140. Modelling Situations with Algebra

Learning Intentions

• model simple situations using algebra
• write expressions from descriptions
• To understand that applying a model requires defining what the variables stand for

Pre-requisite Summary

• Know that a variable or pronumeral stands for a number
• Be able to write simple algebraic expressions such as $3x$ or $x+5$
• Understand that multiplication can be written without the $\times$ sign
• Be able to read word descriptions carefully
• Know that an expression represents a quantity, not a complete equation
• Understand that a model uses algebra to represent a real situation
• Know that variables must be defined clearly before using them

Worked Examples

Worked Example 1

Define a variable and write an expression for each situation:
a) The number of books on a shelf is $4$ more than the number of books in a box
b) A taxi fare is a fixed charge of $\mathrm{\$}3$ plus $\mathrm{\$}2$ for each kilometre travelled

Worked Example 2

Write an algebraic expression from each description:
a) $5$ more than a number $x$
b) $3$ times a number $n$
c) $8$ less than a number $p$

Worked Example 3

Model each situation using algebra, clearly defining the variable:
a) A student has $6$ more stickers than Tom
b) A rectangle has length that is $3$ cm more than its width

Worked Example 4

Write an expression from the description, then state what the variable means:
a) The total cost of buying $m$ muffins at $\mathrm{\$}4$ each
b) The number of seats in $r$ rows with $8$ seats in each row

Worked Example 5

Model each situation with an expression involving two variables:
a) The total number of pencils in $x$ red boxes and $y$ blue boxes, if each box holds $10$ pencils
b) The total cost of $a$ adult tickets and $c$ child tickets, if adult tickets cost $\mathrm{\$}12$ and child tickets cost $\mathrm{\$}7$

Worked Example 6

A model is given. Interpret it and state the variable meaning:
a) $3n+5$
b) $2w+2l$
c) $4p-6$

Problems

Problem 1

Define a variable and write an expression for each situation:
a) The number of pens in a drawer is $7$ more than the number of pens in a bag
b) A cinema ticket costs $\mathrm{\$}10$ plus a booking fee of $\mathrm{\$}2$

Problem 2

Write an algebraic expression from each description:
a) $9$ more than a number $x$
b) $4$ times a number $n$
c) $6$ less than a number $p$

Problem 3

Model each situation using algebra, clearly defining the variable:
a) A child has $5$ more marbles than Sam
b) A rectangle has length that is $4$ cm more than its width

Problem 4

Write an expression from the description, then state what the variable means:
a) The total cost of buying $k$ burgers at $\mathrm{\$}6$ each
b) The number of chairs in $t$ rows with $9$ chairs in each row

Problem 5

Model each situation with an expression involving two variables:
a) The total number of apples in $x$ green bags and $y$ red bags, if each bag holds $12$ apples
b) The total cost of $a$ adult tickets and $c$ child tickets, if adult tickets cost $\mathrm{\$}15$ and child tickets cost $\mathrm{\$}8$

Problem 6

A model is given. Interpret it and state the variable meaning:
a) $5n+2$
b) $2l+2w$
c) $7p-4$

Exercises

Understanding and Fluency

1. State what must be done before using a variable in a model:
   a) define what it stands for
   b) choose whether it represents a quantity or a cost
   c) make sure the meaning matches the situation

2. Write an algebraic expression for each description:
   a) $4$ more than a number $x$
   b) $6$ times a number $y$
   c) $10$ less than a number $n$
   d) the sum of a number $p$ and $3$

3. Write an algebraic expression for each description:
   a) a number $m$ increased by $8$
   b) a number $t$ decreased by $5$
   c) $7$ times a number $r$
   d) $2$ times a number $k$, then add $9$

4. Model each situation using algebra and define the variable:
   a) A bag contains $3$ more oranges than a basket
   b) A movie ticket costs $\mathrm{\$}14$ for each person
   c) A phone plan costs $\mathrm{\$}20$ plus $\mathrm{\$}5$ for each extra gigabyte

5. Model each situation using algebra and define the variable:
   a) A rectangle has width $w$ and length $w+6$
   b) A class has $4$ more girls than boys
   c) A bike hire costs $\mathrm{\$}12$ plus $\mathrm{\$}3$ per hour

6. Write an expression for each real-life situation:
   a) The cost of $n$ notebooks at $\mathrm{\$}4$ each
   b) The total number of legs on $c$ chairs if each chair has $4$ legs
   c) The perimeter of a square with side length $s$

7. Write an expression for each real-life situation:
   a) The total cost of $p$ pencils at $\mathrm{\$}2$ each and $3$ erasers
   b) The number of pages in $b$ books if each book has $120$ pages
   c) The amount left after spending $\mathrm{\$}7$ from $\mathrm{\$}m$

8. For each model, state what the variable could represent in a real situation:
   a) $8x$
   b) $x+12$
   c) $3x+5$
   d) $2a+2b$

Reasoning

9. Explain why a model such as $3x+4$ is incomplete unless the variable $x$ is defined.

10. A student says that the expression for “$5$ less than a number $x$” is $5-x$. Explain the mistake.

11. Noah says that in every model, the variable must stand for a single object rather than a number of objects. Is he correct? Explain.

12. Explain why the same algebraic expression can model different real situations.

13. A student writes the cost of $n$ movie tickets at $\mathrm{\$}12$ each as $n+12$. Describe the error.

Problem-solving

14. A fruit shop sells apples for $\mathrm{\$}3$ each. Write an expression for the cost of buying $a$ apples.

15. A school hall has $r$ rows with $12$ chairs in each row. Write an expression for the total number of chairs.

16. A streaming service charges a fixed fee of $\mathrm{\$}9$ plus $\mathrm{\$}2$ for each movie rented. Write an expression for the total cost if $m$ movies are rented.

17. A garden is rectangular. Its width is $w$ m and its length is $w+4$ m. Write an expression for the perimeter.

18. A student has $\mathrm{\$}50$ and spends $\mathrm{\$}x$ on lunch and $\mathrm{\$}8$ on a drink. Write an expression for the amount left.

19. An event sells adult tickets for $\mathrm{\$}15$ and child tickets for $\mathrm{\$}9$. Write an expression for the total income from $a$ adult tickets and $c$ child tickets.

Potential Misunderstandings

• Students may think a variable always stands for one fixed value instead of a quantity that can vary
• Students may forget to define what the variable represents
• Students may reverse phrases such as “less than” when writing expressions
• Students may confuse addition and multiplication in models such as $4n$
• Students may think a model must always use only one variable
• Students may not realise that the same expression can represent different situations depending on the variable definition
• Students may write an equation when only an expression is needed
• Students may think the letter chosen for a variable changes the mathematics, rather than just labelling the quantity