Planning tool
Year levels
Strands
Expected level of development
Australian Curriculum Mathematics V9: AC9M10A03
Numeracy Progression: n/a
At this level, students connect the graphical representation of exponential functions and their corresponding exponential rule. They use digital tools to explore how systematically making changes to the value of the base, pronumerals or variables changes the shape of the corresponding exponential graph.
Students connect repeated multiplication of a number with exponential number patterns and functions, and how this is displayed as a constant multiplier (ratio) between consecutive values of the dependent variable in a table of values. They also develop an appreciation of how the value of the base number affects whether the functions represent growth or decay.
Exponential equations are solved graphically with the use of digital tools and algebraically by applying the index laws. Students are encouraged to revise the exponent laws and expand their application to working with exponential expressions involving negative exponents and simple fractional exponents.
Introduce problems that involve finding the point of intersection between exponential and linear functions.
Teaching and learning summary:
 Revise exponent laws with integer values and extend to simple fractional exponents.
 Connect the rules of exponential functions with their graphical representations.
 Use tables of values to establish if consecutive values of the dependent variable have a constant ratio, and hence represent an exponential function.
 Use digital tools to explore the graphs of exponential functions and manipulate variables systematically so that patterns in the changes may be observed.
 Solve exponential equations graphically and algebraically using the exponent laws.
Students will:
 connect algebraic and graphical representations of exponential relations
 use tables, graphs and algebra to solve exponential equations
 use digital tools to explore, investigate and solve problems involving exponential equations.
Some students may:
 confuse exponential functions with quadratic functions by having the variable as the base, rather than the exponent. Exponential functions can have different base values; however, the variable is the exponent, for example, 2^{x }= 16.
 underestimate the rapid growth and decay of exponential relations and how this differs from quadratic and linear functions.
 misinterpret y = 2 × 3^{x }as y = (2 × 3 )^{x} rather than y = 2(3)^{x}.
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 to connect repeated multiplication of a number with exponential number patterns and functions.
 We are learning to identify patterns resulting from varying the base number of the exponential function y = b^{x} for 0 < b < 10.
 We are learning to identify patterns resulting from varying the values of a and c in the rules y = a × 2^{x} and y = 2^{x} + c.
Why are we learning about this?
Exponential functions and their corresponding graphs are used to describe and help us understand relationships in the world around us. Some examples are how quickly a virus spreads, depreciation rate of a car, length of time needed to repay a loan, critically evaluating difference alternatives and making predictions about the weather.
What to do
Introductory activities
 Start by refreshing and reviewing the concepts of integer exponents and exponential notation here.
 Then think about how growth can be represented here.
 For each of the following number patterns, decide if it is an increasing (growing) or decreasing (decaying) number pattern. Check your answer by multiplying the next few values in the number pattern using a calculator.
5, 5 × 5, 5 × 5 × 5 …
1.04, 1.04 × 1.04, 1.04 × 1.04 × 1.04 …
0.8, 0.8 × 0.8, 0.8 × 0.8 × 0.8 …
Task 1: Exponential functions with different bases
 Use digital tools such as the GeoGebra graphing tool or a graphics calculator to explore the graphs of exponential functions with different base numbers, and observe the resulting patterns.
 To start, type y = 2^{x} into the rule input box on the top left.
 Next, type y = b^{x} into the next rule. Notice the option to add a slider for b, select ‘b’ to add the slider. The graph corresponding to
b = 1 will be shown, which is a horizontal straight line, as 1 raised to any power will be equal to 1, for example: 1^{3} = 1 × 1 × 1 = 1.  Move slider b to values greater than 1 and compare the resulting graphs to the graph of y = 2^{x}. Observe any differences and similarities, such as x and y intercepts and changes in the shape of the graph.
 Move slider b to values between 0 < b < 1 and compare the resulting graphs to the graph of y = 2^{x}. Observe any differences and similarities, such as x and y intercepts and changes in the shape of the graph.
Task 2: Exploring more graphs of exponential functions
 Use the same tool to explore more graphs of exponential functions.
 Enter the following four rules into separate input boxes and observe any differences and similarities. (Note: y = 2 × 2^{x} can also be written as y = 2(2)^{x}.)
y = 2(2)^{x} y = 5(2)^{x} y = 10(2)^{x} y = (2)^{x}2
 Now enter the following four rules into separate input boxes and observe any differences and similarities.
y = 2^{x} + 5 y = 2^{x} – 8 y = 2^{x} + 0.5 y = 2^{x} – 0.9
Success criteria
 I can connect repeated multiplication of a number with exponential number patterns and functions.
 I can identify patterns resulting from varying the base number of the exponential function y = b^{x} between 0 < b < 10.
 I can identify patterns resulting from varying the values of a and c in the rules y = a × 2^{x} and y = 2^{x} + c.
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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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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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Questioning
A culture of questioning should be encouraged and students should be comfortable to ask for clarification when they do not understand.
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Differentiation
Differentiation involves teachers creating lessons that are accessible and challenging for all students.
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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.

Indices and logarithms
This teacher guide comprises detailed information on indices, exponential graphs and equations, exponential growth and logarithms. There are worked examples and exercises for students, including answers.
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Writing exponential functions
This interactive activity requires students to work backwards to identify the exponential function that matches the graph. Students can work through many examples to make meaningful connections between the graphical and algebraic representations.
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Exponential functions and growth rate
This interactive activity allows students to manipulate the graph coordinates to investigate the effect on the related equation.
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Representing exponential growth
This resource comprises a lesson plan, guidance on exponential growth, lesson materials and practice questions. There is a specific focus on explaining the connections between the equation and the graph, and on showing students how to interpret exponential growth.
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Assessment
By the end of Year 10, students can recognise the connection between algebraic and graphical representations of exponential relations and solve related exponential equations.

Representing exponential growth
This resource comprises a lesson plan, guidance on exponential growth, lesson materials and practice questions. There is a specific focus on explaining the connections between the equation and the graph, and on showing students how to interpret exponential growth.
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Understanding decay
This series of questions asks students to think about exponential decay in tabular and graphical form.
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