Percentages and income
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Introduction
This set of resources provides opportunities for students to learn skills and solve problems involving percentages in the context of earning a commission.
Achievement standard
By the end of Year 7, students represent natural numbers in expanded form and as products of prime factors, using exponent notation. They solve problems involving squares of numbers and square roots of perfect square numbers. Students solve problems involving addition and subtraction of integers. They use all 4 operations in calculations involving positive fractions and decimals, choosing efficient calculation strategies. Students choose between equivalent representations of rational numbers and percentages to assist in calculations. They use mathematical modelling to solve practical problems involving rational numbers, percentages and ratios in financial and other applied contexts, justifying choices of representation. Students use algebraic expressions to represent situations, describe the relationships between variables from authentic data and substitute values into formulas to determine unknown values. They solve linear equations with natural number solutions. Students create tables of values related to algebraic expressions and formulas and describe the effect of variation.
They apply knowledge of angle relationships and the sum of angles in a triangle to solve problems, giving reasons. Students use formulas for the areas of triangles and parallelograms and the volumes of rectangular and triangular prisms to solve problems. They describe the relationships between the radius, diameter and circumference of a circle. Students classify polygons according to their features and create an algorithm designed to sort and classify shapes. They represent objects two-dimensionally in different ways, describing the usefulness of these representations. Students use coordinates to describe transformations of points in the plane.
They plan and conduct statistical investigations involving discrete and continuous numerical data, using appropriate displays. Students interpret data in terms of the shape of distribution and summary statistics, identifying possible outliers. They decide which measure of central tendency is most suitable and explain their reasoning. Students list sample spaces for single step experiments, assign probabilities to outcomes and predict relative frequencies for related events. They conduct repeated single-step chance experiments and run simulations using digital tools, giving reasons for differences between predicted and observed results.
Content descriptions
Use the 4 operations with positive rational numbers including fractions, decimals and percentages to solve problems using efficient calculation strategies. (AC9M7N06)
Use mathematical modelling to solve practical problems involving rational numbers and percentages, including financial contexts; formulate problems, choosing representations and efficient calculation strategies, using digital tools as appropriate; interpret and communicate solutions in terms of the situation, justifying choices made about the representation. (AC9M7N09)
Teacher resources
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Calculating commission
Calculating commission - Solutions
Student learning resources
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Calculating commission
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Suggested activity sequence
This sequence is intended as a framework to be modified and adapted by teachers to suit the needs of a class group.
You may want to have students complete the worksheet after they have played “crumple and shoot”.
Part A: Calculating commission
- Introduce the concept of commission.
Commission is a way of earning income based on the value of items you have sold. Salespeople, such as real estate agents and car salespeople, are often paid a commission. Real estate agents earn a commission on the value properties they have sold. Car salespeople earn a commission based on the value of cars they have sold. The amount of the commission is usually expressed as a percentage. People who earn a commission often also get a ‘retainer’, which is a smaller amount that is paid regularly like a wage or salary and is not related to how much they have sold.
Explicitly teach students how to calculate commission using a gradual release of responsibility model.
To calculate a commission:
𝐶𝑜𝑚𝑚𝑖𝑠𝑠𝑖𝑜𝑛 = 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 × 𝑡𝑜𝑡𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑠𝑎𝑙𝑒𝑠
To calculate the pay of a salesperson:
𝑃𝑎𝑦 = 𝑐𝑜𝑚𝑚𝑖𝑠𝑠𝑖𝑜𝑛 + 𝑟𝑒𝑡𝑎𝑖𝑛𝑒r
Pay Calculations Retainer
(Earnings from hours)
= hours x rate
= 120 x $20.09
= $2,410.80Commission
= percentage x total value of sale
= 1% × (365 000 + 795,000)
= 1% × 1,160,000 (1% = 1/ 100 = 0.01)
= $11,600Total earnings
= retainer + commission
= 2,410.80 + 11,600
= $14,010.80- Explain the worksheet and provide the following instructions:
- Each problem has a points value attached. The problems in the left hand column aren’t as challenging as those in the right hand column and so have a smaller point value.
- Students are to complete at least 15 points worth of problems.
- Students may work in small groups to discuss the problems and how they might tackle them, troubleshoot and compare ideas, but each student is to write their own solution in their own notebook.
- Provide feedback to students on their learning by marking the problems in the left hand column fairly quickly, working through the remaining 6 problems in a more detailed way, gathering student input about their approaches and their final answers. This task is suitable as an assessment item.
Part B: Crumple and shoot
This is a low tech game that gives students the opportunity to practice a maths skill collaboratively. It will take at least half an hour, up to 50 minutes if you haven’t played the game before.
- Organise students in teams of 3 or 4 – make sure they’re mixed ability groups.
- Use the visualiser to explain the game and the rules to students.
- Give each team a stack of half A4 size scrap paper. Each team appoints one scribe.
- Read out the question from the visualiser. Each team works out and agrees on their answer before the scribe writes it (legibly) on their piece of paper.
- When you think they’ve had enough time call out ‘Answers up’, and one member of each team holds their answer in the air.
Quickly go around and check the answers. If the answer is wrong –take the group’s paper. If the answer is right, the group gets to keep their paper and send a team member to the spot where they’re going to crumple it and shoot it into the wastepaper basket to get a point.
Note the answers are included on the visualiser- To get a point teams have to answer the question correctly and shoot their paper into the bin.
- Crumple and shooters line up and take their turn to crumple their answer sheet and shoot it into the bin (put a line of tape on the floor for shooters, and a tape X marking the spot where the bin should be).
- Keep a tally on the board.
Some tips
- Do a practice question before you start if you haven’t played this game previously.
- You can accept answers to the nearest whole dollar or tens if you like.
- Some questions are worth 2 points, so will need to be written on 2 pieces of paper. Think about your willingness to consider carry through errors ahead of time and be ready to defend your policy.
- You could make it compulsory to rotate shooters amongst the team or make the rule that everyone must shoot at least once if you need to level the playing field.
- Add a 2 point shooters line if a leader emerges too soon and other teams need to catch up.
- You might like to add prizes for winning teams.