M4Log-LABmm

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Name: Period: Date:

Project: Modeling with M&M’s

DO NOT EAT THE M&M’s UNTIL YOU ARE DONE COLLECTING ALL DATA

Materials:

Paper plate, scoop of m&m’s, 2 rulers (cm), cups of various sizes, graphing calculator

PART 1

1] Make a chain of m&m’s against the centimeter side of a ruler to gather the data in the table below. Round to the nearest half centimeter.

# of m&m’s 0 3 7 12 14 18 23 27 30

length of m&m chain (cm)

2] Graph your data (scatterplot only) with number of m&m’s on the x-axis and length of the m&m chain on the y-axis.

3] Which dimension(s) is/are being measured in terms of the m&m’s? (Recall that distance is length, area is the product of length and width, and volume is the product of length, width, and height.)

4] Is this data linear, quadratic, cubic, or something else? How do you know? (Fill in the blanks)

The data is _________________ because we are measuring ______________ which is ________ - dimensional.

5] Use a graphing calculator to find the equation of the best fitting model for the data. Round to the nearest hundredth.

6] In words, not interval notation, give the domain of the model in the context of the problem. Explain why it is not all real numbers.

7] Based on your model (do not actually measure this), what is the diameter of one m&m? (Fill in the blanks)

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8] Use the model to predict number of m&m’s needed to make a chain that is 60 cm long. Show your work and round to the nearest whole m&m.

9] Use the model to predict the length of a chain that contains 60 m&m’s. Show your work and round your answer to the nearest tenth.

PART 2

10] Place as many m&m’s as you can inside each circle (not just the ring, but the whole circle), making sure that each m&m is completely inside the circle, laying flat, and no m&m’s are overlapping.

Diameter (cm) 0 2 4 6 8 10

m&m’s

11] Graph your data (scatterplot only) with diameter of the circle on the x-axis and number of m&m’s on the y-axis.

d=2 d=4 d=6 d=8

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15] In words, not interval notation, give the domain of the model in the context of this problem. Explain why it is not all real numbers.

16] Use the model to predict number of m&m’s needed to fill a 10 cm diameter circle. Show your work and round to the nearest whole m&m.

17] Explain how your prediction compared to your actual data result for a 10 cm diameter circle. Do you think your model will be accurate for larger diameter circles?

18] Use the model to predict number of m&m’s needed to fill a 15 cm diameter circle. Show your work and round to the nearest whole m&m.

19] Use the model to predict the diameter of a circle that holds 100 m&m’s. Show your work and round to the nearest half centimeter. (Hint: quadratic formula)

PART 3

20] Use the plastic cups provided and place as many m&m’s as you can inside each cup, making sure that all m&m’s are completely inside the cup and not extending above it.

radius of cup base (cm) 0 1.75 2 2.75 3.25

m&m’s

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25] In words, not interval notation, give the domain of the model in the context of this problem. Explain why it is not all real numbers.

26] Use the model to predict number of m&m’s needed to fill a 5 cm radius cup. Show your work. Is your prediction reasonable? If not, explain why your model is not reliable for extrapolation.

PART 4

27] Place 2 m&m’s in a cup. This is trial number 0. Shake the cup and dump out the m&m’s onto the paper plate. For every m&m with the “m” showing, add another m&m and then record the new population. (Ex. If 5 m&m’s land face up, then you add 5 more m&m’s.) Repeat until you are done with 12 trials OR you run out of m&m’s.

Trial # 0 1 2 3 4 5 6 7 8 9 10 11 12

# of m&m’s 2

28] Graph your data (scatterplot only) with 29] Is the data linear, quadratic, cubic, or something else? the trial number on the x-axis and the number Which type of model does your graph seem to look like? of m&m’s on the y-axis. (Circle one, then fill in the blanks)

Linear Quadratic (length) (length x width = area)

Cubic Exponential

(length x width x height = volume) (“J” curve; growth starts slow and increases rapidly)

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30] Use exponential regression on your graphing calculator to find the exponential model that best fits the data. Round to the nearest hundredth. Based on the format of the equation, why do you think this model is called “exponential”?

31] Using your graphing calculator to find ordered pairs, sketch this model on your scatterplot above.

32] Should your graph touch the x-axis? Why or why not? (Hint: what value of x would make y=0 or any negative number?)

33] Use the model to predict the number of m&m’s there would be in these trials. Round to the nearest whole m&m.

Trial 25 _____________________ Trial 50 _______________________

PART 5

34] Count out 60 m&m’s. This time when you shake the cup and dump out the m&m’s onto the plate, remove the m&m’s with the “m” showing. Record the new m&m population. Continue this process and fill in the table. You are done when you have completed 7 trials –OR– when your m&m population gets to 0. Do NOT record 0 as the population, leave it blank!!!

Trial # 0 1 2 3 4 5 6 7

# of m&m’s 60

35] Sketch the graph (scatterplot only) representing your data.

36] Use your graphing calculator to find the exponential model that best fits the data. Round coefficients to the nearest tenth.

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38] Use the exponential decay model you found to determine the m&m population on the 4th trial? How does this number compare to your actual data for the 4th trial. Are they the same? Are they similar? What are some reasons why your results are different? Explain.

39] The general model for an exponential function is 𝑦 = 𝑎(𝑏)𝑥. Conducting a large number of these types of experiments gives the “perfect” theoretical model for the data in #34 as 𝑦 = 60(0.5)𝑥_.

A] What does a represent in the context of this problem?

B] What does b represent in the context of this problem?

40] Compare the exponential models (both the graphs and equations) you created in Parts 4 and 5. List at least two differences you notice between the models. (Hint: consider the shape of the graph and the b value)

1]

2]

BONUS 