All On The Line - UH



Addressing Algebraic and Geometric Concepts using Manipulatives and the Graphing Calculator

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University of Houston – Central Campus

EatMath Workshop

November 7, 2009

Warm-Up Activity

Puzzle 1: How many squares are in the figure below?

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Puzzle 2: How many squares are in the figure below?

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All On The Line

|Problem 1 – Intersecting Lines |

|Graph y = 2x + 1 and y = x – 2. Press Y= and enter the first equation as Y1 and the |[pic] |

|second as Y2. |[pic] |

|Press zoom and select ZStandard. | |

|1. What is the slope of each line? | |

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|Use the Intersect command to find the intersection point of the two lines. Press 2nd | |

|Calc and select intersect. | |

|Now, use the arrow keys to move the cursor to | |

|the first line, Y1, and press enter | |

|the second line, Y2, and press enter | |

|the guess of the intersection point and press enter | |

|2. What is the intersection point? What does this point represent for the equations? |

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|3. Graph y = [pic]x + 1 and y = –x + 6. What is the slope of each line? |

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|4. What is the point of intersection of the two lines in Question 3? How can you verify that this point on the graph is actually the |

|intersection point? |

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|5. Two lines with different slopes will intersect in one point. |

|( Always ( Sometimes ( Never |

|Problem 2 – Parallel Lines |

|6. What is the slope of y = [pic]x + 4 and y = [pic]x – 1? |

|7. Graph the lines in Question 6. Graph two more sets of equations that have the same slope. Record the equations below. |

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|Parallel lines intersect. ( True ( False |

|8. Solve x + 3y = 1 and x – 3y = 1 for y. What is the slope of each line? |

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|9. The lines x + 3y = 1 and x – 3y = 1 are parallel. |

|( True ( False |

|10. What kind of lines are y = 4 and x = 4? |

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|11. What is another way to describe or name the pair of lines in the question above? |

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|Problem 3 – Same Lines, Infinite solutions |

|12. Solve x + y = 3 and 2x + 2y = 6 for y. What is the slope of each line? |

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|13. How are the two lines related to each other? |

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|14. Consider 3x + y = 3 and 6x + 2y = 6. Are the two lines the same or different? How do you know? |

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|15. The slope of both lines in Question 14 is –3. |

|( True ( False |

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|Problem 4 |

|16. The sum of two numbers is 12. The difference between the numbers is 4. Write two equations that represent this problem. |

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|17. Enter three pairs of numbers that add up to 12 in L1 and L2. What are your three pairs? |

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|18. Graph your equations from Question 16, with a Stat Plot of L1 and L2, and determine the solution. Use the Intersect command if needed. |

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|Problem 5 |

|19. Freddie (x) is 3 years older than Joann (y) and their ages total 19. Write two equations that represent the problem. |

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|20. Enter three pairs of ages into L1 and L2. What are your three pairs? |

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|21. Graph your equations from Question 19, with a Stat Plot of L1 and L2, and determine the solution Use the Intersect command if needed. |

|Dinner Party | |

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|In this activity, you will explore: |[pic] |

|Modeling linear data | |

|Writing the equation of a line | |

|Finding the slope and y-intercept of a line | |

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|Your boss asks you to plan a retirement party for one of your co-workers. You are | |

|comparing the cost of a dinner party at different restaurants. Each restaurant | |

|charges a flat room fee (no matter how many guests attend) and a per plate fee. | |

|Problem 1 – Linear Bistro |

|The chart shows the costs of a party at Linear Bistro for different numbers of |Guests |

|guests. What is the room fee at Linear Bistro? What is the per plate fee? |Cost |

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| |5 |

| |260 |

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| |10 |

| |370 |

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| |20 |

| |590 |

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| |25 |

| |700 |

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| |50 |

| |1250 |

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|To find out, enter the data into L1 and L2 in your calculator. |[pic] |

|Adjust your window settings as shown. |[pic] |

|Make a scatter plot of the data from the chart. |[pic] |

|1. Which are the x-values, the number of guests or the costs? Which are the | |

|y-values? | |

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|2. Look at the points. What do you notice? | |

|The points form a line, so this data is linear. Use the LinReg command to draw a |[pic] |

|line through these points. (Adding Y1 to the end of the command stores the equation | |

|of the line in Y1.) | |

|View the graph. |[pic] |

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|3. Describe the line. Do all the points lie on the line? | |

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|The y-intercept of the line represents the flat room fee at Linear Bistro. | |

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|4. What does the slope of the line represent? | |

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|5. Use the scale on the y-axis to estimate the y-intercept. | |

|We can use a function table to find the exact value of the y-intercept. |[pic] |

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|6. How can you find the y-intercept in a function table? | |

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|View the function table by pressing 2nd Graph | |

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|7. What is the room fee at Linear Bistro? | |

|You can write a row from the function table as a coordinate pair. For example, the first row can be written as (0, 150). |

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|8. Write another row from the function table as a coordinate pair. |

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|9. Use these two points to find the slope of the line. What is the per plate fee at the Linear Bistro? |

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|10. Press Y= to view the equation of the line. What do you notice about the slope, the y-intercept, and the equation? |

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|Problem 2 – Straight Eight’s Restaurant |

|Straight Eight’s Restaurant charges an $80 room fee and $35 per plate. |

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|11. How much would a dinner party for 10 people cost at Straight Eight’s? |

|Write an equation in the form y = mx + b that models the cost of a dinner party at |[pic] |

|Straight Eight’s for x guests. Enter it as Y1 and view its graph. (Remember to turn | |

|your scatter plot from Problem 1 off!) | |

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|View the function table and use it to check your equation. Is the y-intercept | |

|correct? Does the value at x = 10 match your answer to Question 11? | |

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|Problem 3 – First Degree Café |

|The First Degree Café charges $185 for a party of 5 people. The cost per plate is $21. |

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|12. Write an equation in point-slope form, (y – y1) = m(x – x1), that models the cost of a dinner party at the First Degree Café. |

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|Simplify the equation and graph it as Y1. |

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|13. View the function table. Explain how to use it to check your equation. |

Tri This!

|Problem 1 – Systems of equations |

|Graph the following equations. Draw your graph on the screen at the right. |[pic] |

|y = –2x | |

|y = x + 3 | |

|y = 5 | |

|Find the intersection points of each vertex of the triangle formed. Press 2nd Calc and select intersect to find the intersection points. Make|

|sure to select the correct equations each time using the arrow down and arrow up keys accordingly. |

|Label each equation and each intersection point on the graph above. |

|1. Identify the systems of equations and their solution(s). |

|System 1: [pic] Solution(s): ______ |

|System 2: [pic] Solution(s): ______ |

|System 3: [pic] Solution(s): ______ |

|2. Can the point (2, 5) be a solution to the system [pic]? Explain your reasoning. |

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|3. Where is the point (0, 4) in relation to the triangle? Is this point a solution to any of the three systems? Explain your reasoning. |

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|4. How many solutions does each system listed in Question 1 have? |

|5. Are any of the intersection points solutions to the system of equations [pic]? |

|Problem 2 – System of inequalities |

|Change the \ symbol (to the left of Y1, Y2, and Y3) to shade above or shade below |[pic] |

|(by pressing enter) for each equation until the darkest shaded region forms a | |

|triangle. | |

|This will change the equations to inequalities with ( or ( symbols. | |

|Draw your modified graph of the inequalities on the screen at the right. Label each |[pic] |

|equation and each intersection point. | |

|Use the Home Screen to test each vertex in each inequality. |[pic] |

|The first entry at the right shows storing the x- and y-coordinates of the first | |

|vertex. The second entry tests the point in the inequality. The calculator returns 1| |

|if the inequality is true and 0 if the inequality is false. | |

|6. How many of the vertices of the triangle are solutions to the system? |

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|7. Test points inside the triangle as well. How many solutions are there to the system? |

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|8. If the inequalities of the system were changed to < and >, how many of the vertices would be solutions? |

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|9. What differences in the solutions did you find between systems of linear equations representing a triangle and a system of linear |

|inequalities representing a triangle? |

Transformations With Lists

|Problem 1 – Creating a Scatter Plot |

|Open the list editor by pressing STAT Enter. Enter the x-values into list L1 and the y-values into list L2. |

|x |

|2 |

|8 |

|8 |

|12 |

|8 |

|8 |

|2 |

|2 |

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|y |

|3 |

|3 |

|1 |

|5 |

|9 |

|7 |

|7 |

|3 |

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|Create a connected scatter plot of L1 vs. L2. |[pic] |

|Press ( ( and select Plot1. Change the settings to match those shown at the right. | |

|Press Window and adjust the window settings to those shown at the right. |[pic] |

|Press Graph to view the scatter plot. | |

|Sketch the scatter plot. |[pic] |

|Problem 2 – Reflections and Rotations |

|Go back to the list editor. Enter the formula =–L1 at the top of list L3 to create the |[pic] |

|opposite of each of the x-values in L1. | |

|Then, enter the formula =–L2 at the top of list L4 to create the opposite of each of the | |

|y-values in L2. | |

|Graph the following scatter plots using Plot2, one at a time. For each combination of lists, determine what type of reflection occurred. |

|Press ( to view Plot1 and Plot2 together. |

|A: x ( L3 and y ( L2 |B: x ( L1 and y ( L4 |C: x ( L2 and y ( L1 |

|[pic] |[pic] |[pic] |

|(–x, y) ________________ |(x, –y) ________________ |(y, x) ________________ |

|Use Plot2 to create the following scatter plots. For each combination, determine what type of rotation occurred. |

|D: x ( L4 and y ( L1 |E: x ( L2 and y ( L3 |F: x ( L3 and y ( L4 |

|[pic] |[pic] |[pic] |

|(–y, x) ________________ |(–x, –y) _______________ |(y, –x) ________________ |

|Problem 3 – Translations |

|Press (STAT enter) to go back to the list editor. |[pic] |

|In the formula bar for L3, enter =L1–5 to translate the x-values. In the formula bar for | |

|L4, enter =L2+3 to translate the y-values. | |

|Change Plot2 so that the X-list is L3 and the Y-list is L4. Press Graph to view the |[pic] |

|scatter plots. | |

|Where did the image shift? How many units left/right and how many units up/down? | |

|Translate the scatter plot into Quadrant 3 by editing the formula bars for L3 and L4. |[pic] |

|L3 formula: ____________________ | |

|L4 formula: ____________________ | |

|Explain how the image shifted. | |

|Problem 4 – Dilations |

|In the list editor, change the formula for L3 to =0.5*L1 and the formula for L4 to |[pic] |

|=0.5*L2. | |

|Press Graph to view the scatter plots. | |

|Explain what happened to the image. | |

|Dilate the scatter plot into Quadrant 3 by editing the formula bars for L3 and L4. |[pic] |

|L3 formula: ____________________ | |

|L4 formula: ____________________ | |

|Explain what happened to the image. | |

Multiple Representations

Problem 1

|Verbal Description |

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|The cell phone company charges $0.25 per minute and a $20 monthly service fee. |

|Equation |

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|Table Graph |

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1. Find the total monthly charge of 240 minutes of cell phone use.

2. What does $0.25 represent?

3. What’s significant about the $20 monthly service fee?

Problem 2

|Verbal Description |

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|A personal trainer at a local gym charges the following fees for a one-hour training session for 1 to 4 people. |

|Equation |

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|Table Graph |

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|# of people Cost ($) |

|1 20 |

|2 24 |

|3 28 |

|4 32 |

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1. Write an equation that best represents the relationship between n, the number of people, and c, the cost?

2. If this pattern continues, what would be the cost for 15 people in the training session?

3. If the cost is $144, how many people can attend the training session?

Pattern Block Activity 1

1. How many  [pic]  are in  [pic]?

2. How many  [pic] are in   [pic]  ?

3. How many  [pic]  are in   [pic]  ?

4. How many  [pic] are in   [pic]  ?

5. How many  [pic] are in   [pic]  ?

6. How many  [pic] are in   [pic]  ?

Pattern Block Activity 2

1. If     [pic]    = 1,       [pic]  = ___ .

2. If    [pic]    = 1,       [pic]  = ___ .

3. If    [pic]    = 1,    [pic] = ___ .

4. If    [pic]    = 1,    [pic] = ___ .

Pattern Block Activity 3

1. If   [pic]  +   [pic]  = 1,   what is   [pic]?

2. If   [pic]  +   [pic]  = 1,  what is   [pic]+ [pic]?

3. If   [pic]  +   [pic]  = 1,   what is  [pic] + [pic]?

4. If   [pic]  +   [pic]  = 1, what is   [pic]?

5. If   [pic]  -   [pic]  = 1,   what is [pic]+ [pic]?

Summarize the Trapezoids Data and Look for Patterns

Find the pattern.

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|Hexagon Number |Process Column |Trapezoids Used |

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|1 | |2 |

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1. How many trapezoids does it take to build the smallest hexagon using the pattern blocks?

2. How many trapezoids does it take to build the next larger hexagon?

3. How many trapezoids does it take to build the third hexagon?

4. How many trapezoids does it take to build the nth hexagon?

Summarize the Rhombus Data and Look for Patterns

Find the pattern.

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|Hexagon Number |Process Column |Rhombi Used |

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|1 | |3 |

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|n | | |

1. How many rhombi does it take to build the smallest hexagon using the pattern blocks?

2. How many rhombi does it take to build the next larger hexagon?

3. How many rhombi does it take to build the third hexagon?

4. How many rhombi does it take to build the nth hexagon?

Summarize the Triangles Data and Look for Patterns

Find the pattern.

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|Hexagon Number |Process Column |Triangles Used |

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|1 | |6 |

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1. How many triangles does it take to build the smallest hexagon using the pattern blocks?

2. How many triangles does it take to build the next larger hexagon?

3. How many triangles does it take to build the third hexagon?

4. How many triangles does it take to build the nth hexagon?

What relations exist between the rules involving trapezoids, rhombi, and triangles? Is there any reasonable explanation for these relations?

TAKS Related Problems

1. A function is described by the equation f(x) = x2 + 5. The replacement set for the independent variable is {1, 5, 7, 12}. Which of the following is contained in the corresponding set for the dependent variable?

A 0

B 6

C 7

D 15

2. Which equation best describes the relationship between x and y in this table?

|x |y |

|−4 |−11 |

|−1 |−2 |

|2 |7 |

|5 |16 |

A y = [pic]x + 1

B y = [pic]x − 1

C y = 3x − 1

D y = 3x + 1

3. Which of the following cannot be described by a linear function?

A The amount spent on n shirts that cost $20 each.

B The number of miles driven for h hours at a constant speed of 60 miles per hour.

C The total amount saved after making an initial deposit of $100 and depositing $30 a month thereafter for n months.

D The area of a rectangular garden that is x feet wide and has a length equal to twice its width.

4. The function f(x) = {(1, 2), (2, 4), (3, 6), (4, 8)} can be represented in several other

ways. Which is not a correct representation of the function f(x)?

A

B

C x is a natural number less than 5 and y is twice x

D y = 2x and the domain is {1, 2, 3, 4}

5. Which of the following equations represents a graph that is parallel to the graph of the equation 8x + 2y = 10 and that has a y-intercept of 8?

A −4x + y = 5

B 4x + y = 8

C 8x – 2y = 10

D 8x + 2y = 4

6. A series of equations are graphed on the same coordinate grid.

Which equation’s graph would be next in the sequence above?

A y = x + 9

B y = 5x + 8

C y = x + 8

D y = 5x + 9

7. Which graph best represents a line parallel to the line with the equation y = 3x + 4?

8. Virginia graphed the system of equations shown below.

Y = X + 6

Y = 2X + 5

She determined that the solution to the system was (1, 7).

If Virginia translated both lines up 2 units, what would be the solution to the new system of equations?

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Resources Used

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“Accelerated Curriculum for Mathematics Grade 8”. Region IV Education Service Center (2007).

Algebra TEXTEAMS

tea.state.tx.us

education.



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