## Grade 8

## Mathematics Curriculum

**Year 8 Level Description**

The proficiency strands **understanding, fluency, problem-solving,** and **reasoning** are an integral part of mathematics content across the three content strands: number and algebra, measurement and geometry, and statistics and probability. The proficiencies reinforce the significance of working mathematically within the content and describe how the content is explored or developed. They provide the language to build in the developmental aspects of the learning of mathematics. The achievement standards reflect the content and encompass the proficiencies.

At this year level:

**understanding**includes describing patterns involving indices and recurring decimals, identifying commonalities between operations with algebra and arithmetic, connecting rules for linear relations with their graphs, explaining the purpose of statistical measures, and explaining measurements of perimeter and area.**fluency**includes calculating accurately with simple decimals, indices, and integers; recognizing the equivalence of common decimals and fractions including recurring decimals; factorizing and simplifying basic algebraic expressions and evaluating perimeters and areas of common shapes and volumes of three-dimensional objects.**problem-solving**includes formulating and modeling practical situations involving ratios, profit and loss, areas and perimeters of common shapes, and using two-way tables and Venn diagrams to calculate probabilities.**reasoning**includes justifying the result of a calculation or estimation as a reasonable, deriving probability from its complement, using congruence to deduce properties of triangles, finding estimates of means and proportions of populations.

## Number and Algebra

- Number and Place Value
- Real Numbers
- Money & financial mathematics
- Patterns and Algebra
- Linear and non- linear relationships

Use index notation with numbers to establish the index laws with positive integral indices and the zero index (ACMNA182)

- evaluating numbers expressed as powers of positive integers.

Carry out the four operations with rational numbers and integers, using efficient mental and written strategies and appropriate digital technologies (ACMNA183)

- using patterns to assist in finding rules for the multiplication and division of integers.
- using the number line to develop strategies for adding and subtracting rational numbers.

Investigate terminating and recurring decimals (ACMNA184)

- recognizing terminating, recurring and non-terminating decimals and choosing their appropriate representations.

Investigate the concept of irrational numbers, including π (ACMNA186)

- understanding that the real number system includes irrational numbers.

Solve problems involving the use of percentages, including percentage increases and decreases, with and without digital technologies (ACMNA187)

- using percentages to solve problems, including those involving mark-ups, discounts, and GST.
- using percentages to calculate population increases and decreases.

Solve a range of problems involving rates and ratios, with and without digital technologies (ACMNA188)

- understanding that rate and ratio problems can be solved using fractions or percentages and choosing the most efficient form to solve a particular problem.
- calculating population growth rates in Australia and Asia and explaining their difference.

Solve problems involving profit and loss, with and without digital technologies (ACMNA189)

- expressing profit and loss as a percentage of cost or selling price, comparing the difference
- investigating the methods used in retail stores to express discounts.

Extend and apply the distributive law to the expansion of algebraic expressions (ACMNA190)

- applying the distributive law to the expansion of algebraic expressions using strategies such as the area model.

Factories algebraic expressions by identifying numerical factors (ACMNA191)

- recognizing the relationship between factorizing and expanding
- identifying the greatest common divisor (highest common factor) of numeric and algebraic expressions and using a range of strategies to factorise algebraic expressions.

Simplify algebraic expressions involving the four operations (ACMNA192)

- understanding that the laws used with numbers can also be used with algebra.

Plot linear relationships on the Cartesian plane with and without the use of digital technologies (ACMNA193)

- completing a table of values, plotting the resulting points, and determining whether the relationship is linear.
- finding the rule for a linear relationship

Solve linear equations using algebraic and graphical techniques. Verify solutions by substitution (ACMNA194)

- solving real-life problems by using variables to represent unknowns.

## Measurement & Geometry

- Using units of measurement
- Geometric Reasoning

Choose appropriate units of measurement for area and volume and convert from one unit to another (ACMMG195)

- choosing units for an area including mm
^{2}, cm^{2}, m^{2}, hectares, km^{2}, and units for volume including mm^{3}, cm^{3}, m^{3} - recognizing that the conversion factors for area units are the squares of those for the corresponding linear units.
- recognizing that the conversion factors for volume units are the cubes of those for the corresponding linear units.

Find perimeters and areas of parallelograms, trapeziums, rhombuses, and kites (ACMMG196)

- establishing and using formulas for areas such as trapeziums, rhombuses, and kites

Investigate the relationship between features of circles such as circumference, area, radius, and diameter. Use formulas to solve problems involving circumference and area (ACMMG197)

- investigating the circumference and area of circles with materials or by measuring, to establish an understanding of formulas.
- investigating the area of circles using a square grid or by rearranging a circle divided into sectors

Develop formulas for volumes of rectangular and triangular prisms and prisms in general. Use formulas to solve problems involving volume (ACMMG198)

- investigating the relationship between volumes of rectangular and triangular prism

Solve problems involving duration, including using 12- and 24-hour time within a single time zone (ACMMG199)

- identifying regions in Australia and countries in Asia that are in the same time zone.

Define congruence of plane shapes using transformations (ACMMG200)

- understanding the properties that determine the congruence of triangles and recognizing which transformations create congruent figures.
- establishing that two figures are congruent if one shape lies exactly on top of the other after one or more transformations (translation, reflection, rotation), and recognizing that the matching sides and the matching angles are equal.

Develop the conditions for congruence of triangles (ACMMG201)

- investigating the minimal conditions needed for the unique construction of triangles, leading to the establishment of the conditions for congruence (SSS, SAS, ASA, and RHS)
- solving problems using the properties of congruent figures
- constructing triangles using the conditions for congruence

Establish properties of quadrilaterals using congruent triangles and angle properties, and solve related numerical problems using reasoning (ACMMG202)

- establishing the properties of squares, rectangles, parallelograms, rhombuses, trapeziums, and kites.
- identifying properties related to side lengths, parallel sides, angles, diagonals, and symmetry.

## STATISTICS & PROBABILITY

- Chance
- Data representation and interpretation

Identify complementary events and use the sum of probabilities to solve problems (ACMSP204)

- identifying the complement of familiar events
- understanding that probabilities range between 0 to 1 and that calculating the probability of an event allows the probability of its complement to be found.

Describe events using the language of 'at at least', exclusive 'or' (A or B but not both), inclusive 'or' (A or B or both), and 'and'. (ACMSP205)

- posing 'and', 'or' and 'not' probability questions about objects or people

Represent events in two-way tables and Venn diagrams and solve related problems (ACMSP292)

- using Venn diagrams and two-way tables to calculate probabilities for events, satisfying 'and', 'or' and 'not' conditions.
- understanding that representing data in Venn diagrams or two-way tables facilitates the calculation of probabilities.
- collecting data to answer the questions using Venn diagrams or two-way tables.

Investigate techniques for collecting data, including census, sampling, and observation (ACMSP284)

- identifying situations where data can be collected by census and those where a sample is appropriate.

Explore the practicalities and implications of obtaining data through sampling using a variety of investigative processes (ACMSP206)

- investigating the uses of random sampling to collect data.

** **Explore the variation of means and proportions of random samples drawn from the same population (ACMSP293)

- using sample properties to predict characteristics of the population.

Investigate the effect of individual data values, including outliers, on the mean and median (ACMSP207)

- using displays of data to explore and investigate effects.

## ACHIEVEMENT STANDARD

By the end of Year 8, students solve everyday problems involving rates, ratios and percentages. They describe index laws and apply them to whole numbers. They describe rational and irrational numbers. Students solve problems involving profit and loss. They make connections between expanding and factorising algebraic expressions. Students solve problems relating to the volume of prisms. They make sense of time duration in real applications. They identify conditions for the congruence of triangles and deduce the properties of quadrilaterals. Students model authentic situations with two-way tables and Venn diagrams. They choose appropriate language to describe events and experiments. They explain issues related to the collection of data and the effect of outliers on means and medians in that data.

Students use efficient mental and written strategies to carry out the four operations with integers. They simplify a variety of algebraic expressions. They solve linear equations and graph linear relationships on the Cartesian plane. Students convert between units of measurement for area and volume. They perform calculations to determine perimeter and area of parallelograms, rhombuses and kites. They name the features of circles and calculate the areas and circumferences of circles. Students determine the probabilities of complementary events and calculate the sum of probabilities.