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					                               Place Value
Learning Goal: I will understand place value.

Model:

                                                        Shade 1

                          =1




                                                      Shade 1
                                                             10
                          __________ tenths = 1




                                                      Shade 1
                                                             100
                          ___________hundredths = 1




Show the following numbers by shading rectangles.



1.   .3 means    __________________________




2.   .09 means    __________________________




                                                                  1
3.   .3 + .09 equal_____ which means   __________________________




4.   2 means   __________________________




5. Why is it not possible to show 4 below?_________________________




 6. 1.43 means____________ + _____________ + ______________




                         +                     +



7.   1.43 means   __________________________




What is interesting about adding .7 to .3?__________________________




                                                                      2
                          +                       =

.7 + .3 means____________+ ______________ = __________________

Show why .08 is less than .5




Explain____________________________________________________

Show why .4 is greater than .35




Explain___________________________________________________



                        Moving to Symbols
Why don’t we always use models to show numbers?



Why do we use numerals?



Define the word sum.
_____________________________________________________

Think of the values of the numbers to find the sums.




                                                             3
1. .3  .2 equals __________ which means _________________________



2. .04  .01 equals ________ which means ________________________



3. .3  .02   equals ________ which means_________________________



4. 2  .6 equals ____________ which means_______________________



Line up the numbers in order.

.35,          .08        1.02         .90    .37        .9



Make two sentences from your order.

________________________ is less than ________________________



_______________________ is greater than _____________________



_______________________ is the same as ______________________




                                                                    4
                    Homework Practice for Place Value
Parent Signature________________                    Name______________
Goal: Understand place value.
Show the following numbers by shading rectangles if

                                        equals the number 1.



1.   .5 means    __________________________




2.   .02 means    __________________________




3.   .5 + .02 equals_____ which means     __________________________




4.   2.5 means    __________________________




5. 1.37 means     _________ + ______________ + ___________




7.   1.37 means   __________________________. Show its value below.




                                                                       5
Show why .18 is less than .09




Explain____________________________________________________

Show why .7 is greater than .30




Explain____________________________________________________

Think of the values of the numbers to find the sums.

1. .5  .1 equals __________ which means ________________________

2. .07  .02 equals ________ which means _________________________

3. .7  .01 equals ________ which means_________________________

4. 2  .6 equals ____________ which means________________________

Line up the numbers in order.
1.05,        1.08        .92                       .90                2



Make two sentences from your order.

________________________ is less than ________________________
_________________________ is greater than ____________________

Rate your self on the goal of understanding place value_______
      0     Don’t understand place value at all
      1     Understand it a little
      2     I still make a lot of mistakes, but I think I get it.
      3     I make little mistakes only, but I get it.
      4     I could do tougher place value problems than I have seen in class.




                                                                                 6
Line up the following numbers.
       1.45, 4.15, 8.30, 1.95,   .85




Neighbor check _________

What are these numbers?
1.45 _______________________________________
4.15 _______________________________________
8.30 _______________________________________
1.95 _______________________________________
.85 _______________________________________

Make sentences.
_______________________ is smaller than ______________________.

_______________________ is bigger than ______________________.

Eighty-five hundredths is _________________________ eight and three
tenths.

Eight and three tenths is _________________________ eighty-five
hundredths.




Less than                                    Greater than



Make sentences.
_______________________ is less than ______________________.

_______________________ is greater than _____________________.




What is an inequality?



                                                                      7
Inequality symbols




Practice.
Ex.   1.92 ______ .67

   A.     .75 ______ 2.3

   B.     .89 ______ .90

   C.     3.1 ______ 2.74

   D.     .65 ______ 5.60



Write two sentences for each inequality.
Ex.   4.52 > .36
      Four and fifty-two hundredths is greater than thirty-six hundredths.
      Thirty-six hundredths is less than four and fifty-two hundredths.



   A. .59 < .81
      ____________________ is greater than ___________________.

        ____________________ is less than ______________________.

   B. 1.35 > 1.34




   C. .47 < 2.25




                                                                         8
Write the sentences using symbols.
Ex. Six tenths is greater than thirty-one hundredths.
      .6 > .31     OR           .31 < .6



   A. Fifty-four hundredths is less than seventy-seven hundredths.




   B. Three and five tenths is greater than three and twenty-nine
      hundredths.




   C. Nine hundredths is less than seventeen hundredths.




Line up these numbers in order.
A. .85, .90,      .78



      What are these numbers?
      .85 _________________________________________
      .90 _________________________________________
      .78 ________________________________________

      Make a sentence. Use the term “less than” or “greater than”.




      Write your sentence with an inequality symbol.




                                                                     9
Homework                                    Name __________________

What are these numbers?
     .7 ________________________________
     .08 ________________________________
     .70 ________________________________
     .78 ________________________________

Line up the numbers in order.



Make two sentences from your order.

_____________________ is less than _________________________.
_____________________ is greater than ______________________.

What is an inequality?



Use an inequality symbol in each problem.
A.    .65 ______ .64
B.    7.55 ______ 5.7
C.    4.9 ______ 4.92
D.    5.88 ______ 6.1

Write two sentences for each inequality..
Ex.   4.52 > .36
      Four and fifty-two hundredths is greater than thirty-six hundredths.
      Thirty-six hundredths is less than four and fifty-two hundredths.

   A.    6.1 < 7.28




   B.    4.91 > 4.77

                                                        Parent Signature:



                                                                        10
What are these numbers?
     .8 ___________________________
     3.4___________________________
     2.5___________________________
     4.7 ___________________________

Line up the numbers in order.




Number Line

      (Horizontal)




      (Vertical)




Put these prices on a number line.
      $6.50 for two boxes of Lucky Charms cereal,
      $3.00 for a plastic bucket,
      $.80 for a KitKat candy bar,
      $1.50 for a yard of fabric,
      $1.75 for a bunch of bananas,
      $0 for a high-five




                                                    11
In the last example, a plastic bucket was $3.00.

      How much would two cost? _______________

      How much would three cost? _______________

      How much would four cost? _______________

      How much would none cost? _______________



Line up these prices on a number line.




In the last example, a plastic bucket was $3.00.

      How much would two cost? _______________

      How much would three cost? _______________

      How much would four cost? _______________

      How much would none cost? _______________



Line up these prices on a number line.




                                                   12
Homework                                Name ______________________

Line up these numbers in order.
       $.42, $.83,     $.05,    $.17,     $.35




Put these numbers on a number line.




A salad costs $2.50.
      How much would two cost? _______________

      How much would three cost? _______________

      How much would four cost? _______________

      How much would none cost? _______________

Put these numbers on a number line.




                         Parent Signature _________________________



                                                                      13
A Better Way to Organize

      Who is Rene’ Descartes?




Cartesian coordinate system

I can put my data on the ____________________________________.



A blue pen costs $1.50.
      What is the cost of two pens? __________________________
      What is the cost of three pens? ________________________
      What is the cost of four pens? _________________________
      What is the cost of five pens? __________________________
      What is the cost of zero pens? __________________________

Label a coordinate system for this data. Put the cost on the vertical number
line.




Write a sentence to describe the graph.




                                                                          14
A frozen pizza costs $5.05.
      What is the cost of two pizzas? __________________________
      What is the cost of three pizzas? ________________________
      What is the cost of four pizzas? _________________________
      What is the cost of five pizzas? __________________________
      What is the cost of zero pizzas? __________________________

Label a coordinate system for this data. Put the cost on the vertical number
line.




Write a sentence to describe the graph.




                                                                          15
Data collected: 4 corn on the cob for $1.00.

      What is   the   cost of 8 cobs? __________________________
      What is   the   cost of 12 cobs? ________________________
      What is   the   cost of 16 cobs? _________________________
      What is   the   cost of 20 cobs? __________________________
      What is   the   cost of 0 cobs? __________________________

Label a coordinate system for this data. Put the cost on the vertical number
line. How should we label the horizontal number line?




Write a sentence to describe the graph.




How much does one cob cost?




                                                                          16
Homework                                    Name ___________________
Data collected:                       Parent Signature________________

      What is   the   cost of    ? __________________________
      What is   the   cost of    ? ________________________
      What is   the   cost of    ? _________________________
      What is   the   cost of    ? __________________________
      What is   the   cost of    ? __________________________

Label a coordinate system for this data. Put the cost on the vertical number
line. How should we label the horizontal number line?




Write a sentence to describe the graph.




Create this same graph for display in class. Use a fresh sheet of graph
paper.




                                                                          17
A blue pen costs $1.80. If we start with $10.00, how much money will we
have left if we buy:
       One pen?___________________________
       Two pens? __________________________
       Three pens? ________________________
       Four pens? _________________________
       Five pens? __________________________
       Zero pens? __________________________

What is the greatest number of pens that we could buy with $10.00?



Label a coordinate system for this data. Put the money left on the vertical
number line.




Write a sentence to describe the graph.



                                                                          18
We have been given $50 to buy pizzas for a party. If A frozen pizza costs
$4.95, how much money would we have left if we buy:
      one pizza? __________________________
      two pizzas? ________________________
      three pizzas? _________________________
      four pizzas? __________________________
      five pizzas? __________________________
      zero pizzas? __________________________

How many pizzas could we buy for $50?




Label a coordinate system for this data. Put the money left on the vertical
number line.




Write a sentence to describe the graph.




                                                                          19
Data collected: 3 cans of Sprite for $1.29. Given a soda budget of $20, how
much money will be left if we buy:

      3 cans? __________________________
      6 cans? ________________________
      9 cans? _________________________
      12 cans? __________________________
      What is the cost of 0 cans? __________________________

Label a coordinate system for this data. Put the money left on the vertical
number line. How should we label the horizontal number line?




Write a sentence to describe the graph.



How much money will be left if we buy one can?

How many cans of soda could be purchased before going over the $20
budget?



                                                                          20
Homework                                    Name ___________________
                                      Parent Signature________________

Collect data Find a price that is less than $2______________________
Starting with $15.00, how much money will I have left if :
       I buy                           ? __________________________
       I buy                           ? __________________________
       I buy                           ? __________________________
       I buy                           ? __________________________
       I buy                           ? __________________________
       I buy zero                      ?___________________________

Label a coordinate system for this data. Put the money left over on the
vertical number line. What should be the label for the horizontal number
line?




Write a sentence to describe the graph.

Create this same graph for display in class. Use a fresh sheet of graph
paper.



                                                                           21
How to Write a Check




Class Exercise

Our electricity budget for the year is $600.00.
Here is a copy of our bill:




Practice writing a check to Heber Light and Power Company




                                                            22
Create a ledger of your budget for twelve months.

Starting Amount          Month 0         $____________
                         Month 1         $____________
                         Month 2         $____________
                         Month 3         $____________
                         Month 4         $____________
                         Month 5         $____________
                         Month 6         $____________
                         Month 7         $____________
                         Month 8         $____________
                         Month 9         $____________
                         Month 10        $____________
                         Month 11        $____________
                         Month 12        $____________



Use graph paper to put your data on a Cartesian coordinate system.
Include labels and even scale on your graph.




Write sentences to describe your data.
_________________________________________________________________
_________________________________________________________________
_________________________________________________________________




                                                                     23
Given: Communication budget for the year: $1,400.00
Your bill will be the same for each month.




Create a ledger of your budget for twelve months.

Starting Amount          Month 0         $____________
                         Month 1         $____________
                         Month 2         $____________
                         Month 3         $____________
                         Month 4         $____________
                         Month 5         $____________
                         Month 6         $____________
                         Month 7         $____________
                         Month 8         $____________
                         Month 9         $____________
                         Month 10        $____________
                         Month 11        $____________
                         Month 12        $____________



Write checks for each payment. You will write twelve checks and attach
them to your work.




                                                                         24
Use graph paper to put your data on a Cartesian coordinate system.
Include labels and even scale on your graph.




Write sentences to describe your data.
_________________________________________________________________
_________________________________________________________________
_________________________________________________________________




                                                                     25
26
27
Given: Power budget for the year: $500.00
Your bill will be the same for each month.




Create a ledger of your budget for twelve months.

Starting Amount          Month 0         $____________
                         Month 1         $____________
                         Month 2         $____________
                         Month 3         $____________
                         Month 4         $____________
                         Month 5         $____________
                         Month 6         $____________
                         Month 7         $____________
                         Month 8         $____________
                         Month 9         $____________
                         Month 10        $____________
                         Month 11        $____________
                         Month 12        $____________



Write checks for each payment. You will write twelve checks and attach
them to your work.




                                                                         28
Use graph paper to put your data on a Cartesian coordinate system.
Include labels and even scale on your graph.




Write sentences to describe your data.
_________________________________________________________________
_________________________________________________________________
_________________________________________________________________




                                                                     29
30
31
Given: Cell phone budget for the year: $1,300.00
Your bill will be the same for each month.




Create a ledger of your budget for twelve months.

Starting Amount          Month 0         $____________
                         Month 1         $____________
                         Month 2         $____________
                         Month 3         $____________
                         Month 4         $____________
                         Month 5         $____________
                         Month 6         $____________
                         Month 7         $____________
                         Month 8         $____________
                         Month 9         $____________
                         Month 10        $____________
                         Month 11        $____________
                         Month 12        $____________



Write checks for each payment. You will write twelve checks and attach
them to your work.




                                                                         32
Use graph paper to put your data on a Cartesian coordinate system.
Include labels and even scale on your graph.




Write sentences to describe your data.
_________________________________________________________________
_________________________________________________________________
_________________________________________________________________




                                                                     33
34
35
Given: Car budget for the year: $6,000.00
Your bill will be the same for each month.




Create a ledger of your budget for twelve months.

Starting Amount          Month 0         $____________
                         Month 1         $____________
                         Month 2         $____________
                         Month 3         $____________
                         Month 4         $____________
                         Month 5         $____________
                         Month 6         $____________
                         Month 7         $____________
                         Month 8         $____________
                         Month 9         $____________
                         Month 10        $____________
                         Month 11        $____________
                         Month 12        $____________



Write checks for each payment. You will write twelve checks and attach
them to your work.




                                                                         36
Use graph paper to put your data on a Cartesian coordinate system.
Include labels and even scale on your graph.




Write sentences to describe your data.
_________________________________________________________________
_________________________________________________________________
_________________________________________________________________




                                                                     37
38
39
Homework.                                          Name ________________________________
                                                   Parent Signature ________________________

Given: Student loan budget for the year: $900.00
Your bill will be the same for each month.




Create a ledger of your budget for twelve months.
Starting Amount          Month 0          $____________
                         Month 1          $____________
                         Month 2          $____________
                         Month 3          $____________
                         Month 4          $____________
                         Month 5          $____________
                         Month 6          $____________
                         Month 7          $____________
                         Month 8          $____________
                         Month 9          $____________
                         Month 10         $____________
                         Month 11         $____________
                         Month 12         $____________



Write checks for each payment. You will write twelve checks and attach
them to your work.




                                                                                          40
Use graph paper to put your data on a Cartesian coordinate system.
Include labels and even scale on your graph.




Write sentences to describe your data.
_________________________________________________________________
_________________________________________________________________
_________________________________________________________________




                                                                     41
42
43
                              Interpreting Graphs

Use the following graph to answer the questions below.

                                     Total price

  $16.00
  $14.00                                               $13.50
  $12.00
                                              $11.25
  $10.00
                                          $9.00
   $8.00                                                            Total price
                                 $6.75
   $6.00
   $4.00                     $4.50
   $2.00             $2.25
   $0.00         $0.00
             0           2            4            6            8
                     number of 3-ring binders


1. What is the price of a 3-ring binder? _______________________

2. What is the total price of 4 binders? __________________________

3. How many binders could we buy for $11.25_______________________

4. How many binders could we buy for $10.00_______________________

5. Could a straight line be drawn exactly through all of the points that are
plotted on the graph? ___________

6. Why is the line not shown?___________________________________

7. Finish the statement: As the number of binders increases by one, the
total price ______________________________________________.



                                                                               44
Use the following graph to answer the questions below.


                              Money left in my pocket

  $12.00

  $10.00        $10.00

   $8.00            $8.40
                             $6.80                               Money left in my
   $6.00
                                $5.20                            pocket
   $4.00                                 $3.60
   $2.00                                    $2.00

   $0.00                                             $0.40
            0            2           4           6           8
                             Snicker bars


1. How much was in my pocket when I went into the store?

2. What is the price of a Snicker bar?

3. Do I have enough money for 7 Snicker bars?

4. How much money will be left in my pocket if I buy 5 bars?

5. What is the maximum number of bars that I can buy? How much will be
left if in my pocket if I buy as many as possible?



6. Fill in the blanks to complete the statement: Starting with __________,
the amount of money in my pocket decreases by ________________ for
every additional _________________________________.



                                                                           45
Use the graph below to estimate the answers of the questions below.

                         Money left in my pocket

  $8.00
  $7.00
  $6.00
  $5.00
                                                         Money left in my
  $4.00
                                                         pocket
  $3.00
  $2.00
  $1.00
  $0.00
          0          2         4         6         8
                          Sharpies


1. What is a good estimate of the money I started with?_______________



2. Estimate the price of a Sharpie._________________________



3. Estimate how much money that I will have left if I buy 5
sharpies.______________



4. If I walked out of the store with just over $4 in my pocket, how many
sharpies did I buy?_________________________________________

5. Estimate the largest number of sharpies that I can buy. Show your work.




                                                                           46
Homework
                     Interpreting Graphs
                                               Name_________________
                                   Parent Signature_________________
Use the graph to estimate the answers to the questions below.

                                Total cost

  $8.00
  $7.00
  $6.00
  $5.00
  $4.00                                                       Total cost
  $3.00
  $2.00
  $1.00
  $0.00
           0          2          4           6          8
                          Trading Cards


1. Estimate the cost of one package of trading cards._________________
2. How many packages can you buy for $7.00?______________________
3. Estimate the total cost of 5 packages.__________________________
4. Can a straight line be drawn exactly through the
points?__________________

5. Why does the graph not have a line
drawn?___________________________________________________



6. Finish the statement: Starting with a minimum total cost of ________
for __________________________, the total cost _______________
for every additional _________________________________________.



                                                                      47
                      Graphing Continuous Data
                                             Name____________________
                                             Parent Signature____________
The cost of denim fabric $6.00 per yard.
      What is the cost of one yard? __________________________
      What is the cost of two yard? ________________________
      What is the cost of three yards? _________________________
      What is the cost of five yards? __________________________
      What is the cost of zero yards? __________________________

      What is   the   cost of a half of a yard? ______________________
      What is   the   cost of two and one half yards?_________________
      What is   the   cost of one foot (0ne-third of a yard)?______________
      What is   the   cost of three and one-third yards?_________________




Label a coordinate system for this data. Put the cost on the vertical number
line.




Write a sentence to describe the graph.



                                                                          48
The cost of bananas is $1.20 per pound.
      What is the cost of one pound of banana? ____________________
      What is the cost of three pounds of bananas? _________________
      What is the cost of five pounds of bananas? ___________________
      What is the cost of zero pounds of bananas? _________________

      What is   the   cost of a half of a pound of bananas?____________
      What is   the   cost of four and a half pounds of bananas?__________
      What is   the   cost of a fourth of a pound of bananas? _____________
      What is   the   cost of two and one fourth pounds of bananas?________



Label a coordinate system for this data. Put the cost on the vertical number
line.




Write a sentence to describe the graph.




                                                                          49
Data collected: The cost of diesel fuel is $3.00 per gallon.

      What is   the   cost of 8 gallons of diesel fuel? ___________________
      What is   the   cost of 12 gallons of diesel fuel? _________________
      What is   the   cost of 16 gallons of diesel fuel? _________________
      What is   the   cost of 20 gallons of diesel fuel? _________________
      What is   the   cost of 0 gallons of diesel fuel? ___________________

      What is the cost of 0.1 gallons?____________________

      What is the cost of 10.1 gallons?___________________

      What is the cost of 0.4 gallons?____________________

      What is the cost of 5.4 gallons?____________________

Label a coordinate system for this data. Put the cost on the vertical number
line. How should we label the horizontal number line?




Write a sentence to describe the graph.




                                                                          50
Homework                                    Name ___________________
                                      Parent Signature________________

      Data collection: Find the cost of a product that is sold by weight,
volume, or length.
                   Product Name______________________
                   Product Price___________Per_________
                                                (Unit of measure)

The cost of one ______________ of _____________ is _____________
                 (Unit of measure) (Product)       (dollar amount)
The cost of three ____________ of _____________ is _____________
The cost of five ______________ of _____________ is _____________
The cost of zero ______________ of ____________ is _____________

The cost of a half ____________ of ___________ is _____________

The cost of four and a half_____________ of __________ is _________

Label a coordinate system for this data. Put the cost on the vertical number
line. How should we label the horizontal number line?




Write a sentence to describe the graph.



                                                                            51
Homework                               Name _______________________
                                       Parent Signature ____________________________


Look at an advertisement. Choose three items that are priced by the pound.

Find the price of 0,1,2,3,4,5 pounds of each item.



Graph the total costs of each item on the same Cartesian coordinate system.
Be sure to label the vertical and horizontal number lines.
Use different colors to label each graph. List what each color represents.

Is this data continuous?

Is a line graph appropriate? Why or why not?



Which item has the steepest curve?

______________________ data has the steepest curve because _______
_________________________________________________________.

Compare the slopes of the graphs. Use complete sentences.




                                                                                   52
               Graphing From a Starting Cost
                                            Name____________________
                                            Parent Signature____________
      The total cost of going to a movie depends on two costs: the cost of
an $8.00 ticket and the cost of $1.00/bag for bags of popcorn that we buy.

      The total cost of going to the movie if we buy zero bags of popcorn
      is__________.
      The total cost of going to the movie if we buy one bag of popcorn
      is__________.
      The total cost of going to the movie if we buy two bags of popcorn
      is__________.
      The total cost of going to the movie if we buy five bags of popcorn
      is__________.

Label a coordinate system for this data. Put the cost on the vertical number
line.




Write a sentence to describe the graph.




                                                                            53
      The total cost of renting a car depends on two costs: a rental fee of
$30 and the cost of $.50/mile for every mile we drive the car.

      The total cost of renting a car if we drive zero miles is__________.

      The total cost of renting a car if we drive 50 miles is__________.

      The total cost of renting a car if we drive 100 miles is__________.

      The total cost of renting a car if we drive 200 miles is__________.

      The total cost of renting a car if we drive 350 miles is __________.



Label a coordinate system for this data. Put the cost on the vertical number
line.




Write a sentence to describe the graph.




                                                                              54
The total cost of a certain weight-loss program depends on two costs: A
membership fee of $30 and $10/lb. for every pound we lose.

      What is   the   total cost if we   lose   zero pounds?__________
      What is   the   total cost if we   lose   one pound?____________
      What is   the   total cost if we   lose   3 pounds?_____________
      What is   the   total cost if we   lose   a half of a pound?_______
      What is   the   total cost if we   lose   six and a half pounds?_______

Label a coordinate system for this data. Put the cost on the vertical number
line. How should we label the horizontal number line?




Write a sentence to describe the graph.




                                                                                55
Homework                                        Name ___________________
                                          Parent Signature________________

Data collection:

Estimate the cost of one small jar of peanut butter________________
Estimate the cost of one dinner roll_______________

Use your estimates that you listed to find the following total costs:

One jar   of peanut butter   and   zero rolls_________________________
One jar   of peanut butter   and   one roll___________________________
One jar   of peanut butter   and   two rolls___________________________
One jar   of peanut butter   and   three rolls_________________________
One jar   of peanut butter   and   four rolls__________________________
One jar   of peanut butter   and   eight rolls___________________________

Label a coordinate system for this data. Put the total cost on the vertical
number line. How should we label the horizontal number line?




Write a sentence to describe the graph.




                                                                              56
                                   Rules
                                            Name____________________
                                            Parent Signature____________

Use each sentence to create a rule for calculating the total cost.

1. The total cost of a number of plastic buckets, if each bucket costs $3.00




2. The total cost of a number of frozen pizzas, if each costs $5.05




3. The total cost of a number of cobs of corn if 4 corn on the cob sell for
$1.00.




4. The total cost of going to a movie depends on two costs: the cost of an
$8.00 ticket and the cost of $1.00/bag for bags of popcorn that we buy.




5. The total cost of renting a car depends on two costs: a rental fee of $30
and the cost of $.50/mile for every mile we drive the car.




                                                                              57
6. The total cost of a certain weight-loss program depends on two costs: A
membership fee of $30 and $10/lb. for every pound we lose.




7. The cost of denim fabric $6.00 per yard. Find the total cost of denim for
a certain length of fabric.




8. The cost of bananas is $1.20 per pound. Find the total cost for a certain
weight (in pounds) of bananas.




9. The cost of diesel fuel is $3.00 per gallon. Find the total cost given a
certain quantity (in gallons) of diesel fuel.




                                                                              58
Estimate the cost of one small jar of peanut butter________________
Estimate the cost of one dinner roll_______________
Use the estimates to make a rule for the total cost based on number of
rolls.




Use each sentence to create a rule that finds the money left over.

 1. A blue pen costs $1.80. If we start with $10.00, how much money will we
have left if we buy a certain number of pens.




2. We have been given $50 to buy pizzas for a party. If A frozen pizza
costs $4.95, how much money would we have left if we buy a certain number
of pizzas?




3. 3 cans of Sprite for $1.29. Given a soda budget of $20, how much money
will be left if we buy a certain number of cans?




4. Choose the price of a particular item to complete the following sentence:
The price of a ___________ is _____________ a piece.
Starting with $15.00, how much money will I have left if I buy a certain
number?




                                                                          59
Write a rule for the following graphs.
1.

                                           Total price

     $16.00
     $14.00                                                   $13.50
     $12.00
                                                    $11.25
     $10.00
                                                $9.00
      $8.00                                                                 Total price
                                       $6.75
      $6.00
      $4.00                        $4.50
      $2.00           $2.25
      $0.00       $0.00
              0                2            4             6            8
                          number of 3-ring binders




2.

                                   Money left in my pocket

     $12.00

     $10.00       $10.00

      $8.00          $8.40
                               $6.80                                   Money left in my
      $6.00
                                     $5.20                             pocket
      $4.00                                 $3.60
      $2.00                                      $2.00

      $0.00                                             $0.40
              0            2            4           6           8
                               Snicker bars




                                                                                          60
Homework:   Rules                             Name ___________________
                                           Parent Signature______________



1. Data collection: Find the cost of a product that is sold by weight, volume,
or length.
                   Product Name______________________
                   Product Price___________Per_________
                                                 (Unit of measure)

Write a rule that calculates the total cost for any amount that you could
buy.



2. Write a rule for the following graph.

                                 Total cost

  $8.00
  $7.00
  $6.00
  $5.00
  $4.00                                                         Total cost
  $3.00
  $2.00
  $1.00
  $0.00
            0          2          4           6           8
                           Trading Cards




                                                                            61
                            Ordered Pairs
                                            Name____________________
                                            Parent Signature____________
The cost of a notebook is $2.25.
      Write a sentence that describes the total cost in terms of the
number of notebooks that we buy.



      Fill out the chart below using the description that we wrote.
    Number of notebooks                   Total cost
               0
               1
               2
               3
               4
               5

An ordered pair is a combination of two numbers. For example 1, 2.25 is an
ordered pair from the table above. The first number 1 represents 1 notebook
and the second number 2.25 represents $2.25, the cost of 1 notebook.

Ordered pairs can be drawn as points on a graph in the same way that we
have placed points on a graph before.

Label a coordinate system for this data. Put the total cost on the vertical
number line. Label each point with an ordered pair.




                                                                              62
Continuous Data: The cost of apples is $1.60 per pound.

      Write a sentence that describes the total cost in terms of the pounds
of apples that we buy.




      Fill out the chart below using the description that we wrote.
      Pounds of apples                    Total cost
               0
                1
               .5
               3
              4.5
               5

Label a coordinate system for this data. Put the total cost on the vertical
number line. Label each point with an ordered pair.




                                                                              63
Starting Cost: The cost of diesel fuel is $3.00 per gallon. If we buy a bottle
of additive for $5.20 before we fill up, write a sentence that describes the
total cost in terms of the gallons we buy.




      Fill out the chart below using the description that we wrote.
    Gallons of Diesel fuel                Total cost
               0
               .5
              10.5
               20
               30
               50

Label a coordinate system for this data. Put the total cost on the vertical
number line. Label each point with an ordered pair.




                                                                              64
Homework                                    Name ___________________
                                      Parent Signature________________

      Data collection: Find the cost of a product that is sold by weight,
volume, or length.
                   Product Name______________________
                   Product Price___________Per_________
                                                 (Unit of measure)
Write a sentence that describes the total cost in term of the amount that is
bought.



     Fill out the chart below using the description that we wrote.
Quantity of:                               Total cost
                 0
                .5
                 3
                 5
               10.5
                12



Label a coordinate system for this data. Put the cost on the vertical number
line. Label the points with ordered pairs.




                                                                          65
                                  Variables
                                            Name____________________
                                            Parent Signature____________
For each rule:
    Underline the words that represent numbers that can change.
    Write the words that you underlined, and assign a each variable.
    Write a formula for the rule.
    Try the rule out.



   Example: The total distance (in miles) travelled can be found by
   multiplying 45 by the number of hours.

   D= total distance travelled (miles)
   H= number of hours

   D = 45 X H

   45 X 2 = 90     D = 90 miles

1. Since I start with $50 in my pocket, the amount of money I have left will
be 50 minus 4.50 times the number of anchor bolts that I buy.

      Underline the words that represent numbers that can change.
      Write the words that you underlined, and assign each a variable.



      Write a formula for the rule.

      Try the rule out.




                                                                          66
2. The amount of money that Marie spends at the carnival can be found by
   multiplying the number of tickets that she buys by $1.25, then adding
   $5.00 for the entrance fee.

      Underline the words that represent numbers that can change.
      Write the words that you underlined, and assign each a variable.



      Write a formula for the rule.

      Try the rule out.

3. The remaining distance on a trip from Salt Lake City to Heber City can be
found by subtracting the minutes travelled from 44. (Assume an average
speed of 60 mph.)

      Underline the words that represent numbers that can change.
      Write the words that you underlined, and assign each a variable.



      Write a formula for the rule.

      Try the rule out.

4. The class collected and analyzed hand span and height data to create a
model that suggested that the height of a person (in inches) could be
estimated by multiplying the span of their hand (in centimeters) by 3.

      Underline the words that represent numbers that can change.
      Write the words that you underlined, and assign each a variable.



      Write a formula for the rule.

      Try the rule out.




                                                                            67
5. Between 6:00 and 7:00pm, the number of fans in The Nest can be
approximated by multiplying 12 by the number of minutes past 6:00pm plus
253.

                             Underline the words that represent numbers that can change.
                             Write the words that you underlined, and assign each a variable.



                             Write a formula for the rule.

                             Try the rule out.

6. The number of fans who remain in The Nest can be estimated by
subtracting from 989 the number of minutes after the final buzzer times
57.

                             Underline the words that represent numbers that can change.
                             Write the words that you underlined, and assign each a variable.



                             Write a formula for the rule.

                             Try the rule out.



7. Assign variables then write a formula for the following graph.

                             $30.50
                             $30.00       0, $30.00
  Money left in the wallet




                             $29.50                       1, $29.35
                             $29.00
                                                                          2, $28.70
                             $28.50
                             $28.00                                                       3, $28.05
                             $27.50                                                                       4, $27.40
                             $27.00
                                                                                                                          5, $26.75
                             $26.50
                             $26.00                                                                                                       6, $26.10
                             $25.50
                                      0               1               2               3               4               5               6               7
                                                                                  Cans of soda




                                                                                                                                                          68
Homework: Variables                                              Name______________
                                                 Parent Signature___________________



The final price of a shirt (which includes tax) can be found by multiplying
the price on the shirt’s tag by 1.07.
    Underline the words that represent numbers that can change.
    Write the words that you underlined, and assign each a variable.



      Write a formula for the rule.

      Try the rule out.




Write a rule for the graph below.



Assign variables then write a formula.




Try the formula out to see if it works.



                                   total cost of going to the movie

  $18.00
  $16.00                                                                                    $15.50
  $14.00                                                                       $14.25
                                                                  $13.00
  $12.00                                             $11.75
  $10.00                               $10.50
                           $9.25
   $8.00       $8.00
   $6.00
   $4.00
   $2.00
   $0.00
           0           1           2             3            4            5            6            7
                                                bags of popcorn




                                                                                                         69
Slope and Intercept                                                           Name __________________
                                                                              Parent Signature _____________________



                                          Cost of Trip to State Tournament

                 $12.00
                 $10.00                                                                                   5 , 10.50
                                                                                           4 , 9.00
  Cost of Trip




                  $8.00                                                     3 , 7.50
                  $6.00                                      2 , 6.00                                                     Cost
                  $4.00                       1 , 4.50
                              0 , $3.00
                  $2.00
                  $0.00
                          0               1              2              3              4              5               6
                                                 Number of Trips to the Snack Bar


Describe the data.
  1. My data starts at ______________________________________.
      This starting amount is known as the ________________________.
      Write a sentence using this vocabulary and the starting amount.



    2. My data is ________________________________ for every
       __________________________________________________.
       This change is known as the ___________________.
       Write a sentence using this vocabulary and this change.



Write a rule for the data.
3. The cost of the trip _______________________________________
      ____________________________________________________.

                 Circle the words that represent numbers that can change.
                 Write the words that you underlined, and assign each a variable.



                 Write a formula for the rule.

                 Try the rule out.




                                                                                                                                 70
                                                        Fundraiser

               $120.00
                             0, $110.00
               $100.00                    2, $95.00
  Sports Fee



                $80.00                                4, $80
                $60.00                                             6, $65
                                                                                                             Sports Fee
                                                                                8, $50
                $40.00                                                                        10, $35
                $20.00
                 $0.00
                         0           2            4            6            8            10             12
                                          Number of Cookie Dough Packs Sold




Describe the data.
  4. My data starts at ______________________________________.
      This starting amount is known as the ________________________.
      Write a sentence using this vocabulary and the starting amount.



    5. My data is ________________________________ for every
       __________________________________________________.
       This change is known as the ___________________.
       Write a sentence using this vocabulary and this change.



Write a rule for the data.
6. The sports fee _______________________________________
      ____________________________________________________.

               Circle the words that represent numbers that can change.
               Write the words that you underlined, and assign each a variable.



               Write a formula for the rule.

               Try the rule out.




                                                                                                                          71
                                                          Loan Balance

                 2500

                 2000       0, 2000
  Loan Balance


                                      1, 1750
                 1500                           2, 1500
                                                          3, 1250
                 1000                                               4, 1000
                                                                                 5, 750
                  500                                                                         6, 500
                                                                                                           7, 250
                    0                                                                                                   8, 0
                        0         1         2         3         4            5            6            7            8          9
                                                                    Months




Describe the data.
  6. My data starts at ______________________________________.
      This starting amount is known as the ________________________.
      Write a sentence using this vocabulary and the starting amount.



    7. My data is ________________________________ for every
       __________________________________________________.
       This change is known as the ___________________.
       Write a sentence using this vocabulary and this change.



Write a rule for the data.
6. The loan balance _______________________________________
      ____________________________________________________.

                 Circle the words that represent numbers that can change.
                 Write the words that you underlined, and assign each a variable.



                 Write a formula for the rule.



Try the rule out.




                                                                                                                                   72
Homework: Slope and Intercept                 Name __________________
                                              Parent Signature _____________________



  1. A refrigerator is set at 41 degrees. When the power goes out, the
     temperature increases 3.8 degrees every ten minutes.
      Time (minutes)      Refrigerator Temperature
         0                        41 degrees
         10                        __________
         20                        __________
         30                        __________
         40                        __________
         50                        __________


  2. How much does the refrigerator temperature increase every minute?



      This change is known as the ________________.
      Write a sentence using this vocabulary and the change.



  3. What is the starting temperature?



      This starting temperature is known as the __________________.
      Write a sentence using this vocabulary and the starting temperature.



4. Write a rule for the data.
      The refrigerator temperature ____________________________
      ____________________________________________________
      ____________________________________________________

     Circle the words that represent numbers that can change.
     Write the words that you underlined, and assign each a variable.



     Write a formula for the rule.

     Try the rule out.



                                                                                   73
Slope-Intercept Form                   Name __________________
                                       Parent Signature _____________________




When a formula is in slope-intercept form, we can easily find the slope and
intercept of the graph.

      For Example: A = 8 + 3 X P is in slope-intercept form because we can
see that the slope is 3 and the intercept is 8.

For each formula, find the slope and the intercept.

1. For the formula D = 400 + 50 X H:

                   The slope is _____ and the intercept is ______.

2. For the formula M = 50 – 2 X B:

                   The slope is _____ and the intercept is ______.

3. For the formula C = 25 X R + 650:

                   The slope is _____ and the intercept is ______.

4. For the formula T = 5.60 X P

                   The slope is _____ and the intercept is ______.




How do you know which number is the slope?




How do you know which number is the intercept?



Which formula’s graph has the steeper slope? Why?
A. T = 40 + 3 X P       Or         B. T = 20 + 4 X P


                                                                                74
                                                 Weight of the dump truck


                 25000
                                                                                  21000       21750
                 20000                                19500           20250
  Total weight


                             18000       18750
                 15000

                 10000

                 5000

                    0
                         0           1            2               3           4           5           6
                                                         scoops of gravel




5. The intercept is _________ and the slope is __________.
   Assign variables: _____ = Total weight, ______ = Scoops of gravel.

          The formula for the total weight is ___________________________.

6. The Shyster Glass Company deducts $50 for every vase that Roger
   breaks from his $3000 earnings.

                   A) Write a rule that finds what Roger has left of his earnings when
                      he breaks vases.



                   B) Underline the words that represent numbers that can change.
                   C) Assign variables to the words you underlined.




                   D) Write a formula for the rule.

                   E) The slope is ____________
                      because___________________________________________.



                   F) The intercept is __________
                      because___________________________________________.




                                                                                                          75
                   Homework:       Slope-Intercept Form                              Name______________
                                                                            Parent Signature_______________

                                       Gallons that have leaked from a faulty kitchen faucet


                 2030
                                                                                                      2027.5
                 2025
                                                                                           2022
  wasted water




                 2020
                 2015                                                     2016.5
                 2010                                    2011
                 2005                  2005.5
                 2000       2000
                 1995
                        0          1                 2                3                4          5            6
                                                                   days




7. The intercept is _________ and the slope is __________.
   Assign variables: _____ = Total weight, ______ = Scoops of gravel.

          The formula for the gallons of wasted water is __________________.

8. Jerry’s grandmother gave him $500 to start a savings account for
   college and his mother promises to deposits $40 to the account every
   month until he goes to college.

                   A) Write a rule that finds the balance in Jerry’s college fund for a
                      given number of months.

                   B) Underline the words that represent numbers that can change.
                   C) Assign variables to the words you underlined.




                   D) Write a formula for the rule.

                   E) The slope is ____________
                      because___________________________________________.



                   F) The intercept is __________
                      because___________________________________________.


                                                                                                                   76
Make Formulas Mean Something                Name __________________
                                             Parent Signature _____________________


 Often in math classes we practice with formulas that have no specific
meaning. For example, the formula J =70 – 3 X B has no particular meaning
because we don’t know what the variables J and B represent. Let’s see if we
can make up a rule for the formula. We can see two things for sure:
       1. The intercept is 70
      2. The slope is -3.
      The formula suggest that something called J starts at 70 then goes
down in steps of 3 for every one of something called B.

Think about something represented by the variable J that could start at 70
and decrease in steps of 3.
            Example: J = The temperature in Juan’s room.
         1. ____________________________________________
         2. ____________________________________________
         3. ____________________________________________

For each of the J’s we listed let’s find something represented by the
variable B that relates to the decrease in J.
             Example: B = The hours after Juan turns off the heat.
          1. ____________________________________________
          2. ____________________________________________
          3. ____________________________________________

Write a rule for each situation.
             Example: The temperature in Juan’s room started at 700 then
decreased by 30 every hour after he turned off the heat.

1.




2.




3.


                                                                                77
 Graph the formula J = 70 – 3 X B. Label the axes with the words from one
 of the rules on the previous page. Use an appropriate scale and consistent
 spacing of the numbers on each axis.


14




12




10




8




6




4




2




            5          10         15          20




 Useful skills are learned and practiced using formulas or equations that
 have no specific meaning.

 Find the slope and intercept of each equation.
 1. A = 7 + 2 X B       Slope = ____________ Intercept = ___________

 2. F = 5 X T + 4      Slope = ____________ Intercept = ___________

 3. C = 9 – 3 X D      Slope = ____________        Intercept = ___________

 Since the letter X could be used as a variable, we avoid using X for a
 multiplication symbol.
        For example: 5 X B is written 5B or 5 B.
 Find the slope and intercept of each equation.

 4. A = 2 + 3B         Slope = ____________        Intercept = ___________

 5. Y = 7 – 5X         Slope = ____________        Intercept = ___________




                                                                              78
Group Activity

For each formula: (a) find the slope, (b) find the intercept, (c) assign
meaning to each variable, and (d) write a rule that makes sense.
 1. D = 500 – 28 X P
       (a)    The slope = _____________.
       (b)    The intercept is ________________.
       (c)    D = _________________________________________
             P = __________________________________________
      (d) Rule: __________________________________________
             _____________________________________________
             _____________________________________________

2. T = 8.25 + 3 X B
      (a)    The slope = _____________.
      (b)    The intercept is ________________.
      (c)    T = _________________________________________
            B = __________________________________________
     (d) Rule: __________________________________________
            _____________________________________________
            _____________________________________________

3. W = 150 + 2 X G
     (a)    The slope = _____________.
     (b)    The intercept is ________________.
     (c)    W = _________________________________________
           G = __________________________________________
    (d) Rule: __________________________________________
           _____________________________________________
           _____________________________________________

4. P = 1.25 X C
       (a)   The slope = _____________.
       (b)   The intercept is ________________.
       (c)   P = _________________________________________
            C = __________________________________________
      (d) Rule: __________________________________________
            _____________________________________________



                                                                           79
Homework: Make Formulas Mean Something

                                            Name____________________
                                            Parent Signature_______________


For the formula P = 35 – 3T, find the slope, find the intercept, assign
meaning to each variable, and write a rule that makes sense.

       1.   The slope = _____________. Increasing or Decreasing?

       2. The intercept = ________________.

       3. Assign Variables
            P = _________________________________________
            T = __________________________________________

      4. Rule: __________________________________________
            _____________________________________________
            _____________________________________________

Graph the formula J = 35 – 3T. Label the axes with the words from one of
the rule you wrote above. Use an appropriate scale and consistent spacing of
the numbers on each axis.



              14




              12




              10




              8




              6




              4




              2




                           5           10           15           20




                                                                          80
Rational Numbers                     Name ________________________
                                     Parent Signature _____________________


A rational number is any number that
________________________________________________________.



Draw a figure that represents each rational number.
1. 4/5                                       2. 2 1/4

         Ex.




3. 1/9                                     4. 7/12




5. 27/50                                   6. 3 7/11




7. 8/15                                    8. 1 7/8




                                                                         81
Prime Factorization
Ex. 32  2  2  2  2  2



Practice. Write the prime factorization for each number.
      45                       120                      81




Reducing Fractions- Dividing out common factors.
      18 2  3  3 2 1 2
Ex.                   
      63 3  3  7 1  7 7

Practice. Reduce the following fractions. You must show factorization for
each.
12                   15            9                   14
42                   65            45                  42




20                   21            72                 3a 3b 2c
36                   36            30                 12abc 3




                                                                            82
To compare rational numbers, we need to look at pieces that are the same
size. This is called finding a common denominator.




Compare and put the numbers in order.

         ,    ,   ,    ,




We can also compare by changing fractions into their decimal equivalent.

      How?




Terminating

Repeating


                                                                           83
Write the fraction or mixed number as a decimal. Tell whether the fraction
is a terminating decimal or a repeating decimal.

1. 3/5                                            2. 2 2/5




3. 1/6                                            4. 5/12




5. 33/50                                          6. 14 7/16




7. 2/3                                            8. 1 4/5




Order the numbers from exercises 1-8.




                                                                         84
Homework: Rational Numbers             Name________________________
                                       Parent Signature _______________



Draw a figure representing each number. Write the fraction or mixed
number as a decimal. Tell whether the fraction is a terminating decimal or a
repeating decimal.

1. 3/5                                                2. 2 2/5




3. 1/6                                                4. 5/12




5. 2/3                                                6. 1 4/5




Compare and order the numbers from least to greatest.

9.       1 1/8, 1 3/7,          1.1,          1.43,         1 4/15




10.      1/8,   0.3,     1/3,          4/9,           0.7




                                                                           85
Rational Numbers on a Number Line    Name ________________________

Reduce each fraction. Then compare and order the numbers.
      11           19         21          41
         ,            ,          ,
      14           21         28          49




To place these numbers on a number line, we need to think about good
landmarks (scale).




Compare and order these numbers. Place the numbers on a number line.
       2           1         9            2           7
     3 ,             ,          ,        2 ,        1
       5          10         20           5          10




                                                                       86
Compare and order these numbers. Place the numbers on a number line.
      2           1         9            2           7
     1 ,            ,          ,        2 ,        1
      7           7         21           7          14




Compare and order these numbers. Place the numbers on a number line.
      1          1          9          4          7
        ,          ,          ,          ,
      9          3         12          6         12




What is an inequality?



Compare the two numbers. Write two inequalities for the comparison.
     1           3
     8          16




                                                                       87
Compare the two numbers. Write two inequalities for the comparison.
      1           2
     15          10




Compare the two numbers. Place the numbers on a number line. Write two
inequalities for the comparison.
        8                   5
       12                  14




Compare the first two numbers. Place the numbers on a horizontal number
line. The compare the second two numbers and place on a vertical number
line.
                    5          4
       Horizontal:
                   11         12




                    1          2
      Vertical: 2          3
                    3          3




                                                                          88
Homework: Rational numbers on       Name _____________________
     a number line                  Parent Signature _____________



Compare and order these numbers. Place the numbers on a number line.
       3           4         7            2           7
     2 ,             ,          ,        2 ,        1
       5          10         20           5          10




Compare the two numbers. Write two inequalities for the comparison.
      3          5
     8          10




Compare the two numbers. Place the numbers on a number line. Write two
inequalities for the comparison.
        6                   7
       18                  15




                                                                         89
On the SECRET coordinate plane, you should plot the locations of five ships.
The five ships will follow these specifications:

        Aircraft carrier -- 5 points long
        Battleship -- 4 points long
        Submarine -- 3 points long
        Destroyer -- 3 points long
        PT-Boat -- 2 points long

The boats must be either horizontal or vertical.

They may not overlap.

Draw a rectangle around each “boat”.


                                   SECRET




Record the coordinates of each boat below.

AC                    B            Sub       D           PT




                                                                          90
Fold this paper in half so your opponent does not see the contents.




                            Battle Station




You will take turns guessing the location of your opponent’s boats.

Record the coordinate pair of each guess. On your Battle Station coordinate
system, place an O for a “miss” and an X for a “hit”.

If you guess all the points on your opponent’s ship, they are required to say
“You sunk my battleship” or whatever type of boat it is. (The same goes for
your battleships)

Your guesses:




                                                                            91
                                Decreasing to Zero
                                  Horizontal Intercept

                                                  Name____________________
                                                  Parent Signature____________

    You start with $42 on your lunch balance, and are charged $2.50 everyday
    you eat at the cafeteria.

             Write a rule that gives the remaining balance for a given number of
    meals.



          Underline the words that represent numbers that can change then
    define variables for them.

             Write a formula.

    What is the slope?_________ What is the vertical intercept? ______
         Graph the formula and label the axes.




8




6




4




2




                         5                 10                 15


             How many meals can you get before you run out of money?

             How much money will be left in the balance?



                                                                                   92
    Anna has 20 gallons in the tank of her Toyota Tercel, which uses 1/35 gallon
    per mile (35 mpg).

          Write a rule that gives the remaining gallons of fuel in the tank of
    Anna’s Tercel after she drives so many miles.

          Underline the words that represent numbers that can change then
    define variables for them.

          Write a formula.

          What is the slope?______      What is the vertical intercept? ______

          Graph the formula and label the axes.




8




6




4




2




                       5                  10                 15




          How many miles will she be able to travel on the 20 gallons of fuel?



          Find the ordered pair at the horizontal intercept of the graph.

          Find the ordered pair at the vertical intercept of the graph.


                                                                                 93
    Phillip rides an elevator that starts on the 17th floor and goes downward 1
    floor in 3 seconds (or at the rate of 1/3 floor per second).

    Write a rule that gives the number of floors that Phillip is above ground
    level after a given number of seconds.

          Underline the words that represent numbers that can change then
    define variables for them.

          Write a formula.

          What is the slope?______      What is the vertical intercept? ______

          Graph the formula and label the axes.




8




6




4




2




                       5                  10                  15




          How long will it take Phillip to reach the ground level?



          Find the ordered pair at the horizontal intercept of the graph.

          Find the ordered pair at the vertical intercept of the graph.



                                                                                  94
    Rebecca knows that she can average 43 mph (.724 miles per minute) from
    her grandma’s house in Mapleton to her home 38 miles away in Heber City.

    Write a rule that gives the number of miles Rebecca is away from home
    given the number of minutes travelling there.

          Underline the words that represent numbers that can change then
    define variables for them.

          Write a formula.

          What is the slope?______     What is the vertical intercept? ______

          Graph the formula and label the axes.




8




6




4




2




                      5                  10                 15




          How long will it take Rebecca to get home?



          Find the ordered pair at the horizontal intercept of the graph.

          Find the ordered pair at the vertical intercept of the graph.



                                                                               95
    Homework: Decreasing to zero                      Name______________
                                      Parent Signature___________________

    You know that you can mow 20 square yards of lawn in 1 minute, and that
    your whole yard has 1600 square yards of lawn.

    Write a rule that gives the number yards left to mow given the number of
    minutes that you have been mowing the lawn.

          Underline the words that represent numbers that can change then
    define variables for them.

          Write a formula.

          What is the slope?______     What is the vertical intercept? ______

          Graph the formula and label the axes.




8




6




4




2




                      5                  10                 15




          How long will it take to mow the whole lawn?

          Find the ordered pair at the horizontal intercept of the graph.

          Find the ordered pair at the vertical intercept of the graph.


                                                                               96
                          Reaching a Target Value
                                 Graphing a Solution

                                               Name____________________
                                               Parent Signature____________

    Andy wants to drop weight for wrestling. He weighs 187 now, and plans to
    lose 1.5 lbs. per week.

           Write a rule that gives Andy’s weight for a given number of weeks of
    dieting.

          Underline the words that represent numbers that can change then
    define variables for them.

          Write a formula.

          Graph the formula and label the axes.




8




6




4




2




                      5                   10                15




          Draw a horizontal line at 170 lbs.

          How many weeks until Andy weighs 170 lbs?




                                                                               97
    A hot-air balloon starts at 1000 feet above Heber City and rises at a rate of
    5 feet per second.



          Write a rule that gives the balloon’s height at a given number of
    seconds.

          Underline the words that represent numbers that can change then
    define variables for them.

          Write a formula.

          Graph the formula and label the axes.




8




6




4




2




                      5                  10                  15




          Draw a horizontal line at 2000 feet.

          How many seconds until the balloon is 2000 feet above Heber City?



          Write an ordered pair where the two lines cross.



          Write a sentence for the meaning of the ordered pair.



                                                                               98
8




6
                                                         Math can be used to save Lives.




4




2




                             5                      10                                     15



    On August 5, 2010, 33 miners were trapped 2300 feet underground. After
    69 days the men were brought to the surface one at a time in the pod shown
    above. The pod could be pulled up as fast as 92 feet per minute.

           Write a rule that gives the pod’s height from the bottom of the shaft
    at a given number of minutes of a miner’s rescue.



          Underline the words that represent numbers that can change then
    define variables for them.

          Write a formula.

          Graph the formula and label the axes.
          Graph a horizontal line at 2300 feet.
          Write an ordered pair where the two lines cross.
          Write a sentence for the meaning of the ordered pair.



                                                                                            99
    Homework: Reaching a Target Value
                                                   Name____________________
                                              Parent Signature________________

    Miguel knows that it will take $5000 dollars to pay for his first year of
    college, so he set aside the $1200 that he earned this summer in a savings
    account then plans to deposit $200 per month to the account.

    Write a rule that gives Miguel’s college savings for a given number of months
    of deposits

          Underline the words that represent numbers that can change then
    define variables for them.

          Write a formula.

          Graph the formula and label the axes.




8




6




4




2




                      5                  10                   15




           Draw a horizontal line at $5000.
           How many months of saving until Miguel can pay for one year of
    college?

          Label the ordered pair where the lines intersect.
          Write what the point means.


                                                                              100
                                 Inverse Machines
                                                   Name____________________
                                                   Parent Signature____________

         A) In the top machine write what has been done to the variable N.
         B) In the bottom machine write how to reverse what has been done to N.
         C) Try out a number for N to see if the bottom machine reverses what the
         top machine does to the number.
                                                   7
     4



                                                  N-4
 N+7




______                                            ______




                                                 18
 8



                                                N  2
3N




                                                ______
______


                                                                               101
                              10
  5



                            3N - 4
2N + 3




                              15
   8



                            N  3-4
N 2 + 5




_____                        ______
_
                                ____       try a number.
  ____      try a number.



                             (N + 1)  2
 5(N – 1)




                              ______
  ______

                                                           102
         Homework: Inverse Machines
                                                   Name_____________________
                                            Parent Signature___________________


         A) In the top machine write what has been done to the variable N.
         B) In the bottom machine write how to reverse what has been done to N.
         C) Try out a number for N to see if the bottom machine reverses what the
         top machine does to the number.




                                               5N
N-8




______
                                                ______




                                               (N – 1)  3
2N - 3




______                                              ______




                                                                               103
                    Putting Inverse Machines in Reverse
                                                    Name____________________
                                                    Parent Signature____________

          A) In the top machine write what has been done to the variable.
          B) In the bottom machine write how to reverse what has been done.
          C) Try the number in the bottom machine. Try the result in the top
          machine.
                                                  ______
 ______



                                                   J+5
  V-8
                                                                                     7
                                             4




                                                 ______
______



                                                  N 7
 P  5
                                                                                     14
                                         8




                                                                               104
                 _____
 _____
                 _
 _


                 3N - 4
2N + 3
                                           17
            15




_____
_


                 N 4 - 8
N 3 + 1
                                      32
            8




                  _____
 _____
                  _
 _


                  (N + 1)  2
 5(N – 1)
                                       5
            20




                                105
         Homework: Putting Inverse Machines in Reverse
                                                   Name_____________________
                                            Parent Signature___________________


         A) In the top machine write what has been done to the variable.
         B) In the bottom machine write how to reverse what has been done.
         C) Try the number in the bottom machine. Try the result in the top
         machine.
                                              ______
______



                                               H - 14
G+7
                                                                                    5
                                         23




                                               ______
______



                                               (N + 1)  5
3N + 1
                                                                                    3
                                          19




                                                                              106
        Solving Equations with Inverse Machines
                                            Name____________________
                                           Parent Signature____________
The equation 3x - 5 = 22 can be shown as a machine.
If we make the inverse machine of 3x – 5 and put 22 into it, we can find the
x that makes the equation true.

Check this out!



       3x – 5 Times by 3 then reduce by 5
                                                     is 22


                    Increase by 5 then divide by 3
                    ( 22 + 5)  3
                         27  3


         9

How can we see that x = 9 makes the equation true?

Solve the equations using these three steps.
 1. Write what happened to the variable.
 2. Write the inverse by reversing what happened to the variable.
 3. Use the inverse to solve the equation. (You solve an equation when you
find out what makes it true.)
 4. Always check your answer.

Example: Solve 7x – 4 = 38

What happened                         Inverse             Solve          Check
x was multiplied by 7               Add 4 then         ( 38 + 4 )  7   7 6-4
then decreased by 4                 divide by 7             42  7       42 – 4
                                                              6            38
Solve 4B + 3 = 11
What happened                          Inverse           Solve           Check




                                                                            107
Solve the equations using these three steps.
 1. Write what happened to the variable.
 2. Write the inverse by reversing what happened to the variable.
 3. Use the inverse to solve the equation. (You solve an equation when you
find out what makes it true.)
 4. Always check your answer.
Equation        What happened   Inverse        Solve            Check
N – 5 = 11


J + 17 = 25


3T = 21


L  7=9


5V + 2 = 27


Y  3 + 2 = 11


(D – 4)/3 = 7


(G + 2) 5 = 15


3X – 11 = 10


7(R – 3) = 49


D4
     11
 3




                                                                         108
Homework: Solving Equations with Inverse Machines
                                          Name_______________________
                                         Parent Signature_________________
Solve the equations using these three steps.
 1. Write what happened to the variable.
 2. Reverse what happened to the variable to write the inverse.
 3. Use the inverse to solve the equation. (You solve an equation when you
find out what makes it true.)
 4. Always check your answer.
Equation        What happened   Inverse          Solve          Check
Q + 3 = 12      Increase by 3   Decrease by 3    12 – 3         9 + 3 = 12
                                                 9                  12 = 12

J - 17 = 2


6T = 54


L  4=7


4V + 3 = 27


K 3 + 2 = 23


(M + 4)/2 = 7


(G -7) 2 = 10


3P – 17 = 10


8(R + 3) = 40




                                                                        109
Putting it all Together                   Name ___________________
http://www.youtube.com/watch?v=LcWyyyms07w

In 1940, about 40 Siberian tigers existed in the wild. Russian anti-poaching
controls have helped to increase the population over the last 70 years. On average,
the tiger population increased by 5 tigers each year.

Write a rule that represents the number of Siberian tigers in the wild for any year after
1940.



Identify variables.



Write a formula for the Siberian tiger population.



Gather data.
       In 1940, the population was _______________.
       After 1 year, the population was ___________.
       After 2 years, the population was ___________.
       After 10 years, the population was ___________.
       After 20 years, the population was ___________.
       After 50 years, the population was ___________.
       After 70 years, the population was ___________.


Graph the data. Label your number lines and use an appropriate scale.



       8




       6




       4




       2




                               5                     10                    15




                                                                                        110
Write the formula for the Siberian tiger population again.



Is the data increasing or decreasing?



What is the slope?



What is the intercept?



How can we determine when the population will be at 500 tigers?



Write a specific equation.

Describe the equation in words.




Write the inverse by reversing what happened to the variable.




Use the inverse to solve the problem.




Check your solution.




                                                                  111
Homework: Putting it all together                Name ___________________



Write the formula for the Siberian tiger population again.


How can we determine when the population will be at 450 tigers?



Write a specific equation.

Describe the equation in words.




Write the inverse by reversing what happened to the variable.




Use the inverse to solve the problem.




Check your solution.




                                                                      112
                Positive and Negative Integers
                                              Name____________________
                                              Parent Signature____________
Solve the equation using the inverse.

B + 9= 5

What happened to B         Write the inverse            Solve




Is it possible to solve the equation B + 9 = 5?

If so what kind of number is the solution?



Can two numbers add to zero?

Give some examples.



We are going to model adding and subtracting Integers with blocks. You and
a partner will be given two different colored stacks of 8 blocks. Decide
which stack will represent positives and which stack will represent negatives.

Positive color___________________ Negative color________________




                   +                                            -

                Positive                                    negative




What number is shown by                 -      ?
                               +




                                                                          113
Start with        +
                        -        to represent zero, then model each


                  +     -

                  +     -

problem with blocks. Draw a picture to show each answer.

1.   5+3                2.   5–3                 3.   3–5




4.   5 + -3             5. -2 + 3                6. -4 + 3




7. -2 + -3              8. -2 – 4                9. 4 – (-2)




10. -4 – (-3)           11. 2 + (-5)             12. 0 – (-2)




                                                                      114
Solve the equations using these steps.
 1. Write what happened to the variable.
 2. Write the inverse by reversing what happened to the variable.
 3. Use the inverse to solve the equation. (You solve an equation when you
find out what makes it true.)
 4. Always check your answer.
Equation        What happened       Inverse             Solve       Check
N – 5 = -3


J – 17 = -8


T - 10 = -21


L+9=7


5V + (-2) = 23


Y + (-5) = -11


D  2 +(-1) = 7


G – (-5) = 2


H – (-6) = -3


7T – (-4) = 11


D   5 
           2
   3




                                                                         115
Putting It All Together (B)            Name ________________________


Cal scoops ice cream at the Mooseum. He looks in the tip jar at 6:30 and
there are $4.50. Every 20 minutes, he looks and there are $2.00 more in
the jar.

1. On average, how much money gets added per minute?




2. Find a rule that gives the amount of money in the tip jar for a certain
number of minutes.




3. Define variables for the numbers that change.




4. Write a formula.



5. Use your formula to fill out the chart below.

         Minutes after 6:30                     Amount in the Tip Jar
                  0
                 10
                 20
                 30
                 60




                                                                             116
    6. Graph the data. Use an appropriate scale, and label the axes.



8




6
                                                                 The slope is ______________

                                                                 The intercept is ___________
4




2




                  5             10           15            20




    7. Write an equation for when the amount in the tip jar is $10.00.
    _____________________________________

    8. Write the equation in words
    ______________________________________________

    9. Write the inverse in words
    _______________________________________________



    10. Use the inverse to solve it.




    11. Check your answer.




                                                                              117
The Distributive Property             Name _______________________

I want to give each of the 28 students in class 6 M&Ms. There are 20 boys
and 8 girls. How many M&Ms do I need?
      Ideas from Partner Discussion:




      Ideas from Class Discussion:




      Which idea you like best?




One way to represent the product of two numbers is to draw a rectangular
array.
For example the product 3  2 can be illustrated with the following array:

                             3


                     2




This array shows that 3 groups of 2 are the same as 2 groups of 3. The
Product is 6.



How could we use an array to solve the M&M problem?




                                                                         118
Some people do two-digit multiplication using a similar method. For example,
consider the product: 15  38


                                          38

                                30                       8

            10
  15

            5


Fill in the chart with the appropriate products.


What is the total area? Use the diagram to find the product 15  38 .


Partner Practice. Use the arrays to find the products.

1. 27(15)                                          2. 82x17




                                                                         119
3. 61(41)                                         4. 6(132)




Class Practice.
5. 4(y)                                           6. 3h(4)




7. 9(7m)                                          8. 12(6v)




9. d(3d)                                          10. 5g(5g)




                              (w+9)
Sometimes we will need to multiply groups. For example, 4(w+9).
                              w 9
                          4

             Product:


                                                                  120
Practice:

11. 3(x-10)         12. .5(y+8)




Partner Practice.
13. (h+7)11         14. 6(9+c)




15. 2v(v+12)        16. 5z(z-3+w)




17. a(a + b + c)    18. 10(x + 2y +z)




                                        121
Homework: The Distributive Property   Name ___________________
                                      Parent Signature _______________



Use an array to multiply.
1. 19(73)                             2. 56(13)




3. 85(14)                             4. 910(32)




5. 8(x+7)                             6. 12(y-5)




7. 7(b+4+c)                           8. 3(x+y+5z)




                                                                   122
Match Game                                     Match Game
        Name:                                          Name:

3  (5  2) 2                                   11-20
                                                      1
     2                                            2




                                                [16+3(4-1)]
[14  2(7  1)  3(5  1)]  9                              6
                                                    5




                     1 2                        12 1
13  0(2  19)  23                               3 2 160
                     2 1                         3 4




Reduce                                          Reduce
64                                               96
24                                               36




40                                              56
55                                              77




Descrive the equation.                     Describe the equation.
Write the inverse. Then use it to solve.   Write the inverse. Then use it to solve.
3 x  9  21                               10 x  5  35

                                                                         123
                      Decomposing Numbers
                                              Name____________________

A sum is the result of addition. For example the sum of 3 and 4 is written
( 3 + 4 ), which can be written as 7. The sum of Q and 8 can only be written
as ( Q + 8 ) if we do not know the value of Q.

Write each number as 3 different sums.

1.   45      a)                         b)                         c)
2.   73      a)                         b)                         c)
3.   -5      a)                         b)                         c)
4.   0       a)                         b)                         d)

A product is the result of multiplication. For example the product of 3 and
7 is written  3  7  , which can be written as 21. The product of 5 and B can
only be written as 5B or 5  B unless you know the value of B.

Write each number as 3 different products.

5.   24      a)                         b)                         d)
6.   18      a)                         b)                         d)
7.   20      a)                         b)                         d)
8.   72      a)                         b)                         d)




Some products are easy to find.
Explain why the following numbers are easy to multiply.

9. ( 300 X 5000)



10. ( 13 X 1,000)




                                                                             124
     When you multiply with arrays, make sure you write each number as a sum of
     easy numbers to multiply. If you have to use a calculator for any step, you
     probably need to use easier numbers.

     For example the product 47 X 58 can be found in the following way

                                      58

                                 50                        8




              40
47


               7




     Fill in the array with the appropriate products. What is the total area?

     What is the product 47 X 58?




     Use arrays to find the products.

     11.     35 X 18                  12. 52(33)                     13. 17 . 83




                                                                                125
Use arrays to find the products.

14. 62 X 35                    15. 19(63)   16. 24 . 95




17. 120 . 37                   18. 537(8)   19. 7 X 214




20. 325 X 271




                                                     126
               Starting Blocks Charge Model
                             Integer Addition

Model each problem by drawing the appropriate positive and negative blocks
above the starting blocks. Strike out blocks to show your result.

1. 5 – 7                 2. 4 – (-2)                    3. – 2 – 5




4. 5 – 3                 5. -4 + -3                     6. 1 – (-5)




7. -6 + 5                8. -4 – (-5)                   9. -5 – (-2)




                                                                       127
Homework: Multiplying with Arrays                    Name______________
          Starting Blocks                     Parent Signature_____________




Use arrays to find the products.

1. 38 X 37                     2. 53(41)                       3. 23 . 75




Model each problem by drawing the appropriate positive and negative blocks
above the starting blocks. Strike out blocks to show your result.

4. -4 – (-2)                   5. -3 – (-5)                    6. -3 - 4




                                                                           128
Match Game 2                                   Match Game 2
Name:                                          Name:

7  2  22  30 
                                                              9 2
                                                 4  2  2     40   8  3 
                                                              3 1




16  4                                                20  2
        5 3  2                                            8  6 
                                                                        2

  6                                                    5 1




                                                   23  1 1 35 2
3  2  3  1  6                                         
              2

                                                     3    7 1 5




                                                    16+5  4  1 2    2 
                                                                            
 2  8-5   6   3
                                                             6




Reduce                                          Reduce
81                                               48
54                                               60




Describe the equation.                     Describe the equation.
Write the inverse. Then use it to solve.   Write the inverse. Then use it to solve.
7 x  4  32                               8x  3  35                   129
Decomposing Numbers B                         Name ______________

Write each number as 3 different sums.

1.   56        a)                    b)                c)
2.   61        a)                    b)                c)
3.   -3        a)                    b)                c)
4.   0         a)                    b)                c)

Write each number as 3 different products.

5.   32        a)                    b)                c)
6.   16        a)                    b)                c)
7.   40        a)                    b)                c)
8.   64        a)                    b)                c)

Use arrays to find the products.

9. 64 X 37                     10. 17(43)              11. 44 . 75




12. 130 . 27                   13. 526(419)            14. 8 X 317




                                                                130
Algebra 1                                      Name ________________________

Vocabulary

Write the words for each number.
      1                                   1
1.                                  2.
      6                                   7



      1                                   1
3.                                  4.
      3                                   5



      1                                   1
5.                                  6.
      2                                   4



      1                                   1
7.                                  8.
      8                                   9



       1
9.                                  10.   .1
      10



       1
11.                                 12.   .01
      100


Write a symbol for each term.
13. 4 cards           14. 2 kings   15. 5 sweaters        16. 3 numbers




                                                                          131
Write using symbols and simplify if you can.
   17. 2 cards plus 4 sweaters plus 5 numbers plus 3 cards plus 10 kings plus 2 sweaters
        plus 4 numbers plus 8 cards.




Draw a picture and use symbols for each situation.
18. 2 groups of 3 chickens                   19. 4 groups of 2 camels




20. 2 groups of 3chickens and 7 eggs         21. 3 groups of 2 camels and 1 tent




Write a situation for the given symbols.
22. 4m                                       23. 5d




24. 7a + 8r




   25. 6(2d + 3e)




   26. 3(2y + 4f)




                                                                                       132
Write the symbol for each word.
27. is                28. plus               29. times   30. of




31. decreased by                      32. quotient       33. added to




Write a sentence for each equation.
   34. 2x + 3h = 14




   35. 4(5a + 7) = 13




         1      1
   36.      y   z  20
         3      5




                                                                        133
                    Match Game 3:   Combining Like terms
Goal: Be able to combine like terms.
Steps to get there:
      A) Practice order of operation.
      B) Practice adding and subtracting fractions.
      C) Practice adding and subtracting decimals.
      D) Practice adding and subtracting integers.
      E) Practice using the distributive property.
      F) Learn basic factoring, change the order with the distributive
property.
      G) Combine like terms.

Name____________________                   Name____________________
Match Game (3)                                  Match Game (3)
A) Order of operations
    3  23  1 0                                10  20  6
1.            7                             1.              6  32
        4                                            3




2. 5 + 2(3 - (7 - 5) + 1)                    2. 3 + 3( 8 – (14 – 7) + 1)




3. 5  3  1                                3. 20  7  5
                2                                             2




     4  2[3  5 12  11]                       7  1  2  3 18  17  
                                                                          
4.                                           4.
               5                                              3




                                                                               134
Name___________________               Name_____________________
B) Adding and subtracting fractions (no calculators – show every step)

       3 2                                        1 4
5.                                        5.      
      11 11                                      11 11




      3 2                                        4 3
6.                                        6.     
      5 5                                        5 5




      1 3                                        1 1
7.                                        7.     
      5 10                                       3 6




      2 1                                        3 1
8.                                        8.     
      3 6                                        5 10




      1 1                                         2 2
9.                                        9.      
      5 3                                         3 15




      5 1                                        1 1
10.                                       10.    
      8 4                                        4 8




                                                                         135
C) Adding and subtracting decimals (no calculators)

Name_______________________ Name_______________________

11. 5.03 + 7.204                            11. 2.102 + 10.132




12. 11.037 + 2.142                          12. 7.153 + 6.026




13. 8.876 – 2.451                           13. 9.766 – 3.341




14. 3.5 + 5.8                               14. 2.4 + 6.9




15. 12.05 – 3.75                            15. 13.15 – 4.85




                                                                 136
D) Adding and subtracting integers.
Model each problem by drawing the appropriate positive and negative blocks
above the starting blocks. Strike out blocks to show your result.

Name____________________             Name_____________________

16. -5 + -2                                       16.   -6 + -1




17. 5 – (-3)                                      17. 2 – (-6)




18. -5 – (-7)                                     18. -1 – (-3)




                                                                       137
E) The distributive property
Use arrays to find the products

Name______________________        Name___________________

19. 52(64)                              19. 128(26)




20. 12(x + 3)                           20. 4(3x + 9)




21. G(5 + 2) use an array               21. G(3 + 4) use an array




22. ( 7 + 5)T                           22. ( 2 + 10)T




23. Show that the arrays for      23. Show that the arrays for
   (2 + 9)x and 11x are equal.       (4 + 7)x and 11x are equal.




                                                               138
F) Basic Factoring: The distributive property

Rewrite each product using the distributive property. Show that both ways
are equal.
Example: 5(2) + 7(2) can be written (5 + 7)(2)
           10 + 14                   (12)(2)
              24                        24
Name____________________               Name_______________________

24. 7(3) + 5(3)                                  24. 3(3) + 9(3)




25. 5(10) + 6(10)                                25. 8(10) + 3(10)




26. 3(x) + 11(x)                                 26. 7(x) + 7(x)



Check with x = 2                                 Check with x = 2



27. 13(y) - 11(y)                                27. 28(y) - 26(y)




Check with x = 7                                 Check with x = 7



28. 5(x2) + 4(x2)                                28. 2(x2) + 7(x2)

Check with x = 3                                 Check with x = 3



                                                                       139
G) Now you are ready to combine like terms!

Name________________________ Name________________________

Simplify each sum or difference by combining like terms.
29. 3x + 2x                                        29. x + 4x



30. 5w + 8w                                       30. 2w + 11w



31. 11R – 5R                                      31. 8R – 2R



      5   1                                             3   3
32.     x x                                      32.     x x
      8   8                                             8   8


      1    3                                             1   2
33.     y y                                      33.      y y
      7   14                                            14   7



34. 1.5K + 3.4K                                   34. 4.2K + .7K




35. 9.102x + 1.310x                               35. 3.211x + 7.201x



36. 5x2 + 3x2                                     36. 2x2 + 6x2



37. -3x + 7x                                      37. -8x + 12x



38. 5x – (-2x)                                    38. 3x – (-4x)



39. 2x + 5x + 3y + 7 y                            39. x + 6x + 4y + 6y


                                                                         140
Homework:      Combining Like Terms (1)
                                             Name___________________
                                           Parent Signature_______________
No Calculator on any problem.

A) Order of operations
      7  32                                    2  3  2  3 13  10  
                                                                        
1.            20                          2.
        4                                                   2




B) Adding and subtracting fractions (show every step)

      5 2                                       1 1
3.                                        4.    
     13 13                                      4 3




C) Adding and subtracting decimals
5. 5.03 + 7.204                            6. 2.102 + 10.132




D) Adding and subtracting integers.
Model each problem by drawing the appropriate positive and negative blocks
above the starting blocks. Strike out blocks to show your result.
7. -2 – (-3)                                       8. 6 - 8




                                                                             141
E) The distributive property
Use arrays to find the products

9. 32(63)                                         10. 141(25)




F) Basic Factoring: The distributive property

Rewrite each product using the distributive property. Show that both ways
are equal.
Example: 5(2) + 7(2) can be written (5 + 7)(2)
           10 + 14                   (12)(2)
              24                        24

11. 2(8) + 5(8)                                   12. 3(x) + 9(x)



                                                  Check with x = 2



G) Combine like terms.

Simplify each sum or difference by combining like terms.

13. 2x + 8x                                       14. 6x - x



15. -5w + 8w                                      16. -2w + -11w



      3   1
17.     T T                                      18. 3.5R – 2.1R
      5   5



                                                                       142
        Combining Like-Terms to Solve Equations
The Commutative Property of Addition:

            Generalizaton: a + b =
Show that the commutative property can make addition easier.

1. 7 + 38 + 3                          2. 25 + 157 + 5




3. 21 + 1229 + 29




The Inverse Property of Addition says that every number but zero has an
opposite, and when you add a number to its opposite you get 0.

Use n’s for negative counters and p’s for positive counters to show examples
that
Subtracting is adding the opposite. a) use subtraction b) use addition.

4. 4 – 3                 5. 3 – (-2)         6. 2 – 5           7. -3 – (-1)
a)                       a)                  a)                       a)




b)                       b)                  b)                        b)




                                                                            143
8. Explain why addition is commutative but subtraction is not commutative.
Show how subtraction by adding the opposite and the commutative
property make these problems easier.

9. 23 + 88 – 13                       10. 58 + 275 – 48




11. 1255 + 2749 - 1254




More on combining like terms

Remember 5x means x+x+x+x+x and 2x means x+x so 5x + 2x means
(x+x+x+x+x) + (x+x) = 7x
Can you think of another way to show that 5x + 2x = 7x?




Explain why 3x + 2y cannot be simplified.



Explain why 3ab + 8ab equals 11ab.




                                                                       144
Multiplying integers

Explain how 4(5) can be written as an addition problem using only 5s.




What does 4(-3) mean in terms of groups of -3?




What does -3(2) mean in terms of groups of 2?




Explain what -4(-5) means?




Finish the following generalizations

a(b) =                      a(-b) =          -a(b) =              -a(-b) =



Examples
3(4)=              3(-4)=                    -3(4)=             -3(-4)=



What does the product (1/2)(2) equal?

How about (5) (1/5)?

½ and 2 are reciprocals. Likewise 5 and 1/5 are reciprocals. What is the
reciprocal of 17?



What happens when you multiply a number by its reciprocal?




                                                                             145
The Inverse Property of Multiplication says that every number but zero has
a reciprocal, and that when you multiply a number by its reciprocal you get 1.

                                   b           1
Finish the generalization:            1,   b   ________
                                   b           b

This means that dividing by 2 is the same as multiply by ½.

It also means that dividing by ½ is the same as multiplying by 2. The same
goes for any number and its reciprocal.

Big Idea: “Dividing is the same as multiplying by the reciprocal.”
Generalization: a  b 

Change each division problems to a multiplication problem.

                                                                            1
12. 5  7                            13. 15  3                  14. 12 
                                                                            2



Complete the following generalizations.

a b               a   b                    (a)  b       a    b 




The Commutative Property of Multiplication means:

               Generalization: ab =
Show that the commutative property can make multiplication easier.
             1                                              1
15. 15 11                                            16.     40 12
             3                                              6




                                                                                146
Explain why multiplication is commutative but division is not commutative.



Change the division problems to multiplication problems then evaluate.
        1                                     2                           3
17. 5                                18.                       19.  3  
        7                                    1                          5
                                             
                                             3



                                                                        2
                                                                         
                                             5
                                                                    20.  
      1 1                                                                3
20.                                    21.     5
      3 2                                    8                        4
                                                                         
                                                                        9




Solving Equations by Balancing

Solving equations by balancing is very similar to solving with inverses.

Solve 4x + 3 = 11

          Inverses                                      Balancing

x was multiplied by 4 then increased           Undo but keep the equation
 by 3 to get 11                                balanced at all times.
11 can be decreased by 3 then
divided by 4 to get x                   equation balanced at all times.
(11 – 3)/4
                                        4x + 3 = 11
8/4
                                                -3 -3 subtract 3 from both
                                               sides
      2
                                        4x + 0 = 8      (identity property of +)




                                                                              147
4x = 8

4x = 8       divide by sides by 4

4     4

1x = 2     (identity property of X)

x = 2 the variable is isolated
To isolate the variable means:

Combine like terms then solve equation by balancing. Show every step.
Check your answer by substituting it back into the equation to see if it
makes a true statement.

21. 3x  9  4  2x  3                      22. 5  x  3  x  3
         23. 5x  2  3x 11  4x 1




                                                                           148
Homework:   Combining Like Terms to Solve Equations

                                            Name____________________
                                 Parent Signature____________________
Combine like terms then solve equation by balancing. Show every step.



1. x + 5 + 2x = 14                                       2. 5 + 3x – 3
+ 2x = 22




3. 3 – 4x + 11 + x = -1                                  4. 3x + 1 =
4x




5. 3(2x + 1) – 4x = 17                                   6. 6 + 2(x –
1) + x = 12 - x




                                                                  149
                            Introduction to Factoring
                              (The Distributive Property)
                                                            Name____________________

Simplify.

1. aaaaaa ________           2. a  a  a  a  a  a ________      3. sstttt _________

4. a 2 a 4 ________          5. 2a  4a __________                  6. st  st 3 _________

7. b  b  b  bbb ________ 8. aaa  aaa _________                  9. ccc  cccc _______


Find the greatest common factor.

1. 5, 35 _________           2. 14, 42 _______              3. 33, 55 _________

4. x, 5x _________           5. t 2 , t 3 ________          6. 4r , rb _________

7. 7 x,14 x5 _______         8. 12tw,9w2 _________          9. 18c 2 d , 24cd 4 _________


Use an array to multiply.

1. 5 1  7                 2. 7  2  6                  3. 11 3  5




4. x 1  5                 5. t 2 1  t                 6. r  4  b




7. 7 x 1  2 x 4           8. 3w  4t  3w               9. 6cd  3c  4d 3 




                                                                                       150
Factor (rewrite using the distributive property)

1. 6t 12                     2. 5x  7 x2                   3. 12a  16b



4. 28x  7 xy                 5. 54a3b  63a 2b2             6. 13x  26 x 2



Solve by balancing. Check your answer.

1. 3x  11  8                2. 2x  3  x  2  x  5             3. 2  3 x  2  x  1




Check:                        Check:                                 Check:



Sylvia has made 3 birthday invitations in 2 minutes.
Write a formula that she could use to find the total number of invitations given the
number of minutes she continues to work. (hint: What is the intercept?______ What is
the slope? ____________)



Graph the function.




Write an equation then solve it by balancing to find the number of minutes that it will
take Sylvia to make a total of 16 invitations.




                                                                                          151
Factoring Revisited
(The Distributive Property)
                                                  Name____________________

Simplify.

1. ggggg                  2. g  g  g  g  g            3. mmmvv



4. g 2 g 3                5. 2 g  3g                     6. mv 2  mv 4




7. c  c  c  c  cccc   8. nn  nn  nn                  9. bbb  bb  b




Find the greatest common factor.

1. 6, 30                         2. 18, 45                 3. 22, 77




4. a, 8a                         5. h 2 , h 3              6. 5k , 2kb




7. 6 x,12 x3                     8. 21hy ,14h 2            9. 20 w2 x,30 wx 3




                                                                                152
Use an array to multiply.

1. y  2  4               2. w2  3  w         3. a 9  c 




4. 5x 1  4 x3            5. 2w  7d  5w       6. 3cd  2c  6d 4 




Factor (rewrite using the distributive property)

1. 7t 14                   2. 7 x  6 x2          3. 16a  32b




4. 27 x  9 xy              5. 56a 2b  64a3b2     6. 15 x  30 x3




                                                                          153
Solve by balancing. Check your answer.

1. 2 x  13  17         2. 4x  6  x  3  x  15   3. 1  4  x  3  2x 1




Check:                   Check:                           Check:




Matt’s company can produce 5 snowmobiles in 2 hours.
How many snowmobiles can the company produce in 1 hour? __________
What is the slope of the data? __________
Matt starts the work day with 16 snowmobiles in storage.
What is the intercept of the data? _________
Write a formula that he could use to find the total number of snowmobiles
given the number of hours worked.



How many snowmobiles in the shop for each time given?
0 hours?                2 hours?                 6 hours?

Graph the function. Label your axes and scale.




Write an equation and solve to find the number of hours that it will take
Matt’s company to have a total of 42 snowmobiles in storage.




                                                                              154
Homework:   Introduction to factoring
                                                    Name___________________________
                                                    Parent Signature___________________
Factor (rewrite using the distributive property)

1. 2t  8                      2. 5u  7u 2                  3. 24a  36c


4. 8 y  32 xy                 5. 50a3b  45a 4b3            6. 17 x3  34 x2


Solve by balancing. Check your answer.

1. 7 x  1  16                2. 3x  3  2x  2  x  9           3. 2  5  x  3  x  8




Check:                         Check:                                Check:


Rose has made 5 pinwheels in 20 minutes.
Write a formula that she could use to find the total number of pinwheels given the
number of minutes she continues to work. (hint: What is the intercept?______ What is
the slope? ____________)



Graph the function.




Write an equation then solve it by balancing to find the number of minutes that it will
take Rose to make a total of 23 pinwheels.




                                                                                          155
Perimeter and Area                          Name ___________________

Perimeter:



Use a ruler to find the dimensions of each rectangle. Write the dimensions
on the rectangle.
Then calculate the perimeter.




Practice worksheet

Given the algebraic dimensions on each rectangle, calculate the perimeter.


              2x



 x+3
                                                                   3w + 7


                                                  9 - 2w




                                                                         156
Area:




Write the dimensions of each rectangle below. (Hint: you already measured
these).
Use the dimensions to calculate the area.




Practice Worksheet.

Use the algebraic dimensions to calculate the area of each rectangle.


              2x



 x+3
                                                                     3w + 7


                                                   9 - 2w




What is the difference between perimeter and area?



                                                                            157
Homework: Perimeter and Area                Name ___________________
                                            Parent Signature ______________________



Given the dimensions, find the perimeter and the area of each rectangle.


              16                                     12


 11                                 5




               5x



 2x + 4
                                                                        4w + 3


                                                   10 - 4w




                                                          w + 3z
                3x + 4y
 2y                                                                     w + 3z




                                                                               158
Slope-Intercept Form                                Name ___________________
Exponents, Factoring

Carl graduates from college with student loans totaling $20,000. He sets up a payment plan
where each month he pays $250.
        What is the intercept? ________
        Is this amount increasing or decreasing?
        What is the slope? ________
        Define variables.

       Write a formula to determine how much Carl owes.

       Collect some data.




       Graph




From a different story, we might find the formula
       Q = 14 + 3B

       What is the intercept? ________ What does that mean?

       What is the slope? ________ What does that mean?

       Collect data and graph.




                                                                                      159
From a different story, we might find the formula
       A = 315 – 22C

       What is the intercept? ________ What does that mean?
       What is the slope? ________ What does that mean?
       Collect data and graph.




From a different story, we might find the formula
       Y = 20x + 10

       What is the intercept? ________ What does that mean?
       What is the slope? ________ What does that mean?
       Collect data and graph.




Quick Practice
1. M = 90 -11K                               2. Y = 14x + 35
   Intercept _________                           Intercept __________
   Slope ________                                Slope _________
   Graph                                         Graph




                                                                        160
Exponents

What does x5 mean?



What does 7y mean?



Simplify
1. m3(m2)                  2. (4ab)3                   3. (2x)(5x)2




4. 5y(12xy)                5. 120                      6. (22a)(a2b)




Combine Like Terms
7. 5g + 3g + 19            8. 4m + -3y + 2m + -8y      9. 3x + 5x2




10. -11a2 + 9a2 +3a + 8a   11. 16rg + -2r + 3g + -5r   12. 10x2 – 5xx




Multiply (Distribute)
13. 3(x + 7)               14. 8k(2k – 5)              15. 3xy(y2 +2)




                                                                     161
Factoring

Factor the following.
1. 32                        2. 60                        3. 88x2




Find the greatest common factor for each.
4. 16, 30                                   5. 20ab2 , 35abc




Factor the following.
6. 12x + 36                                 7. 3a2 + 2a




8. 5d2k – 20dk                              9. 18w3 + 12w2 + 24w




Graphing coordinate practice worksheet.
       (             ,           )




                                                                    162
Homework: slope-intercept form,                          Name __________________
Exponents, Factoring                                     Parent Signature __________

1. From a story, we might find the formula
       H = 25 + 4M

       What is the intercept? ________ What does that mean?

       What is the slope? ________ What does that mean?

       Collect data and graph.




2. 3y(7xy)                              3. 80                                4. (15a3)(a2b)




5. -8a2 + 3a2 +15a + 6a                 6. 10rg + -3r + 2g + -7r             7. 9x2 – 3xx




Multiply (Distribute)
8. 4(x + 11)                     9. 3a(2a + 3y)                       10. 5fb(b + 3f)




Factor
11. 4x + 24                                       12. 3a2 + 5ab




                                                                                        163
Slope-Intercept Form                                  Name ___________________
Exponents, Factoring B

The Fillmans bought a new hot tub and filled it with water from the garden hose. Initially,
the temperature of the water was 56 degrees. When they turned on the tub, the
temperature rose about 4 degrees every 40 minutes.
        What is the intercept? ________
        Is this amount increasing or decreasing?
        How many degrees does the temperature change every minute?
        What is the slope? ________
        Define variables.

       Write a formula to determine the temperature of the tub.

       Collect some data.




       Graph




From a different story, we might find the formula
       Q = 12 + 7B

       What is the intercept? ________ What does that mean?

       What is the slope? ________ What does that mean?

       Collect data and graph.




                                                                                         164
From a different story, we might find the formula
       A = 285 – 15C

       What is the intercept? ________ What does that mean?
       What is the slope? ________ What does that mean?
       Collect data and graph.




From a different story, we might find the formula
       Y = 15x + 22

       What is the intercept? ________ What does that mean?
       What is the slope? ________ What does that mean?
       Collect data and graph.




Quick Practice
1. M = 130 - 9K                              2. Y = 8x + 16
   Intercept _________                           Intercept __________
   Slope ________                                Slope _________
   Graph                                         Graph




                                                                        165
Exponents

What does x6 mean?



What does 4y mean?



Simplify
1. m5(m3)                 2. (2abc)3                  3. (3x)(5x)2




4. 4y(10xy)               5. 880                      6. (10a)(a2b3)




Combine Like Terms
7. 4g + 9g + 14           8. 2m + -5y + 8m + -7y      9. 2x + 11x2




10. -8a2 + 4a2 +7a + 5a   11. 15rg + -3r + 2g + -8r   12. 31x2 – 6xx




Multiply (Distribute)
13. 4(x + 5)              14. 6k(3k – 7)              15. 2xy(y2 + 5)




                                                                     166
Factoring

Factor the following.
1. 54                        2. 40                         3. 30y2




Find the greatest common factor for each.
4. 14, 98                                   5. 12ab2 , 32a3bc




Factor the following.
6. 10x + 24                                 7. 5a2 + 30a




8. 6d2k – 21dk                              9. 15w3 + 35w2 + 20w4




Find the Area of each Rectangle


                 3x +8                                               2a+3b


 x+9                                               a-5




                                                                             167
Homework: slope-intercept form,                          Name __________________
Exponents, Factoring B                                   Parent Signature __________

1. From a story, we might find the formula
       H = 27 + 3M

       What is the intercept? ________ What does that mean?

       What is the slope? ________ What does that mean?

       Collect data and graph.




2. 2y(8x2y)                             3. 90                                4. (4a3)(7a2b)




5. -3a2 + 8a2 +11a + 7a                 6. 11rg + -2r + 6g + -9r             7. 5x2 – 2xx




Multiply (Distribute)
8. 3(x + 10)                     9. 5a(7a + 6y)                       10. 3fb(b + 2f2)




Factor
11. 7x + 21                                       12. 8a2 + 12ab




                                                                                         168
Slope-Intercept, General Form                         Name ___________________
Exponents, Factoring

Elmo lost some extra weight to look really good for his 5-year class reunion. Tipping the
scales at 195 pounds, he successfully lost 2 pound per month until he reached his target
weight.
        What is the intercept? ________
        Is this amount increasing or decreasing?
        What is the slope? ________
        Define variables.

       Write a formula to find how much Elmo weighed.

       Collect some data.



       Graph




From a different story, we might find the formula
       Q = 23 + 2B

       What is the intercept? ________ What does that mean?

       What is the slope? ________ What does that mean?

       Collect data and graph.




                                                                                            169
Quick Practice
1. M = 83 + 5K                                       2. Y = -3x + 18
   Intercept _________                                Intercept __________
   Slope ________                                     Slope _________
   Graph                                              Graph




General Form: AX + BY = C
Tanya has $20 to buy food at a roadside stand. She can buy F pounds of fruit at $2 per
pound or N pounds of nuts at $5 per pound. If Tanya spends all $20, the equation
        2F + 5N = 20
describes all of the possible weight combinations of nuts and fruit that she could buy.

Use the equation 2F + 5N = 20 to answer the following questions.

1. How many pounds of nuts can Tanya buy if she buys 10 pounds of fruit?



2. How many pounds of fruit can she get if she buys 2 pounds of nuts?



3. How many pounds of nuts can Tanya buy if she buys 2 pounds of fruit?



4. How many pounds of nuts can she buy if she buys 0 pounds of fruit?



Use the data you collected in problems 1-4 to graph 2F + 5N = 20.




                    Lbs of Nuts




                                                             Lbs. of Fruit




                                                                                          170
Exponents

1. What is the difference between 7x and x7?



Simplify
2. m3(m+m+m)                                            3. (3a2)3




4. (3x3)(2x)2                                           5. (5y(7xy))0




6. Why is 3x2 not the same as (3x)2?



Combine Like Terms
7. 5g + 3t + 7g                        8. 4y + -y + 2m + -8y + m               9. 17x3 + x(x2)




Multiply (Distribute)
10. 3(x + 7)                           11. 8k(2k – 5)                          12. 3xy(y2 +2)




13. Distribute then combine like terms.                 14. Distribute then solve.
     5( x + 3) + 7x + 8                                     8( b – 3) = 3( b + 2)




                                                                                           171
Factoring

Factor the following.
1. 80x                          2. 16x2                       3. 8xy




Find the greatest common factor for each.
4. 16x2, 80x                                  5. 80x, 8xy




Factor the following.
6. 15x + 25                                   7. 16x2 + 8xy + 80




Use the given arrays to factor.

8.    x2 + 5x + 6 = (      )(             )   9. x 2 + 8x + 15 = (      )(   )




     x2        3x                                     x2           3x



     2x         6                                     5x           15




                                                                                 172
Due (Wednesday, January 20)
Homework: slope-intercept form, General Form       Name __________________
Exponents, Factoring                               Parent Signature __________

1. T = 8 + 3K                                      2. y = -2x + 15
   Intercept _________                              Intercept __________
   Slope ________                                   Slope _________
   Graph                                            Graph




Simplify
3. (114)0                                  4. (3a3)(a2b)2




5. -a2 + 3a2 + 4a + 6a2                    6. 5 + 3gx + 2 + 4xg




7. Distribute then simplify.               8. Distribute then solve.
     4(x + 1) + 2                                 5(2a + 3) = 4a + 3




Factor
11. 4x + 24                                12. a2 + 7a + 12 = (        )(        )



                                                 a2      3a

                                                 4a      12




                                                                                     173
Factoring, General Form,                      Name _____________________
Solving Equations

Use arrays to factor each of the following.


      x2 + 9x + 20                                    x2 + 12x + 32




     x2               5x                         x2             8x

     4x               20                          4x            32




     x2 + 12x + 27                                          x2 + 13x +40



    x2                                                   x2

                      27                                              40




Exponent Practice
          x y
                  4
           3
                                              (3ab 2 )(4a) 2




                                                                           174
General Form: AX + BY = C

For her book club, Courtney wants to have crackers and cheese on hand. She has $45 to
buy mozzarella and gouda cheese. Courtney already has crackers. She can buy M pounds of
mozzarella at $5 per pound or G pounds of gouda at $8 per pound. If Courtney spends all
$45, the equation
        5M + 8G = 45
describes all of the possible weight combinations of mozzarella and gouda that she could
buy.

Use the equation 5M + 8G = 45 to answer the following questions.
1. How many pounds of gouda can Courtney buy if she buys 9 pounds of mozzarella?



2. How many pounds of mozzarella can she get if she buys 3 pounds of gouda?



3. How many pounds of gouda can Courtney buy if she buys 2 pounds of mozzarella?



4. How many pounds of gouda can she buy if she buys 0 pounds of mozzarella?



Use the data you collected in problems 1-4 to graph 5M + 8G = 45 .




                 Lbs of Mozzarella




                                                            Lbs. of Gouda



Exponent Practice
      5 gh(2 g  9h)                         4m(3mn 2 )(2m)




                                                                                     175
Homework Due Friday January 21st                         Name ___________________
Factoring, General Form, Solving Equations

   1.   Use arrays to factor the following.

        x2 + 16x + 64                                          x2 + 15x +56




   x2                 8x                                     x2

         8x           64                                               56



   2. From a story, we get an equation 4x + 3y = 12. Use the equation to answer the
      following questions.
          a. If we buy 3 of the x’s, how many y’s can we buy?



              b. If we buy 4 of the y’s, how many x’s can we buy?



              c.   If we buy 2 of the x’s, how many y’s can we buy?



              d. Use your data from a-c to graph 4x + 3y = 12.




                                          y




                                                                  X

   3. Solve the equation.
              7(2x + 1) = 2x +4



                                                                                      176
Linear graphs, solving equations,                       Name ___________________
Factoring

For their wedding, Elvis and Diana created their own blend of M&Ms. Diana’s choice of
purple with the message “Celebrate” costs $19 per pound. Elvis’s choice of silver with the
message “Love” costs $15 per pound. They have a total budget of $1,900 for the candy.
The equation 19D + 15E = 1900 can be used to help determine different combinations to buy.
http://www.youtube.com/watch?v=mlyM_janCSE


1. If they decide to buy 100 pounds of Diana’s choice, how many pounds of
Elvis’s choice can be purchased?



2. If they decide to buy 48 pounds of Elvis’s choice, how many pounds of Diana’s choice can
be purchased?



3. If they decide to buy 30 pounds of Diana’s choice, how many pounds of
Elvis’s choice can be purchased?



Use the data you collected from problems 1-3 to graph the different combinations.




       Elvis’s
       Choice
       (pounds)




                              Diana’s Choice (pounds)



   4. Is the data increasing or decreasing?

   5. Can you determine the slope?



   6. Using your graph, what is the intercept?




                                                                                        177
Solving Equations

Solve.
1. 3x + 5 = 22                 2. 8x + 10 = -14




3. 5(2x + 1) = 12              4. 3(5x – 8) = 2(4x + 12)




5. 7x + 9 + 3x = 12 – 8 - 2x   6. 18 + 2x – 9 = 8x + 3x + 1




7. 9(2x – 4) + 8 = 17          8. 19 - 4x = 3(8x + 11) + 2




Exponent Practice.
1. (5 xy )                     2. (2ab)(3a )
             3                                 2




                                                              178
Factoring Practice.
Use arrays to factor.
1. 8ab + 6b                                           2. 9x2 + 12x




Use arrays to factor the trinomials.
3. x  17 x  72                                      4. a  3a  2
    2                                                        2




5. d  6d  8                                         6. k  11k  30
     2                                                       2




Paste here for the sorting activity. Explain why you sorted the way you did.




Explanation:                                  Explanation:


                                                                               179
Cut Up the Following Equations and Graphs. Sort everything into two
categories and tape them on the previous page. Explain how you sorted.

                                                                          4
M  17  3H                          g  5x2                                  12
                                                                          xy


2 D  8W  17                        y  80  14 x                         a 2  b 2  22


(3af ) 4  2  19                    9q  12b  20                       r  3 x  11




Homework                             Name ________________________
Linear graphs, solving equations,
Factoring

Solve.
1. 4x + 2 + 9x = 12 – 4 + x                    2. 3(2d + 7) = 25




Factor.
3. x 2  6 x  9                               4. x2  14 x  49




                                                                         180
Linear Equations, Graphing,                          Name ___________________
Solving Equations, Factoring

Label each equation as LINEAR or NONLINEAR.




As a class, choose one LINEAR equation to graph. ___________________
If we know the graph is a line, we need _______ ordered pairs.
Choose ANY value for x: _______
Find the y coordinate that completes the ordered pair.
(        ,          )




Choose ANY value for y: ________
Find the x coordinate that completes the ordered pair.
(        ,         )




                                                                          181
As a class, choose another LINEAR equation to graph. ___________________
If we know the graph is a line, we need _______ ordered pairs.
Choose ANY value for x: _______
Find the y coordinate that completes the ordered pair.
(        ,          )




Choose ANY value for y: ________
Find the x coordinate that completes the ordered pair.
(        ,         )




For the LINEAR equation 3x + 7y = 21, find two ordered pairs and graph.
Choose any value for x: ________
Find the y coordinate that completes the ordered pair.
(        ,         )




Choose ANY value for y: ________
Find the x coordinate that completes the ordered pair.
(        ,         )




                                                                           182
The equations have been collected from different stories. Solve each equation.
1. 5x + 7 = 23                               2. 4 = 3a – 14




3. 2y + 5 = 19                               4. 6 + 5c = -29




5. 5x – 3 = 13- 3x                           6. -4c – 11 = 4c + 21




7. 9(4b – 1) = 2(9b + 3)                     8. 3(6+5y) = 2(-5 + 4y)




9. 3(d – 8) – 5 = 9(d + 2) + 1               10. 2(a – 8) + 7 = 5(a + 2) – 3a – 19




                                                                                     183
Factor each trinomial using an array.




                                        184
Homework: Linear Equations and graphs,               Name ________________________
Solving equations, Factoring

1. For the LINEAR equation 2x + 5y = 20, find two ordered pairs and graph.
Choose any value for x: ________
Find the y coordinate that completes the ordered pair.
(        ,         )




Choose ANY value for y: ________
Find the x coordinate that completes the ordered pair.
(        ,         )




Solve each equation.
2. 7x + 18 = 22                                      3. 8x + 2 + 4x = 27




4. 3(2x + 8) -5 = 14 + 2x                            5. 5(x – 7) = 9x + -3




Factor the trinomial using an array.
6. x  13x  42                                      7. x  5 x  84
    2                                                     2




                                                                               185
Introduction to Radicals

Vocabulary:          a is called a radical. The number a is called the radicand.


Evaluate the radicals

1.       36                                 2.   81                       3.       49 



4. Graph the radicals on a number line.

     8                       48                          24                      3


            1           2         3          4       5        6         7             8     9




Factor then split the factors into equal groups (if possible).

5. 100                                       6. 18                          7. 64




8. 24                                        9. 36                          10. 42




Why is          4  9 the same as     36 ?


What does          a = c mean?                                    What does      b  d mean?


Use c’s and d’s to show why           a  b always equals     ab ?


                                                                                                186
Simplify.

11.         2 5                         12.        3 7           13. 2 16




14.         5 5                         15.    27 3              16.   5 15




Put the numbers from problems 11-16 into two groups.




How are the numbers different?


A radical is simplified if the radicand has no square factors.

 15 is simplified, 12 is not simplified. Why?




     26 is simplified,   75 is not simplified. Why?




Simplify the radicals.

1.      8                               2.    27                       3.     32




                                                                                   187
Like Radicals                                       Not Like Radicals
                                2                                               1
     2    5 2     -7 2 .3 2       2                  5     3    -2 2 -2 7         11
                                5                                               7


Numbers are like radicals if:_________________________________________________


Just like       5x + 3x = 8x,    5 3  7 3  12 3

Simplify by combining like radicals.

1. 3 5  4 5                            2. 4 3  3                      3.    2 34 2




4.       8 3 2                         5.   12  27                    6. 2 18  3 8




Solve the equations.

1. 3x  7  11                  2. 5x  2  3x  7  x          3. 7  x  3  5  4  2 x




Factor

1. 4 x2  12 x                          2. 15ax 12x                    3. 14a 2b  28ab2


Factor using arrays
1. a 2  12a  32               2. c 2  10c  16               3. x 2  3x  54




                                                                                              188
Find the intercepts of the linear equation then sketch the graph

1. 3x + 2y = 12                2. x – 7y = 14                 3. 3x + 5y = 10




Find any two points that make the linear equation true then sketch the graph

1. 3x – y = 4                  2. y = 3 – x                   3. 3y = x




Solve each equation for b.

1. b + 3 = x                   2. 8b = v                      3. b – 1 = g




     b                                                                 b5
4.     y                      5. 7b + r = a                  6.           t
     5                                                                  2




                 3a b                                    4x y   2x 
                    3      3                                  2    3        2
Simplify 7.                                          8.


                                                                                189
Homework: Introduction to Radicals                           Name___________________

Simplify

1.   50                              2.   20                      3.   48



Add like radicals

4. 3 7  5 7                                5. 2 3  5 3  3 5



Simplify

6.   84 2                                  7.   18  7 3




Find two points that make the equation true then graph

8. 3x  5 y  15                            9. x  2 y  8




Solve for g
10. g  5  h                               11. 4 g  y  15




                                                                                 190
Linear equations, graphing, solving                Name ___________________
Perimeter and Area

From a story about tollbooths, we get the equation T = 250 + 50C.
      What is the intercept? ___________
      What is the slope? __________
      Is the data increasing or decreasing?
      Collect some data
              C       250+50(C)             T             (C,T)
              0




      Graph your data.




From a story about pizza, we get the equation 6w + 5h = 30.
      Is it easy to see the intercept?             The slope?
      Collect some data.
      W               h                    (w,h)




      Graph your data.




                                                                        191
Solving Equations.
7 x  8  100                              4 x  3  2 x  9  8x  5  1




4(2 x  1)  15                            2(9w  3)  10  3( w  1)




Which equations are linear?
Write Linear or Non-linear next to each.
G  3  4b                                 3x  7 f  14


7 x2  y  4                               y  12 xy  8



8  15d  y                                7 x  .3 y  12



Simplify the radicals.
 50                             80                            242




Use a calculator to approximate each radical above.




                                                                            192
Like Radicals.
Circle the like radicals in each expression and add when possible.
4 2 3 7 5 7 8 3




9 3  48  3




Factoring.
Use an array to factor each expression.
h2  5h  6                                            x2  18x  81




3dg  6 g 2                                            r 3 q  14rq




Simplify.
4w(3w)(2w)                             10 x 2 (3x) 2                   (132 xyz )0




                                                                                     193
Perimeter:

        How do we calculate perimeter?

Find the perimeter of each figure.

             8.3cm     6.3cm                            2w      14

    11.2cm                   5.5cm              4x +7
                                                                11+4w

           6.3cm     7.4cm                               9x


Perimeter=                                         Perimeter=




Area:

     When we calculate the area of a rectangle, we _______________
_______________________________________________________.

Find the area of each rectangle.


             5.8in                               d+7

 1.4in                                   3d




             16cm                                 g+3

  16cm                                    g+3




                                                                        194
Finding the dimensions of squares.
For each square, the area is given. Find the dimensions.



               144                                    484


      Area = 144                             Area =
       144  122  12
      Each side length = 12                  Each side length =




               289                                    361


      Area =                                 Area =

      Each side length =                     Each side length =




               81                                     12


      Area =                                 Area =



      Each side length =                     Each side length =




                                                                  195
Homework: Solving equations,                 Name _________________
Simplifying radicals

Solve.
1. 8x + 10 = 18                       2. 5x + 3 -2x = 14 + 1 + x




3. 4(2x + 7) = 19                     4. -3(x + 9) + 1 = 5(4x – 8)




Simplify.
5. 60                                 6.   125




Given the area of the square, find the dimensions.

                         Area =

     576
                         Side length =




                                                                     196
Pythagorean Theorem, Like Radicals,                Name ______________
Solving Equations

Draw square areas to help find the missing dimensions of the right triangles.




          9                                                      7


      c           12                                                     24
                                                             c




          8                                                          6

                  11                                          c           5
        c




                                                                 10
          15

                  a                                          14           b
      25




                                                                          197
Radicals

Simplify. Then use a calculator to get an approximation.
1. 121                          2. 36                            3.   40




4.   28                         5.   64                          6.   45




Like radicals have ______________________________________.

      Ex.

Simplify. Do not use a calculator approximation.

7. 3 2  5  7 2                            8. 8 5  2 11  11




9. 5 3  12                                 10.    20  45




Distribute (Multiply)
11. 5(3x + 7)                               12. 4(2x – 3)




                                                                           198
Solve each equation.
13. 14 = 8x + 9           14. 22 = 7 + 3x




15. x + 7x + 8 = 4x + 2   16. 10x – 5 – 3x = 12x + 3 + 2




17. 4(x + 3) = 25         18. 2(6x + 13) = 5(x – 1)




Clearing Fractions.

              3x                4 3x
        Ex.      5       Ex.       2
              7                 5 5




      5x      2                 x4
19.      1              20.       2
       3      3                  6




                                                       199
Radicals, Pythagorean Theorem,                  Name ___________________
Slope-Intercept Form

Simplify. Then use a calculator to approximate.
1. 24                           2. 75                          3.    45




4. 2 7  63                                     5.   20  72




Find the missing side of each right triangle.
6.                                      7.

                                                                10
              15

                     a                                         14         b
           25




8. Find the perimeter of a triangle whose side lengths are 3x, x + 8, x – 1.




9. Find the perimeter of a square whose side length is 2a+b.




                                                                              200
10. The area of a hexagon can be calculated using the formula
            Area = .5ap               Find the area if a = 3.7 and p=27.4.




11. Find the value of 9gk + d2 if g=4, k=3.8, and d=5.




Slope-Intercept Form
      Ex.                                     *Make sure one variable is alone



For each, find the intercept and slope. Then create a graph.
12. y = 16 + 3x                             13. y = 22 – 4x
      Intercept _______                              Intercept _______
      Slope ______                                   Slope _______




14. y = -(3/4)x + 5                           15. y = 4x + -3
      Intercept _______                              Intercept _______
      Slope ______                                   Slope _______




                                                                             201
                        Given the following graphs, find the intercept and the slope. Write an
                        equation.
                                      8
                                                                Intercept:
                                                                Increasing or Decreasing?
                                      6

                                                                How much?
                                                                Slope:
                                      4



                                      2
                                                                Equation:
                                                                       y=
-15         -10           -5                      5       10        15        20


                                      -2



                                      -4



                                      -6



                                      -8



                                     -10

                                            8
                                                                    Intercept:
                                            6                       Increasing or Decreasing?
                                            4
                                                                    How much?
                                                                    Slope:
                                                                    Equation:
                                            2




      -15         -10          -5                     5        10        15y=      20


                                            -2



                                            -4



                                            -6



                                            -8



                                           -10
                                             8
                                                                    Intercept:
                                                                    Increasing or Decreasing?
                                                                    How much?
                                             6



                                             4
                                                                    Slope:
                                             2                      Equation:
                                                                           y=
      -15         -10           -5                    5        10        15        20


                                             -2



                                             -4



                                             -6



                                             -8



                                           -10




                                                                                                 202
Factor
16. w2  6w  8                       17. b2  15b  56




18. 9ab  12bc                        19. 2d  18d 2




Simplify.
20. 4x(3x + 7g)                       21. 7f(2 – 3r)




22. (x + 8)(x +4)                     23. (k + 3)(k – 2)




Put each equation in slope-intercept form. Find the intercept and slope.
24. y + 3 = 5x                         25. y + 2x = 4




26. 8y + x = 24                       27. 3x + 2y = 12



                                                                           203
Homework: CRT B concepts
#1-5,7,10
                                                             Name___________________

Estimate the square roots. Check your estimate with a calculator.

1.   18                               2.   98                       3.   51

Guess__________                       Guess__________               Guess__________

Check__________                       Check__________               Check__________

Classify each number as rational or irrational. Explain

4. 4.0087 Rational Irrational Why?______________________________________

5. 3 49     Rational Irrational Why?______________________________________

6. .3232… Rational Irrational Why?______________________________________

7.   14    Rational Irrational Why?______________________________________


Simplify

8. 3 50  6 2                                        9.   54  24




                                                                x
10. Find the perimeter of a triangle whose sides measure: 3x,     , and x  3 if x = 6.
                                                                3

Write the equation in slope-intercept form by solving for y then identify the slope and the
y-intercept.
11. 3x + 2y = 5                       12. x – 4y = 6                13. 2 x  y  4




Slope_______                          Slope_______                  Slope_______

y-intercept________                   y-intercept_________          y-intercept_______


                                                                                          204
Slope, Intercepts                                     Name __________________
Solving Equations, Linear and Quadratic

Mikayla sells Girl Scout cookies for a fundraiser. Her grandma gave her $25
to begin. Each additional box that she sells will raise $3.50.
          How much money did Mikayla have to start? _________
          What is the intercept? _______
          Is the data increasing or decreasing?          How much? ______
          What is the slope? _______
          Write a formula to describe how much money Mikayla has:

          How much money did Mikayla have after selling one box? ________
          How much money did Mikayla have after selling two boxes? ________
          How much money did Mikayla have after selling ten boxes? ________
          Graph your data.




From another story, we produced the following graph:
                  15




                  10

           $ in
           Pat's
                   5
           account

    -10                      10
                                  Weeks   20         30        40         50            60



                   -5
                                                     What is the intercept?______
                                                     Is the data increasing or
                  -10                                decreasing?
                                                     How much? ______
                  -15                                What is the slope? _______
                                               Write an equation describing the data.




                                                                                        205
I-tunes collected some data from a new release. This data is shown in the table
below.
                                   0      .99
                                   1      1.98
                                   2      2.97
                                   3      3.96
                                   4      4.95
                                   5      5.94

       What is the intercept? _________
       Is the data increasing or decreasing?     How much? __________
       What is the slope? _________
       Write an equation describing the data.



Avalanche Control suggests staying off terrain with a slope greater than 1. Would
the following pitch be considered safe?
                                                Find the slope to explain your
                                                answer.




Is this slope safe?                             Find the slope to explain your
                                                answer.




                                                                                  206
Solve each equation.
1. 5x + 9 = 32                         2. 4(x + 1) = 8.3




3. 5 + 3x + 1 = 2x + 4x + 10           4. 12(x + 1) = 3(2x – 1)




         x     1                            x 1     5
5.         3                         6.        4
         5     5                              2      2




Is the data closer to linear?                 Is the data close to linear?
Is it increasing or decreasing?               Is it increasing or decreasing?
Correlation:                                  Correlation:
     6


                                                6



     4

                                                4




     2
                                                2




                       5          10                     15       5      20       10   25   15




                                                -2
 -2




                                                -4

 -4



                                                -6


 -6
                                                                                207
                                                -8
Which equations are linear?

G  3  4b                                 3x  7 f  14


7 x2  y  4                                   y  12 xy  8



8  15d  y                                7 x  .3 y  12




Factor
x 2  7 x  12                                        a 2  11a  30




Solving quadratics by factoring and using the zero product property.

x 2  7 x  12  0




a 2  11a  30




Find the missing side of the right triangle.
                                                      24


                                                               7




                                                                       208
Homework: Slope, solving equations              Name __________________

1. Given the following data:
           0 17
           1 15
           2 13
           3 11
           4 9
What is the intercept? _________
Is the data increasing or decreasing?
                            22            How much?
What is the slope?          20



                            18



2. Given the following data:16



                            14



                            12



                            10



                             8



                             6



                             4



                             2




                       -5            5   10    15     20          25       30   35     40   45   50


                            -2


What is the intercept? _________
                            -4


Is the data increasing or decreasing?     How much?
What is the slope?

3. Solve each equation.
       5x + -3 + 8x = 2 – 9 + 11x                      -2(x +7) = 3x -8




4. Factor and solve each equation.
       x 2  7 x  10  0                                  a 2  9a  20




                                                                                     209
Zero Product, Inequalities,
Slope, and Intercept                                       Name________________________

Write each equation in slope-intercept form (Solve for y) to identify the slope and
intercept.

1. y  5 x  4                                             2. 7 x  y  2



Slope________                                              Slope_________

Intercept_____                                             Intercept______

3. 3x  2 y  6                                            4. x  2 y  4




Slope________                                              Slope_________

Intercept_____                                             Intercept______




Write a generalization with the variables a and b to show:
 “If we get a zero by multiplying two numbers then one of the numbers has
to be zero.”



If p  6  0 then what is the value of p ?

If q  r  0, and r  7, then what is the value of q?

If  x  2 8  0, then  x  2  must equal what value?
What is the value of x ?

If  x  3 y  0, and y  5 , then what is the value of x ?




                                                                                      210
Solve using the zero product rule.

1.    a  3b  1  0                             2.    x  9 x  4  0



Solve the quadratic equations by factoring then using the zero product rule.

3. x2  3x  40  0                                  4. x 2  7 x  12  0




5. x 2  8x  12  0                                 6. x 2  12 x  36  0




7. x2  2 x  35  0                                 8. x2  x  56  0




                                                                                 211
Station 1: Simplifying radicals                               Expert_________________

Simplify

1.    75          initials______                    2.       32  18           initials______




3.   12  4 27            initials______            4. 3 2  20  7 5 initials______




Station 2: Pythagorean Theorem                                Expert_________________
Solve for the unknown side of the right triangle.


                 ?                                                     13
6                                                        5


                 8                                                     ?




initials______                                                initials______



                                                                                            212
Station 3: Solving Equations                              Expert__________________

Solve.

1. 3x  4  11          initials______           2. 4  2 x  7           initials______




                                                      x     3
3. 3 x  4  2  x initials______             4.     3               initials______
                                                      2     2




Station 4: Slopes and Intercepts from graphs              Expert________________

Find the slope and intercept.




1.                                         2.
                       initials______                             initials______
Slope________                              Slope_______

Intercept______                            Intercept______

Equation_____________________              Equation___________________________


                                                                                       213
Homework: Stations A                                            Name__________________

Solve using the zero product rule.
1.  a  8 b  4  0                                2.    x  2 x  5  0



Solve the quadratic equations by factoring then using the zero product rule.

3. x2  2 x  63  0                                   4. x 2  9 x  18  0




5. x 2  10 x  16  0                                 6. x 2  10 x  25  0




7. Simplify 3 5  45  7 2                             8. Solve 3x  2( x  1)  7




9. Find the slope and y-intercept of 2 x  5 y  200




                                                                                     214
Systems,                               Name ___________________
Solving Equations

System of Equations:
When we solve a system, we are looking for ________________________
________________________________________________________.

For example,




Practice.
Find the solution to the system.
1.                                     2.




                                                  Solution:
               Solution:

3.                                     4.




               Solution:
                                                  Solution:




                                                                 215
Systems with algebra
      These two equations describe two sets of linear data. We want to
know which data point they have in common.
      3x + y = 12 and y = 7




                                              Solution:

Try this system:
      x + 4y = 8 and y = 2




                                              Solution:

Practice:
1. Solve the system: 2x + y = 11 and y = 4




                                              Solution:

2.Solve the system: x + 3y = 9 and x =2




                                              Solution:

3. Solve the system: 2x + y = 10 and y = 3x




                                              Solution:



                                                                         216
Station 1: Linear Equations                        Expert__________________

   1. Explain which of the following equations are NOT linear. Initial ____
      a. A  3  7B      b. y  4 x 2  7    c. 3xw  9  w

                                                        5
         d. y  4 x  7   e. .4 x  .9 y  13      f.      g  12
                                                        f

   2. Which equations are in slope-intercept form? Initial ____
      a. 8w+3z=10       b. y = 13+2x        c. x = -15 -2y

         d. y = 11x +5    e. 8x + y = 2            f. y = 15 – x/2



   3. Put each equation in slope-intercept form. Initial ____
      a. 3x + y =12             b. 4y = 12 + 24           c. 2x + 3y = 6




         Initial _____            Initial ______                     Initial ______


Station 2: Solving Quadratic Equations             Expert __________________

Factor and solve using the zero product property.
1. x 2  12 x  35  0                2. w2  12w  36  0




Initial _____                             Initial _____
3. d  12d  11  0
     2
                                          4. x 2  3x  40  0




Initial _____                             Initial _____



                                                                                      217
Station 3: Pythagorean Theorem                 Expert ________________

Find the missing side of the right triangle.




                                                               ?
     6           10                             12



                 ?                                             7




          Initial _____                               Initial _____




Station 4: Simplifying Radicals                Expert ________________

1.       40      Initial _____                 2.    45  80          Initial ____




3.       18      Initial _____                 4.    12  75  5      Initial ____




Use a calculator to approximate:
5. 17        Initial _____                     6.    22        Initial ____




                                                                               218
Homework: Station B                         Name _________________

1. Solve the system.




2. Solve the system.
            3x + y = 8 and y = 1x




3. Put in slope-intercept form. Then find the intercept and slope.
      7x + y = 9
                          Intercept ______
                          Slope _____




4. Factor. Then use the zero product property to solve.
      x 2  9 x  18  0




                                                                     219

				
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