Make Sense of Number to Solve Problems
In order to meet the demands of being a worker, a learner and a family and community member, adults need to be able to solve operational problems with numbers.
Three of the progressions in the Make Sense of Number to Solve Problems strand focus on number strategies, and three focus on key aspects of number knowledge. Number strategies are the processes that learners use to solve operational problems with numbers – strategies that make it easier to solve number problems with understanding. Rather than a single strategy for subtracting (or any operation), the most appropriate strategy can and should change flexibly as the numbers and the content or problem change. The three number knowledge progressions describe the key items of knowledge that people need to understand. They include number sequences, place value and number facts. The strategy and knowledge progressions are viewed as interdependent, with strategies creating new knowledge through use and knowledge providing the foundation for developing new strategies. 4
Number strategies can be grouped into counting strategies and partitioning strategies.
Counting strategies involve counting in ones to solve problems, often with the support of objects (such as fingers).
Counting all the objects
This involves joining or separating sets to solve addition or subtraction problems. Learners count all the objects in both sets to find the answer.
Learners count on or back to solve addition or subtraction problems. For example, instead of counting all objects to solve 8 + 5, a learner counts on from 8: 9, 10, 11, 12, 13.
Partitioning strategies are based on using knowledge of number properties to split numbers (partitioning) and combine them again in ways that make it easier to reach the solution. Partitioning strategies include the following strategies.
Deriving from known facts
Learners derive unknown information from a known fact. A learner may solve 25 + 26 by using what they know (25 + 25 = 50), then adding 1 to reach 51. Similarly, if a learner knows 6 x 7 = 42, they can solve 6 x 70 = 420.
Learners break the numbers into ones, tens and hundreds, add numbers of the same place value together and then combine these numbers. For example:
63 + 35 can be solved as (60 + 30) + (3 + 5) = 98.
45 x 6 can be solved as (40 x 6) + (5 x 6) = 270.
Using tidy numbers with compensation
Learners round a number to the nearest ten or hundred, then compensate for what has been added or subtracted. For example,
73 – 29 can be solved as 73 – 30 + 1 = 44
and 64 + 28 can be solved as 64 + 30 – 2 = 92.
Learners change a subtraction problem into an addition problem in order to have an easier route to the solution. For example,
66 – 48 becomes 48 + ? = 66; 48 + 2 + 16 = 66, so 66 – 48 = 18.
Halving and doubling (or dividing by 3 and trebling)
Learners use knowledge of number doubling and trebling. For example,
16 x 8 can be solved as 2 x (8 x 8) = 2 x 64 = 128.
16 x 4 can be solved as 8 x 8 = 64.
3 x 27 can be solved as 9 x 9 = 81.
In order to be an informed citizen, employee and consumer, an adult needs to be able to reason statistically.
Preparing, analysing and interpreting data
The amount of statistical information available to help people make decisions in business, politics, research and everyday life is vast. For example, consumer surveys guide the development and marketing of products, experiments evaluate the safety and efficacy of new medical treatments and statistics sway public opinion on issues and represent (or misrepresent) the quality and effectiveness of commercial products.
Current thinking in statistics education emphasises the need for learners to undertake statistical investigations themselves in order to understand statistics and use them wisely. There are two main types of statistical investigation. In the first type, learners pose questions, gather data and use the data to answer the questions. In the second type of investigation, learners look for patterns and trends in existing data sets and generate questions to be answered. It is the second type of investigative approach that is addressed in these learning progressions. The decision to focus on existing data sets reflects the fact that most adults are seldom engaged in data collection but often need to consider data that has already been collected and presented.
Probability impacts on people’s everyday decision-making in such varied contexts as buying a Lotto ticket, purchasing a car, taking medicine or taking an umbrella to work. (What are the chances of winning Lotto, surviving a crash in that particular model of car, experiencing one of the listed side effects of a certain drug or of the forecasted rain eventuating?) Human nature means that we don’t always make decisions based on facts; intuition and wishful thinking often influence decisions that could be assisted by using a basic knowledge of probability. Terms often used for probability include chance, luck, likelihood, odds, percentage and proportion. Probability is often counterintuitive in the way it operates, so it is important that people do not assume that their initial assessment of a probability situation takes all the relevant factors into account and relates them correctly.
Measure and Interpret Shape and Space
Measures are a cornerstone of mathematics and of our lives: it is difficult to think of anything that is not measured.
Baxter et al., 2006, page 6
Measurement exists in all human cultures along with counting, locating, designing, explaining and playing.
Understanding what a measurable attribute is and becoming familiar with the units and processes that are used in measuring attributes is a major emphasis in the measurement progression. Experiences with measurement build understanding, making adults more aware of the dimensions of the world.5 Measurement also offers an opportunity for learning and applying other mathematics, including number operations, geometric ideas and statistical concepts.
The approach to measurement is a practical one, like the approach taken in the other numeracy progressions. The measurement progression emphasises the appropriateness and precision of the measure to the particular measurement problem or task.
An understanding of geometry and a sense of space are fundamental components of numeracy. Adults use ideas of shape and space when representing and solving problems in real-world situations
and in other areas of mathematics. Geometric representations can help people make sense of area and fractions, while the shapes and patterns in histograms and scatter plots can give insights about data.
Adults use spatial reasoning when following maps, planning routes, designing floor plans and creating art. The Shapes and Transformations progression and the Location progression are more about describing relationships and reasoning than about definitions and theorems.