Mathematics overview: Stage 10 (Modified) Unit
Hours
Calculate with roots and integer indices
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Manipulate algebraic expressions by expanding the product of two binomials
• 7
Manipulate algebraic expressions by factorising a quadratic expression of the form x² + bx + c
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Understand and use the gradient of a straight line to solve problems
7
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Plot and interpret graphs of quadratic functions Solve problems involving similar shapes
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Calculate exactly with multiples of π
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Apply Pythagoras’ theorem in two dimensions
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Use geometrical reasoning to construct simple proofs
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Use tree diagrams to list outcomes
12
Constructions
6
Assessment 1 Calculating Space (Segments and Sectors) Calculating Space (trigonometry) Proportional reasoning Assessment 2 Algebraic Proficiency - Visualising
7
Understanding Risk
4
Assessment 3
Calculations, Checking & Rounding Indices, Roots, Reciprocals, Hierarchy Factors, Multiples, Primes, Std Form & Surds Algebra:basics,setting up,rearranging,solving EOY exams Further consolidation of topics needed
BAM indicators •
Algebraic Proficiency I and II
Essential knowledge • Know how to interpret the display on a scientific calculator when working with standard form • Know the difference between direct and inverse proportion • Know how to represent an inequality on a number line • Know that the point of intersection of two lines represents the solution to the corresponding simultaneous equations • Know the meaning of a quadratic sequence • Know Pythagoras’ theorem • Know the definitions of arc, sector, tangent and segment • Know the conditions for congruent triangles
AlgebraicProficiency I Key concepts
The Big Picture: Algebra progression map
• Rearranging Formulas – development form Stage 8/9
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Understand the notation of algebra Manipulate algebraic expressions Evaluate algebraic statements
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Factorise an expression by taking out common factors including quadratic expressions Simplify an expression involving terms with combinations of variables (e.g. 3a²b + 4ab2 + 2a2 – a2b) Multiply brackets up to (a + b)(c + d)(e + f) Know the multiplication (division, power, zero) law of indices Understand that negative powers can arise Substitute positive and negative numbers into formulae Know the meaning of the ‘subject’ of a formula Change the subject of a formula when one step is required Change the subject of a formula when two or more steps are required
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Algebraic Proficiency II Key concepts
The Big Picture: Algebra progression map
• solve, in simple cases, two linear simultaneous equations in two variables algebraically • derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution • find approximate solutions to simultaneous equations using a graph Return to overview Possible learning intentions • • •
Possible success criteria
Solve simultaneous equations Use graphs to solve equations Solve problems involving simultaneous equations
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Prerequisites • • • •
Solve linear equations Substitute numbers into formulae Plot graphs of functions of the form y = mx + c, x ± y = c and ax ± by = c) Manipulate expressions by multiplying by a single term Book Ref CGP3: 152-153 10 Ticks: L7/8 p4 5-11 Got it: p13-15 q16-19
Understand that there are an infinite number of solutions to the equation ax + by = c (a ≠ 0, b ≠ 0) Understand the concept of simultaneous equations Find approximate solutions to simultaneous equations using a graph Understand the concept of solving simultaneous equations by elimination* Target a variable to eliminate Decide if multiplication of one equation is required Decide whether addition or subtraction of equations is required Add or subtract pairs of equations to eliminate a variable Find the value of one variable in a pair of simple simultaneous equations Find the value of the second variable in a pair of simple simultaneous equations Solve two linear simultaneous equations in two variables in very simple cases (no multiplication required) Solve two linear simultaneous equations in two variables in simple cases (multiplication of one equation only required) Derive and solve two simultaneous equations Interpret the solution to a pair of simultaneous equations
Mathematical language Equation Simultaneous equation Variable Manipulate Eliminate Solve Derive Interpret
Pedagogical notes Other UCC resources
Pupils will be expected to solve simultaneous equations in more complex cases in Stage 10. This includes involving multiplications of both equations to enable elimination, cases where rearrangement is required first, and the method of substitution. NCETM: Glossary Common approaches Pupils are taught to label the equations (1) and (2), and label the subsequent equation (3) Teachers use graphs (i.e. dynamic software) to demonstrate solutions to simultaneous equations at every opportunity
Reasoning opportunities and probing questions • • •
Show me a solution to the equation 5a + b = 32. And another, and another … Show me a pair of simultaneous equations with the solution x = 2 and y = -5. And another, and another … Kenny and Jenny are solving the simultaneous equations x + 4y = 7 and x – 2y = 1. Kenny thinks the equations should be added. Jenny thinks they should be subtracted. Who do you agree with? Explain why.
Suggested activities
Possible misconceptions
KM: Stick on the Maths ALG2: Simultaneous linear equations NRICH: What’s it worth? NRICH: Warmsnug Double Glazing NRICH: Arithmagons
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Learning review KM: 9M5 BAM Task
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Some pupils may think that addition of equations is required when both equations involve a subtraction Some pupils may not multiply all coefficients, or the constant, when multiplying an equation Some pupils may think that it is always right to eliminate the first variable Some pupils may struggle to deal with negative numbers correctly when adding or subtracting the equations
Visualising and constructing
6 hours
Key concepts
The Big Picture: Properties of Shape progression map
• use the standard ruler and compass constructions (perpendicular bisector of a line segment, constructing a perpendicular to a given line from/at a given point, bisecting a given angle) • use these to construct given figures and solve loci problems; know that the perpendicular distance from a point to a line is the shortest distance to the line • construct plans and elevations of 3D shapes Return to overview Possible learning intentions • • •
Possible success criteria
Know standard mathematical constructions Apply standard mathematical constructions Explore ways of representing 3D shapes
Prerequisites • • • •
Plans/elevations 10Ticks L6 p8 29/30 Constructions/Loci CGP3 166-174 10Ticks L7/8 p5 3-10
Use compasses to construct clean arcs Use ruler and compasses to construct the perpendicular bisector of a line segment Use ruler and compasses to bisect an angle Use a ruler and compasses to construct a perpendicular to a line from a point (at a point) Understand the meaning of locus (loci) Know how to construct the locus of points a fixed distance from a point (from a line) Identify when to use the locus of points a fixed distance from a point (from a line) Identify when a perpendicular bisector is needed to solve a loci problem Identify when an angle bisector is needed to solve a loci problem Choose techniques to construct 2D shapes; e.g. rhombus Combine techniques to solve more complex loci problems Know how to deal with a change in depth when dealing with plans and elevations Construct a shape from its plans and elevations Construct the plan and elevations of a given shape
Mathematical language
Measure distances to the nearest millimetre Create and interpret scale diagrams Use compasses to draw circles Interpret plan and elevations Book Ref
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Got it: p21-24 q26-28, q31
Compasses Arc Line segment Perpendicular Bisect Perpendicular bisector Locus, Loci Plan Elevation
Pedagogical notes Other UCC resources
Ensure that students always leave their construction arcs visible. Arcs must be ’clean’; i.e. smooth, single arcs with a sharp pencil. NCETM: Departmental workshops: Constructions NCETM: Departmental workshops: Loci NCETM: Glossary Common approaches All pupils should experience using dynamic software (e.g. Autograph) to explore standard mathematical constructions (perpendicular bisector and angle bisector).
Reasoning opportunities and probing questions • • • •
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(Given a single point marked on the board) show me a point 30 cm away from this point. And another. And another … Provide shapes made from some cubes in certain orientations. Challenge pupils to construct the plans and elevations. Do groups agree? If this is the plan show me a possible 3D shape. And another. And another. Demonstrate how to create the perpendicular bisector (or other constructions). Challenge pupils to write a set of instructions for carrying out the construction. Follow these instructions very precisely (being awkward if possible; e.g. changing radius of compasses). Do the instructions work? Give pupils the equipment to create standard constructions and challenge them to create a right angle / bisect an angle
Suggested activities
Possible misconceptions
KM: Construction instruction KM: Construction challenges KM: Napoleonic challenge KM: Circumcentre etcetera KM: Locus hocus pocus KM: The perpendicular bisector KM: Topple KM: Gilbert goat KM: An elevated position KM: Solid problems (plans and elevations) KM: Isometric interpretation (plans and elevations)
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Learning review KM: 9M8 BAM Task •
When constructing the bisector of an angle some pupils may think that the intersecting arcs need to be drawn from the ends of the two lines that make the angle. When constructing a locus such as the set of points a fixed distance from the perimeter of a rectangle, some pupils may not interpret the corner as a point (which therefore requires an arc as part of the locus) The north elevation is the view of a shape from the north (the north face of the shape), not the view of the shape while facing north.
Calculating space
13 hours
Key concepts • • • •
The Big Picture: Measurement and mensuration progression map
identify and apply circle definitions and properties, including: tangent, arc, sector and segment calculate arc lengths, angles and areas of sectors of circles calculate surface area of right prisms (including cylinders) calculate exactly with multiples of π Return to overview
Possible learning intentions • • •
Possible success criteria
Solve problems involving arcs and sectors Solve problems involving prisms Investigate right-angled triangles
Prerequisites • • • •
Know and use the number π Know and use the formula for area and circumference of a circle Know how to use formulae to find the area of rectangles, parallelograms, triangles and trapezia Know how to find the area of compound shapes Book Ref
Arc length & sector area 10 Ticks L9-10 p4 pg9
Surface area prisms CGP3: 192-193 10Ticks L7/8 p6 17-18
Got it: p23 q30 P25-26 q32-33 P28/29 q35
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Know the vocabulary of circles Know how to find arc length Calculate the arc length of a sector when radius is given Know how to find the area of a sector Calculate the area of a sector when radius is given Calculate the angle of a sector when the arc length and radius are known Know how to find the surface area of a right prism (cylinder) Calculate the surface area of a right prism (cylinder) Calculate exactly with multiples of π
Mathematical language
Pedagogical notes
Circle, Pi Other UCC resources Radius, diameter, chord, circumference, arc, tangent, sector, segment (Right) prism, cylinder Cross-section Notation π Abbreviations of units in the metric system: km, m, cm, mm, mm2, cm2, m2, km2, mm3, cm3, km3
This unit builds on the area and circle work form Stages 7 and 8. Pupils will need to be reminded of the key formula, in particular the importance of the perpendicular height when calculating areas and the correct use of πr2. Note: some pupils may only find the area of the three ‘distinct’ faces when finding surface area. Pupils must experience right-angled triangles in different orientations to appreciate the hypotenuse is always opposite the right angle. NCETM: Glossary Common approaches Pupils visualize and write down the shapes of all the faces of a prism before calculating the surface area. Every classroom has a set of area posters on the wall.
Reasoning opportunities and probing questions • •
Suggested activities
KM: The language of circles Show me a sector with area 25π. And another. And another … Always/ Sometimes/ Never: The value of the volume of a prism is less KM: Stick on the Maths: Right Prisms NRICH: Curvy Areas than the value of the surface area of a prism. NRICH: Changing Areas, Changing Volumes Learning review KM: 9M10 BAM Task, 9M11 BAM Task
Possible misconceptions • • • •
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Some pupils will work out (π × r)2 when finding the area of a circle Some pupils may use the sloping height when finding cross-sectional areas that are parallelograms, triangles or trapezia Some pupils may confuse the concepts of surface area and volume Some pupils may not include the lengths of the radii when calculating the perimeter of an arc
Investigating properties of shapes
12 hours
Key concepts • • • •
The Big Picture: Properties of Shape progression map
make links to similarity (including trigonometric ratios) and scale factors know the exact values of sinθ and cosθ for θ = 0°, 30°, 45°, 60° and 90°; know the exact value of tanθ for θ = 0°, 30°, 45° and 60° know the trigonometric ratios, sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse, tanθ = opposite/adjacent apply it to find angles and lengths in right-angled triangles in two dimensional figures Return to overview
Possible learning intentions • • • •
Possible success criteria
Investigate similar triangles Explore trigonometry in right-angled triangles Set up and solve trigonometric equations Use trigonometry to solve practical problems
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Bring on the Maths: GCSE Higher Shape Investigating angles: #5, #6, #7, #8, #9
Prerequisites • • •
Understand and work with similar shapes Solve linear equations, including those with the unknown in the denominator of a fraction Understand and use Pythagoras’ theorem
Appreciate that the ratio of corresponding sides in similar triangles is constant Label the sides of a right-angled triangle using a given angle Choose an appropriate trigonometric ratio that can be used in a given situation Understand that sine, cosine and tangent are functions of an angle Establish the exact values of sinθ and cosθ for θ = 0°, 30°, 45°, 60° and 90° Establish the exact value of tanθ for θ = 0°, 30°, 45° and 60° Know how to select the correct mode on a scientific calculator Use a calculator to find the sine, cosine and tangent of an angle Know the trigonometric ratios, sinθ = opp/hyp, cosθ = adj/hyp, tanθ = opp/adj Set up and solve a trigonometric equation to find a missing side in a right-angled triangle Set up and solve a trigonometric equation when the unknown is in the denominator of a fraction Set up and solve a trigonometric equation to find a missing angle in a right-angled triangle Use trigonometry to solve problems involving bearings Use trigonometry to solve problems involving an angle of depression or an angle of elevation
Mathematical language
Pedagogical notes
Similar Opposite Adjacent Hypotenuse Trigonometry Function Ratio Sine Cosine Tangent Angle of elevation, angle of depression
Ensure that all students are aware of the importance of their scientific calculator being in degrees mode. Ensure that students do not round until the end of a multi-step calculation This unit of trigonometry should focus only on right-angled triangles in two dimensions. The sine rule, cosine rule, and applications in three dimensions are covered in Stage 11. Note that inverse functions are explored in Stage 11. NRICH: History of Trigonometry NCETM: Glossary
Notation sinθ stands for the ‘sine of θ’ sin-1 is the inverse sine function, and not 1÷ sin
Common approaches All students explore sets of similar triangles with angles of (at least) 30°, 45° and 60° as an introduction to the three trigonometric ratios The mnemonic ‘Some Of Harry’s Cats Are Heavier Than Other Animals’ is used to help students remember the trigonometric ratios
Reasoning opportunities and probing questions • • •
Show me an angle and its exact sine (cosine / tangent). And another … Convince me that you have chosen the correct trigonometric function (When exploring sets of similar triangles and working out ratios in corresponding cases) why do you think that the results are all similar, but not the same? Could we do anything differently to get results that are closer? How could we make a final conclusion for each ratio?
Suggested activities
Possible misconceptions
KM: From set squares to trigonometry KM: Trigonometry flowchart NRICH: Trigonometric protractor NRICH: Sine and cosine
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Learning review GLOWMaths/JustMaths: Sample Questions Both Tiers GLOWMaths/JustMaths: Sample Questions Higher Tiers KM: 10M10 BAM Task
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Some students may not appreciate the fact that adjacent and opposite labels are not fixed, and are only relevant to a particular acute angle. In situations where both angles are given this can cause difficulties. Some students may not balance an equation such as sin35 = 4/x correctly, believing that the next step is (sin35)/4 = x Some students may think that sin-1θ = 1 ÷ sinθ Some students may think that sinθ means sin × θ
Proportional reasoning
9 hours
Key concepts
The Big Picture: Ratio and Proportion progression map
• revise direct and inverse proportion including graphical and algebraic representations • change freely between compound units (e.g. density, pressure) in numerical and algebraic contexts • use compound units such as density and pressure Return to overview Possible learning intentions • • •
Possible success criteria
Solve problems involving different types of proportion Investigate ways of representing proportion Know and use compound units in a range of situations
Prerequisites • • •
Direct/inverse proportion CGP3 87-90 My Maths 2C 274-283 10Ticks L7/8 p1 35-36, L9/10 p1 11-16
Know the difference between direct and inverse proportion Recognise direct (inverse) proportion in a situation Know the features of a graph that represents a direct (inverse) proportion situation Know the features of an expression (or formula) that represents a direct (inverse) proportion situation Understand why speed, density and pressure are known as compound units Know the definition of density (pressure, population density, speed) Solve problems involving density (pressure, speed) Convert between units of density
Mathematical language
Find a relevant multiplier in a situation involving proportion Understand the meaning of a compound unit Convert between units of length, capacity, mass and time
Book ref
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Similar and congruent CGP3 213-216 Compound measures CGP3 95-97 10Ticks L6 p5 9-20
Direct proportion Inverse proportion Multiplier Linear Compound unit Density, Population density Pressure Notation Kilograms per metre cubed is written as kg/m3
Pedagogical notes Other UCC resources
Pupils have explored enlargement previously. Use the story of Archimedes and his ‘eureka moment’ when introducing density. Up-to-date information about population densities of counties and cities of the UK, and countries of the world, is easily found online. NCETM: The Bar Model NCETM: Multiplicative reasoning NCETM: Departmental workshops: Proportional Reasoning NCETM: Departmental workshops: Congruence and Similarity NCETM: Glossary Common approaches All pupils are taught to set up a ‘proportion table’ and use it to find the multiplier in situations involving direct proportion
Got it: p18-20 q22-25 P27 q34
Reasoning opportunities and probing questions
Suggested activities
Possible misconceptions
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Show me an example of two quantities that will be in direct (inverse) proportion. And another. And another … Convince me that this information shows a proportional relationship. What type of proportion is it? 40 3 60 2 80 1.5 Which is the greatest density: 0.65g/cm3 or 650kg/m3? Convince me.
KM: Graphing proportion NRICH: In proportion NRICH: Ratios and dilutions NRICH: Similar rectangles NRICH: Fit for photocopying NRICH: Tennis NRICH: How big?
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Learning review KM: 9M7 BAM Task, 9M9 BAM Task •
Many pupils will want to identify an additive relationship between two quantities that are in proportion and apply this to solve problems The word ‘similar’ means something much more precise in this context than in other contexts pupils encounter. This can cause confusion. Some pupils may think that a multiplier always has to be greater than 1
Algebraic proficiency: visualising
12 hours
Key concepts
The Big Picture: Algebra progression map
• recognise, sketch and interpret graphs of quadratic functions • recognise, sketch and interpret graphs of simple cubic functions and the reciprocal function y = 1/x with x ≠ 0 • plot and interpret graphs (including reciprocal graphs) and graphs of non-standard functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration Return to overview Possible learning intentions • • • •
Explore graphs of quadratic functions Explore graphs of other standard non-linear functions Create and use graphs of non-standard functions Solve kinematic problems
Prerequisites • •
Possible success criteria
Recognise graphs of simple quadratic functions Plot and interpret graphs of kinematic problems involving distance and speed
Book Ref CGP3: 136-148, MyMaths 2B: 98-105 Got it: p8-12 q11-15
Reasoning opportunities and probing questions
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Plot graphs of quadratic (cubic, reciprocal) functions Recognise and interpret the graphs of quadratic (cubic, reciprocal) functions Sketch graphs of quadratic (cubic, reciprocal) functions Plot and interpret graphs of non-standard functions in real contexts Find approximate solutions to kinematic problems involving distance, speed and acceleration
Mathematical language Function, equation Linear, non-linear Quadratic, cubic, reciprocal Parabola, Asymptote Gradient, y-intercept, x-intercept, root Rate of change Sketch, plot Kinematic Speed, distance, time Acceleration, deceleration Notation y = mx + c Suggested activities
Pedagogical notes Other UCC resources
This unit builds on the graphs of linear functions and simple quadratic functions work from Stage 8 and 9. Where possible, students should be encouraged to plot linear graphs efficiently by using knowledge of the y-intercept and the gradient. NCETM: Glossary Common approaches ‘Monter’ and ‘commencer’ are shared as the reason for ‘m’ and ‘c’ in y = mx + c and links to y = ax + b Students plot points with a ‘x’ and not ‘‘ Students draw graphs in pencil All student use dynamic graphing software to explore graphs
Possible misconceptions
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Convince me the lines y = 3 + 2x, y – 2x = 7, 2x + 6 = y and 8 + y – 2x = 0 are parallel to each other. What is the same and what is different: y = x, y = x2, y = x3 and y=1/x ? Show me a sketch of a quadratic (cubic, reciprocal) graph. And another. And another … Sketch a distance/time graph of your journey to school. What is the same and what is different with the graph of a classmate?
KM: Screenshot challenge KM: Stick on the Maths: Quadratic and cubic functions KM: Stick on the Maths: Algebraic Graphs NRICH: Diamond Collector NRICH: Fill me up NRICH: What’s that graph? NRICH: Speed-time at the Olympics NRICH: Exploring Quadratic Mappings NRICH: Minus One Two Three
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Learning review KM: 9M4 BAM Task, 9M6 BAM Task •
Some pupils do not rearrange the equation of a straight line to find the gradient of a straight line. For example, they think that the line y – 2x = 6 has a gradient of -2. Some pupils may think that gradient = (change in x) / (change in y) when trying to equation of a line through two given points. Some pupils may incorrectly square negative values of x when plotting graphs of quadratic functions. Some pupils think that the horizontal section of a distance time graph means an object is travelling at constant speed. Some pupils think that a section of a distance time graph with negative gradient means an object is travelling backwards or downhill.
Understanding risk
4 Hours
Key concepts
The Big Picture: Probability progression map
• calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions • enumerate sets and combinations of sets systematically, using tree diagrams • understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size Return to overview Possible learning intentions • • •
Understand and use tree diagrams Develop understanding of probability in situations involving combined events Use probability to make predictions
Prerequisites • • • • •
Possible success criteria
Add fractions (decimals) Multiply fractions (decimals) Convert between fractions, decimals and percentages Use frequency trees to record outcomes of probability experiments Use experimental and theoretical probability to calculate expected outcomes Book Ref 10Ticks L7/8 p1 15-18 L9/10 p2 25-32 Got it: p33-35 q39-41
Reasoning opportunities and probing questions
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Review how to label a tree diagram with probabilities Label a tree diagram with probabilities when events are dependent Use a tree diagram to calculate probabilities of independent combined events Use a tree diagram to calculate probabilities of dependent combined events Understand that relative frequency tends towards theoretical probability as sample size increases
Mathematical language Outcome, equally likely outcomes Event, independent event, dependent event Tree diagrams Theoretical probability Experimental probability Random Bias, unbiased, fair Relative frequency Enumerate Set Notation P(A) for the probability of event A Probabilities are expressed as fractions, decimals or percentage. They should not be expressed as ratios (which represent odds) or as words Suggested activities
Pedagogical notes Other UCC resources
Tree diagrams can be introduced as simply an alternative way of listing all outcomes for multiple events. For example, if two coins are flipped, the possible outcomes can be listed (a) systematically, (b) in a two-way table, or (c) in a tree diagram. However, the tree diagram has the advantage that it can be extended to more than two events (e.g. three coins are flipped). NCETM: Glossary Common approaches All students carry out the drawing pin experiment Students are taught not to simply fractions when finding probabilities of combined events using a tree diagram (so that a simple check can be made that the probabilities sum to 1)
Possible misconceptions
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Show me an example of a probability problem that involves adding (multiplying) probabilities Convince me that there are eight different outcomes when three coins are flipped together Always / Sometimes / Never: increasing the number of times an experiment is carried out gives an estimated probability that is closer to the theoretical probability
KM: Stick on the Maths: Tree diagrams KM: Stick on the Maths: Relative frequency KM: The drawing pin experiment Learning review KM: 9M13 BAM Task
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When constructing a tree diagram for a given situation, some students may struggle to distinguish between how events, and outcomes of those events, are represented Some students may muddle the conditions for adding and multiplying probabilities Some students may add the denominators when adding fractions
GCSE
1aH. Calculations, checking and rounding (N2, N3, N5, N14, N15)
Teaching time 4 hours
OBJECTIVES By • • • • •
the end of the sub-unit, students should be able to: Add, subtract, multiply and divide decimals, whole numbers including any number between 0 and 1; Put digits in the correct place in a decimal calculation and use one calculation to find the answer to another; Use the product rule for counting (i.e. if there are m ways of doing one task and for each of these, there are n ways of doing another task, then the total number of ways the two tasks can be done is m × n ways); Round numbers to the nearest 10, 100, 1000, the nearest integer, to a given number of decimal places and to a given number of significant figures; Estimate answers to one- or two-step calculations, including use of rounding numbers and formal estimation to 1 significant figure: mainly whole numbers and then decimals.
POSSIBLE SUCCESS CRITERIA Given 5 digits, what is the largest even number, largest odd number, or largest or smallest answers when subtracting a two-digit number from a threedigit number? Given 2.6 × 15.8 = 41.08 what is 26 × 0.158? What is 4108 ÷ 26?
Teaching time 4 hours
1bH. Indices, roots, reciprocals and hierarchy of operations (N3, N6, N7) OBJECTIVES By • • • •
the end of the sub-unit, students should be able to: Use index notation for integer powers of 10, including negative powers; Recognise powers of 2, 3, 4, 5; Use the square, cube and power keys on a calculator and estimate powers and roots of any given positive number, by considering the values it must lie between, e.g. the square root of 42 must be between 6 and 7; Find the value of calculations using indices including positive, fractional and negative indices;
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Recall that
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Understand that the inverse operation of raising a positive number to a power n is raising the result of this operation to the power
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Use index laws to simplify and calculate the value of numerical expressions involving multiplication and division of integer powers, fractional and negative powers, and powers of a power; Solve problems using index laws; Use brackets and the hierarchy of operations up to and including with powers and roots inside the brackets, or raising brackets to powers or taking roots of brackets;
• •
n = 1 and n = 0
–1
1
n
for positive integers n as well as, n
1 2
= √n and
n
1 3
= 3√n for any positive number n;
1
x y, x y , brackets;
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Use an extended range of calculator functions, including +, –, ×, ÷, x², √x, memory,
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Use calculators for all calculations: positive and negative numbers, brackets, powers and roots, four operations.
POSSIBLE SUCCESS CRITERIA What is the value of 25? Prove that the square root of 45 lies between 6 and 7. −
2
Evaluate (23 × 25) ÷ 24, 40, 8 3 . Work out the value of n in 40 = 5 × 2n.
1
n
;
1cH. Factors, multiples, primes, standard form and surds (N3, N4, N8, N9)
Teaching time 7 hours
OBJECTIVES By • • • • • • • • • • •
the end of the sub-unit, students should be able to: Identify factors, multiples and prime numbers; Find the prime factor decomposition of positive integers – write as a product using index notation; Find common factors and common multiples of two numbers; Find the LCM and HCF of two numbers, by listing, Venn diagrams and using prime factors – include finding LCM and HCF given the prime factorisation of two numbers; Solve problems using HCF and LCM, and prime numbers; Understand that the prime factor decomposition of a positive integer is unique, whichever factor pair you start with, and that every number can be written as a product of prime factors; Convert large and small numbers into standard form and vice versa; Add, subtract, multiply and divide numbers in standard form; Interpret a calculator display using standard form and know how to enter numbers in standard form; Understand surd notation, e.g. calculator gives answer to sq rt 8 as 4 rt 2; Simplify surd expressions involving squares (e.g. √12 = √(4 × 3) = √4 × √3 = 2√3).
POSSIBLE SUCCESS CRITERIA Know how to test if a number up to 120 is prime. Understand that every number can be written as a unique product of its prime factors. Recall prime numbers up to 100. Understand the meaning of prime factor. Write a number as a product of its prime factors. Use a Venn diagram to sort information. Write 51080 in standard form. Write 3.74 x 10–6 as an ordinary number. Simplify √8. Convert a ‘near miss’, or any number, into standard form; e.g. 23 × 107.
2aH. Algebra: the basics, setting up, rearranging and solving equations (N1, N3, N8, A1, A2, A3, A4, A5, A6, A7, A17, A20, A21)
Teaching time 10 hours
OBJECTIVES By the end of the sub-unit, students should be able to: • Use algebraic notation and symbols correctly; • Know the difference between a term, expression, equation, formula and an identity; • Write and manipulate an expression by collecting like terms; • Substitute positive and negative numbers into expressions such as 3x + 4 and 2x3 and then into expressions involving brackets and powers; • Substitute numbers into formulae from mathematics and other subject using simple linear formulae, e.g. l × w, v = u + at;
4x 2
= 2x;
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Simplify expressions by cancelling, e.g.
• • • • • • • • • • • • • • •
Use instances of index laws for positive integer powers including when multiplying or dividing algebraic terms; Use instances of index laws, including use of zero, fractional and negative powers; Multiply a single term over a bracket and recognise factors of algebraic terms involving single brackets and simplify expressions by factorising, including subsequently collecting like terms; Expand the product of two linear expressions, i.e. double brackets working up to negatives in both brackets and also similar to (2x + 3y)(3x – y); Know that squaring a linear expression is the same as expanding double brackets; Factorise quadratic expressions of the form ax2 + bx + c; Factorise quadratic expressions using the difference of two squares; Set up simple equations from word problems and derive simple formulae; Understand the ≠ symbol (not equal), e.g. 6x + 4 ≠ 3(x + 2), and introduce identity ≡ sign; Solve linear equations, with integer coefficients, in which the unknown appears on either side or on both sides of the equation; Solve linear equations which contain brackets, including those that have negative signs occurring anywhere in the equation, and those with a negative solution; Solve linear equations in one unknown, with integer or fractional coefficients; Set up and solve linear equations to solve to solve a problem; Derive a formula and set up simple equations from word problems, then solve these equations, interpreting the solution in the context of the problem; Substitute positive and negative numbers into a formula, solve the resulting equation including brackets, powers or standard form;
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Use and substitute formulae from mathematics and other subjects, including the kinematics formulae v = u + at, v2 – u2 = 2as, and s = ut +
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Change the subject of a simple formula, i.e. linear one-step, such as x = 4y; Change the subject of a formula, including cases where the subject is on both sides of the original formula, or involving fractions and small powers of the subject; Simple proofs and use of ≡ in “show that” style questions; know the difference between an equation and an identity; Use iteration to find approximate solutions to equations, for simple equations in the first instance, then quadratic and cubic equations.
POSSIBLE SUCCESS CRITERIA Simplify 4p – 2q2 + 1 – 3p + 5q2. Evaluate 4x2 – 2x when x = –5. Simplify z4 × z3, y3 ÷ y2, (a7)2,
(
8 x6 y 4
)
1 3
.
1 2
at2;
Expand and simplify 3(t – 1) + 57. Factorise 15x2y – 35x2y2. Expand and simplify (3x + 2)(4x – 1). Factorise 6x2 – 7x + 1. A room is 2 m longer than it is wide. If its area is 30 m2 what is its perimeter? Use fractions when working in algebraic situations. Substitute positive and negative numbers into formulae. Be aware of common scientific formulae. Know the meaning of the ‘subject’ of a formula. Change the subject of a formula when one step is required. Change the subject of a formula when two steps are required.