# Stage 4 - algebra - algebraic techniques

Use a real-life situation to apply the concepts of variables and constants; use a spreadsheet model to analyse a real- life problem and link spreadsheet formulae to algebraic expressions; form equivalent algebraic expressions in context

## Strategy

Students can:

- use a real-life situation to apply the concepts of variables and constants
- use a spreadsheet model to analyse a real-life problem and link spreadsheet formulae to algebraic expressions
- form equivalent algebraic expressions in context

### Activity 1: variables and constants

The teacher poses a problem to the students in terms of a scenario:

**Harry Potter exclusive Part 1**: You have decided to set up a discount book operation at your local shopping complex. To begin with, you’ve been given exclusive licence to the next Harry Potter book, Harry Potter and the Mad Mathematicians. It’s even thicker than the last book and at 975 pages, you’ve decided that a selling price of $35 will be more than reasonable. The centre management agreed to give you a special one year lease of $500 to set up your stall. Griffin Press in Australia have given you the exclusive rights in your district for one year. This licence will cost you $200 for one year and they will provide you the books for $20 each. Over the course of the project we’ll use some mathematics to analyse the profitability of this deal.

Harry Potter exclusive worksheet (PDF 383.38KB)

**Income and cost formulas**

The teacher introduces the use of letters to stand for numbers through the use of patterns in tables. The terms **constant** and **variable** are gradually brought into the discussion as simple formulae are developed for income and cost.

Students are asked to consider the meanings of the terms *Income and Cost*. They are to describe **patterns** associated with the income derived from selling the books as well as the costs incurred. Students will translate formulae (or rules) from words into symbols.

Stated in **words**, the rules are:

### Income = $ x number of books sold

### Cost = $ x number of books +

Complete the following table of values for the * Income*:

### Income = $ x number of books sold

Number of books | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

Income | ^{enter answer} |
^{enter answer} |
^{enter answer} |
^{enter answer} |
^{enter answer} |

Complete the following table of values for the * Cost*.

### Cost = $ x number of books +

Number of books | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

Cost | ^{enter answer} |
^{enter answer} |
^{enter answer} |
^{enter answer} |
^{enter answer} |

Recall that the **formula** is a shorter way of writing a rule. **Algebra** uses letters and symbols to stand for numbers and this allows us to write rules simply and easily. In our project, we can let:

**N**stand for the number of books bought or sold**I**stand for the income earned from the sale of N books**C**stand for the total cost of the books

N, I and C are **pronumerals** standing in the place of numbers. They are also called **variables** because their values can vary.

The symbols and represent values that don't change. Because they remain the same they are called constants. An example of constant is the boiling point of water (100 degrees Celsius).

- Can you name other examples of constants?

Re-write these rules for * Income* and

*in symbols:*

**Cost**Rule in words: Income = $ x Number of books sold

Rule in symbols:

____________________________ (Leave out the multiplication sign)

Rule in words: **Cost** = $ x number of books +

Rule in symbols:

____________________________

Once the students understand the meaning of the Income and Cost formulae, they should practice substituting values into them by answering the questions below:

### Questions

- How much income is made by selling 1 book?
- How much does it cost to order 1 book?
- How much profit/loss is made on 1 book?
- How much income is made by selling 2 books?
- How much does it cost to order 2 books?
- How much profit/loss is made on 2 books?
- How much income is made by selling 5 books?
- How much does it cost to order 5 books?
- How much profit/loss is made on 5 books?
- How much income is made by selling 20 books?
- How much does it cost to order 20 books?
- How much profit/loss is made on 20 books?
- What you notice about the answers that you have obtained?

The next part of the activity provides students with the opportunity to apply algebraic reasoning to solve a problem by using spreadsheets. Students can work in pairs or individually. They enter the formulae for** Income and Cost** to calculate the profit/loss in multiple rows.

Profit/loss can be calculated using Income-Cost. A positive answer is profit and a negative answer is loss

**Poses the question – How many books must we sell before we start making a profit?**

Students set up a spreadsheet with column headings

- Number of books sold
- Income
- Cost
- Profit

**Note**:

- Format the 'Profit' column as currency to display a negative amount in red with a minus sign (you may wish to use conditional formating)
- Use the fill-down feature (or similar) in the 'income', 'cost' and 'profit columns to save time

Ten rows should be sufficient to enter a range of values in the 'number of books sold' column in order to arrive at the number needed to be sold for a break-even situation. Break-even occurs when income-cost equals zero.

Example spreadsheet with formulae

**Questions**

Use the spreadsheet to find the answers to the following questions:

- The formulae in the income column all have the same number (35) and a cell reference such as A3. Which one is the variable and which one is the constant and what do they each represent?
- The Profit column contains formulae like = B3 – C3. Explain what this formula is calculating. Will the value in this cell always be the same no matter what value is entered in the cell A3? Is the value in cell D3 a variable or a constant?
- Enter a value of 30 in the top row of the first column. For what variable (I, C or N) is this number being substituted?
- How much profit/loss do you make if you sell 30 books?
- How much profit/loss do you make if you sell 100 books?
- How many books do you need to sell to make a profit of $1000?
- How many books do you need to sell to make a profit of $500?
- So, How many books must we sell before we start making a profit?
**Challenge Question: Is there a better (quicker) more direct way of finding the answer?**

### Activity 2 – equivalent algebraic expressions

**Part 2**: Business is booming! Your business has picked up and you decide to sell accessories including Harry Potter bookmarks (which cost you $2 each and will sell for $4), Harry Potter stickers that come in two sizes. The small stickers cost you $1 each (and sell for $2 each) while the large stickers cost $2 each and sell for $3 each.

### Guided activity

Pose the following :

1. A customer buys one Harry Potter book, 3 bookmarks and 2 small stickers. Students are to write at least four equivalent expressions that give the total bill for this purchase.

possible response | 35 + 3 x 4 + (2 x 2) | 3 x 4 + 35 + 2 x 2 | 35 + (4+4+4) + (2 + 2) |
---|---|---|---|

possible response | 35 + 4 + (2 x 4) + (2 x 2) | 4 + (2 x 4) + (2 x 2) + 35 | 35 + (4 x 3) + 2 + 2 |

2. Another customer buys 2 books and p large stickers. Write four equivalent expressions for the total bill.

possible response | 35 x 2 + 3p | 35 + 35 + 3p | 70 + 3p |
---|---|---|---|

possible response | 3p + 2 x 35 | p + 70 | 35 + 2p + p + 35 |

possible response | p + p + p + 35 + 35 | 70 + p + p + p | p + p + p + 2 x 35 |

3.How many equivalent expressions can be used to calculate the total bill for a purchase of **m** bookmarks and **y **large stickers?

possible response | 4m + 3y | 4m + y + y + y | m + m + m + m + y + y + y |
---|---|---|---|

possible response | 2(m + m) + 3y | 2m + 2m + y + 2y | 2(m + y) + m + m + y |

Students are then asked to explain the meaning of some of the expressions and why they are equivalent.

For example, 4m + y + y + y may be interpreted as buying four bookmarks in one purchase and three large stickers purchased one at a time. This is the same as buying four bookmarks and three large stickers at once.

Guided activity - Algebraic techniques Stage 4 (PDF 467.66KB)

### Group activity

- Students work in small groups to make up equivalent algebraic expressions based on the above scenario, for other students to interpret in words and to modify.
- Students create similar real-life situations upon which to base simple algebraic modelling problems. These may link to other curriculum areas.

## References

### Australian curriculum

CMNA176: Create algebraic expressions and evaluate them by substituting a given value for each variable.

### NSW syllabus

MA4-8NA: Generalises number properties to operate with algebraic expressions.

### Teacher resources

Lesson plans and activities