Breakfast Biscuits

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Purpose

This unit seeks to connect learning outcomes across all five content strands: number, geometry, statistics, and measurement, with a particularly strong connection being made to the use of measurement and statistics.

Achievement Objectives
GM4-1: Use appropriate scales, devices, and metric units for length, area, volume and capacity, weight (mass), temperature, angle, and time.
GM4-6: Relate three-dimensional models to two-dimensional representations, and vice versa.
S4-3: Investigate situations that involve elements of chance by comparing experimental distributions with expectations from models of the possible outcomes, acknowledging variation and independence.
Specific Learning Outcomes
• Construct nets.
• Measure mass, capacity, length, and temperature using scales and devices.
• Conduct investigations and present results.
• Carry out experiments and systematically record the results.
Description of Mathematics

The main focus of this unit is the measurement of breakfast biscuits by length and width, and by weight. Students find the weight needed to crush a breakfast biscuit, measure the amount of milk a breakfast biscuit absorbs, consider the 3-dimensional layout of the biscuits, and design a net for the packet. Each task involves basic metric units of measurement, cm, mL, g, and kg. Through these measurement tasks, students utilise their number knowledge and strategies.

Students also work with circles and ellipses (optional) which requires calculation of a suitable radius given dimensions of the breakfast biscuits. They gather data about a probability experiment and use methods for find all the possible outcomes for theoretical probability.

Opportunities for Adaptation and Differentiation

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:

• providing scaffolding to students who need experience with the units and tools of the metric measurement system
• allowing students to physically perform measurement tasks
• encouraging trial and improvement
• using calculators to reduce the burden of calculation
• providing opportunities for students to work in collaborative groupings to allow them to share ideas, questions, and benefit from peer scaffolding and extension
• organising groups of students to include students with varying levels of mathematical confidence and knowledge to encourage tuakana-teina (peer learning)
• challenging students to extend the tasks (e.g. design an elliptical, four-biscuit bowl is a good challenge).

The context for this unit is breakfast biscuits (i.e. Weetbix) which should be familiar to most students, as the biscuits are a popular cereal product in Aotearoa New Zealand. Ensure food is not wasted during investigations. ‘Used’ Weetbix should be eaten, not discarded. Some problems in the unit could be adapted for use with other packaged goods, such as boxes of non-edible items like tennis balls, toilet paper, firestarters, and stationery.

Te reo Māori kupu such as ine (measure), papatipu (mass), kītanga (capacity), roa (long, length), whānui (wide, width), paemahana (temperature), and ine-taumaha (scale) could be introduced in this unit and used throughout other mathematical learning.

Required Resource Materials
• 375g boxes of breakfast biscuits (e.g. Weetbix TM)
• Milk
• Measurement containers
• Thermometers
• Plates
• Cardboard
• Rulers
• Scissors
• Paper bags
• Cards cut from Copymaster One
• Bathroom scales
• Books
• Bricks (optional)
Activity

Session 1

1. Organise students into groups. 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. It may be necessary for you to provide individualised, small-group, and/or whole-class teaching directed around each of the concepts explored in the questions (e.g. around how to measure length).
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. Encourage the students to seek relevant information from their own experiences 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. As a class, answer the question and review any misconceptions that arise through discussion.
5. Pose the following provocations to the class, and have students discuss them in pairs.
What object in the classroom weighs about 20 grams?
Show with your hands how long 9.5cm is (length of a Weetbix).
6. 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. If your students are unfamiliar with the concept of a 'net' you could show them some pictures of nets, or perhaps the net of another folded-out container. You might make connections to nets that reflect other, relevant contexts (e.g. containers found in the classroom).
7. 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.
How much milk is half a glass? (1 glass is about 250mL so ½ a glass is about 125 mL).
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” (absorbed) 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. Ensure your students understand how to carry out each investigation, and have adequate time, space, and resources (e.g. measuring equipment, graphic organisers) to enable the successful completion of these investigations. As part of this, you should model each investigation and provide support to ensure students are carrying out the investigation correctly.
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. Consider whether your students need explicit teaching around the different types of displays that could be used, or whether they might benefit from being directed to use a specific type of display. Each report should also contain a conclusion. Consider providing a model or some sentence starters to support students in writing this. 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. Tell the students that 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 mL) of milk. They want you to design the bowl. It needs to be circular and have enough depth to hold the biscuits and milk with no fear of overflow. Therefore, we need to work out the measurements to give the person making the bowl.
2. Give the students a suitable time to investigate the problem before bringing the class back together.
3. Students may have noticed in session one that the biscuits are close to twice as long as they are wide. 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.
4. 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. You can also use plastic milk bottles filled with water (2L bottle weighs about 2kg). If you find that books or bricks aren’t heavy enough try people! (Our experiments showed that a biscuit could withstand the mass of 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?
Do you discard (truncate) the highest and lowest measure?

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. You might model for students how to set up a table to record this data, or provide them with a graphic organiser to use.
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.

Session 6

1. Provide time for your students to compile their learning from throughout the unit and create a presentation demonstrating the key understandings developed. Encourage students to present their learning in a manner that is appropriate, meaningful, and engaging. Links could be made here with explanation writing.
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Level Four