MATHS300
Lesson 58
Chocolate Chip Cookies
Lesson Plan
Summary:
In this lesson the challenge of preparing chocolate chip cookies with at least a minimum number of
yummy chocolate chips is unashamedly used to encourage construction of a simulation, and explore the
uses of the statistics which describe and limit it. Apart from mathematical outcomes there is the
important realisation for students of the way mathematics is used in manufacturing industries.
Resources required:
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Packet(s) of chocolate chip cookies
Investigation Sheet
Random number generator (cards, dice, spreadsheet, graphic calculator)
Software is available from the Software Library.
Content Outcomes\Links To Curriculum Documents
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empirical simulation as a problem solving strategy
design and modification of a simulation
stem & leaf plots
split stem & leaf plots
probability
natural variability
confidence levels
Lesson Stages
1.
2.
3.
4.
5.
6.
7.
8.
9.
Posing the problem
Estimations for the 10:7 problem
Designing a simulation model
Running the simulation of the 6:3 problem
Simulating with the software
Interpreting the results
Exploring variations
Modifying constraints
Writing a report
Software: Contributions To Learning
In this lesson the software is an example of the type of program a mathematician would have to design
in order to solve this problem. The user can choose the number of cookies in the batch and the
minimum number of chocolate chips they want each cookie to have.
The software the allows students to test hypotheses without spending money on ingredients to try
combinations which could prove unsatisfactory (even inedible!), and without expending unnecessary
energy in the kitchen.
However, there is no 'right answer' to the question, so there will still be need for an obligatory bake-off
and taste test.
The visual aspect of the software is very powerful and a wonderful backup for the introductory hands-on
experimenting that the lesson requires. Students are often amazed at how long it takes in real time to
'fill' each cookie in a batch with the required minimum number of chocolate chips. The software
confirms that their physical experience is not a one-off situation as they see the software place chips
with the same degree of randomness which they experienced.
Further the visual presentation of data from a 100 trial 'batches' as Stem & Leaf Plots or Split Stem &
Leaf Plots confirms that the range of results can be quite broad.
Lesson Notes
1. Posing The Problem
With chocolate chip cookies packet in hand ask the students to:
Imagine we are cooks wanting to make a batch of these cookies for a fussy
family. We put some chocolate chips into the mixture and stir. How can we be
sure each cookie gets several chips?
For my senior students I used the context of a manufacturer whose
reputation and sales are dependent on delivering what is advertised, in the
context of the economic imperative to keep costs to a minimum.
If you put in too few you have angry customers, but if you put in too many
you are wasting money.
Distribute the cookies, but hold back on the eating for now, if possible!
At this stage keep the problem vague. An important part of the modelling process is trying to clarify the
problem so that it can be tackled using mathematical methods. This presentation tries to engage the
students in the process in a way similar to the way mathematicians work.
Discuss the variables and agree that the key factors are:
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the number of cookies in each batch
the range of the number of chips per cookie
For my Grade 6 class I had to be very specific. "If ten chips were stirred into
the mixture to make ten cookies, do you think every cookie would get a
chip?" They intuitively recognised that natural variability almost certainly
meant no! So the next question was: "Well would eleven be enough?"
Agree on initial conditions such as:
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ten cookies in the batch (any number could be chosen)
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'a chip acceptability level' of seven chips per cookie
2. Estimations for the 10:7 Problem
Tell the students that the recipe produces ten cookies. If the chips get randomly distributed in the
stirring, how many chips are needed in the mixture so that each of the ten cookies will have at least
seven chips?
Estimating at this stage has two immediate advantages:
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it provides an insight into students' initial perceptions
it provides motivation as students then want to find out if their guess is correct.
Ask students to come to the board and record their initial guess on a stem and leaf graph. How many
chips would they stir into the mixture to ensure at least 7 chips in each cookie?
Write two headings on the board:
10:7 First Guess
10:7 Changed Guess?
7
8
9
10
11
12
13
14
15
7
8
9
10
11
12
13
14
15
The initial reaction of many students to this question is 70! (Ten cookies with 7 chips in each.) After
further discussion the idea develops that there may not be a perfectly equal spread of chips among the
cookies.
I moved discussion along at this point by asking the students to carefully
munch into their cookie and try to count the number of chips.
Ask students to record this number on the board under a heading:
No. of chips found
In my class the point of this was immediately obvious - that there is
significant variability. For the brand of cookie we chose, the average number
of chips was about 11, but we had a range from 4 to 17.
In the light of this observation, ask students, if they want to change their initial estimates for the 10:7
problem. Most do and most will increase their guess. Once again, record guesses on the Stem & Leaf
plot.
10:7 First Guess
10:7 Changed Guess?
7
8
9
10
11
12
13
14
15
0, 5, 0, 2, 0, 5, 0, 9, 0, 5
5, 5, 0, 5, 4, 9, 0, 5
2, 6, 9
0, 0, 0, 0, 8
7
8
9
10
11
12
13
14
15
0, 5, 5, 8, 0, 5
5, 0, 5, 5, 0, 7, 5
0, 0, 5, 5, 0, 6, 0, 5, 9
0, 0
0
0
(For some students this might be their first exposure to Stem & Leaf. In this case it uses the tens digit as
the 'stem' and the ones digit as the 'leaves'. So the first four entries in the first row of the first guess
stand for 70, 75, 70 & 72.)
3. Designing A Simulation Model
Students may suggest baking lots of batches would be a good way to find out the number of chips
needed in the mixture to ensure that each cookie in a batch of ten gets at least seven chips. However
reference to the time and costs associated with this will suggest that another way needs to be sought.
Is there some way we can use something else to represent the way the chips
go into the cookies at random? This would enable us to work on a simulation
of the problem.
Split the class into groups of 3 or 4 and provide time to grapple with the problem of designing a
simulation. Keep reminding the students that they are looking for a way to represent the randomness of
the distribution of cookies.
Would a coin help? ... What else can you think of that produces random
results?
Groups will come up with different models, and this is a chance to discuss the merits of each. Perhaps
you can give them all a try.
Some possible simulations are:
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Use packs of cards (without the pictures) so that all numbers 1 through 10 are equally likely.
These represent the cookies. Shuffle and cut one. Add one 'chip' to the 'cookie' with the number
that turns up on the card. Repeat until all cookies have at least seven chips.
Use a ten sided dice
Use a graphic calculator or a spreadsheet to generate random numbers from 1 to 10.
Note: These ideas are related to the deliberate choice of ten for the number of cookies in the batch. So,
if you chose other size batches, you could use:
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A coin for batches of two cookies.
Card suits for batches of four cookies.
Normal cube dice for batches of six cookies.
Or pairs of randomly generated numbers for batches of one hundred cookies.
4. Running a Simulation of the 6:3 Problem
Based on the discussion about models select one simulation which the class can try together. This will
mean that:
1. All students will be using a method which will work
2. Results can be combined to get more data
3. A class discussion can develop as the simulation progresses
The simulation I chose is the 6:3 problem. That is, six cookies and we need
at least three chips in each cookie.
I chose the 6:3 problem for three reasons:
1. It highlights the problem solving strategy of solving a simpler
problem.
2. It is quicker to carry out yet still provides plenty of data for analysis.
3. The choice of 6 allows the use of a standard dice.
I want each pair to carry out at least one simulation. Keep going until every
cookie has at least 3 chips. Record the results on your worksheet and the class
results on the Stem & Leaf on the board. When you have finished, do it again
using a separate table.
My students were a bit unsure what was required, so I gathered them
around. We sketched 6 large circles on a piece of paper (to represent the
cookies). Then we rolled a dice - the number 4 came up. "That means one
choc chip goes to Cookie 4." We put a small disc in that circle. After two or
three more rolls they soon realised what the trial was about and went off to
run their own trials.
Having the Task Centre version of this activity was a big advantage at this
point. I asked a couple of kids who had used it in the Task Centre to explain
what it was about. Then I did a quick model with the equipment and
everyone was soon off doing their own simulation.
Groups will finish at different times due to the natural variability in results, but you will need about 40
trials, so early finishers can simply do a third trial. This is sufficient data to see the distribution and raise
the problems of interpretation that are to come.
Given their earlier guesses, students are often quite surprised at the results so far. With student help
the data can quickly be re-ordered to show the distribution from lowest result to highest. The graph will
provide opportunity to discuss and compare natural variability and theoretical probability.
To generate discussion you might ask questions like:
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What is the highest number of chips in any cookie?
What is the range of the number of chips allocated to cookies?
Does it seem that all outcomes are equally likely?
Did anyone 'score' the minimum of 18?
Revising Estimates for the 10:7 You could ask the students to record this graph and round off this section
by asking whether the results indicates that they would like to refine their guess for the original 10:7
problem.
5. Computer Simulation
The software uses a random number generator to establish its simulation. Since computers are designed
to carry out huge numbers of calculations in milliseconds, it is able to provide an amazing amount of
simulation data extremely quickly.
The main initial use of the software is to provide data for the 10:7 problem.
Option 1 simulates one batch at a time as follows. In this example, 112 chips were needed in the batch
before each cookie had at least 7 chips.
Option 2 simulates a run of 100 batches and displays results on a stem and leaf graph. In the following
10:7 simulation the Stem & Leaf shows that one batch took only 84 chips, the maximum needed was
214 chips, the most common group was between 110 and 119.
Note: for some simulations, such as the 6:3 example below, the software needs to display the results on
a Split Stem & Leaf Graph.
By convention the 0 to 4 leaves are shown on the top stem number and the 5 to 9 leaves are shown on
the bottom one.
I let the students take turns to use the computer to explore different
options and trial numbers. They noticed that even though the software
could get more data much faster than our model, the problems of
interpretation remained. I related this to the situation professional
mathematicians face.
One interesting side problem we noted was estimating the chance that a
batch might need exactly 70 chips. That is, a result of exactly 7 chips in each
cookie. In a computer laboratory we ran hundreds of groups of 100 batches.
Only rarely did we get a minimum of under 80, and we were not even
remotely close to 70. We concluded from the empirical data that the
chances were less than one in a million!!!
6. Interpreting the Results
The students will soon notice that the shape of the distribution is a long 'tail off' towards the higher
numbers. This is called a positively skewed distribution. Several unusually large observations occurred
and we talked about these outliers, and how likely they were.
Are the extreme values errors? Why did they occur? Are such large values
improbable or impossible?
Encourage students to analyse further with questions like:
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The minimum number of chips in our model (ten cookies with a minimum of seven chips) is
seventy. What is the possible maximum? (infinite).
How many chocolate would you recommend that the cook puts in the mixture? Why?
Ask each pair to discuss this and record a response. Briefly follow this with a class presentation of all the
suggestions. Usually students suggest the mean, median or the maximum. If anyone wants to discuss
the practical problems of adding a very large number of chocolate chips to the mixing bowl, defer the
question until later when the model will be modified.
If you are recommending the median, roughly what percentage of your
cookies will have fewer than the seven recommended chips?
(The answer will have to be 50% since the median is the data point which has half the data above it and
half below.)
If you are recommending the maximum, roughly what percentage of your
cookies will have fewer than the seven recommended chips?
(The answer is 0% - certainty. The only trouble is that we don't know which number will be the
maximum for a given batch!)
Other points which could come from the discussion are:
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There is no single correct answer; it depends on how often you are prepared to find a single
cookie with less than the desired number of chips.
If we want to be absolutely certain that every cookie has seven chips, an awful lot of chocolate
chips need to go into the mixture.
If we cannot be 100% certain, then how certain do we want to be?
This last point introduces the idea of degree of confidence in an answer, ie: what percentage of identical
times do you want to be right?
Suppose we elected to be 90% or 95% certain. What does this mean in practical terms? How could be
determine the effect of this on the actual number of chocolate chip cookies with less than seven
chocolate chips?
Using the sample data above for the 10:7 case, 90% of the results are under 142. So a value of 142
corresponds to being 90% certain that every cookie in the batch has at least seven chips. That is that in
nine out of ten batches, all the cookies will contain, at least, the recommended number of cookies.
(It is perhaps worth mentioning at this point that 90% or 95% confidence levels may be acceptable in
this simulation, but may not be in others. Few people would want to fly if aeroplanes only had a 95%
chance of arriving at their destination. However, many terminally ill patients would take a new drug
even if it had a 50% chance of curing them.)
Reveal the students' original guesses at the number of chips needed for a 10:7 batch, and compare
those with the negotiated solution from the simulation. As a quick way to indicate that the number of
cookies in a batch does matter, ask:
If there was only one cookie in a batch, how many chips would need to be
added to the mixture to be sure that each cookie had seven chips?
7. Exploring Options with The Computer
The software also allows us to examine 'What if ...' questions in the same way a mathematician does.
For example:
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What if ... we change the recommended number of cookies?
What if ... we change the number of cookies in a batch?
What if ... we reverse the problem and require a maximum number of chips per cookie and no
minimum, ie: guarantee that at least one cookie has seven chips.
My students enjoyed exploring a variety of scenarios. For example, 8
cookies with 4 chips in each, then 8 cookies with 5 in each etc. In each case
the challenge is to predict how the distribution might look and to decide
"How many chips would you recommend that the manufacturer include?"
8. Modifying Constraints
Consider the assumptions in the model; the ideas either acknowledged or taken for granted. A change in
any of them represents a change in the model:
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Cookies should be the same size and weight.
Chocolate chips are identical.
Chocolate chips are independent, ie: each goes 'its own way' and they don't stick together.
The chips are equally likely (1 in 10 chance in our 10:7 problem) to end up in any of the cookies
in the batch.
There is no upper limit to the number of chips any cookie can have.
This last one is likely to have been pointed out already by several students. It is physically impossible for
each cookie to hold large numbers of chips. Some may ask about the ability of the stirring process to
produce an even distribution of chips through the mixture.
These totally valid and sensible objections can be approached in several ways. The ones listed here have
all been suggested by students:
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Stop adding chips to a 'cookie' in the simulation when you reach a prescribed maximum.
Such an approach is consistent with the reality of real life model building. The limit would be set
after practical experience with the size, weight, texture and taste of several trial batches.
When the highest chip number in one batch gets too much above the lowest, stop adding to the
cookie with the highest chip number (or add only to the lowest), so that the lowest has a chance
to catch up. This effectively gives a limit to the maximum.
Start with the same number of chips in each cookie being a number just under the required
minimum, rather than zero as is implied in the model to date, so there is less opportunity for
large differences to occur. This is another way to reduce the spread of results, but it doesn't
impose a fixed limit to the number of chips.
Note that in order to investigate the effect of a fixed upper limit, it is not necessary to generate new
data. The new totals can be generated from current data as shown by the table below. All numbers over
an agreed maximum are simply replaced by the maximum, as shown in this table.
Sample Simulation Results
Stop adding chips when the maximum per cookie (shown in brackets) is reached.
Cookie Number
Original Model
Modified Model
(10)
Modified Model
(14)
Modified Model
(18)
1
14
10
14
14
2
15
10
14
15
3
17
10
14
17
4
10
10
10
10
5
16
10
14
16
6
21
10
14
18
7
13
10
13
13
8
7
7
7
7
9
17
10
14
17
10
9
9
9
9
Totals
139
96
123
136
Arrange for groups to explore various changes to the model and report back. Reports may be presented
orally but would include a stem and leaf graph, and justification of the value which provides for
confidence that 90% of the cookies in a batch will have at least seven chips.
Compare and discuss the effects of the different modifications on the total number of chips that need to
be added to the mixture.
8. Writing a Report
If you choose to, this
is the appropriate
time to encourage
'food for thought' by
arranging for a baking
session. Following
this, students will
have sufficient
experience to write a
report of the entire
project. This may be in
the form of a
publication like:
MATHS300 is a living site. This lesson will be enriched through further teacher development in
classrooms across the world. You are invited to contribute to that process by submitting:
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variations
extensions
inspirations
photos
student work
Please email material to the address below. If it is included it will be acknowledged. You can review
current contributions in the Classroom Contributions folder for this lesson.
email: Doug.Williams@curriculum.edu.au
Worksheet: Chocolate Chip Cookies
Reproducible Page ... © Maths300, Curriculum Corporation
Results of Simulation 1
Cookie
Tally of chips in each cookie. (Stop when every cookie has got ...)
Number
1
2
3
4
5
6
Results of Simulation 2
Cookie
Tally of chips in each cookie. (Stop when every cookie has got ...)
Number
1
2
3
4
5
6
Record the class Stem & Leaf graph below or on the back.