Basic College Math

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Basic College Math

• The Distributive Property

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Getting ready for algebra!

• Now that were ready to study algebra, were
  going to need more than just the order of
  operations to keep us in line!

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A brand new property!

• Were going to need a new property called the
• distributive property of multiplication over
  addition.
• It also has a sister called the
• distributive property of multiplication over
  subtraction.
• Collectively, they are sometimes just called
• The distributive property.

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What will we learn?

• We are going to see how the property works with
  a numerical example.
• Were going to see how it works with some
  algebraic examples.
• Were going to map out the potential pitfalls of
  working with this property.
• Were going to read a story to help us remember
  how to use it correctly!

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Remember the order of operations?

• If you have a problem like 5( 6 8 ), the order
  of operations tells you to do what is in
  parentheses first.
• 5( 6 8 ) becomes 5( 14 ) which equals 70.
• Remember the 5 directly against the open
  parentheses means to multiply by 5.

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The Distributive Property of Multiplication over
Addition tells us . . .

• If we have something that is multiplying the sum
  of two or more numbers like 5( 6 8 ). . .
• We have an alternate way to calculate the answer.

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Practical Example

• Suppose you hire the neighbors daughter for
  5.00 an hour to help you rake leaves. She is all
  excited about the job because she wants to buy a
  stereo for her room.

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Raking in the profits

• Day 1, she works 6 hours
• thats 5(6) or 30 toward her stereo.
• Day 2, she works 8 hours
• thats another 5(8) or 40 toward her stereo.
• WOW!
• 30 40 equals 70 for her stereo!

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Paying the bill

• You go to the bank to get the cash to pay her.
• All you care about is that she worked 14 hours.
• 5(14) 70.
• You both get the same answer even though you did
  the problem differently.
• The Distributive Property tells you that it is
  OK.
• You can multiply and then add (like she did)
• or add and then multiply (like you did).

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Some algebraic examples

• 5( x 2) 5( x ) 5( 2 ) 5x 10
• 2( 3x 6 ) 2( 3x ) 2( -6 ) 6x - 12

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A more complicated problem

• Take the problem - 5( x 7 ) 30
• You first distribute and then combine like terms.
• Answer - 5( x 7 ) 30 - 5( x ) - 5( 7 )
  30
• - 5x -
  35 30
• - 5x - 5

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Pitfall 1 Negative numbers

• If the number in front of the parentheses is
  negative, you must multiply each item in the
  parentheses by the negative number.
• Watch your signs!
• - 6(x 4) - 6( x ) - 6( - 4 ) - 6x 24
• -2(2x 5) -2( 2x ) - 2( 5 ) -4x -10
• -3(-3x 4) -3( - 3x) - 3(- 4) 9x 12

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Pitfall 2 Oops - forgot the second guy!

• 5( x 7 ) 5x 7 WRONG!
• You get the 5x right because its staring you in
  the face.
• In your haste to get the right answer, you forget
  all about the poor 7.
• 5(x 7) 5( x ) 5( 7 ) 5x 35 RIGHT!

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Pitfall 3 Over enthusiasm

• 5(x 7) 30 5( x ) 5( 7 ) 5( 30 )
• 5x 35 150 5x
  - 115 WRONG
• You get so excited about distributing the 5 that
  you distribute him to everybody even the
  numbers that arent in the parentheses!
• 5( x 7 ) 30 5( x ) 5( 7 ) 30
• 5x 35 30 5x
  5 RIGHT!

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Pitfall 4 Its backwards!

• ( x 7 )(5) What the heck?
• The commutative property of multiplication tells
  us that we can re-write the problem because order
  doesnt matter when you multiply.
• 5( x 7 ) Ah, thats better!

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Let me tell you a story . . .

• The first time I ever went to Sea World, my
  friends and I decided to go to the Shamu show.
• When we entered the area, I noticed that there
  were seats right down at the front that were
  painted red instead of blue, like the other
  seats.
• There were no Reserved signs on them, so we
  decided to go sit there and move if we were told
  we had to.

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THE SEATS WERE GREAT!

• We could see Shamu swimming around through the
  windows in the side of his tank.
• At last, the show started and Shamu did a mighty
  leap out of the water right in front of us.
• It was amazing!
• I had my camera out, ready to shoot the picture
  of a lifetime. . .

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Perfect!

• . . . and then he re-entered the water . . .

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. . . and that is how Shamu can help you remember
the rules.

• When you are simplifying an expression like
• 5 ( x 7 ) 30
• The 30 is in the blue seats high and dry and
  out of harms way.
• The x and the 7 are in the red seats definitely
  the splash zone!
• And the 5 is Shamu!

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The End

• Splash photo courtesy of Mike Sandells
  www.mikejs.com