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**Working with Algebra Tiles Part II**

Jim Rahn

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Balancing Equations

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**Actions with Balanced Scales**

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.

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unbalanced Three red squares are added to the right side. One yellow and one red square are added to the right side. One yellow squares is removed from the left side and one yellow square is removed from the right side. Two red squares are added to the right side of the scale and two yellow squares are added to the left side. Multiply the number of items on each side by two. Two red squares and two yellow squares are added to the left side of the scale. A red square is added to the left and a yellow square is removed from the right. balanced balanced unbalanced balanced balanced balanced

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

Two yellow squares are removed from the left and two yellow squares are added to the right side of the scale. Two red squares are removed from the left and two red squares are removed from the right side of the scale. One zero pair is added to the left side and one zero pair is added to the right side of the scale. Two yellow squares are added to the right side and two red squares are added to the left side of the scale. One red square is added to each side of the scale. Double the number of squares on the left and divide the number of squares on the right by two. balanced unbalanced balanced balanced unbalanced balanced unbalanced

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**Actions that Balance a Scale**

One yellow and one red square are added to the right side. One yellow squares is removed from the left side and one yellow square is removed from the right side. Multiply the number of items on each side by two. One red square is added to each side of the scale. Two red squares and two yellow squares are added to the left side of the scale

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**Actions that Balance a Scale**

A red square is added to the left and a yellow square is removed from the right. The number of items on each side is cut in half. Two red squares are removed from the left and two red squares are removed from the right side of the scale. One zero pair is added to the left side and one zero pair is added to the right side of the scale.

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**Actions that Balance a Scale**

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. =

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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 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**

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**

Use the algebra models to represent x+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**

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? =

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**Use the algebra models to represent. on the equation balance**

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? =

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**Use the algebra models to represent. on the equation balance**

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? =

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**Use the algebra models to represent. on the equation balance**

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? =

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**Use the algebra models to represent. on the equation balance**

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? =

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**Factoring with Algebra Tiles**

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**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.

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**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.

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**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.

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**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.

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**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.

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**Form a rectangle that represents x2 -3x-4.**

What are the factors of this polynomial?

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**Form a rectangle that represents x2 -x-6.**

What are the factors of this polynomial?

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Perfect Squares

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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?

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

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**Study each of the following trinomials**

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

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**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. 36

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

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.

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**Suppose the trinomial was**

. Label this square with the appropriate area and dimensions. x x2 8x x 8x 64

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Suppose 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 22 2x 1+ 3 3

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

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**Suppose the trinomial was**

Set up the Completing the Square Template and solve this x. x2 -c

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**Working with Recursive Sequences**

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**Studying Recursive Patterns with the Graphing Calculator**

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**Explain your reasoning.**

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. (1250/102)x80=980

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

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 Floor Number Basement (0) 1 2 3 4 … 10 25 Height (ft) -6 7 20 33 215

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**How can we model this on the graphing calculator?**

Method 1 Method 2 Floor Number Basement (0) 1 2 3 4 … 10 25 Height (ft) -6 7 20 33 215

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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.

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**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.

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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

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**Using Constant Rate of Change in Distance-Time Problems**

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?

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**Make a table to record the distance from Cape May for**

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 the intervals to 10 minute intervals until you have covered 4 hours.

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**Create a graph of the values in your chart.**

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.

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**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?

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**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?

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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?

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**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?

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**Learning to Write Phrases and Equations**

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**What do you get if you start with ___?**

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.

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**Try it Again! 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.

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**Try it One More Time! 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.

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**Have you noticed anything about your answer?**

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**Let’s pick three numbers this time!**

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.

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**Place the Description-Expression Template in your Communicator**

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**Organize a Chart Description Expression Starting Value**

Add 3 to the starting number Multiply by 2 Make 2 groups Add 10 Make 4 groups Divide by 4 Keep 1 group Subtract 4

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**Try Another Number Trick**

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.

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**Let’s pick three numbers this time!**

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.

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**Learning to Write an Expression**

Description Expression Starting Value x

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Description Expression Starting Value x Add 2 x+2

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Description Expression Starting Value x Add 2 x+2 Divide by 4

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Description Expression Starting Value x Add 2 x+2 Divide by 4 Add 4

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Description Expression Starting Value x Add 2 x+2 Divide by 4 Add 4

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Description Expression Starting Value x Add 2 x+2 Divide by 4 Add 4 Multiply by 2

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Description Expression Starting Value x Add 2 x+2 Divide by 4 Add 4 Multiply by 2 Subtract 9

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Description Expression Starting Value x Add 2 x+2 Divide by 4 Add 4 Multiply by 2 Subtract 9 Multiply by 4

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**Try this Number Trick 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?

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Try this Number Trick 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.

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Organize a Chart Description Expression Starting Value x

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**Multiply the starting number by 2**

Organize a Chart Description Expression Starting Value x Multiply the starting number by 2 2x

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**Multiply the starting number by 2**

Organize a Chart Description Expression Starting Value X Multiply the starting number by 2 2x Add 6 2x+6

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**Multiply the starting number by 2**

Organize a Chart Description Expression Starting Value x Multiply the starting number by 2 2x Add 6 2x+6 Divide by 2

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**Organize a Chart 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

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**Pick a number Multiply the result by 3. Subtract 6.**

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

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**Complete a chart to Determine an Algebraic Expression**

Description Expression Pick a Number x

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**Complete a chart to Determine an Algebraic Expression**

Description Expression Pick a Number x Add 9 x+9

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**Complete a chart to Determine an Algebraic Expression**

Description Expression Pick a Number x Add 9 x+9 Multiply by 3 3(x+9)

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**Complete a chart to Determine an Algebraic Expression**

Description Expression Pick a Number x Add 9 x+9 Multiply by 3 3(x+9) Subtract 6 3(x+9) - 6

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**Complete a chart to Determine an Algebraic Expression**

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

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**Complete a chart to Determine an Algebraic Expression**

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 Compare your answers

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**Evaluating an expression on the calculator**

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?

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**Try Another One 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?

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Organize a Chart Description Expression Starting Value x

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Organize a Chart Description Expression Starting Value x Add 1 x+1

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**Organize a Chart Description Expression Starting Value x Add 1 x+1**

Multiply by 2 2(x+1)

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**Organize a Chart Description Expression Starting Value x Add 1 x+1**

Multiply by 2 2(x+1) Subtract 4 2(x+1)-4

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**Organize a Chart Description Expression Starting Value x Add 1 x+1**

Multiply by 2 2(x+1) Subtract 4 2(x+1)-4 Divide by 2

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**You Write the Number Trick for this Expression**

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**Try this one! The expression above describes a new number trick.**

Given The expression above describes a new number trick. Write in words, the description for each step. Record it in the chart under description and sequence. Show the expression for each step. Using your calculator, test the number trick to be sure you get the same result no matter what number you choose. Which operations that undo previous operations make this number trick work?

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**Now You Try It 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.

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**Now You Try It Placing your answer in the bottom right hand box.**

Exchange your chart with another student.

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

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**Solving Equations is Just Undoing Operations**

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

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**Solve Equations is Just Undoing Operations**

Description Sequence Expression Undo Result Pick a number X Divide the number by 4 /4

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**Solve Equations is Just Undoing Operations**

Description Sequence Expression Undo Result Pick a number X Divide the number by 4 /4 Add 7 +7

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**Solve Equations is Just Undoing Operations**

Description Sequence Expression Undo Result Pick a number X Divide the number by 4 /4 Add 7 +7 Multiply by 2 x2

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**Solve Equations is Just Undoing Operations**

Description Sequence Expression Undo Result Pick a number X Divide the number by 4 /4 Add 7 +7 Multiply by 2 x2 Subtract 8 -8

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**Solve Equations is Just Undoing Operations**

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

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**Solve Equations is Just Undoing Operations**

Description Sequence Expression Undo Pick a number X Divide the number by 4 /4 Add 7 +7 Multiply by 2 x2 Subtract 8 -8 44 x4 11 -7 18 /2 36 28 +8

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**Try another One! Set up the problem: Description Sequence Expression**

Undo Result Pick a number X

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**Conclusions 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?

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Description Undo Results Pick x -6.5 ÷(-4) •(-4) 26 +2 -2 28 ÷3 •3 84 ÷4 •4 21 -7 +7 14 Check -6.5, by going back to the graphing calculator and performing the 5 steps in order to see the result is 14.

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**Working with Algebra Tiles Part II**

Jim Rahn

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**Solve Equations is Just Undoing Operations**

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

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**Solve Equations is Just Undoing Operations**

Description Sequence Expression Undo Pick a number X Divide the number by 4 /4 Add 7 +7 Multiply by 2 x2 Subtract 8 -8 x4 -7 /2 +8

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Alge-Tiles For all Alge-Tile work it is essential to remember that RED means minus And Any other colour means plus.

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