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Level Four > Geometry and Measurement

Breakfast Biscuits

Purpose:

This unit seeks to connect learning outcomes across all five content strands, number, geometry, statistics, and measurement. However the context lends itself strongly to use of measurement and statistics. In the unit students investigate questions that involve them in experiment design, data collection and display, and drawing conclusions from data. Students investigate the measurements of breakfast biscuits, how much milk they absorb, how strong they are and the hidden truth about collecting cards.

Specific Learning Outcomes:
• construct nets
• measure masses, capacities, lengths, and temperatures using scales and devices.
• conduct investigations and present results.
• carry out experiments and systematically record the results.
Description of mathematics:

Number Framework

Required Resource Materials:
375g boxes of breakfast biscuits (e.g. Weetbix TM) , milk, measurement containers, thermometers, plates, cardboard, rulers, scissors, paper bags and cards cut from Copymaster One, bathroom scales, books, bricks (optional)
Key Vocabulary:

net mass, grams, kilograms, dimensions, absorption, scatter plot, ellipse, elliptical, circular, chance, prediction, ramdom, possibilities, tree diagram, probability

Activity:

Session 1

1. Provide each group of students with a full 375 gram box of breakfast biscuits.

2. Ask the students to solve the following problems without opening the packet. All the information they need is on the packet.
What is the mass (weight) of a single biscuit? (at least 20 grams)
What is the length, width, and height of a single biscuit?
How are the biscuits arranged in the packet?

3. The students will need to seek relevant information from their own experience and from the data on the packet. You may need to scaffold the tasks for some students. Suggestions like the following may be helpful:
How many biscuits are in the whole packet?
What is the net mass (less the packet) of all the biscuits?
What do the pictures on the packet tell you about the length, width and height of a biscuit?
What edge measurement of the packet is most likely to be the length of a biscuit? Why?
How wide and high will one biscuit be then?

4. After you have answered the questions collectively ask the students to predict what they think the packet will look like when pulled apart and laid flat. Ask them to sketch the net they believe forms the packet.  Get students to share their ideas, concentrating the discussion on how the features of the packet, like faces, angles, and edges, match the nets. Discuss how the packet must have tabs in order to be glued together. Ask them to use the measurements of the packet to create an exact net for the packet, including a sketch of the orientation of the designs. Students will need to measure the sides and angles accurately.  The packets can be pulled apart later to check the accuracy of the students’ nets.

Session 2

1. The information on the packet suggests that one serving consists of two breakfast biscuits, and that this should be accompanied by ½ a glass of milk.

2. Pose this problem to the students:
How thirsty is a breakfast biscuit?
Is a biscuit thirstier if the milk is cold or hot?

3. Ask the students to discuss how they might investigate this problem, and make a list of materials they will need. After an appropriate period of group discussion bring the class back together.

4. Ask each group to report back. Focus on whether or not the students have considered what thirsty means, in this case how much milk the breakfast biscuit can absorb. Encourage them to look at the variables that must be considered, such as:
Does it matter how much milk you put in with the biscuit? Will it always “drink” the same amount?
Does it matter how long you leave the biscuit to drink? How long is it before it cannot absorb anymore?
At what temperature is the milk to be called hot or cold?
Will two biscuits drink twice as much milk as one biscuit?

Invite the students to predict the answers to these questions before they investigate.

5. Discuss how the amount of milk “drunk” by the biscuit can be found. One method is to measure the milk put into the plate with the biscuit then remove and measure the extra milk after a suitable time. This non-absorbed milk can be poured off into a different vessel. Subtracting the non-absorbed milk will give how much milk the biscuit has “drunk”.

6. Given all the possible variables it may be necessary to assign different investigations to different groups of students. For example, one group might investigate the effect of heating the milk while another might explore how much milk is absorbed by the biscuit with different times.

7. It is important that you encourage the students to present their findings using appropriate displays. For example, a group presenting data on the effect of temperature on how much milk is absorbed might use a scatterplot to show their results:

Each report should contain a conclusion. Remind the students that finding no difference in “thirstiness” is a worthwhile conclusion in itself.

8. After the students have reported back their findings provide some supplementary problems such as:
If you pour 125 ml of cold milk (5°C) over your breakfast biscuit, how much will be drunk by the biscuit and how much will be free?
How much hot milk (50°C) should you pour over your breakfast biscuit to have the same quantity of milk free?
Some athletes eat as many as 30 breakfast biscuits per day. How much milk would they take in at the same time?

Session 3

1. The breakfast biscuit company have decided to have a new promotion. For every three packets a person buys they get a free breakfast bowl that is designed to hold only two biscuits and half a glass (125 mls) of milk.  They want you to design the bowl. It needs to be circular or oval (elliptical) shaped and have enough depth to hold the biscuits and milk with no fear of overflow.

2. Work out the measurements you will need to give the person making the bowl.

3. Give the students a suitable time to investigate the problem before bringing the class back together.

4. Students may have noticed in session one that the biscuits are twice as long as they are wide (85mm x 42.5mm). Discuss why they might have been created like this. Note that two biscuits can be put together to form a square that will fit snugly into a circular bowl. Designing a four biscuit elliptical bowl is a good challenge.

1. Ask the students to compare the size of the bowl they have designed with breakfast plates from home. Discuss why most breakfast plates are circular and what advantages that might have. Why is a square bowl not a practical option?

Session 4

1. Breakfast biscuits are designed to be strong so they don’t crumble easily. The question is, “How strong is a breakfast biscuit?”

2. Place a biscuit flat between two pieces of paper. Lay it flat on the ground. Pile heavy books or bricks on it until the biscuit crushes. If you find that books or bricks aren’t heavy enough try people! (Our experiments showed that a biscuit could withstand the mass a small child)

3. Ask the students to repeat the experiment with one, two, and four biscuits laid flat. Pose the problem, “For breakfast biscuits, is there strength in numbers?” Get the students to report their findings using tables or graphs.  For example:

 Number of Biscuits Crush Masses (kg) 1 25, 30, 21, 28, 26 2 32, 37, 24, 33, 46 4 39, 43, 50, 47, 41

1. Note that students may take several samples at different numbers of biscuits. This raises issues of which measure to take. Do you take the average or median or do you present all the scores as a range?

Session 5

1. Inside every packet of breakfast biscuits you will find two collector cards. There are three superheroes to collect in the set (Copymaster One: Wondergirl, Antboy, Superslayer). If you buy two packets of breakfast biscuits you will get four cards but there might be many of the same card and none of the others you want. The question to answer is, How many packets do you have to buy to get a full set?

2. Discuss how the cards might be put into a packet so they are quite mixed up. This is a random selection process that might be replicated by having a collection of equal numbers of each card in a paper bag and drawing out two cards.

3. Ask the students to predict how many packets they will need to buy. Get them to use the paper bag technique to find out how many packets need to be bought before a full set of three superheroes is obtained. Students will need to record what comes out of the bag carefully.

4. Discuss how many trials (attempts) each group of students should complete. Three trials with ten different groups usually provides sufficient data.

5. Students will be surprised to find that getting a full set from four cards happens about half the time. Discuss why they think the chances are not as high as they first appear. They are likely to suggest that cards get duplicated more as you get more. Each of the possible outcomes could be presented on a tree diagram but this becomes very complex (there are 27 different outcomes).

6. Ask the students whether buying three packets instead of two is likely to increase the chances. They may like to carry out trials to see how much difference this makes to getting a full set of superheroes. Selecting six cards increase the chances of a full set to nearly three-quarters, although this is very difficult to calculate mathematically.

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