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NA5-1: Reason with linear proportions.

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Elaboration on this Achievement Objective

This means students will explore linear proportions in a variety of contexts. Linear proportions apply to situations which can be modelled using equivalent fractions, that is, a/b = c/d where a,b,c, and d are integers (usually whole numbers). So proportional reasoning pervades many of the outcomes in all three strands and includes many of the following contexts:

  • Using multiple ways to represent the same number in operator situations, for example 45% x 52 = can be seen as 0.45 x 52 and 45/100 x 52, and comparing the potential results of operator situations by suspending calculation and thinking relationally, 30% x 34 = 60% x 17 (by doubling and halving) or 1.3 x 3.3 < 3.9 x 1.2 (since 1.3 x 3.3 = 3.9 x 1.1).
  • Comparing the results of sharing situations which involve fractional quotients, for example 3 pizzas shared among 5 boys (3/5 pizza each) results in a lesser share than 2 pizzas shared among 3 girls (2/3 pizza each), and find the difference in shares (2/3 -3/5=1/15) .
  • Comparing the size of two fractions, decimals or percentages, using benchmark fractions or equivalence, give the difference between the fractions, and name a fraction between two fractions. For example, 4/7 > 5/9 since 4/7 is 1/14 greater than 1/2 and 5/9 is 1/18 greater than 1/2 or 4/7 = 36/63 and 5/9= 35/63 so the difference between 4/7and 5/9 is 1/63.
  • Reasoning qualitatively about the size effect on a fraction as the numerator, denominator, or both numbers are changed. For example, given the fraction 5/11, reason that 5/10 and 6/11will be greater, 4/11 and 5/12 will be less, and comparing it with 4/10 and 6/12 will require further investigation.
  • Measuring one fraction with another either by converting to equivalent forms or scaling the result of the same divisor acting on one. For example, each trip takes 3/4 of a full tank of petrol. You have 2/5 of a tank. What fraction of a trip can you make? As 3/4 = 15/20 and 2/5 = 8/20 so 8/20 is 8/15 of 15/20 (2/5 ÷3/4 = 8 /15) or 1 ÷3/4 = 4/3 (1 1/3 trips on a full tank) so 2/5 x 4/3 = 8/15 trips with two-fifths of a tank.
  • Other examples of reasoning with linear proportions are discussed through the other achievement objectives.

 

 

Teaching resources for this Achievement Objective
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A neutral solution

Level Five
Number and Algebra
This unit provides an opportunity for students to explore an integrated mathematics and science context: chemical reactions involving acids and bases. Within this, students develop their use of rates and ratios through exploration of linear relationships using tables and graphs. Students also...
  • use rates, ratios, and reasoning with linear proportions to solve a problem.
  • make connections between representations of linear data in tables and graphs.
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Faceprints

Level Four
Level Five
Integrated
Figure It Out activities
This is a level 4-5 activity from the Figure It Out series. A PDF of the student activity is included.
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Fuel for a Commute

Level Five
Number and Algebra
The purpose of this unit is to engage students in an integrated mathematics/technology context: technical knowledge (focused on fuel efficiency and vehicle running costs) through which they can apply their understanding of rates and proportions to solve problems involving distance and time. In turn...
  • Use rates and proportions to compare data.
  • Find and describe a rate of change, for data given from a linear trend.
  • Find and describe a linear trend.
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Fuel for thought

Level Five
Number and Algebra
Units of Work
The purpose of this unit is to engage students in applying their knowledge and skills about rates to solve problems that involve, distance and time within the context of fuel efficiency, depreciation, and speed. This unit integrates with the technology learning area.
  • Use tables, graphs, double number lines, and equations to solve problems with rates.
  • Develop and execute calculator algorithms to solve rate problems.
  • Solve problems where a measure is missing but the rate is given.
  • Solve problems where two measures are given but the rate is unknown.
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Gains and Losses

Level Five
Number and Algebra
Figure It Out activities
This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework. A PDF of the student activity is included.

investigate fractions and proportions

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Mocktails

Level Five
Integrated
Units of Work
This unit provides an opportunity for students to learn and apply knowledge of measurement of capacity and weight, ratios, and money (decimal numbers) in an engaging and authentic food-technology context.
  • Develop a problem solving strategy to work out the unit cost for a produced item (a drink).
  • Measure and compare weight and capacity accurately, recognising that weighing water is equivalent to measuring its capacity in ml.
  • Convert volume measurements to weight measurements.
  • Calculate and apply ratios.
  • M300
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Ratios

Level Five
Integrated
  • Convert between millilitres and litres.
  • Convert ratios to fraction and percentage expressions.
  • Find equivalent ratios.
  • Compare ratios.
  • Reduce ratios to simplest form using common factors.
  • Combine ratios.
  • Solve problems involving rates.
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Ratios

Level Five
Integrated
Units of Work
In this unit students work with ratios in the context of mixing fruit drinks and other situations.
  • Convert between millilitres and litres.
  • Convert ratios to fraction and percentage expressions.
  • Find equivalent ratios.
  • Compare ratios.
  • Reduce ratios to simplest form using common factors.
  • Combine ratios.
  • Solve problems involving rates.
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RJ's Fabulous Juice Bar

Level Five
Number and Algebra
Figure It Out activities
This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework. A PDF of the student activity is included.

solve problems involving linear proportions

find fractions of whole numbers (Level 4)

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Shopping for data

Level Five
Number and Algebra
Rich learning activities
The purpose of this activity is to engage students in using proportional reasoning to find a solution to a problem in context.
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Skateboard Factory

Level Five
Number and Algebra
Figure It Out activities
This is a level 5 number link activity from the Figure It Out series. It relates to Stage 8 of the Number Framework. A PDF of the student activity is included.

solve problems involving proportions

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Spot prizes

Level Five
Number and Algebra
Rich learning activities
The purpose of this activity is to engage students in using proportional thinking to solve a problem.
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Washing detergent proportions

Level Five
Number and Algebra
Rich learning activities
The purpose of this activity is to engage students in finding equivalent rates and interpreting the rates in context.
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Wise Investments

Level Five
Number and Algebra
The purpose of this unit is to engage students in applying their understanding of decimals and percentages to problems within the context of financial literacy. This develops an appreciation of the calculations necessary to make informed decisions when investing.
  • Apply understanding of currency and percentages to problem solving contexts. 
  • Use spreadsheets to display balance, interest, and deposit data. 
  • Apply numerical reasoning involving decimals, currency, and percentages.