Integers

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Integers

• Mathematical Systems
• 7th grade Pre-Algebra

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Teacher Page

• Content Mathematical Systems Integers
• Grade Level 7th grade
• Creator Mary Anne Burton
• Objective Students will compute the sums,
  differences, products or quotients using integers
  in equation format.
• Process MA-1 1.6, 1.8 2.2, 3.3

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Addition of Integers

• You are probably familiar with a number line.
  Traditionally, zero is placed in the center.
  Positive numbers extend to the right of zero and
  negative numbers extend to the left of zero. In
  order to add positive and negative integers, we
  will imagine that we are moving along a number
  line.
• Draw a number line on your paper.

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Adding Integers on a Number Line

• If asked to add 8 and -2, we would start by
  moving eight units to the right of zero. Then we
  would move two units left from there because
  negative numbers make us move to the left side of
  the number line. Since our last position is six
  units to the right of zero, the answer is 6.

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Adding Integers on a Number Line

• If asked to add -13 and 4, we start by moving
  thirteen units to the left of zero. Then we move
  four units to the right. Since we land up nine
  units to the left of zero, the answer is -9.

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Adding Integers on a Number Line

• If asked to add -6 and -5, first move six units
  to the left of zero. Then move five units further
  left. Since we are a total of eleven units left
  of zero, the answer is -11.

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Rules for Addition

• Positive Positive Positive
• Positive Negative Depends
• Negative Positive Depends
• Negative Negative Negative

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How to solve positive negative or negative
positive

• Absolute Value This is the number of spaces from
  zero on the number line.
• Example the absolute value of 10 is 10
• the absolute value of -10 is
  10
• Once you find the absolute value of each number
  you subtract and take the sign of the greater
  number.

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Subtraction with Integers

• The technique for changing subtraction problems
  into addition problems is extremely mechanical.
  There are two steps
• Change the subtraction sign into an addition
  sign.
• Take the opposite of the number that immediately
  follows the newly placed addition sign.

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Subtraction with Integers

• Let's take a look at the problem 3 4 x
• According to step 1, we have to change the
  subtraction sign to an addition sign.
• According to step 2, we have to take the
  opposite of 4, which is -4.
• Therefore the problem becomes
• 3 (-4) x
• Using the rules for addition, the answer is -1.

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Subtraction with Integers

• Here is another problem -2 8 x
• Switching the problem to an addition problem, it
  becomes -2 (-8) x, which is equal to -10.
• 6 - (-20) x is equal to 6 20, which is 26.
• -7 - (-1) x is the same as -7 1, which is -6.

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Rules for Multiplication

• Positive x Positive Positive
• Positive x Negative Negative
• Negative x Positive Negative
• Negative x Negative Positive

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Understanding Multiplication Rules

• The first rule is the easiest to remember because
  we learned it so long ago. Working with positive
  numbers under multiplication always yields
  positive answers. However, the last three rules
  are a bit more challenging to understand.

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Multiplication with Integers

• The second and third rules can be explained
  simultaneously. This is because numbers can be
  multiplied in any order. -3 x 7 has the same
  answer as 7 x -3, which is always true for all
  integers. This example demonstrated the
  commutative property.

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Multiplication with Integers

• The fourth rules follows the pattern that when
  two numbers of the same sign are multiplied the
  answer is always positive.
• -8 x 3 -24
• -8 x 2 -16
• -8 x 1 -8
• -8 x 0 0
• -8 x -1 8
• -8 x -2 16
• -8 x -3 24

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Here are some examples

• 5 x -10 ?
• 0 x 8 ?
• -10 x 9 ?
• 6 x 6 ?
• 7 x -6 ?

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Division with Integers

• The rules for division are exactly the same as
  those for multiplication. If we were to take the
  rules for multiplication and change the
  multiplication signs to division signs, we would
  have an accurate set of rules for division.

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Rules for Division with Integers

• Positive Positive Positive
• Positive Negative Negative
• Negative Positive Negative
• Negative Negative Positive

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Division with Integers

• -9 3 -320 (-4) -5
• -18 (-3) 6 Now you try the ones
  below
• 36 (6) ?
• 8 (-7) ?
• -63 (-9) ?
• 18 (-2) ?
• -70 (-10) ?

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Now You Are the Teachers

• Group Presentation demonstrating your knowledge
  of the rules of Integers.
• 2-3 students per group
• Choose one of the following presentations
• Cartoon Creations
• Demonstration Dilemma
• Sing Us A Song
• Thriving Thespians

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Cartoon Creations

•  This project must contain a comic strip that
  demonstrates your understanding of Integers. It
  should consist of eight panels, minimum. Elements
  such as artistic quality and mathematical
  accuracy will be checked. Cartoons that contain
  those items as well as a creative component will
  receive more points.

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Demonstration Dilemma

• This project is appropriate for students who
  enjoy showing people how to do new things and/or
  creating models.
• It may also include a physical explanation on
  how to solve problems with Integers using
  mathematical tools. Students may also explain how
  and where Integers are used in our everyday
  world. This project, although open-ended, must
  involve student acquired props and models. The
  more involved the class gets, the more points
  will be awarded. It must involve at least a five
  minute presentation to the class.

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Sing Us A Song

• This project is appropriate for those who enjoy
  using their voice, It involves creating lyrics,
  possibly an original work or a parody of
  something already in existence (like the theme to
  a television program), that demonstrates your
  understanding of Integers. The song must be
  presented to the teacher as a hardcopy and must
  be at least a page in length.     The actual song
  will be sung in class. The singer may choose to
  supplement this presentation with recorded music
  or a played instrument, which may be a team
  effort -- two people maximum. Part of this grade
  will be based on the emotional delivery of the
  piece.

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Thriving Thespian

• A skit will be produced and it involves creating
  a script, possibly an original work or a play on
  something already in existence pun intended,
  that demonstrates your understanding of Integers.
  The script must be presented to the teacher as a
  hardcopy and must be at least four pages in
  length.     The actual presentation must be at
  least five to ten minutes. This project can
  involve a team effort, up to three people
  maximum. The team may choose to supplement this
  presentation with props and music. Part of this
  grade will be based on the emotional delivery of
  the skit.

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Scoring GuideProjects will be worth a maximum of
100 points!

• Factor Weak ----Strong
• Creativity
  0   1   2   3   4   5
• Math 0 1 2 3
  4 5
• Content   0  1   2   3   4   5
• Neatness
  0   1   2   3   4   5
• Presentation 0   1   2   3   4  
   5
• Once the factors are determined they will be
  added together and then multiplied by 5 to
  generate the points for the project