### Task 197 ... Years 4 - 8

#### Summary

Closely matching problems that face real manufacturers, this investigation asks for the number of chocolate chips that have to be in a mixture to guarantee that each cookie has 3 chips. In this example, the mixture is designed to make a batch of six cookies but it begs questions such:
• What happens if we want more/fewer cookies in the batch?
• What happens if we want more/fewer chips in each cookie?
Again matching the requirements of real business, the students are asked to act as consultants and, based on the data they gather, advise the company about the number of chocolate chips to put in the mixture.

#### Materials

• Playing board
• One dice
• About 70 counters as pretend chocolate chips

#### Content

• average
• mathematical modelling
• probability experiences
• recording mathematics
• statistics, analysing data
• statistics, collecting & organising data
• statistics, confidence levels
• statistics, frequency
• statistics, inference
• statistics, mode / median / mean

#### Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

This task leads naturally into one way mathematics is applied in business. A question is posed, data is collected and analysed and decisions are made. The card suggests doing the experiment 'several times' - deliberately not defining several because a critical question for mathematicians is How much data is enough?. We want the students to carry out the trials and record results so that they get a feel for the random elements of the situation and so that they have sufficient data to make a reasoned judgement. Here are the results for 10 trials of making a batch of 6 cookies with a minimum of 3 chips per cookie.

38, 24, 31, 30, 35, 30, 27, 26, 27, 24
Remember these figures are the number of chips to put in the mixture for a batch of six cookies so that the minimum is 3 chips per cookie. That will be the advertising claim - 'At least 3 chocolate chips in every cookie!'.
• Is ten trials enough data on which to make a decision?
• How should the data be displayed to help guide the decision?
• What statistics (range, mode, median, mean) could be used to analyse the data?
• Should outliers (the very largest and very smallest results) be ignored?
• If we choose a figure likely to give three chips per cookie, then will be at least one cookie in the batch of six that has three chips. But what number might be in the other cookies? For example, the 38 result above produced cookies with 8, 3, 5, 9, 9 & 4 chips. Is it possible to make a cookie with 9 chips, or would it basically be a 'piece of chocolate'?
• How much do chocolate chip cookies cost anyway? This would be part of the manufacturing costs. Whatever the best mathematical result may be, can the client afford to use that number of chips per batch?
In a sense, it doesn't matter what the students choose as the best number to put into the batch, provided they can justify the decision on the basis of the data and can make intelligent comment about some of the related questions.

#### Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

You will need one dice for each pair of students, counters or something else to be used as chocolate chips and scrap paper to use as a tray of 6 cookies. If you have one Poly Plug for each student in your class and they each use their red board, you will have plenty of 'chocolate chips' for the 6 cookies/3 chips trials.

One way to start this whole class lesson is from the journals of students who have used this task in your regular task session. This means you will have to wait until a few pairs of students have tried the task and recorded their results and thoughts. This might take three or four weeks of making sure that the task is used regularly.

(If you know that you will be using the journal results, it's a good idea to make a point of checking entries for this task at the end of the relevant sessions. Another smart idea is to photocopy the various reports as they are completed so that you have the collected data in your possession when you begin the whole class investigation.)

Invite students who have used the task to introduce it to the class. This can work quite well in a 'fishbowl situation' at a central table using the board supplied with the task. The introduction will produce one outcome.

So, in this trial we found that there needed to be ... chocolate chips in the mixture to get a batch of six cookies to have at least 3 chips in every cookie. Do you think that would be the result if we did another trial?
At this point build a list on the board of student predictions about the best number of chips to put into a 6/3 mixture. Using a stem and leaf plot is a good idea, in fact, the data can be used to introduce this tool. Obviously the few students who have used the task will (probably) have a better idea than those who have only seen one trial, but the purpose of making this list is to see how an increasing amount of data effects predictions. If you plan your board space appropriately, you can record this set of predictions, and the ones later in the session as a back to back stem and leaf plot.

(This is a stem and leaf plot of the ten trials listed above.

0 |
10 |
20 | 4, 7, 6, 7, 4
30 | 8, 1, 0, 5, 0
40 |
The tens part of the number is the stem and the units part is the leaf. A back to back stem and leaf plot would have a second set of trial results as leaves to the left of the stem.)

In a second part of the board start a stem and leaf plot headed Actual Data and enter the one result from the introduction.

That's not much data to predict from. Let's get some more.
Ask each pair to fold their paper into six sections numbered 1 to 6 and carry out one trial. As they finish they add their result to the stem and leaf plot. Soon there will be 10 or more results.
Our challenge is to advise the company about the number of chocolate chips to put in the mixture for 6 cookies, so that we can tell our customers that every cookie has at least three chocolate chips. If you only had this data, what advice would you give?
Discuss whether this is enough data. Encourage further discussion with a question like: Would we tell the company to put 18 chips in each batch? or Would we tell the company to put (maximum so far) into each batch? When it is suggested that more data is needed, include the data from one of the journal records.
Now we have more data, I want you to think again about your prediction and come out to the board and put it here...
This is where you use the Back to Back plot, or use a third part of the board to gather the data under the heading Prediction 2. Discuss what has influenced any changes when compared with the first prediction.
We are fortunate that some of our workers have already carried out more trials when they were using the task. I am going to ask them to add their results to the Actual Data and while they are doing that every group can run one more trial and include their data as well.
Discuss ways to analyse this data including such statistics as range, mode, median and mean. Also, if appropriate to the age, discuss how to deal with outliers. On the basis of this final set of data and the discussion, challenge each pair to prepare a report, with evidence, to the general manager of D-Licious Cookie Company. This report could be submitted for assessment, in which case you might want to include discussion of a rubric for assessment. Rubric and other assessment tools can be found in the Assessment section of the site.
• What happens if the machinery in the factory changes and can now make cookies in batches of 10, or if the boss decides to advertise a different minimum number of chips in each cookie?

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 58, Chocolate Chip Cookies. This lesson plan suggests using a real packet of chocolate chip cookies as the stimulation for the lesson. After initial prediction, the students carefully eat their way through a cookie and count the chips. The lesson is also supported by software that can run many trials of many different No. of cookies / Minimum number of chips scenarios.

#### Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.