## Educators' Guide for Pedagogy and Assessment

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### Learning Area: Mathematics

# Mathematics > LEVEL 8

### Learning Area Outcome: I understand the structure of the number system and the relationship between numbers.

## Subject Focus: Number - The number system

1] I am able to use the number line to illustrate simple inequalities.

2] I can write multiples of numbers using power notation.

3] I can identify all the common factors of up to three numbers.

4] I can identify the highest common factor (HCF) of up to three numbers.

5] I can express any integer as a product of prime factors. I can use prime factorization to work out the square root of large numbers and to work out the LCM and HCF.

6] I can write numbers in standard form and vice-versa.

WRITING

7] I can find the reciprocal of a number.

2] I can write multiples of numbers using power notation.

3] I can identify all the common factors of up to three numbers.

4] I can identify the highest common factor (HCF) of up to three numbers.

5] I can express any integer as a product of prime factors. I can use prime factorization to work out the square root of large numbers and to work out the LCM and HCF.

6] I can write numbers in standard form and vice-versa.

WRITING

7] I can find the reciprocal of a number.

### Learning Area Outcome: I understand the structure of the number system and the relationship between numbers.

## Subject Focus: Number - The number system (Assistive Technology & Other Resources)

1] I can use assistive technology (

*e.g. tablets, computers & calculators)*and other learning resources to learn about numbers and their properties.### Learning Area Outcome: I can calculate mentally and using pencil and paper and assistive technology. I can calculate to the most appropriate level of accuracy. I can check the reasonableness of the answers obtained in calculations by rounding numbers and making rough approximations.

## Subject Focus: Number - Numerical calculations (Whole Numbers, Decimal Numbers & Fraction Numbers - The Four Operations)

1] I can use rounded numbers to make rough approximations.

COGNITIVE LEARNING

2] I can work out mental calculations involving powers and roots,

COGNITIVE LEARNING

3] I can round any decimal number to a given number of significant figures.

COGNITIVE LEARNING

2] I can work out mental calculations involving powers and roots,

*e.g. √1600, 90², 20³, ³√8000.*COGNITIVE LEARNING

3] I can round any decimal number to a given number of significant figures.

### Learning Area Outcome: I can calculate mentally and using pencil and paper and assistive technology. I can calculate to the most appropriate level of accuracy. I can check the reasonableness of the answers obtained in calculations by rounding numbers and making rough approximations.

## Subject Focus: Number - Numerical calculations (Percentages)

1] I can work through simple situations involving percentage increase and decrease (

2] I can work out reverse percentage calculations.

COGNITIVE LEARNING

3] I can work through situations involving successive percentage changes using a multiplying factor.

4] I can work out the simple interest, the principal, the rate, time or the amount.

5] I can use the simple interest formula.

6] I can work out compound interest, appreciation and depreciation.

7] I can use the appreciation and depreciation formulae.

8] I can use the trial and error method to determine the number of years in compound growth and decay situations.

9] I can work out the number of repayments needed to repay a loan.

10] I can work out the total accrued yearly value of an investment.

*e.g. cost and selling price ; discounts ; profit and loss, and percentage error)*.2] I can work out reverse percentage calculations.

COGNITIVE LEARNING

3] I can work through situations involving successive percentage changes using a multiplying factor.

4] I can work out the simple interest, the principal, the rate, time or the amount.

5] I can use the simple interest formula.

6] I can work out compound interest, appreciation and depreciation.

7] I can use the appreciation and depreciation formulae.

8] I can use the trial and error method to determine the number of years in compound growth and decay situations.

9] I can work out the number of repayments needed to repay a loan.

10] I can work out the total accrued yearly value of an investment.

### Learning Area Outcome: I can calculate mentally and using pencil and paper and assistive technology. I can calculate to the most appropriate level of accuracy. I can check the reasonableness of the answers obtained in calculations by rounding numbers and making rough approximations.

## Subject Focus: Number - Numerical calculations (Money & Consumer Mathematics)

1] I know the difference between selling rate and buying rate in currency exchange rates. I can use buying rate and selling rate to convert currencies.

2] I can work through complex situations involving personal and household finance (

2] I can work through complex situations involving personal and household finance (

*e.g. earnings, loans, simple interest, compound interest, income tax and VAT).*## Subject Focus: Number - Numerical calculations (Ratio & Proportion)

1] I can use ratio to solve complex problems.

2] I can draw and interpret simple scale drawings involving 2D plans, bearings, angles of elevation and angles of depression.

WRITTING

3] I can work through situations that involve direct proportion with or without the use of the unitary method.

4] I can use the rules for multiplying and dividing integer powers of numbers (positive, negative and zero indices)

5] I can apply the four rules on numbers in standard form.

2] I can draw and interpret simple scale drawings involving 2D plans, bearings, angles of elevation and angles of depression.

WRITTING

3] I can work through situations that involve direct proportion with or without the use of the unitary method.

4] I can use the rules for multiplying and dividing integer powers of numbers (positive, negative and zero indices)

*e.g.*7^{8}x 7^{-5}= 7^{3}, 6^{5}÷ 6^{-2}= 6^{7}, (2^{3})^{5}= 2^{15}, 9^{0}= 15] I can apply the four rules on numbers in standard form.

## Subject Focus: Number - Numerical calculations (Assistive Technology & Other Resources)

1] I can use assistive technology (

*e.g. tablets, computers and calculators)*and other resources appropriate to this level to calculate and to learn about numerical calculations.### Learning Area Outcome: I can recognise and describe patterns and relationships in various mathematical ways and can use algebraic manipulations.

## Subject Focus: Algebra – Fundamentals of Algebra

1] I can generate the terms of a sequence given the

2] I can use expressions to describe the

3] I can use algebraic notation to represent two or more unknown values in expressions involving +, −, x, ÷ and squares.

4] I can simplify non-linear algebraic expressions by collecting like terms.

5] I can simplify non-linear algebraic expressions by multiplying a single term over a bracket.

6] I can expand two brackets of the form

7] I can simplify any algebraic expression by expanding brackets and collecting like terms.

8] I can add, subtract and simplify algebraic fractions with numerical denominators.

9] I can simplify algebraic fractions with linear algebraic denominators

10] I can evaluate non-linear expressions by substituting directed numbers.

11] I can change the subject of the formula that includes squares and square roots and when the same subject letter occurs more than once.

12] I can write down and solve linear equations involving an unknown and integers or fractions on both sides.

COGNITIVE LEARNING

13] I can solve simple linear inequalities in one variable and represent the solution on the number line.

14] I can factorise expressions by using the common factor method.

15] I can factorise quadratic expressions involving trinomials and difference of two squares .

16] I can solve algebraically two simultaneous linear equations.

17] I can solve quadratic equations by factorisation.

18] I can solve quadratic equations by using the formula.

19] I can solve equations involving algebraic fractions with linear denominators.

20] I can find approximate solutions to equations for which there is not a simple method of solution using the trial and improvement method.

MANAGING LEARNING

21] I can construct tables of values for quadratic functions.

22] I can plot the graph of a quadratic function from a table of values.

23] I can find the gradient of a line from the coordinates of two points on the line.

24] I can write the equation of a straight line given a set of co-ordinates or the line graph.

MANAGING LEARNING

25] I can compare and contrast information from two or more straight line graphs concerning real life situations.

26] I can solve graphically two simultaneous linear equations.

27] I can solve graphically two simultaneous equations: one linear and one quadratic.

28] I can draw quadratic graphs and identify maxima/minima.

MANAGING LEARNING

29] I can use quadratic graphs to find the value of a coordinate given the other.

30] I can use quadratic graphs to solve quadratic equations.

31] I can draw and interpret linear and non-linear graphs arising from real life situations.

32] I can use the function notation. e.g.

33] I can use the rules for multiplying and dividing integer powers (positive, negative and zero indices).

e.g. a

*n*^{th}term.2] I can use expressions to describe the

*n*^{th}term of a linear sequence*e.g.*-3*n*- 1.3] I can use algebraic notation to represent two or more unknown values in expressions involving +, −, x, ÷ and squares.

4] I can simplify non-linear algebraic expressions by collecting like terms.

5] I can simplify non-linear algebraic expressions by multiplying a single term over a bracket.

6] I can expand two brackets of the form

*(ax ± b)(cx ± d)*and*(ax ± b)*^{2 }and can simplify by collecting like terms.7] I can simplify any algebraic expression by expanding brackets and collecting like terms.

8] I can add, subtract and simplify algebraic fractions with numerical denominators.

9] I can simplify algebraic fractions with linear algebraic denominators

*e.g. Simplify*^{4}/_{(2x + 4)}10] I can evaluate non-linear expressions by substituting directed numbers.

11] I can change the subject of the formula that includes squares and square roots and when the same subject letter occurs more than once.

12] I can write down and solve linear equations involving an unknown and integers or fractions on both sides.

COGNITIVE LEARNING

13] I can solve simple linear inequalities in one variable and represent the solution on the number line.

14] I can factorise expressions by using the common factor method.

15] I can factorise quadratic expressions involving trinomials and difference of two squares .

16] I can solve algebraically two simultaneous linear equations.

17] I can solve quadratic equations by factorisation.

18] I can solve quadratic equations by using the formula.

19] I can solve equations involving algebraic fractions with linear denominators.

20] I can find approximate solutions to equations for which there is not a simple method of solution using the trial and improvement method.

*e.g. Solve for x: x*^{3}*-*2*x =*100MANAGING LEARNING

21] I can construct tables of values for quadratic functions.

22] I can plot the graph of a quadratic function from a table of values.

23] I can find the gradient of a line from the coordinates of two points on the line.

24] I can write the equation of a straight line given a set of co-ordinates or the line graph.

MANAGING LEARNING

25] I can compare and contrast information from two or more straight line graphs concerning real life situations.

26] I can solve graphically two simultaneous linear equations.

27] I can solve graphically two simultaneous equations: one linear and one quadratic.

28] I can draw quadratic graphs and identify maxima/minima.

MANAGING LEARNING

29] I can use quadratic graphs to find the value of a coordinate given the other.

30] I can use quadratic graphs to solve quadratic equations.

31] I can draw and interpret linear and non-linear graphs arising from real life situations.

32] I can use the function notation. e.g.

*f*(*x*) = 5*x*- 3.33] I can use the rules for multiplying and dividing integer powers (positive, negative and zero indices).

e.g. a

^{8}× a^{-5}= a^{3}; y^{5}÷ y^{-2}= y^{7 }; (c^{3})^{5}= c^{15}### Learning Area Outcome: I can recognise and describe patterns and relationships in various mathematical ways and can use algebraic manipulations.

## Subject Focus: Algebra – Fundamentals of Algebra (Assistive Technology & Other Resources)

1] I can use assistive technology, (

*e.g. tablets and computers)*and other resources, (e.g. algebra blocks) appropriate to this level to learn about the fundamentals of algebra.### Learning Area Outcome: I understand and can use forms of measurement and can make reasonable estimations.

## Subject Focus: Shape, space and measures – Measures (Angles)

1] I can show and label the sixteen main compass points.

2] I can interpret and use three-figure bearings.

3] I can define the trigonometric ratios (sine, cosine and tangent) as the ratios of sides in a right-angled triangle.

4] I can use trigonometric ratios to find unknown lengths and angles in right-angled triangles.

5] I can use trigonometric ratios in situations that involve angles of elevation and/or depression and bearings.

6] I am able to find missing angles in diagrams by forming and solving algebraic equations.

2] I can interpret and use three-figure bearings.

3] I can define the trigonometric ratios (sine, cosine and tangent) as the ratios of sides in a right-angled triangle.

4] I can use trigonometric ratios to find unknown lengths and angles in right-angled triangles.

5] I can use trigonometric ratios in situations that involve angles of elevation and/or depression and bearings.

6] I am able to find missing angles in diagrams by forming and solving algebraic equations.

### Learning Area Outcome: I understand and can use forms of measurement and can make reasonable estimations.

## Subject Focus: Shape, space and measures – Measures (Length, Area, Volume, Mass & Capacity)

1] I can calculate the area of a sector of a circle

2] I can calculate the area of compound shapes that include sectors of circles.

3] I can calculate the length of an arc of a circle.

4] I can derive and use the formula for the surface area and volume of a prism or cylinder.

COGNITIVE

5] I can calculate:(i) the surface area of a square-based right pyramid; (ii) the surface area of a right circular cone; (iii) the surface area of a sphere; (iv) the volume of a square-based right pyramid; (v) the volume of a square-based right frustum of a pyramid; (vi) the volume of a right circular cone (vii) the volume of a frustum of a right circular cone; (viii) the volume of a sphere; (ix) the volume of compound shapes.

6] I can rearrange formulae for surface area/volume of solids to find the radius, height and to find the slant height of a cone.

2] I can calculate the area of compound shapes that include sectors of circles.

3] I can calculate the length of an arc of a circle.

4] I can derive and use the formula for the surface area and volume of a prism or cylinder.

COGNITIVE

5] I can calculate:(i) the surface area of a square-based right pyramid; (ii) the surface area of a right circular cone; (iii) the surface area of a sphere; (iv) the volume of a square-based right pyramid; (v) the volume of a square-based right frustum of a pyramid; (vi) the volume of a right circular cone (vii) the volume of a frustum of a right circular cone; (viii) the volume of a sphere; (ix) the volume of compound shapes.

6] I can rearrange formulae for surface area/volume of solids to find the radius, height and to find the slant height of a cone.

### Learning Area Outcome: I understand and can use forms of measurement and can make reasonable estimations.

## Subject Focus: Shape, space and measures – Measures (Assistive Technology & Other Resources)

1] I can use assistive technology (

*e.g. tablets computers and calculators)*and other resources (*e.g. 2D and 3D plastic shapes, measuring instruments)*appropriate to this level to learn about measures.### Learning Area Outcome: I can recognise and describe the properties of shapes. I can use these properties to construct shapes using appropriate mathematical instruments and to prove given geometric statements.

## Subject Focus: Shape, space and measures – Euclidean geometry (Lines & Lines Segments)

1] I can distinguish between lines and line segments.

### Learning Area Outcome: I can recognise and describe the properties of shapes. I can use these properties to construct shapes using appropriate mathematical instruments and to prove given geometric statements.

## Subject Focus: Shape, space and measures – Euclidean geometry (Triangles)

1] I can interpret and use Pythagoras’ Theorem in 2D shapes.

COGNITIVE LEARNING

2] I can interpret and use the Converse of Pythagoras’ Theorem in 2D shapes. I can deduce that a triangle whose sides are in the ratio of 3:4:5 or in the ratio of 5:12:13 is a right angled triangle.

COGNITIVE LEARNING

2] I can interpret and use the Converse of Pythagoras’ Theorem in 2D shapes. I can deduce that a triangle whose sides are in the ratio of 3:4:5 or in the ratio of 5:12:13 is a right angled triangle.

### Learning Area Outcome: I can recognise and describe the properties of shapes. I can use these properties to construct shapes using appropriate mathematical instruments and to prove given geometric statements.

## Subject Focus: Shape, space and measures – Euclidean geometry (Quadrilaterals)

1] I can follow a proof that the angle sum of a quadrilateral is 360˚.

## Subject Focus: Shape, space and measures – Euclidean geometry (Polygons)

1] I can describe the properties of regular polygons related to sides, angles and diagonals, and can describe their symmetrical properties.

READING AND UNDERSTANDING

2] I can calculate and use the sums of the interior and exterior angles of regular and irregular polygons.

READING AND UNDERSTANDING

2] I can calculate and use the sums of the interior and exterior angles of regular and irregular polygons.

## Subject Focus: Shape, space and measures – Euclidean geometry (3D Shapes)

1] I can draw the plan, front elevation and side elevation of a given simple 3D shape.

2] I can draw the net of a cube, a cuboid, a triangular prism and a square-based right pyramid.

2] I can draw the net of a cube, a cuboid, a triangular prism and a square-based right pyramid.

## Subject Focus: Shape, space and measures – Euclidean geometry (Circles)

1] I can identify a chord, a tangent, an arc, a sector and a segment of a circle.

2] I can interpret and apply the circle theorems. I can also follow a proof for each circle theorem.

The circle theorems are:

(i)The angle in a semicircle is a right angle.

(ii) The angle which an arc of a circle subtends at the centre is twice that which it subtends at any other point on the remaining part of the circumference.

(iii) Angles in the same segment of a circle are equal.

(iv) The opposite angles of a cyclic quadrilateral are supplementary. (Angles in opposite segments are supplementary).

(v) The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

(vi)The angle between the radius and the tangent at the point of contact is a right angle.

(vii) Equal chords are equidistant from the centre.

(viii) Chords which are equidistant from the centre of a circle are equal.

(ix) The perpendicular bisector of a chord passes through the centre.

(x) A straight line drawn from the centre of a circle to bisect a chord is at right angles to the chord.

(xi) If two tangents are drawn to a circle from a point outside the circle, then (a) the tangents are equal in length; (b) the angle between the tangents is bisected by the line joining the point of intersection of the tangents to the centre; and (c) this line also bisects the angle between the radii drawn to the points of contact.

(xii) If a straight line touches a circle, and from the point of contact a chord is drawn, the angle which the chord makes with the tangent is equal to the angle in the alternate segment.

2] I can interpret and apply the circle theorems. I can also follow a proof for each circle theorem.

The circle theorems are:

(i)The angle in a semicircle is a right angle.

(ii) The angle which an arc of a circle subtends at the centre is twice that which it subtends at any other point on the remaining part of the circumference.

(iii) Angles in the same segment of a circle are equal.

(iv) The opposite angles of a cyclic quadrilateral are supplementary. (Angles in opposite segments are supplementary).

(v) The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

(vi)The angle between the radius and the tangent at the point of contact is a right angle.

(vii) Equal chords are equidistant from the centre.

(viii) Chords which are equidistant from the centre of a circle are equal.

(ix) The perpendicular bisector of a chord passes through the centre.

(x) A straight line drawn from the centre of a circle to bisect a chord is at right angles to the chord.

(xi) If two tangents are drawn to a circle from a point outside the circle, then (a) the tangents are equal in length; (b) the angle between the tangents is bisected by the line joining the point of intersection of the tangents to the centre; and (c) this line also bisects the angle between the radii drawn to the points of contact.

(xii) If a straight line touches a circle, and from the point of contact a chord is drawn, the angle which the chord makes with the tangent is equal to the angle in the alternate segment.

## Subject Focus: Shape, space and measures – Euclidean geometry (Constructions)

1] I can construct the perpendicular bisector of a line segment, the perpendicular from a point to a line and the angle bisector of a pair of intersecting lines using a straight edge and compasses only and by using a dynamic geometry software package.

2] I can construct triangles given various conditions using a ruler and compasses and dynamic geometry software.

3] I can construct regular polygons using ruler, protractor and compasses and a suitable programming environment such as Logo.

2] I can construct triangles given various conditions using a ruler and compasses and dynamic geometry software.

3] I can construct regular polygons using ruler, protractor and compasses and a suitable programming environment such as Logo.

## Subject Focus: Shape, space and measures – Euclidean geometry (Congruency & Similarity)

1] I can explain the concept of congruency. I can identify congruent shapes.

2] I can prove two triangles are congruent using SSS, SAS, ASA and RHS. I can use the fact that two triangles are congruent in order to find the lengths of missing sides and angles.

COGNITIVE LEARNING

2] I can prove two triangles are congruent using SSS, SAS, ASA and RHS. I can use the fact that two triangles are congruent in order to find the lengths of missing sides and angles.

COGNITIVE LEARNING

## Subject Focus: Shape, space and measures – Euclidean geometry (Assistive Technology & Other Resources)

1] I can use assistive technology (

*e.g. tablets and computers)*and other resources (*e.g. 2D and 3D plastic shapes)*appropriate to this level to learn about properties of shapes.### Learning Area Outcome: I can describe position and movement of shapes in a plane.

## Subject Focus: Shape, space and measures – Transformation geometry (Reflections)

1] I can deduce that reflections preserve length and angle.

COGNITIVE LEARNING

COGNITIVE LEARNING

### Learning Area Outcome: I can describe position and movement of shapes in a plane.

## Subject Focus: Shape, space and measures – Transformation geometry (Rotations)

1] I can find the centre of rotation by inspection and/or by construction.

COGNITIVE

2] I can deduce that rotations preserve length and angle.

COGNITIVE LEARNING

COGNITIVE

2] I can deduce that rotations preserve length and angle.

COGNITIVE LEARNING

### Learning Area Outcome: I can describe position and movement of shapes in a plane.

## Subject Focus: Shape, space and measures – Transformation geometry (Translations)

1] I can deduce that translations preserve length and angle.

### Learning Area Outcome: I can describe position and movement of shapes in a plane.

## Subject Focus: Shape, space and measures – Transformation geometry (Enlargements)

1] I can enlarge a shape given the centre of enlargement using positive and negative scale factors (integral and fractional).

2] I can deduce that enlargements preserve angle but not length.

COGNITIVE LEARNING

2] I can deduce that enlargements preserve angle but not length.

COGNITIVE LEARNING

### Learning Area Outcome: I can describe position and movement of shapes in a plane.

## Subject Focus: Shape, space and measures – Transformation geometry (Combining Transformations)

1] I can create tessellating shapes and draw a tessellation.

2] I can transform 2D shapes by a combination of transformations.

COGNITIVE LEARNING

2] I can transform 2D shapes by a combination of transformations.

COGNITIVE LEARNING

### Learning Area Outcome: I can describe position and movement of shapes in a plane.

## Subject Focus: Shape, space and measures – Transformation geometry (Assistive Technology & Other Resources)

1] I can use assistive technology (

*e.g. tablets and computers)*and other resources*(e.g. 2D and 3D plastic shapes)*appropriate to this level to learn about transformation geometry.### Learning Area Outcome: I can collect, analyse, interpret and communicate statistical information.

## Subject Focus: Data handling and chance – Statistics

1] I can explain the difference between discrete and continuous data.

2] I can construct a frequency table using grouped or ungrouped continuous data.

PRACTICAL

3] I can interpret a histogram with equal class intervals.

4] I can construct a histogram with equal class intervals.

5] I can find the mean of a set of grouped or ungrouped data from a frequency table.

6] I can find the median of a set of ungrouped data from a frequency table.

7] I can find the mode of a set of ungrouped data from a frequency table.

8] I can find the range of a set of ungrouped data from a frequency table.

2] I can construct a frequency table using grouped or ungrouped continuous data.

PRACTICAL

3] I can interpret a histogram with equal class intervals.

4] I can construct a histogram with equal class intervals.

5] I can find the mean of a set of grouped or ungrouped data from a frequency table.

6] I can find the median of a set of ungrouped data from a frequency table.

7] I can find the mode of a set of ungrouped data from a frequency table.

8] I can find the range of a set of ungrouped data from a frequency table.

### Learning Area Outcome: I can collect, analyse, interpret and communicate statistical information.

## Subject Focus: Data handling and chance – Statistics (Assistive Technology & Other Resources)

1] I can use assistive technology (

*e.g. tablets and computers)*and other learning resources to learn about statistics.### Learning Area Outcome: I understand ideas of chance and uncertainty.

## Subject Focus: Data handling & chance – Probability

1] I can work out the probability of an event from a frequency table.

2] I can work out the probability of mutually exclusive events.

3] I know the difference between dependent and independent events.

4] I can work out the probability of independent and dependent events.

5] I can construct and use a probability tree (tree diagram) to work out the probability of independent and dependent events.

2] I can work out the probability of mutually exclusive events.

3] I know the difference between dependent and independent events.

4] I can work out the probability of independent and dependent events.

5] I can construct and use a probability tree (tree diagram) to work out the probability of independent and dependent events.

### Learning Area Outcome: I understand ideas of chance and uncertainty.

## Subject Focus: Data handling & chance – Probability (Assistive Technology & Other Resources)

1] I can use assistive technology (

*e.g. tablets, computers and calculators)*and other learning resources to learn about probability.