Problem One

Drawing a table and working through the clues systematically to eliminate possibilities is a good way to solve this problem. Put an X in a square where the person doesn’t play the game. Tania and Marie play team sports, so put an X against them in the golf and archery columns.

Sonny and Tania hit the ball, which eliminates netball and archery for both of them. There is now only one space left in Tania’s row, so she must play hockey. As a result, no one else can play hockey, and Sonny must play golf.

This shows that Marie plays netball and Jerome must be the archer.

See also Problem Four, page 16, in Problem Solving, Figure It Out, Level 3.

Problem Two

In this problem, the students need to work out how many times 12 goes into 30. 30 ÷ 12 = 5/2. Multiply the amount of each ingredient by 5/2, as shown below.
Flour: 11/2 x 5/2 = 3/2 x 5/2 = 15/4 = 3 3/4 cups
Milk: 3/4 x 5/2 = 15/8 = 1 7/8 cups
Eggs: 2 x 5/2 = 5
Baking powder: 1 x 5/2 = 5/2 = 21/2 teaspoons
Oatmeal: 1/2 x 5/2 = 5/4 = 11/4 cups.

Problem Three

Ensure that the students read this problem carefully. It’s easy to make the mistake of assuming that one boy can fill one bin in 1 hour, two boys can fill two bins in 2 hours, three boys can fill three bins in 3 hours, and so on. This is not correct.
If six boys can fill six bins in 6 hours, they can fill 12 bins in 12 hours and 12 boys can fill twice as much in the same time. So 12 boys can fill 24 bins in 12 hours.
If four girls can fill four bins in 4 hours, they can fill eight bins in 8 hours and 12 bins in 12 hours.
So eight girls can fill 24 bins in that time and 12 girls can fill 36 bins in that time.
Altogether, 24 + 36 = 60, so 60 bins of apples were filled.

Problem Four

This is another problem that the students have to approach carefully. It’s easy to miss out the last bottle.
If the leftover liquid from five bottles can fill one bottle, then the leftover liquid from 25 bottles can fill five bottles. But the leftover liquid in the five refilled bottles can be used to fill another bottle, so altogether, six bottles were refilled (with a little bit left over).

Hints for Students

1. Make a table.
2. How many times does 12 go into 30?
3. Work through this problem systematically and carefully.
How many bins can six boys fill in 12 hours?
4. You could use equipment or a drawing. Check that you have counted all the bottles.

Answers to Problems

1. Tania: hockey
Jerome: archery
Sonny: golf
Marie: netball
2. 3 3/4 cups of flour, 1 7/8 cups of milk, 5 eggs, 2 1/2 teaspoons of baking powder, 1 1/4 cups of oatmeal
3. 60 bins
4. 6

Context
For rural students, the raising of lambs and calves will be very familiar. City students may need some background information about raising calves to help them in interpreting the context. Calf raising is basically about taking 3-day-old calves and raising them until they are weaned and able to survive on grass, water, and
perhaps meal for food. The calves are then on-sold to beef farms that raise them until they are mature or to dairy farms as potential milk cows. Some dairy farms do their own calf raising, sometimes on blocks called “run-offs” dedicated to the purpose.
Students who are unfamiliar with the context may struggle to come up with reasons why the price of calves might fluctuate from year to year. You can help them with this by connecting Jessica’s scenario to the earlier scenarios in the book where price was determined by supply and demand. Ask the students to hypothesise why these might vary in respect to calves. The supply of calves depends mostly on the fertility of cows and the interest of dairy farmers in having calves born. Poor weather can reduce fertility, and poor forecasts of the price for calves can reduce farmers’ interest in producing them. Demand for calves is also determined by weather and projections of prices. Dry winters and springs can mean that feed on the farm will be short. This
also affects the ability of beef-raising farms to take on stock and lowers the demand for 4-month-old calves. For many potential calf rearers, particularly small-volume businesses, poor projections of returns mean the returns do not justify the risks and work required. The price of different breeds of calves is dependent on demand. Some breeds, like Herefords, are more popular because of their superior potential as beef cattle.
Financial language
The following financial terms occur during the story:
Agent/broker: A person who arranges exchanges of goods and services for a fee
Savings: The money people put aside from their income for future needs
Percentages: Fractions out of 100 that are commonly used to compare and to scale amounts, for example, comparing shooting statistics in sport or interest rates on savings
Profit: The amount of money left over from sales once the costs of producing the goods and services are met
Loss: The opposite of profit: the shortfall in money when a business’s costs are higher than its revenue (money made from sales)
Spreadsheet: A computer ledger used to keep financial accounts and other calculations
Account: A contract with a bank to hold or borrow money
Balance: The amount of money either in an account or borrowed from it (negative balance)
Trends and patterns: Happenings over time that allow future events to be predicted, for example, prices of stock
Investment: Savings or the purchase of an asset with the hope of future returns, for example, buying a rental property or business.
Financial understanding
Jessica’s goals show that she is becoming financially responsible. She understands that her decisions now will impact on her future choices. By earning income and saving now, she will be able to finance her own learning at university. Jessica is also demonstrating business understanding and financial maturity as she investigates the financial side of her business.
Jessica is matching her personal goals and capabilities to an undertaking. She demonstrates initiative and drive by running this business to earn money to pay for her university study, thus taking responsibility for managing herself. She uses the opportunities available to her through her farming background, including the use of facilities and the advice and expertise of her parents and her stock agent. She plans
her transactions carefully, particularly monitoring her cash flow and managing resources so that she can raise the greatest number of calves possible for her. She works in a small team with her father and communicates effectively with him to identify and solve potential problems.
 

Activity

Mathematics and statistics
To calculate the answers to question 2c, students need to calculate a percentage of an amount. For example, the price of a Friesian calf in year 3, including the agent’s fee, was $115 ($1,150 ÷ 10). Jim Hogan projects that prices will be 10 to 15 percent higher. Important knowledge for students is that percentage means “out of one hundred”, so the 10 or 15 is applied to every 100 dollars. Ten out of 100 is
one-tenth, so 10 percent of $115 is one-tenth of $115. Division by 10 is also a critical idea here. The effect of dividing by 10 can be shown on a calculator by keying in 1150 ÷ 10 = = = (some calculators may require ÷ 10 to be repeated). The resulting number has digits shifted one place to the right.

Using this idea, 10 percent of $115 can be calculated as $115 ÷ 10 = $11.50. Since 5 percent is half of 10 percent, 5 percent of $115 is half of $11.50 = $5.75. Another important idea is that the calf prices in year 4 will be between 110 percent and 115 percent of the year 3 prices. Percentages of the same amount can be added, so 115 percent of something is 100% + 10% + 5% of that something. This process can be
shown on a double number line as shown below:

With these representations, students should be able to calculate the potential prices for calves in year 4. Question 2d requires them to budget for purchasing 50 calves for no more than $5,500. Important questions you might ask the students include:
What will the average price per calf be? That is, the price if each calf cost the same, the middle price. $5,500 ÷ 50 = $110
What does that tell you about how many of each breed Jessica can buy? Friesian and Hereford–Friesian cross calves are dearer than this average, so a purchase of one of these must be balanced by purchasing Friesian–Jersey cross calves that are cheaper than the average. Jessica will need to buy more Friesian– Jersey cross calves to balance her budget.
How might you use a table or spreadsheet to organise the possibilities to help meet the budget?
A spreadsheet can be set up that allows you to try “if–then” scenarios.

Using this spreadsheet, different scenarios can be explored, such as increases in calf prices by 15 percent and buying different numbers of each breed.
Question 2e involves a rate problem, that is, a multiplicative relationship between two measures. In this case, the measures are cost, in dollars, and number of calves. Most proportional reasoning problems like this can be represented by double number lines. This visual representation helps students to summarise the conditions of the problem and solve it using progressive steps. A double number line for 2e is given below.

Some students are likely to need support in determining how to structure the operations to solve the problems in question 3. Diagrammatic support can help. Question 3a is a rate problem and can therefore be represented using a double number line or ratio table. The rate involved is between litres of milk and
kilograms of powder. Students will need to use the metric conversion: 1 kilogram equals 1 000 grams (kilo means 1 000). As a ratio table, the problem can be represented progressively as:

The problem can be solved efficiently by dividing 20 000 grams by 135 grams, the unit rate, that is, 1 litre : 135 grams. However, as both ratio tables and double number lines allow students to express the problem even if they are unsure of a strategy, they can make progress without using division. The students’ strategies are also more likely to involve mental calculation rather than rote procedures.
Question 4 involves comparison of two fractions as percentages. It is common to express two or more fractions as equivalent fractions with a denominator of 100, otherwise known as percentages. Both Jessica’s and her father’s scours rates are easily determined by a scaling operation.

This can also be expressed as fractions, as shown in the Answers:
4/50 = 8/100                      21/300 = 7/100
Formulating the operations to perform is the biggest obstacle to students solving questions 4b and 4c. Encourage the students to draw diagrams if they need to think through which operations are involved. Examples might be:

So 4 x 6 = 24 or 4 x 8 = 32 litres per day.
Question 4c can be modelled in a similar way:

So 4 x 1 200 g = 4 800 grams = 4.8 kg (per day).
3 x 4.8 = 14.4 kg of powder over the 3 days.

Question 5 also requires some knowledge of the metric system. One millilitre is one-thousandth of a litre. Five millilitres is about the contents of one level teaspoon. Question 5a is a double rate problem that relates calves to doses and millilitres to days. As such, it can be modelled by a ratio table:

So Jessica is 5 millilitres of penicillin short. If 100 millilitres cost $26.00, then 5 millilitres is one-twentieth of that because 20 x 5 mL = 100 mL. $26.00 ÷ 20 can be performed mentally in a number of ways:
$26 ÷ 10 = $2.60, $2.60 ÷ 2 = $1.30; or $26 ÷ 2 = $13, $13 ÷ 10 = $1.30; or $1.00 ÷ 20 = $0.05 (5 Cents), so $26.00 ÷ 20 = $26 x $0.05 = $1.30.
Question 6 involves expressing a ratio of fraction terms. To do that, it is important to identify the parts and the whole.
The ratio of calves is 50:300 (Jessica’s to Dad’s). So diagrammatically, the herd of calves looks like this:

So Jessica’s fraction of the whole herd is one-seventh (not one-sixth, as is the common error). As well as feeding her own calves, Jessica should probably help her father feed his calves in return for her use of the farm facilities. The degree of compensation is discretionary, but the students might want to consider on
which days of the week Jessica might be more able to help her father. For example, apart from holidays, Monday to Friday are school days (and she still will have to feed her own calves on those days), so it may be easier for her to help on the weekends.
Social Sciences Links
Achievement objectives:
• Understand how producers and consumers exercise their rights and meet their responsibilities (Social Studies, level 4)
• Understand how people seek and have sought economic growth through business, enterprise, and innovation (Social Studies, level 5)
Have the students discuss:
– How is Jessica seeking economic growth?
– What flow-on economic growth impacts will Jessica’s enterprise have?

Financial understanding
In this activity, Ethan learns that there are lots of aspects to consider when starting up a small business and that even though he can sell his products to help meet his goal (saving enough money to buy a mountain bike), there are expenses to cover as well as risks to think about. Risks need to be considered before making a financial
decision.
Ethan matches his capabilities with setting up a business that involves something he enjoys doing, and he identifies, assesses, and manages risks associated with the business. With this approach, Ethan is more likely to be successful and meet his goals.
Ensure that the students know the meaning of terms such as costs, income, and profit as well as the numerical value of a dozen, the meaning of @, and how many days/weeks are in a month.
As an introduction, the class could share their knowledge of the different types of eggs available, for example, barn/battery/free range, and discuss why people pay higher or lower amounts depending on their choice of egg type and why the eggs are priced differently. (An Internet search engine using “eggs + price” or “eggs + cost”
may provide some interesting points for debate.)
Mathematics and statistics
This activity involves a fair amount of calculating, quite apart from the financial literacy discussion. If you are using this activity with your whole class, the students at stages 2 and 3 will need teacher support, materials, and calculators to complete mathematical aspects and may also need the numbers modified for the mathematics to be meaningful. The students at stage 4 will also need teacher assistance, materials, and the use of calculators to complete the mathematical tasks.
The students could use numeracy money for question 1a, but a more efficient strategy would be to use doubles.
The students may require materials to complete question 1b; money would be helpful, as would paper strips with money amounts written on them. The students could also use diagrams to show how they are working it out, although stage 6 students could be using working form.
Question 3 is a multi-step problem; you may need to elaborate on these steps before the students attempt to solve it. As an extension, able students could work out how much difference there is in the totals if they use the number of weeks there actually are in 6 months as opposed to the 4 weeks in a month given as a guide in the
question.
Reflective question
Financial understanding
The reflective question should prompt some lively discussion. It’s important for the students to understand that not all pets, hobbies, and interests can or should make money for their owners. Many people choose to spend some of their earnings on pets, hobbies, and interests without expecting or wanting to earn money from them.
Others may earn some money, for example, selling plants from their garden or fish from their tanks, but they usually spend that money on more stock, equipment, or improvements and are not involved just for profit or to make money to pay other living expenses. People who win prizes through amateur sports, such as horse shows
or golf, have usually spent more than the prize value in entry fees, travel, and accommodation.
Further investigations and tasks
The students could:
• research who can have hens and who cannot. Why? What about roosters? Who makes these rules? How can they be changed? Why are people who work with poultry not allowed to keep poultry at home?
• research what they could do/make from home that could make income for them, for example, set up a lemonade stall, grow plants, make bird tables, organise games, make compost, and so on. If they were to earn money doing something that they were interested in, what might they do with that money?
• discuss with grandparents or elderly neighbours whether they and/or their parents had poultry or produced fruit and vegetables when they were growing up.
• discuss ways for Ethan to replace his hens after 3–4 years. What would be the best way for him to plan for this event? Would it be better for him to buy in pullets or to get some fertile eggs and hatch them? (You could use a plus, minus, interesting [PMI] evaluation for these options.) What could Ethan do if he were allowed to have a rooster?

Social Sciences Links
Achievement objective:
• Understand how people make choices to meet their needs and wants (Social Studies, level 2)
The students could discuss what impact Ethan’s “want” or goal has on his choices. They could also discuss what the hens’ needs and wants are.
Other Cross-curricular Links
Technology achievement objective:
• Brief development: Explain the outcome they are developing and describe the attributes it should have, taking account of the need or opportunity and the resources available (Technological Practice, level 2)
The students could:
– design and make a container for Ethan’s eggs
– suggest ways to recycle or reuse egg cartons. You could use different coloured plastic pots to make this fun: the students’ suggestions go into one pot (for example, a purple pot) and are then pulled out and categorised, using other coloured pots, as good, bad, problematic, and so on.

Science achievement objective:
• Life processes: Recognise that all living things have certain requirements so they can stay alive (Living World, level 2)
The students could investigate:
– what hens require in order to be happy, healthy creatures
– the nutritional value of eggs.
Health and Physical Education achievement objective:
• Safety management: Identify risk and use safe practices in a range of contexts (Personal Health and Physical Development, level 2)
The students could investigate what diseases can be passed from humans to poultry and vice versa and how to protect both species from the spread of disease.

Answers to Activity

1. a. $144. ($40 + $60 + $24 + $20)
b. i. $24. ($144 ÷ 6)
ii. $6. ($24 ÷ 4 or $144 ÷ 24 weeks)
2. a. i. $21. ($3 + [3 x $6])
ii. $15. ($21 – $6)
iii. Answers may vary. Ethan will be eating some of the eggs, and his parents have
paid for all the set-up costs and food for the first 6 months.
b. Answers will vary. Hens lay fewer eggs in winter, so Ethan won’t be able to sell as
many over that time. Hens can get sick or die. His customers may not always want to
buy eggs every week. Ethan will need to trade off his free time to look after the hens
and collect and deliver the eggs.
3. Based on 4 weeks per month = 24 weeks, Ethan could earn approximately $312. From 4 dozen eggs a week (including the dozen to his mother), he will earn $504 ($21 x 24 weeks). His costs will be $192 ($6 per week x 24 weeks for running costs = $144 plus $8 a month towards replacing the hens, $8 x 6 = $48.
$504 – $192 = $312
Reflective question
Answers will vary. You may think about what you are good at, for example, baking, fixing things, breeding pets, and what other people need or want and would be prepared to pay a price for. If there is a match and the costs are less than the income,
then a talent, hobby, or interest could be used to set up a business and make a profit. Costs include set-up and materials used. The risks are that it may cost you more to set up, make the product, or run the business than you get from sales or as payment.
You also need to take into account the time spent trying to make money instead of enjoying free time, such as after school or school holidays

1. Write a decimal number on the board (to two decimal places). 
2. Brainstorm different ways to represent it.
   For example 3.81 could be expressed as;

• Three point eight one
• 3 + 0.8 + 0.01
• (3 x 1) + (8 x 0.1) + (1 x 0.01)
• 381 / 100
• 4 – 0.19
• Three and eighty one hundredths
• 7.62 / 2

3. Give each student a decimal number and ask them to represent in as many ways as they can.  Depending on the level of individual students they could be to one, two, or three decimal places.  Less confident students could work in pairs.
4. Students could present their work on a sheet of A3 paper to be displayed on the wall.
5. As the students work encourage them to use a variety of approaches (not just a whole list of addition sums).