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Jim Rahn

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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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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 balanced

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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 balanced unbalanced

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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

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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.

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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.

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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. 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? =

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Use the following pieces: one x 2 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 x 2 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 x 2 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 x 2 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 x 2 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 x 2 -3x-4. What are the factors of this polynomial?

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Form a rectangle that represents x 2 -x-6. What are the factors of this polynomial?

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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 x 2 +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. 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. Dont 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. x2x2 8x 64 x + 8

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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: x2x2 2x

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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. x2x2 -c

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

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Floor Number Basement (0) 1234…10……25 Height (ft) …215… A 25-story building has floors at the described heights. What recursive sequence can describe the heights? Find the height of the 4 th and 10 th floors? Which floor is 215 feet above ground? How high is the 25 th floor? Explain your reasoning

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How can we model this on the graphing calculator? Floor Number Basement (0) 1234…10……25 Height (ft) …215… Method 1Method 2

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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 20 th and 30 th 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) 1234…10……25 Height (ft) …215…

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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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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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Write a recursive sequence would model each cars 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 vehicles 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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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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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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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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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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DescriptionExpression 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

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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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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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DescriptionExpression Starting Valuex

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DescriptionExpression Starting Valuex Add 2x+2

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DescriptionExpression Starting Valuex Add 2x+2 Divide by 4

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DescriptionExpression Starting Valuex Add 2x+2 Divide by 4 Add 4

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DescriptionExpression Starting Valuex Add 2x+2 Divide by 4 Add 4

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DescriptionExpression Starting Valuex Add 2x+2 Divide by 4 Add 4 Multiply by 2

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DescriptionExpression Starting Valuex Add 2x+2 Divide by 4 Add 4 Multiply by 2 Subtract 9

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DescriptionExpression Starting Valuex Add 2x+2 Divide by 4 Add 4 Multiply by 2 Subtract 9 Multiply by 4

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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 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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DescriptionExpression Starting Valuex

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DescriptionExpression Starting Valuex Multiply the starting number by 22x

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DescriptionExpression Starting ValueX Multiply the starting number by 22x Add 62x+6

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DescriptionExpression Starting Valuex Multiply the starting number by 22x Add 62x+6 Divide by 2

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DescriptionExpression Starting Valuex Multiply the starting number by 22x Add 62x+6 Divide by 2 Subtract the starting number or 3

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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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DescriptionExpression Pick a Numberx

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DescriptionExpression Pick a Numberx Add 9x+9

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DescriptionExpression Pick a Numberx Add 9x+9 Multiply by 33(x+9)

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DescriptionExpression Pick a Numberx Add 9x+9 Multiply by 33(x+9) Subtract 63(x+9) - 6

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DescriptionExpression Pick a Numberx Add 9x+9 Multiply by 33(x+9) Subtract 63(x+9) - 6 Divide this answer by 3

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DescriptionExpression Pick a Numberx Add 9x+9 Multiply by 33(x+9) Subtract 63(x+9) - 6 Divide this answer by 3 Subtract your original number Compare your answers

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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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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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DescriptionExpression Starting Valuex

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DescriptionExpression Starting Valuex Add 1x+1

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DescriptionExpression Starting Valuex Add 1x+1 Multiply by 22(x+1)

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DescriptionExpression Starting Valuex Add 1x+1 Multiply by 22(x+1) Subtract 42(x+1)-4

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DescriptionExpression Starting Valuex Add 1x+1 Multiply by 22(x+1) Subtract 42(x+1)-4 Divide by 2

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You Write the Number Trick for this Expression

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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? Given

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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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Placing your answer in the bottom right hand box. Exchange your chart with another student.

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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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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 DescriptionSequenceExpressionUndoResult Pick a number XX

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DescriptionSequenceExpression UndoResult Pick a number XX Divide the number by 4 /4

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DescriptionSequenceExpression UndoResult Pick a number XX Divide the number by 4 /4 Add 7+7

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DescriptionSequenceExpression UndoResult Pick a number XX Divide the number by 4 /4 Add 7+7 Multiply by 2x2

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DescriptionSequenceExpression UndoResult Pick a number XX Divide the number by 4 /4 Add 7+7 Multiply by 2x2 Subtract 8-8

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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 DescriptionSequenceExpressionUndoResult Pick a number XX

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DescriptionSequenceExpressionUndo Pick a number XX Divide the number by 4 /4 Add 7+7 Multiply by 2x2 Subtract /2 -7 x4

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Set up the problem: DescriptionSequenceExpressionUndoResult Pick a number XX

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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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DescriptionUndoResults Pick x (-4) +2 3 ÷ ÷3 -2 ÷(-4) 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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Jim Rahn

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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 DescriptionSequenceExpressionUndo Pick a number XX

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DescriptionSequenceExpressionUndo Pick a number XX Divide the number by 4 /4 Add 7+7 Multiply by 2x2 Subtract /2 -7 x4

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