jim rahn james.rahn@verizon.net

Jim Rahnwww.jamesrahn.com

James.rahn@verizon.net

Tell what would happen to the balanced scale below if each of the actions listed are taken.

Remember, the scale is reset after each action.

1. Three red squares are added to the right side.

2. One yellow and one red square are added to the right side.

3. One yellow squares is removed from the left side and one yellow square is removed from the right side.

4. Two red squares are added to the right side of the scale and two yellow squares are added to the left side.

5. Multiply the number of items on each side by two.

6. Two red squares and two yellow squares are added to the left side of the scale.

7. A red square is added to the left and a yellow square is removed from the right.

balanced

unbalanced

unbalanced

balanced

balanced

balanced

balanced

8. The number of items on each side is cut in half.9. Two yellow squares are removed from the left and

two yellow squares are added to the right side of the scale.

10. Two red squares are removed from the left and two red squares are removed from the right side of the scale.

11. One zero pair is added to the left side and one zero pair is added to the right side of the scale.

12. Two yellow squares are added to the right side and two red squares are added to the left side of the scale.

13. One red square is added to each side of the scale.14. Double the number of squares on the left and divide

the number of squares on the right by two.

balanced

unbalanced

unbalanced

balanced

balanced

balanced

unbalanced

1. One yellow and one red square are added to the right side.

2. One yellow squares is removed from the left side and one yellow square is removed from the right side.

3. Multiply the number of items on each side by two.

4. One red square is added to each side of the scale.

5. Two red squares and two yellow squares are added to the left side of the scale

5. A red square is added to the left and a yellow square is removed from the right.

6. The number of items on each side is cut in half.

7. Two red squares are removed from the left and two red squares are removed from the right side of the scale.

8. One zero pair is added to the left side and one zero pair is added to the right side of the scale.

9. One red square is added to each side of the scale.

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Use Algebra Tiles and the Balance Scale template to determine the answer to each of the problems below. Give the value for x and explain how you determined the answer.

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Use Algebra Tiles and the Balance Scale template to determine the answer to each of the problems below. Give the value for x and explain how you determined the answer.

Use Algebra Tiles and the Balance Scale template to determine the answer to each of the problems below. Give the value for x and explain how you determined the answer.

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Use Algebra Tiles and the Balance Scale template to determine the answer to each of the problems below. Give the value for x and explain how you determined the answer.

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Use Algebra Tiles and the Balance Scale template to determine the answer to each of the problems below. Give the value for x and explain how you determined the answer.

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Use Algebra Tiles and the Balance Scale template to determine the answer to each of the problems below. Give the value for x and explain how you determined the answer.

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Use Algebra Tiles and the Balance Scale template to determine the answer to each of the problems below. Give the value for x and explain how you determined the answer.

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Use Algebra Tiles and the Balance Scale template to determine the answer to each of the problems below. Give the value for x and explain how you determined the answer.

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Use the algebra models to represent x + 3 = 4

on the equation balance. Find the value of x by doing the same thing to both sides of the balance until you have the x (green rectangle) by itself. Does this value make sense?

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Use the algebra models to represent 2x+4 = 8

on the equation balance. Find the value of x by doing the same thing to both sides of the balance until you have the x (green rectangle) by itself. Does this value make sense?

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Use the algebra models to represent

on the equation balance. Find the value of x by doing the same thing to both sides of the balance until you have the x (green rectangle) by itself. Does this value make sense?

2 4 2 3x x

=

Use the algebra models to represent

on the equation balance. Find the value of x by doing the same thing to both sides of the balance until you have the x (green rectangle) by itself. Does this value make sense?

3 1 2 6x x

=

Use the algebra models to represent

on the equation balance. Find the value of x by doing the same thing to both sides of the balance until you have the x (green rectangle) by itself. Does this value make sense?

2 3 1 5x x

=

Use the algebra models to represent

on the equation balance. Find the value of x by doing the same thing to both sides of the balance until you have the x (green rectangle) by itself. Does this value make sense?

2 4 3 2 3 3x x x

=

Use the algebra models to represent

on the equation balance. Find the value of x by doing the same thing to both sides of the balance until you have the x (green rectangle) by itself. Does this value make sense?

2 1 3 2x x

=

Use the following pieces: one x2 piece, four x pieces, and three unit pieces.

Form a rectangle from these eight pieces. What polynomial is represented by the rectangle? Describe the polynomial represented by these eight

pieces. Describe the dimensions of your rectangle.

Use the following pieces: one x2 piece, four x pieces, and four unit pieces.

Form a rectangle from these nine pieces. What polynomial is represented by the rectangle? Describe the polynomial represented by these nine

pieces. Describe the dimensions of your rectangle.

Use the following pieces: one x2 piece, five x pieces, and four unit pieces.

Form a rectangle from these ten pieces. What polynomial is represented by the rectangle? Describe the polynomial represented by these ten

pieces. Describe the dimensions of your rectangle.

Use the following pieces: one x2 piece, one x pieces, and two red unit pieces.

Form a rectangle from these four pieces. What polynomial is represented by the rectangle? Describe the polynomial represented by these four

pieces. Describe the dimensions of your rectangle.

Use the following pieces: one x2 piece, three x pieces, and four red unit pieces.

Form a rectangle from these eight pieces. What polynomial is represented by the rectangle? Describe the polynomial represented by these eight

pieces. Describe the dimensions of your rectangle.

Form a rectangle that represents x2 -3x-4. What are the factors of this polynomial?

Form a rectangle that represents x2 -x-6. What are the factors of this polynomial?

Numbers like 64 are called perfect squares because they are the squares of integers, in this case 8 or -8. The trinomial x2+6x+9 is (x+3)2. So it is also called a perfect square.

Which of these trinomials are perfect squares?

Use the Rectangle Diagrams Template to show the area model for each of the trinomials that are perfect squares

Study each of the following trinomials. Draw the model for each trinomial. If a number is missing, use your model to figure out the missing number.

2 2 1x x

2 ___ 36x x

2 14 ____x x

What is the connection between the middle term and the last term that makes it possible to form a square?

Trinomials can also be placed in an equation such as .

Use the Completing the Square Template to show this equivalence.

Represent the trinomial in the square on the left and the constant in the square on the right.

If the two squares have the same area then their dimensions must be equal also.

Find the dimensions of each square.

Don’t forget both the negative and positive values.

2 6 9 36x x

2 6 9x x 36

2 2( 3) 6x

2 23 6

3 6

3 6

6 3

3 9

x

x

x

x

x or

Solve each of the following equations using the Completing the Square Template.

2 8 16 25x x

2 14 49 64x x

Study the square on the left in each of the problems. Describe any patterns you notice in the area of each section and the coefficients from the trinomial.

Suppose the trinomial was

.

Label this square with the appropriate area and dimensions.

2 16 64x x x2

8x

8x 64

x + 8

x +

8

22 16 64 8x x x

Suppose 2 4 1 22x x

Begin to set up the two squares, but notice on the left that the area of the small square is too small. •How much should it be? •How much can we add to both sides of this equation? •Since this is an equation, add the amount of area you need to both sides. Now complete the problem.•After adding 3 square units to both sides, the shape on the left will be a square and you will have:

x2 2x

2x 1+

22

3 3

2

2

2

4 1 3 22 3

4 4 25

2 25

2 5

2 5

x x

x x

x

x

x

Solve each of these problems using the Completing the Square Template:

2 6 1 28x x

2 18 60 43x x

2 4 1 0x x

2 33 2 8

4x x

Suppose the trinomial was

Set up the Completing the Square Template and solve this x.

2 0x bx c x2

2b

x

2b

x

-c2

2b

2

2b

2 2

2 2b b

x c

2 2 42 4 4b b c

x

2 42 4b b c

x

2 42 4b b c

x

The Empire State Building has 102 floors and is 1250 feet high. How high are you when you are reach the 80th floor?

Explain your reasoning.

Floor Number

Basement (0)

1 2 3 4 … 10 … … 25

Height (ft)

-6 7 20 33 … 215 …

• A 25-story building has floors at the described heights. What recursive sequence can describe the heights?

• Find the height of the 4th and 10th floors?

• Which floor is 215 feet above ground?• How high is the 25th floor? • Explain your reasoning

• How can we model this on the graphing calculator?

Floor Number

Basement (0)

1 2 3 4 … 10 … … 25

Height (ft)

-6 7 20 33 … 215 …

Method 1 Method 2

• Make figure 1-3• Determine how many toothpicks it takes to

make each figure.• Determine the number of toothpicks on each

perimeter.• Make figures 4-6.• Collect a table of data about each picture.• What is a rule for finding the number of

toothpicks in each figure.• What is the rule for find the perimeter of each

figure.• Make figure 10.

• Write a recursive sequence.• Confirm your table values by writing a

recursive procedure on the calculator.• Use the graphing calculator to predict the

number of toothpicks in the 20th and 30th Figures.

• Can you determine the figure that 100 toothpicks in the perimeter? Explain your reasoning.

• Can you determine the figure that has 100 total toothpicks in the figure? Explain your reasoning.

These are examples of real life problems where we are talking about a constant rate of change. Explain why this term makes sense.

Floor Number

Basement (0)

1 2 3 4 … 10 … … 25

Height (ft)

-6 7 20 33 … 215 …

A sports car leaves High Point and heads south for Cape May.

At the same time an overloaded van leaves Atlantic City and a pickup truck leaves Cape May and head north toward High Point.

The sports car is traveling at 72 mph, the pickup truck is traveling 66 mph, and the overloaded van is traveling at 48 mph.

When and where will they pass each other?

Change the rates of change to miles per minute so we can study the problem in smaller increments.

Make a table to record the distance from Cape May for

each vehicle every minute.

After completing the first couple of rows, change theintervals to 10 minute intervals until you have covered 4hours.

Write a recursive sequence would model each car’s distance from Cape May?◦ Sports Car◦ Overloaded Van◦ Utility Truck

If x and y were the variables used in this problem, what would x represent and what would y represent? Label these on your chart.

Create a graph of the values in your chart.

What do you notice about the points that represent each vehicle?

What is the starting position of each vehicle? ◦ Where is this on the table? ◦ Where is it on the graph?

How does the vehicle’s speed effect the graph?

How can you tell which line represents the van?

Where are the vehicles when the van meets the first vehicle heading north?

How can you tell if the pickup truck or sports car is traveling faster from the graph?

Which vehicle arrives at its destination first? How much later do the other vehicles reach

their destination? Are you making any assumptions about each

of the vehicles as you answer the questions?

Write an equation that represents each vehicles distance from Cape May by referring to your recursive sequence.◦ Sports Car◦ Overloaded Van◦ Utility Truck

Enter these equations in your graphing calculator. How do these graphs compare to your paper graph?

By doing a lesson like this, are you engaging the students in a lesson that is ◦ Beginning to get them to think about the steepness of

a line?◦ Beginning to get them to think how the rate of change

affects the steepness?◦ Beginning to show students that some lines increase

and others decrease?◦ Showing them the difference between different types

of slope?◦ Helping them write equations for real situations?◦ Beginning to see how the constant term describes

where a vehicle begins the trip?

Using the graphing calculator and the home screen.

Clear the home screen. Pick a number. Enter it in the calculator and

press ENTER. Add 3 to your number. Press ENTER. Multiply the result by 2 and press ENTER. Add 10 to the answer. Press ENTER. Divide your answer by 4. Press ENTER. Subtract 4 from your answer. Press ENTER. Write your answer on your communicator.

Clear the home screen. Pick a different number. Enter it in the

calculator and press ENTER. Add 3 to your number. Press ENTER. Multiply the result by 2 and press ENTER. Add 10 to the answer. Press ENTER. Divide your answer by 4. Press ENTER. Subtract 4 from your answer. Press ENTER. Write your answer on your communicator.

Clear the home screen. Pick a different number. Enter it in the

calculator and press ENTER. Add 3 to your number. Press ENTER. Multiply the result by 2 and press ENTER. Add 10 to the answer. Press ENTER. Divide your answer by 4. Press ENTER. Subtract 4 from your answer. Press ENTER. Write your answer on your communicator.

Clear the screen. Enter the three numbers in a brace: {a,b,c} Press ENTER. Add 3 to your numbers. Press ENTER. Multiply the result by 2 and press ENTER. Add 10 to the answer. Press ENTER. Divide your answer by 4. Press ENTER. Subtract 4 from your answer. Press ENTER. Write your answer on your communicator.

Description Expression

Starting Value

Add 3 to the starting number

Multiply by 2

Add 10

Divide by 4 Make 4 groups

Subtract 4

Keep 1 group

Make 2 groups

Pick a number. Enter it in the calculator and press ENTER.

Add 2 to your number. Press ENTER. Divide the result by 4 and press ENTER. Add 4 to the answer. Press ENTER. Multiply your answer by 2. Press ENTER. Subtract 9 from your answer. Press ENTER. Multiply your answer by 4. Write your answer on your communicator.

Clear the screen. Enter the three numbers in a brace: {a,b,c} Press ENTER. Add 2 to your number. Press ENTER. Divide the result by 4 and press ENTER. Add 4 to the answer. Press ENTER. Multiply your answer by 2. Press ENTER. Subtract 9 from your answer. Press ENTER. Multiply your answer by 4. Write your answer on your communicator.

Description Expression

Starting Value x

Description Expression

Starting Value x

Add 2 x+2

Description Expression

Starting Value x

Add 2 x+2

Divide by 4 24

x

Description Expression

Starting Value x

Add 2 x+2

Divide by 4

Add 4

24

x

24

4x

Description Expression

Starting Value x

Add 2 x+2

Divide by 4

Add 4

24

x

24

4x

Description Expression

Starting Value x

Add 2 x+2

Divide by 4

Add 4

Multiply by 2

24

x

24

4x

22 4

4x

Description Expression

Starting Value x

Add 2 x+2

Divide by 4

Add 4

Multiply by 2

Subtract 9

24

x

24

4x

22 4

4x

2

2 4 94

x

Description Expression

Starting Value x

Add 2 x+2

Divide by 4

Add 4

Multiply by 2

Subtract 9

Multiply by 4

24

x

24

4x

22 4

4x

2

2 4 94

x

24 2 4 9

4x

Pick any number. Multiply the starting number by 2 Then add 6, Divide this result by 2, Then subtract your original number. What did you get?

Try some other numbers. Remember the number trick says:◦ Pick a number◦ Multiply the starting number by 2◦ Then add 6, ◦ Divide this result by 2, ◦ Then subtract your original number.

Description Expression

Starting Value x

Description Expression

Starting Value x

Multiply the starting number by 2 2x

Description Expression

Starting Value X

Multiply the starting number by 2 2x

Add 6 2x+6

Description Expression

Starting Value x

Multiply the starting number by 2 2x

Add 6 2x+6

Divide by 2 2 6or 3

2x

x

Description Expression

Starting Value x

Multiply the starting number by 2 2x

Add 6 2x+6

Divide by 2

Subtract the starting number or 3

2 6or 3

2x

x

2 62

xx

Using the graphing calculator and the home screen.

Try this trick. Pick a number, write it down, and enter that

number. Add 9 Multiply the result by 3. Subtract 6. Divide this result by 3. Subtract your original number. Compare your answers

Description Expression

Pick a Number x

Description Expression

Pick a Number x

Add 9 x+9

Description Expression

Pick a Number x

Add 9 x+9

Multiply by 3 3(x+9)

Description Expression

Pick a Number x

Add 9 x+9

Multiply by 3 3(x+9)

Subtract 6 3(x+9) - 6

Description Expression

Pick a Number x

Add 9 x+9

Multiply by 3 3(x+9)

Subtract 6 3(x+9) - 6

Divide this answer by 3 3( 9) 63

x

Description Expression

Pick a Number x

Add 9 x+9

Multiply by 3 3(x+9)

Subtract 6 3(x+9) - 6

Divide this answer by 3

Subtract your original number

3( 9) 63

x

3( 9) 63

xx

Compare your answers

On the homescreen try entering several numbers in a matrix:◦ {-3, 15, 20}◦ Add 9◦ Multiply by 3◦ Subtract 6◦ Divide this answer by 3◦ Subtract the original number.

What do you notice? Why is this?

3( 9) 63

xx

Pick a number Add 1 Multiply by 2 Subtract 4 Divide by 2 Compare your answers to others near you. Can you determine why you ended up with

your number?

Description Expression

Starting Value x

Description Expression

Starting Value x

Add 1 x+1

Description Expression

Starting Value x

Add 1 x+1

Multiply by 2 2(x+1)

Description Expression

Starting Value x

Add 1 x+1

Multiply by 2 2(x+1)

Subtract 4 2(x+1)-4

Description Expression

Starting Value x

Add 1 x+1

Multiply by 2 2(x+1)

Subtract 4 2(x+1)-4

Divide by 2 2( 1) 42

x

74 5 13

4x

x

Choose a secret number. Now choose four more nonzero

number and in any order, ◦ add one of them, ◦ multiply by another, ◦ subtract another, ◦ and divide by the final

number. Complete the Description

column, the sequence of steps, and the expression for each step.

Test your expression by picking a value for x. You can use the graphing calculator to find your answer.

Placing your answer in the bottom right hand box.

Exchange your chart with another student.

When you receive the chart your task is to decide how you can determine what number the previous student placed in their expression.

Use the Undo Chart complete the following number trick. Complete the first three columns only.

Pick a number Divide the number by 4 Add 7 Multiply the result by 2 Subtract 8

Description Sequence Expression Undo Result

Pick a number

X X

Description Sequence Expression Undo Result

Pick a number

X X

Divide the number by 4

/4

4x

Description Sequence Expression Undo Result

Pick a number

X X

Divide the number by 4

/4

Add 7 +7

4x

74x

Description Sequence Expression Undo Result

Pick a number

X X

Divide the number by 4

/4

Add 7 +7

Multiply by 2 x2

4x

74x

2 74x

Description Sequence Expression Undo Result

Pick a number

X X

Divide the number by 4

/4

Add 7 +7

Multiply by 2 x2

Subtract 8 -8

4x

74x

2 74x

2 7 84x

Suppose the answer to the problem was 28. What number did we begin with? Place 28 in the bottom box in the last column. Complete the Result Column.

Pick a number Divide the number by 4 Add 7 Multiply the result by 2 Subtract 8

Description Sequence Expression Undo Result

Pick a number

X X

Description Sequence Expression Undo

Pick a number

X X

Divide the number by 4

/4

Add 7 +7

Multiply by 2 x2

Subtract 8 -8

4x

74x

2 74x

2 7 84x

28

36

18

11

44

+8

/2

-7

x4

Set up the problem:

3 2( 4)6 11

5x

Description Sequence Expression Undo Result

Pick a number

X X

How do you look at this equation differently?

Do you now think about how it is put together?

Do you now know how to take it apart? Does solving an equation have new

meaning? Can you solve this by using a few columns

in the undo chart?

3 2 47 14

4

x

Description Undo Results

Pick x

3 2 47 14

4

x

•(-4)

+2

•3

÷4

-7 +7

•4

÷3-2

÷(-4)

14

21

84

28

26

-6.5

Check -6.5, by going back to the graphing calculator and performing the 5 steps in order to see the result is 14.

Jim Rahnwww.jamesrahn.com

James.rahn@verizon.net

Can you start at the bottom of the chart and undo each operation? Write the description in the Undo column.

Pick a number Divide the number by 4 Add 7 Multiply the result by 2 Subtract 8

Description Sequence Expression Undo

Pick a number

X X

Description Sequence Expression Undo

Pick a number

X X

Divide the number by 4

/4

Add 7 +7

Multiply by 2 x2

Subtract 8 -8

4x

74x

2 74x

2 7 84x

+8

/2

-7

x4