental
  strategies (such as adding tens first)
• Do emphasize the commutative (order) property -
  only 55 facts need to be memorized
• Do emphasize the associative (grouping) property
  such as in adding 28 37 2

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Subtraction

• Can occur in at least three ways
• Take away (Juan has 12. He buys a piece of
  candy which costs 3. How much money does he
  have now?)
• Comparison (Juan has 12 pets. Thomas has 3. How
  many more pets does Juan have than Thomas?)
• Completion (It takes 12 eggs to fill the carton.
  Patty has collected 3 so far. How many more does
  she need to collect to fill the carton?)
• Prerequisite skills
• Counting and place value
• Basic addition facts
• Concepts to be learned
• Counting up/down
• Basic subtraction number facts
• Relationship between subtraction and addition
• Algorithm for subtracting large numbers

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Example

• It is 22 miles from Randys house to school. It
  is 14 miles from Johnas house to school. How
  many more miles does Randy have to ride than
  Johna to come to school?

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Example

• A new stereo costs 2,013. John wants one and
  has saved 148. How much more does he need to
  save?

200113 - 1 4 8
1 9 10
200113 - 1 4 8
2013 - 148
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1 8 6 5
5
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Subtraction Dos and Donts

• Do give ample practice with all three types of
  subtraction problems
• Do present the concept in a problem context (not
  just fact practice)
• Dont use the terms borrow or markout- use
  regroup or trade in instead
• Do encourage students to develop their own mental
  strategies (such as counting up)
• Do require students to check subtraction problems
  by adding

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Basic /- Fact Practice

• Do the practice after the concept has been
  thoroughly presented in problem situations
• Use games frequently (Card games like Salute,
  Wheel of Fortune, Math Tic-Tac-Toe, etc.)
• Do the practice daily, even when /- are not the
  lessons of the day
• Practice the tougher facts (involving 6, 7, 8,
  and 9)
• Do some sort of mental math every day

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Multiplication

• Can be defined in at least three ways
• Beans in cups (Sarah has 5 cups. Into each cup
  she puts 4 beans. How many beans in all does she
  have?)
• Repeated addition (Add 5 5 5 5. Also add 4
  4 4 4 4)
• Array (Henry has arranged his blocks into a
  rectangle. His rectangle has 4 blocks across the
  top and 5 blocks down the side. How many blocks
  in all does he have?)
• Prerequisite skills
• Counting
• Basic addition facts
• Concepts to be learned
• Basic multiplication number facts
• Properties of multiplication commutative
  (order), associative (grouping)
• Algorithm for multiplying larger numbers

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Model 1 Balls in Cups

• One common way we think of multiplication is like
  putting balls in cups. For example, if I put 4
  balls in 5 cups, how any balls in all would I
  have?

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Heres a similar example

• If there are four tricycles in a room, how many
  wheels are there?

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This model can be shown as a tree diagram.

• For example, if three students each carry four
  books, how many books in all are there?

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In this model

• Note there are two distinct objects in each
  example (balls-cups, tricycles-wheels,
  students-books). In the multiplication, one
  factor is the number of items of the first object
  (balls, tricycles, students) and the other factor
  is the number of identical second items (cups,
  wheels, books)
• This model leads to an understanding of the
  Fundamental Counting Principle demonstrated in
  problems like this If you can select from 5
  types of sandwiches and 3 types of drinks, how
  many meals could you have consisting of a
  sandwich and a drink?
• It is not as useful in multiplication of
  fractions and decimals such as 1/2 x 1/3 or 0.2 x
  0.3. It also gives the student the false idea
  that the answer to a multiplication problem is
  always more than either factor.

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Model 2 Repeated addition

• 4 4 4 4 4 can be written as 5 x 4
• 6 6 6 can be written as 3 x 6

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In this model

• This differs from the last model in that there
  are not two distinct objects. Here one factor is
  the number to be repeatedly added and the other
  factor is the number of times it appears.
• This model ties in directly with skip counting
  which students do in the primary grades.
• Like the first model, it is not as useful in
  multiplication of fractions and decimals such as
  1/2 x 1/3 or 0.2 x 0.3. It also gives the
  student the false idea that the answer to a
  multiplication problem is always more than
  either factor.
• Another issue is that the commutative property
  isnt as obvious to the student. Does 3 3 3
  3 3 3 3 equal 7 7 7 ?

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Heres a skip counting/multiplication activity
that shows a connection to geometric patterns
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Repeatedly add 2sthat is, multiply by 2
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Repeatedly add 3sthat is, multiply by 3
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Here are the 4s
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What pattern do the 5s construct?
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Next are the 6swith a surprising result.
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What pattern will the 7s make?
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And the 8s
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Even the 9s...
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Model 3 The array model

• Think of multiplication as always creating a
  rectangle. For example, 4 x 5 means to create
  a rectangle 4 units long and 5 units tall. How
  many blocks would you have?

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Multiplication means to make a rectangle
Heres 2 x 7

• Heres 7 x 5

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TheMultiplicationGrid
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TheMultiplicationGrid
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TheMultiplicationGrid
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This model helps in explaining our traditional
multplication algorithm
Consider 13 x 14
13 X 14 12 40 30 100 182
13 X 14 52 130 182
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In this model

• In this model, each factor is one of the
  dimensions of the rectangle.
• This model leads to an understanding of area
  calculations.
• With this model, it is easy to see the
  commutative property. For example, a 7 x 3
  rectangle has the same number of blocks in it as
  a 3 x 7 one.
• Unlike the other models, it can be useful in
  multiplication of fractions and decimals such as
  1/2 x 1/3 or 0.2 x 0.3. For 1/2 x 1/3, we just
  need to construct a rectangle with these
  dimensions (see next slide)
• It also leads the student to an understanding of
  algebra (see following slides)

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Now lets review some Algebra...

• Multiply (x 1)(x 1)
• You were probably taught to use FOIL
• x(x) (x)1 1(x) 1(1)
• x2 2x 1

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Now lets see the same problem done as a
multiplication grid
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Another Algebra problem...

• Multiply (x 1)(x 2)
• Use FOIL again
• x(x) (x)2 1(x) 1(2)
• x2 3x 2

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Now lets see the same problem done as a
multiplication grid
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Multiplication Dos and Donts

• Do give ample practice with all three types of
  multiplication problems
• Do present the concept in a problem context (not
  just fact practice)
• Dont use the term carry - use regroup
  instead
• Do encourage students to develop their own mental
  strategies
• Do emphasize the commutative property -- only 55
  facts to learn
• Dont spend too little time with the easy facts
  involving 0 and 1. These are important for later
  understanding of algebra
• Do use calculators for multiplying larger numbers

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Division

• Means to sort items into equal sets
• Sharing There are 15 cookies to be shared by 5
  students. How many cookies does each person get
• Sharing with left over There are 28 M Ms to
  be shared by 6 students. How many does each
  person get? How many are left over?
• Prerequisite skills
• Counting
• Basic addition, subtaction, and multiplication
  facts
• Concepts to be learned
• Basic division number facts
• Concept of factors
• Relationship between multiplication and division
• Algorithm for dividing with larger numbers

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Division Dos and Donts

• Do draw on students experiences with sharing
• Do present the concept in a problem context (not
  just fact practice)
• Dont make the long division algorithm the most
  important thing youve got to teach students
• Do use mnenomic devices like Dad, Mother,
  Sister, Brother for divide, multiply, subtract,
  bring down
• Do use calculators for dividing larger numbers

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Basic x/ Fact Practice

• Do the practice after the concept has been
  thoroughly presented in problem situations
• Use games frequently (Card games like Salute,
  Wheel of Fortune, Factor Game, etc.)
• Do the practice daily, even when x/ are not the
  lessons of the day
• Practice the tougher facts (involving 6, 7, 8,
  and 9)
• Emphasize multiplication facts of 10
• Do some sort of mental math every day

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Closing thought

• Being able to compute with numbers is obviously
  very important in elementary school mathematics.
  But it should not be the pinnacle of what we
  attempt to teach because for 2.00 a student can
  purchase help for calculating whereas no amount
  of money can buy for him understanding of place
  value, patterns, geometry, estimation,
  probability, statistics, problem solving, etc.