Planning tool
Year levels
Strands
Expected level of development
Australian Curriculum Mathematics V9: AC9M7A03
Numeracy Progression: Number patterns and algebraic thinking: P7
At this level, students will solve linear equations with one variable where the solution is a natural number. Begin with solving singlestep equations to unpack the basic principles before moving to twostep equations. To challenge students, have them apply the same principles to solve equations that begin with variables on both sides. This reinforces the application of arithmetic rules to pronumerals.
Students can work backwards by substituting their solution to see if they are correct. This is a good lesson in extended independent practice, while reinforcing the general principle of inverting or ‘undoing’ operations and to reverse the order of operations.
Introduce a variety of materials to help explain what balancing and substitution really means. Algebra tiles are excellent concrete and representational tools in the classroom. Support students’ understanding by emphasising that equations are required to be balanced on each side of the equation. Use balance scales to demonstrate this and question students about what happens when weights are taken away or added, or what needs to happen for the scales to rebalance again.
Teaching and learning summary:
 Reinforce the concept that equations need to be balanced on both sides of the equals sign.
 Demonstrate how to verify solutions using substitution.
 Show students how to check their answer by working backwards or ‘undoing’ the equation, linking the order of operations and inverses.
 Use algebra tiles and representational tools to aid learning.
 Encourage correct algebraic notation.
 Give students multiple opportunities to practise.
Students:
 write an expression or equation from a word problem using pronumerals
 substitute a value for a pronumeral into an expression (or formula)
 manipulate pronumerals in expressions mathematically
 solve twostep equations with variables on one side.
Some students may:
 try to guess the answers to simple equations but then find more complicated equations difficult when guessing is not possible.
 not realise that pronumerals are standing in place of a number.
 confuse the variable x with the multiplication symbol.
 not respect the ‘grammar’ of mathematics – algebra is a language with its own rules of grammar and there is a specific way to ‘read’ or ‘write’ an expression.
 think that 3x + 2x = 5x^{2}. To address this, be very specific with your language when reading expressions and equations. Say: ‘three lots of x is added to two lots of x to give a total of five lots of x’. And that 5x^{2} should be said as ‘five lots of x squared’.
The Learning from home activities are designed to be used flexibly by teachers, parents and carers, as well as the students themselves. They can be used in a number of ways including to consolidate and extend learning done at school or for home schooling.
Learning intention
 We are learning how equations were first used to solve important everyday challenges.
 We are learning how to solve linear equations.
Why are we learning about this?
Being able to solve equations makes us much more powerful mathematicians. If we want to use mathematics to answer important questions, we will first need to solve an equation. In finding the answer to our equation, we will find the answer to our realworld question. We begin with simple equations that you might be able to solve simply by looking at them. But wait, not so fast! Equations can become much more complicated. We are starting here to sharpen our skills so that we can use them to solve more difficult equations.
What to do
Part A
We are going to transport ourselves to an ancient, bustling marketplace. There are lots of colours, smells and noises. In some ways it is very similar to the markets you might be used to in our world of today, but there are some important differences. We have some nifty tools at our disposal such as our modern scales. Let us imagine a marketplace we have seen before, but without any of its modern technology.
You will need the following:
 produce from your fridge or pantry to sell – something that does not state its weight, for example, a strawberry, a lettuce leaf or some grapes
 balance scales, for example, old kitchen scales, or borrow some at school
 weights, for example, M&Ms, Smarties, dried beans or black peppercorns – something that you have lots of, and are all the same shape and size.
A customer (maybe one of your siblings or friends) comes along and asks to buy some of your grapes. They have a budget of 30 Smarties, so they plonk down their 30 Smarties on one side of your scales and start filling the other side with your grapes.
But the grapes and Smarties do not balance evenly because 7 grapes are too light and 8 are too heavy. What can you do?
You add a couple Smarties to the side of the scales with the 7 grapes until it balances perfectly. Now you have a Babylonian equation.
7g + 2 = 30
On the scales there are 7 grapes and 2 Smarties on one side and 30 Smarties on the other. This means your customer will receive 2 Smarties as change plus their 7 grapes.
Part B
Your customer wants to know how much they paid per grape. An important question that we can only answer by solving our equation above. Let’s work it out in steps.
 7g + 2 = 30; we subtract 2 from both sides to ‘undo’ the +2
 7g = 28; we divide both sides by 7 to ‘undo’ the ×7.
 g = 4; so each grape cost 4 Smarties.
Now it is your turn to create your own equations that you solve from the transactions at your market stall.
Hint: place a set amount of weight on one side of the scales and then place a combination of produce and weights on the other until they balance. Never remove a weight.
Here is another example to guide you:
 3 cherry tomatoes and 4 weights on one side and 25 weights on the other.
3t + 4 = 25
∴ t = 7
Extension task A
Try putting produce on both sides of your scales. This will make your equations more complex. For example, 3 cucumbers and 20 weights on one side and 5 cucumbers and two weights on the other.
3c + 20 = 5c + 2
How can we solve this? The best strategy would be to remove the 3 cucumbers on the left and remove 3 cucumbers from the right. Mathematically, we have subtracted 3c from each side of the equation. Our equation is now:
20 = 2c + 2
You already know how to solve that. Did you find that each cucumber must weigh 9 units?
Extension task B
You might have noticed that in your market stall all the equations looked quite similar and had to be solved in the same way; that is, you needed to subtract a number from both sides and then divide both sides by a number.
To extend our skills, we need to practise with a variety of equations.
Try to solve the equations below using the same technique, but keep in mind that you will not always be subtracting and dividing, you might need to apply a different set of operations.
1. c5 + 6 = 11 
2. 3f – 4 = 11 
3. g7 – 12 = –4 
4. 5q4 + 8 = 3v 
5. 8b – 5 = 5b – 8 
Success criteria
 I can generate and solve equations that model realworld scenarios.
 I can ‘undo’ all four operations in the correct order.
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Teaching strategies
A collection of evidencebased teaching strategies applicable to this topic. Note we have not included an exhaustive list and acknowledge that some strategies such as differentiation apply to all topics. The selected teaching strategies are suggested as particularly relevant, however you may decide to include other strategies as well.

Explicit teaching
Explicit teaching is about making the learning intentions and success criteria clear, with the teacher using examples and working though problems, setting relevant learning tasks and checking student understanding and providing feedback.
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Concrete, Representational, Abstract (CRA)
The CRA model is a threephased approach where students move from concrete or virtual manipulatives, to making visual representations and on to using symbolic notation.
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Mathematics investigation
By giving students meaningful problems to solve they are engaged and can apply their learning, thereby deepening their understanding.
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Worked examples
A worked example is not just a preworked question that is given to the students. There are several types of worked examples and ways of using them.
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Teaching resources
A range of resources to support you to build your student's understanding of these concepts, their skills and procedures. The resources incorporate a variety of teaching strategies.

Introduction to algebra
From the AMSI Supporting Australian Mathematics Project, this resource gives detailed explanations for teachers and students of the curriculum content, worked examples and assessment questions. This resource focuses on introducing algebra and the use of pronumerals.
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The laws of arithmetic and their use in algebra
From the AMSI Supporting Australian Mathematics Project, this resource gives detailed explanations for teachers and students of the curriculum content, worked examples and assessment questions. This resource focuses on how to manipulate pronumerals in algebraic expressions.
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Always a multiple?
Work in groups or as a class and think mathematically to solve this clever puzzle. The number trick is fun for students, yet they need to think algebraically to work through the possible reasons for why the trick works out. There can be multiple approaches to solve this problem, so encourage students to communicate their method to the class.
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Assessment
By the end of Year 7, students can solve linear equations involving one variable with natural number solutions. Students can work backwards using substitution to verify solutions.

Using equations to solve problems
This resource is an online question bank of simple equations to solve. Although many can be solved intuitively, guide students to identify variables, formulate equations and solve problems with substitution. This helps students to connect their mathematical instincts to finding answers algebraically.
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Comparing relationships with tables
This series of tasks asks students to gather information from data presented in tables. Students could work independently or in pairs to consider scenarios to identify the relationships in the data, how to represent it, how to solve it and how further manipulation of the data changes or does not change the relationship between variables.
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