domino effect barbie bungee - mangham math

Created by Lance Mangham, 6th grade math, Carroll ISD

Linear Equations Activities

1. Domino Effect

2. Barbie Bungee

3. Slippery Slopes

4. Feed Me!


Activity A-1: Domino Effect Name:

Adapted from Mathalicious.com

DOMINO EFFECTDOMINO EFFECTDOMINO EFFECTDOMINO EFFECT

Domino’s pizza is delicious. The company’s success is proof that people enjoy their pizzas. The company is also tech savvy as you can order online and they even have a pizza tracker so you can keep tabs on your delivery. Domino’s does not tell you how much the component pieces cost; they only tell you an item’s final price after you build it. In this lesson we will use linear equations to find the base price and cost per additional topping. PART 1 Below are prices for a medium 2-topping pizza and a medium 4-topping pizza from Domino’s in Washington, DC.

Medium (12”) Hand Tossed Pizza

Pepperoni, Green Peppers Price: $13.97

Medium (12”) Hand Tossed Pizza

Pepperoni, Italian Sausage, Onions, Green Peppers Price: $16.95

1. Plot the two pizzas on the included graph, draw a line, and then use the information to answer the following questions.

2. Based on the information above how much do you think Domino’s is charging for each topping?

3. A medium, 3-topping pizza costs $15.46. What would it means if Dominos charged the prices above, but charged $17 for a 3-topping pizza.

4. For the 2-topping pizza, how much in total are you spending on toppings?

5. For the 4-topping pizza, how much in total are you spending on toppings?

6. If you wanted to order a medium cheese pizza (no toppings), how much would you expect to spend? Explain.

7. Now write an equation for the price of a medium pizza and explain what the equation means. Use C to represent the total cost and t to represent the number of toppings.

8. Rewrite your equation above using the commutative property of addition.

9. Which variable is the independent variable and which is the dependent variable?



10. Does a pizza with 12 toppings cost twice as much as a pizza with 6 toppings? Why or why not?

11. Is your graph linear? How do you know?

12. Does your graph represent a proportional or non-proportional relationship? How do you know?

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PART 2 Below are prices for two small pizzas and two large pizzas from Domino’s in Washington, DC.

Small (10”) Hand Tossed Pizza

Pepperoni Price: $9.99

Large (14”) Hand Tossed Pizza

Pepperoni, Italian Sausage, Onions, Green Peppers Price: $19.75

Small (10”) Hand Tossed Pizza

Pepperoni, Mushrooms, Green Peppers Price: $11.99

Large (14”) Hand Tossed Pizza

Pepperoni Price: $14.68

1. Plot the small and large pizzas on the same graph as the medium pizza, draw the lines, and then use the information to answer the following questions.

2. How much does Domino’s appear to be charging for each topping on a small pizza?

3. How much does Domino’s appear to be charging for each topping on a large pizza?

4. How much would a small pizza with no toppings cost?

5. How much would a large pizza with no toppings cost?

6. Now write an equation for the price of a small pizza and explain what the equation means. Use C to represent the total cost and t to represent the number of toppings.

7. Now write an equation for the price of a large pizza and explain what the equation means. Use C to represent the total cost and t to represent the number of toppings.

8. Which graph – small, medium, or large – is the steepest, and why do you think this is?



9. Which graph has the lowest starting value and is this what you’d expect? Explain.

10. Look at the graphs of how much Domino’s really charges for pizza in Washington, DC. How is the actual situation different than what you expected?

The y-intercept is the point where the line of the graph crosses the y-axis.

Use your original information to answer the questions below.

11. What is the y-intercept of the small pizza graph? What does this number represent?

12. What is the y-intercept of the medium pizza graph? What does this number represent?

13. What is the y-intercept of the large pizza graph? What does this number represent?

The slope of a straight line shows how steep the straight line is. The slope is calculated by the following:

2 1

2 1

slope rate of changey yy

x x x

−∆= = =

∆ −

14. What is the slope of the small pizza graph? What does the number represent?

15. What is the slope of the medium pizza graph? What does the number represent?

16. What is the slope of the large pizza graph? What does the number represent?

17.

Often times linear equations are written in the format: y mx b= +

y is the dependent variable.

x is the independent variable. m is the slope of the line.

b is the y-intercept of the line.

Rewrite your three pizza equations in the form y mx b= + .



18.

According to the graph everything looks normal until the fourth topping. At this point Domino’s appears to stop charging for additional toppings. What is the slope of the line between topping 4 and topping 10?

19. What might be some reasons (not necessarily mathematical) why it appears that Domino’s does not charge for toppings 5-10?

20. What might be some reasons (not necessarily mathematical) why it appears that Domino’s does not allow you to order more than 10 toppings?

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EXTENSION Pizza chains like Domino’s charge different prices for toppings depending on the size of the pizza: a topping for a small pizza costs less than a topping for a medium pizza, which costs less than a topping for a large pizza. This makes sense: the larger the pizza, the more topping you get. How do the differences in topping prices compare to the differences in topping amounts? If pepperoni costs $1 on a small pizza and $1.49 on a medium are you really getting $0.49 more worth of pepperoni?

The large pizza has a 14 inch diameter including a 1 inch crust all around. The medium pizza has a 12 inch diameter including a 1 inch crust all around. The small pizza has a 10 inch diameter including a 1 inch crust all around.

1. Using the formula for the area of a circle, 2

A rπ= , determine the interior area (area available for toppings) of a large pizza. Round to the nearest whole number.

2. Determine the interior area of a medium pizza. Round to the nearest whole number.

3. Determine the interior area of a small pizza. Round to the nearest whole number.

4. Complete the table below.

5. Based on your results do you think the cost of an additional topping is fair compared to the amount of topping you get in each case? Why or why not?

Percent Increases

Percent more area Percent higher price

Small to Medium

Medium to Large

Small to Large


Activity B-1: Barbie Bungee Name:

Adapted from www.mathlab.com and Laying The Foundation.

Barbie Bungee The consideration of cord length is very important in a bungee jump—too short, and the jumper doesn’t get much of a thrill; too long, and splat! In this lesson, you model a bungee jump using a Barbie® doll and rubber bands. You will find the relationship between the number of rubber bands and the distance that Barbie falls. You will be creating a bungee jump for a Barbie® doll. Your objective is to give Barbie the greatest thrill while still ensuring that she is safe (and alive after the activity is complete). This means that she should come as close as possible to the ground without hitting the floor. You will conduct an experiment, collect data, and then use the data to predict the maximum number of rubber bands that should be used to give Barbie a safe jump from the top of the high school bleachers (15 feet). Procedure: Before you conduct the experiment, formulate a conjecture: I believe that _____ is the maximum number of rubber bands that will allow Barbie to safely jump

from the top of the high school bleachers.

Now, conduct the experiment to test your conjecture. Complete each step below. Materials needed: Barbie, rubber bands, yardsticks, recording page Assign roles to each group member:

1 Rubber band expert 1 Barbie dropper 2 Data recorders Create a double-loop to wrap around Barbie’s feet. A double-loop is made by securing one rubber band to another with a slip knot, as shown below left.

Wrap the open end of the double-loop tightly around Barbie’s feet, as shown above right.



Attach a second rubber band to the first one, again using a slip knot, as shown below.

We will measure Barbie’s jump using a yardstick. Hold the yardstick near a wall with zero being the highest number. One person should be holding the yardstick. With two rubber bands now attached, hold the end of the rubber bands at the jump line (top of yardstick) with one hand, and drop Barbie from the line with the other hand. Have the other two group member determine the lowest point (not the final position) that Barbie reaches on this jump. Perform the experiment 3 times with the 2 rubber bands attached and record the values in the data table. Then find the average distance of the three jumps to ensure accuracy. Accuracy is important—Barbie’s life could depend on it! Repeatedly attach two additional rubber bands for each new jump, measure the jump distance, and record the results in the data table. When you’ve completed the data table, you may begin to answer the questions.

x Jump #1 Jump #2 Jump #3 y

Number of Rubber Bands

Distance Bungeed (in inches)



Average of 3 Jumps (in inches)

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Rewrite your x and y-values below from the data. Then plot the points (x, y) on the graph below.

Points to plot (x, y)

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Plot the points on the axes below.

Sketch a line that best fits your data.

To do this, pick TWO points that best represent the data and draw a straight line through these two

points. Extend the line to the edges of the graph. Some points should be above the line and some

points should be below the line.



1. Do you think the length of the cord and the size of the person matters when bungee jumping? Why?

2. Would it be smart to lie about your height or weight? Why or Why not?

3. Why is an accurate estimate of height and weight important to conduct a safe bungee jump?

4.

Choose two points that best represent the data that you have collected. Use the SAME TWO points you used to draw your line on the graph! The two points are ( ___, ___ ) and ( ___, ___ ).

5.

Using these two points, find the slope of the line through them. Simplify your answer to the nearest tenth.

Recall: 2 1

2 1

change in rise

change in run

y y ym

x x x

−= = =

−

6. What does the slope represent in this context?

7. Based on your line of best fit what is the y-intercept of the line?

8. What does the y-intercept represent in this context?

9.

Write an equation in the form of bmxy += for your Barbie bungee jumps.

______________ += xy

(slope) (y-intercept)

10. Number of rubber bands and the height of Barbie’s fall: Which is the independent variable and which is the dependent variable in this experiment?

11. Based on your data, what would you predict is the maximum number of rubber bands so that Barbie could still safely jump from the top of the high school bleachers (15 feet)?

12. How confident are you in your prediction above? Justify your answer. Be sure to consider your methods of collecting, recording, and plotting data.

13. What is the minimum height from which Barbie should jump if 25 rubber bands are used? (You should use the line of best fit to determine an answer.)



14. If some weight were added to Barbie, would you need to use more or fewer rubber bands to achieve the same results? Explain your reasoning.

15. Use your equation to determine what distance Barbie would fall using 50 rubber bands. (Take a moment and think -- will 50 be an x-value or a y-value?)

16. Use your equation to determine how many rubber bands you would need to use to have Barbie plunge to a distance of 150 inches. (Once again think – is 150 going to be an x-value in your equation or a y-value?)

17. Barbie wants to bungee off the Eiffel Tower. It is 986 feet tall. How many rubber bands will you need so Barbie just brushes her hair (hopefully not her head) on the ground?

We are now at the moment of truth. Which team has best predicted the maximum number of rubber bands required to give Barbie the bungee jump of a lifetime?

Let the Official Barbie Bungee jumps begin!!!

Our team’s minimum Barbie distance from the ground: _____________________

18. How many rubber bands were actually needed for Barbie to safely jump from the top of the playground equipment?

19. How do your predictions above compare to the conjecture you made before doing the experiment? What prior knowledge did you have (or not have) that helped (or hindered) your ability to make a good conjecture?


Activity C-1: Slippery Slope Name:

Adapted from schools.nyc.gov

Slippery Slope Every day on her way home from school Jennifer uses $1 to buy a bag of chips. The chips cost $0.75 and she saves the remaining quarter in her change jar.

1. Jennifer buys 5 bags of chips per week. At this rate, how long will it take her to save $10.00?

2. Write an equation to show how much money, y, Jennifer will save after any number of weeks, x.

3.

Create a graph below to represent this relationship.

4. Mark is also trying to save money. He has already saved $3.00 and he puts away $1.00 per week in his piggy bank. Write an equation to show how much money, y, Mark will save after any number of weeks, x.

5. Will Jennifer or Mark save $20.00 first? Prove it.

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6.

Create a graph below to represent the equation for Mark.

7. Explain why Mark’s line does not start at the origin and Jennifer’s line does start at the origin.

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Brandon and Madison use different triangles to determine the slope of the line shown below.

Brandon started at (0,-1) and drew a right triangle going up 2 units and right 3 units. Madison started at (-3,-3) and drew a right triangle going up 6 units and right 9 units.

8. Draw and label both triangles on the graph above.

9. Describe how the 2 triangles are related.

10. Find the slope of the line using Brandon’s triangle. Show your work.

11. Find the slope of the line using Madison’s triangle. Show your work.

12. Justify how the triangles relate to the slope of the line. Why can you find the slope using any two points on the line?

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Entrance to an amusement park is $20 and games are $3 each.

13. Write an equation that gives you the total cost of admission and games.

14. Create a graph to represent the situation.

15. What is the slope of the line?

16. How does the slope compare to the unit price of a game (cost per game)? Prove your answer.

Ben is spending his summer driving across the country. He is going to spend the first day just driving west. The table below shows the distance traveled as a function of time.

Distance (miles)

55 110 165 220 275 330

Time (hours)

1 2 3 4 5 6

17. Graph this relationship on a coordinate plane.

18. Find the slope of the line.

19. What information about Ben’s speed can you obtain from the slope?

20. Write an equation to represent the distance traveled after driving a given number of hours. Label each variable.

21.

A second car is traveling at 50 miles per hour. Suppose you make a graph showing the distance traveled by the second car as a function of time. How would the graph for the second car compare with the graph of Ben’s car? Explain your thinking.

22. Would it make sense for a line representing this situation to start anywhere but the origin? Use mathematical reasoning in your response.



23. Find the slope of the line below using the points B and C.

24. Find the slope of the line below using points A and D.

25. Use the properties of similar triangles to explain why any two points on a line can be used to calculate slope.

A (0, -3)

B (2, 0)

C (4, 3)

D (6, 6)

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Activity D-1: Feed Me! Name:

Adapted from Ms. Minervini’s Math Wiki https://wrhsalgebra2.wikispaces.com/file/view/Linear+Equations+Project+Based+Learning.pdf

Feed Me!

Write and evaluate two-variable expressions corresponding to the following situations. Make sure to define your variables. I was soooooo hungry last night at the restaurant! I ordered several Happy Meals and a ton of sodas. Each Happy Meal costs $3.20 and each soda costs $1.75.

1. Write an expression or function to show how much money I spent.

2. Evaluate that expression for: 4 meals and 3 sodas

3. Evaluate that expression for: 10 meals and 2 sodas

4. Evaluate that expression for: 3 meals and no sodas

I got my sister really mad at me by teasing her! If I teased her and started running, and she started running at the same time (chasing me), then the distance between us would be my distance minus her distance (I am a faster runner!). Remember that distance equals rate times time.

5. If we’ve been running for 15 seconds write an expression for the distance between us given my unknown rate and her unknown rate.

6. How much distance is between us if: I run 13 m/sec and she runs 12 m/sec






Mr. Mangham’s class wanted to start a business selling bagels in the entryway of the school in the mornings. They plan to charge $0.50 per bagel.

10. Write an equation that represents the revenue curve (compares the amount of bagels sold with the total money received).

11. Graph this equation on the coordinate plane below by first completing the table of values and then plotting them on the graph.

Bagels (x) Equation Revenue (y)

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Now the class wants to use a graph to predict their total cost associated with selling bagels. Suppose that the fixed cost of the business is $10 and that the variable cost per bagel is $0.25 per bagel.

12. Write an equation that represents the cost curve (compares the amount of bagels sold with the total cost).

13. Graph this equation on the same coordinate plane as your first graph by first completing the table of values and then plotting them on the graph.

Bagels (x) Equation Cost (y)

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14. What is happening when your line that represents revenue goes above your line for cost of business?

15. Find the point of intersection on the graph.

16. What does the point of intersection represent?



By this point you should have an idea about the elements of a linear equation, but let’s review and make sure you could graph a linear equation in any form.

y mx b= +

The y-intercept is where you place your first point. Remember: If there is no y-intercept, it is simply 0. From this point your slope tells you where to move from there.

risem

run=

Examples:

1

2m = − Move down 1 from the y-intercept and 2 units to the right.

5

3m = Move up 5 from the y-intercept and 3 to the right.

3m = − Move down 3 from the y-intercept and 1 to the right.

Graph 2

43

y x= − + First identify the slope

(_______) and y-intercept (_______). Plot the y-intercept on the y-axis. This is the point (0,4). Next, move according to the slope. Move DOWN 2 from the intercept and then RIGHT 3. Connect your points and make your line.

slope y-intercept

The number of spaces you move UP or DOWN

The number of spaces you move RIGHT



What if my equation isn’t in y mx b= + form? You’ll have to make it look like something you know

how to graph, so solve for y first. Example: Graph the equation 5 4 16x y− =

5 4 16

5 5

4 5 16

4 4

5 4

4

x y

x x

y x

y x

− =

− −

− − +=

− −

= −

SNACK SHACK

You decide to try your luck as an entrepreneur and open up the Snack Shack, which caters to the residents of Southlake who love grilled cheese. Through research you find that the cheapest prices for your supplies are as follows:

Bread: $2.88/loaf, which makes 12 sandwiches Cheese: $4.50/bag, which makes 10 sandwiches Butter: $3.60/lb, which makes 40 sandwiches Paper towels: $1.20/roll, used for 120 sandwiches Using the information above find the unit cost of each item (cost per one sandwich).

Bread Butter

Cheese Paper towels

1. How much does it cost to make 1 sandwich?

2. Your total cost is $20 for startup supplies plus the cost per sandwich. Write the equation you can use to calculate your cost below.

3. You charge $1.59 per sandwich. Write the equation for total revenue below.

4. Graph both equations on the next page.

5. How many sandwiches do you have to sell to break even?

6. At what point will your profit exceed costs by $100?



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domino effect barbie bungee - mangham math

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