8th Grade Math Checklist: What Your Child Should Know

Geometry

Your child is developing advanced spatial skills — working with 3D shapes and nets, using coordinate grids with negative numbers, calculating angles, and understanding geometric transformations like reflection and translation.

• Coordinates (age 12+)

  Understand similarity as a relationship where one shape is an enlargement of another; construct similar shapes by enlargement with a given scale factor and centre, with and without coordinate grids

  • Enlarge a shape by a given scale factor from a specified centre of enlargement
  • Determine the scale factor between two similar shapes by comparing corresponding sides
  • Explain why corresponding angles in similar shapes are equal while sides are in proportion

• Circles: Circumference & Area

  Calculate the circumference and area of circles using the formulae C = πd (or 2πr) and A = πr², and solve problems involving perimeters and areas of composite shapes that include circular parts

  • Calculate the circumference and area of a circle given its radius or diameter
  • Find the perimeter or area of a composite shape made from rectangles and semicircles
  • Explain the relationship between π, diameter, and circumference informally

• Angles in triangles (age 11+)

  Derive and apply formulae for the area of triangles, parallelograms, and trapezia, and for the volume of cuboids and other prisms (including cylinders), connecting each formula to its geometric reasoning

  • Calculate the area of a trapezium using A = ½(a + b) × h and explain why the formula works
  • Find the volume of a triangular prism by calculating cross-sectional area × length
  • Derive the formula for the volume of a cylinder as π × r² × h by reasoning from the prism formula

• Trigonometry basics

  Use the trigonometric ratios sin, cos, and tan in right-angled triangles to find unknown sides and angles, including setting up the correct ratio for a given problem

  • Identify which trigonometric ratio to use (SOH CAH TOA) based on the known and unknown sides
  • Calculate an unknown side using sin, cos, or tan and a known angle
  • Find an unknown angle using the inverse trigonometric function (e.g. tan⁻¹)

• Types of angles (age 13+)

  Apply Pythagoras’ Theorem (a² + b² = c²) to calculate unknown side lengths in right-angled triangles, including in real-world and coordinate-geometry contexts

  • Calculate the hypotenuse of a right-angled triangle given the two shorter sides
  • Find a shorter side given the hypotenuse and the other short side
  • Use Pythagoras’ Theorem to find the distance between two points on a coordinate grid

• Angle sums in triangles and polygons

  Derive and use the angle sum in a triangle (180°), use it to deduce the angle sum in any polygon ((n−2) × 180°), and calculate interior and exterior angles of regular polygons

  • Calculate a missing angle in a triangle by subtracting the known angles from 180°
  • Find the sum of interior angles in a hexagon by dividing it into triangles
  • Calculate each interior and exterior angle of a regular polygon given the number of sides

• Coordinate Transformations

  Identify properties of translations, rotations, and reflections; describe and perform these transformations on given figures, and understand that the image is congruent to the original

  • Reflect a shape in a given mirror line (including diagonal lines) and state the coordinates of the image
  • Rotate a shape about a given centre by 90° or 180° and describe the result
  • Translate a shape by a given vector and verify that lengths and angles are preserved

• Angles in triangles (age 12+)

  Know and use the criteria for triangle congruence (SSS, SAS, ASA, RHS), use standard labelling conventions for sides and angles of triangle ABC, and determine whether two triangles are congruent

  • State which congruence criterion (SSS, SAS, ASA, or RHS) applies to a given pair of triangles
  • Use the notation △ABC ≅ △DEF and match corresponding sides and angles
  • Determine whether given measurements produce a unique triangle, more than one, or none

• Types of angles (age 11+)

  Use conventional geometric terms and notation to describe, sketch, and draw points, lines, parallel and perpendicular lines, right angles, regular polygons, and reflectively/rotationally symmetric polygons

  • Use correct notation for line segments (AB), angles (∠ABC), and parallel lines (AB ∥ CD)
  • Sketch a regular hexagon and describe its rotational and reflective symmetry
  • Identify and label perpendicular and parallel lines in a given figure using standard symbols

• Properties of triangles and quadrilaterals

  Derive and illustrate properties of triangles, quadrilaterals, and circles using appropriate language, including interior angles, diagonals, symmetry, and relationships between side lengths

  • List and verify properties of a parallelogram (opposite sides parallel/equal, opposite angles equal, diagonals bisect each other)
  • Explain why a square is a special case of both a rectangle and a rhombus
  • Derive the relationship between the radius, diameter, and circumference of a circle

• Measuring angles (age 11+)

  Draw and measure line segments and angles accurately using ruler and protractor, and interpret scale drawings to extract real measurements

  • Draw a triangle accurately given two sides and the included angle (SAS)
  • Measure angles in a geometric figure to the nearest degree using a protractor
  • Read a scale drawing to determine actual lengths, explaining the scale used

• Understanding angles (age 11+)

  Apply the properties of angles at a point (360°), on a straight line (180°), and vertically opposite angles to find unknown angles in multi-step problems

  • Find a missing angle using the fact that angles on a straight line sum to 180°
  • Use vertically opposite angles are equal to set up and solve an equation for an unknown
  • Combine multiple angle facts in a single diagram to find several missing angles step by step

• Understanding angles (age 12+)

  Use ruler and compasses to perform standard constructions: perpendicular bisector of a line segment, perpendicular to a line from or at a given point, and bisecting an angle

  • Construct the perpendicular bisector of a line segment using compasses and a straight edge
  • Construct an angle bisector and verify by measuring both halves
  • Construct a perpendicular from a point to a line using compasses only

• Angles with parallel lines

  Understand and use the relationship between parallel lines cut by a transversal: corresponding angles, alternate interior angles, and co-interior (same-side interior) angles; use these to find unknown angles

  • Identify alternate, corresponding, and co-interior angle pairs in a diagram with parallel lines
  • Explain why alternate angles are equal using the concept of translation along the transversal
  • Find missing angles in a parallel-line diagram using a combination of angle relationships

• 3-D shapes (age 11+)

  Use the properties of faces, surfaces, edges, and vertices of 3-D shapes (cubes, cuboids, prisms, cylinders, pyramids, cones, and spheres) to solve problems, including visualising cross-sections

  • Count and describe the faces, edges, and vertices of a triangular prism and a square-based pyramid
  • Describe the 2-D cross-section produced by slicing a cylinder horizontally or a cone vertically
  • Use properties of 3-D shapes to determine whether a given net will fold into a specified solid

Algebra

Your child is beginning to work with algebra — using letters to represent unknown numbers, creating and following rules for number patterns, and translating word problems into mathematical equations.

• Linear Function Graphs

  Recognise that a linear function produces a straight-line graph, understand the relationship between an equation of the form y = mx + c and its graphical representation, and interpret gradient and y-intercept in context

  • Explain that changing m in y = mx + c alters the steepness and direction of the line
  • Identify the y-intercept of a line from its equation and from its graph
  • Determine whether a given equation will produce a straight line or a curve

• Plotting Linear Graphs

  Plot linear graphs by generating a table of values, reduce a two-variable linear equation to the form y = mx + c, and calculate gradients from two points on a line

  • Generate a table of (x, y) values for a linear equation and plot the points accurately
  • Rearrange an equation such as 2x + 3y = 12 into the form y = mx + c
  • Calculate the gradient between two coordinate points using rise over run

• Estimating answers (age 13+)

  Use graphs of linear and quadratic functions to estimate output values for given inputs, find approximate solutions to equations, and interpret graphical information in real-world contexts

  • Read off an approximate y-value for a given x from a plotted curve
  • Find approximate solutions to an equation by identifying where a graph crosses the x-axis or another line
  • Interpret a real-world graph (e.g., distance–time) to answer contextual questions

• Quadratic Graphs

  Recognise that quadratic functions produce curved (parabolic) graphs, distinguish them from linear graphs, and use plotted quadratic graphs to estimate values and find approximate solutions

  • Explain why x² produces a U-shaped curve rather than a straight line
  • Plot a simple quadratic such as y = x² − 4 from a table of values
  • Read approximate solutions from a quadratic graph (e.g., where y = 0)

• Simultaneous Equations

  Understand that two linear equations can be solved simultaneously by finding the point where their graphs intersect, and interpret this graphically and algebraically as the pair of values satisfying both equations

  • Explain that the intersection of two lines represents values satisfying both equations
  • Find the approximate solution to a pair of simultaneous equations by reading a graph
  • Verify a proposed solution by substituting into both equations

• Numbers on a number line

  Understand inequalities as statements comparing expressions, represent solutions on a number line, and solve simple linear inequalities using the same inverse-operation methods as equations

  • Write an inequality from a worded constraint (e.g., 'must be at least 12' → x ≥ 12)
  • Represent the solution set of an inequality on a number line with open or closed circles
  • Solve a one-step or two-step inequality such as 3x + 1 < 10

• Solving Linear Equations

  Use algebraic methods to solve linear equations in one variable, including equations that require rearrangement, expanding brackets, and collecting terms on both sides; solve equations with rational number coefficients

  • Solve one-step and two-step linear equations in one variable
  • Solve equations requiring expansion of brackets and collection of like terms
  • Solve equations with the unknown on both sides and with fractional or negative coefficients

• Expanding Single Brackets

  Expand (multiply out) a single term over a bracket using the distributive property, e.g., 3(2x + 5) = 6x + 15; expand expressions involving negative multipliers

  • Multiply a single positive term over a bracket to expand the expression
  • Multiply a negative term over a bracket, correctly handling signs
  • Combine expanding brackets with collecting like terms to simplify fully

• Algebraic Transformations

  Model situations or procedures by translating them into algebraic expressions or formulae and by using graphs; move between word problems, algebraic representations, tables, and graphical representations

  • Translate a word problem or real-world situation into an algebraic expression or formula
  • Construct a table of values from an algebraic rule
  • Plot the corresponding graph and interpret it in the context of the problem

• Simple formulae

  Understand and use standard mathematical formulae; rearrange formulae to change the subject, performing inverse operations to isolate a different variable

  • Identify the subject of a given formula
  • Use inverse operations to rearrange a formula to make a different variable the subject
  • Rearrange formulae involving multiple steps including fractions and powers

• Nth-Term Rules

  Find the nth-term expression for an arithmetic sequence by identifying the common difference and the zero-term, and use it to determine any term in the sequence or test whether a given number belongs to the sequence

  • Derive the nth-term rule for an arithmetic sequence such as 3, 7, 11, 15, … as 4n − 1
  • Use an nth-term formula to find the 50th or 100th term without listing all preceding terms
  • Determine whether a given number (e.g., 99) is a term in a specified arithmetic sequence

• Generating Sequences

  Generate terms of a sequence from a term-to-term rule (e.g., 'add 3 each time') or a position-to-term rule (e.g., '2n + 1'), and identify whether a sequence is arithmetic, geometric, or neither

  • Continue a sequence given a term-to-term rule involving addition, subtraction, or multiplication
  • Generate the first five terms from a position-to-term formula such as 3n − 2
  • Classify sequences as arithmetic (constant difference), geometric (constant ratio), or other

• Factorising Expressions

  Factorise algebraic expressions by taking out common factors — identify the highest common factor of all terms and write the expression as a product, e.g., 6x + 9 = 3(2x + 3)

  • Identify the highest common factor of all terms in an expression
  • Write the factorised form as a product of the HCF and a bracket
  • Check factorisation by expanding the brackets back out

• Expanding Double Brackets

  Expand products of two or more binomials, e.g., (x + 3)(x - 2) = x² + x - 6, using the grid method or FOIL; simplify the result by collecting like terms

  • Expand the product of two binomials using a systematic method (grid or FOIL)
  • Collect like terms after expansion to simplify the result
  • Expand products involving negative terms correctly

Ratio & Proportion

Your child is learning to compare quantities and solve problems involving ratios, proportions, percentages, and scale — essential skills for understanding relationships between numbers in real-world contexts.

• Compound Units

  Use compound units such as speed (distance ÷ time), unit pricing (cost ÷ quantity), and density (mass ÷ volume); solve problems involving compound units

  • Calculate average speed given distance and time, converting between km/h and m/s if needed
  • Compare unit prices of two products sold in different quantities to find the better deal
  • Use the density formula to find mass, volume, or density given the other two values

• Proportion

  Recognise and solve problems involving direct proportion (as one quantity increases, the other increases at a constant rate) and inverse proportion (as one increases, the other decreases), including graphical and algebraic representations

  • Identify whether a real-world relationship is direct or inverse proportion and justify the choice
  • Set up and solve a direct-proportion equation (e.g. if 4 pens cost £6, find the cost of 10)
  • Sketch graphs showing direct proportion (straight line through origin) and inverse proportion (curve)

• Ratio Notation and Relationships

  Understand that a multiplicative relationship between two quantities can be expressed as a ratio; use ratio notation; simplify ratios

  • Explain why 'for every 2 red beads there are 5 blue beads' can be written as 2:5 or as 2/5 of the blue count
  • Identify the multiplicative relationship in a table of values (e.g. y is always 3 times x)
  • Connect the ratio a:b to the fraction a/b and to the linear function y = (a/b)x

• Scale and similar shapes (age 11+)

  Use scale factors to interpret and create scale diagrams and maps, calculating real-life distances from map measurements and vice versa

  • Calculate a real-life distance from a map measurement using a given scale (e.g. 1:25,000)
  • Draw a scale diagram of a room using a chosen scale factor
  • Convert between map distance and actual distance in problems involving different units

• Percentages (age 12+)

  Solve problems involving percentage increase, percentage decrease, finding the original value after a percentage change, and calculating simple interest

  • Calculate a 15% increase on £240 using a decimal multiplier (× 1.15)
  • Find the original price before a 20% discount resulted in a sale price of £64
  • Calculate simple interest on £500 at 3% per annum for 4 years

• Percentages as Fractions

  Define percentage as 'number of parts per hundred'; interpret percentages and percentage changes as a fraction or a decimal; express one quantity as a percentage of another; compare quantities using percentages; work with percentages greater than 100%

  • Convert fluently between percentages, fractions, and decimals
  • Express one quantity as a percentage of another
  • Calculate percentage increases and decreases and interpret percentages greater than 100%

• Dividing Quantities by Ratio

  Divide a given quantity into two parts in a given part:part or part:whole ratio, and express the division as a fraction of the whole

  • Share £60 between two people in the ratio 2:3 by finding the value of one part
  • Express the result of sharing 24 sweets as 10 and 14 in ratio form as 5:7
  • Solve problems involving three-part ratios (e.g. divide 180 in the ratio 1:2:3)

• Proportional Reasoning Vocabulary

  Know and use advanced vocabulary of multiplicative reasoning — direct proportion, inverse proportion, ratio, rate, unit rate, compound unit, scale factor — accurately in problem-solving contexts

  • Distinguish between 'direct proportion' and 'inverse proportion' with real-world examples
  • Use 'rate', 'speed', and 'density' correctly and explain what units they are measured in
  • Calculate using compound measures — e.g. work out the speed of a car that travels 120 miles in 2 hours

• Proportion Graphs

  Represent proportional relationships using double number lines (two parallel number lines aligned at 0) and ratio tables; recognise that equivalent ratios generate straight lines through the origin when graphed

  • Plot pairs of proportional values on a coordinate grid and draw the straight line through the origin
  • Explain why a directly proportional relationship always passes through (0,0)
  • Read a proportion graph to find an unknown value — e.g. 'If 3 kg costs £6, how much does 5 kg cost?'

Probability

• Tree diagrams

  Generate theoretical sample spaces for single and combined events using listing, tables, and tree diagrams, and calculate theoretical probabilities as the number of favourable outcomes divided by the total number of equally likely outcomes

  • List all 36 outcomes when rolling two dice and find P(total = 7)
  • Draw a tree diagram for two successive events and multiply along branches for combined probabilities
  • Calculate the probability of a combined event using a sample space diagram and simplify the fraction

• Complementary events

  Understand and apply the rule that probabilities of all mutually exclusive outcomes sum to one; use this to find the probability of a complementary event (P(not A) = 1 − P(A))

  • Verify that probabilities of all outcomes listed for a spinner sum to 1
  • Calculate the probability of NOT rolling a 6 as 1 − 1/6 = 5/6
  • Identify an error in a probability distribution where the values do not sum to 1

• The Probability Scale

  Understand probability as a measure on a scale from 0 (impossible) to 1 (certain); use the language of probability including likely, unlikely, certain, and impossible

  • Place everyday events on a 0-to-1 probability scale with justification
  • Explain why a fair six-sided die gives each number a probability of 1/6
  • Distinguish between equally likely outcomes (fair coin) and unequally likely outcomes (biased spinner)

• Sets & Venn Diagrams

  Enumerate sets and their unions and intersections systematically using tables, grids, and Venn diagrams to organise and count outcomes

  • List the elements in the union and intersection of two sets using a Venn diagram
  • Use a two-way table to enumerate all possible outcomes of two combined events
  • Shade regions of a Venn diagram to represent A ∪ B, A ∩ B, and A′ (complement)

• Venn Diagrams and Counting Outcomes

  Construct and interpret Venn diagrams with two or three sets to organise and count outcomes; use systematic listing and the product rule for counting to enumerate all possible outcomes of combined events

  • Draw a two-circle Venn diagram to sort 30 students by whether they like football, like cricket, or like both
  • Shade the intersection A ∩ B and the union A ∪ B on a Venn diagram and explain what each region represents
  • Use a completed Venn diagram to calculate P(A), P(B), P(A ∩ B), and P(A ∪ B)

• Experimental probability

  Record, describe, and analyse the frequency of outcomes from probability experiments to develop an understanding of relative frequency as an estimate of probability

  • Conduct a coin-toss experiment, record results in a frequency table, and calculate relative frequencies
  • Compare experimental results with theoretical probability and explain discrepancies
  • Predict that relative frequency approaches theoretical probability as the number of trials increases

Number Representation & Place Value

Your child is mastering large numbers and decimals — reading, writing, and comparing numbers up to 10 million, understanding how decimal places work, and using negative numbers in real-world contexts like temperature and money.

• Number Sets & Infinity

  Appreciate the infinite nature of the sets of integers, real numbers, and rational numbers; position integers on a number line and distinguish between rational and irrational numbers

  • Explain that rational numbers can be written as a fraction of two integers and have terminating or repeating decimals
  • Give examples of irrational numbers and explain why their decimal expansions neither terminate nor repeat
  • Appreciate that between any two numbers there are infinitely many other numbers

• Decimal place value (age 11+)

  Round numbers and measures to an appropriate degree of accuracy including to a specified number of decimal places or significant figures

  • Round numbers to a given number of decimal places
  • Round numbers to a given number of significant figures
  • Choose an appropriate degree of accuracy for a given context

• Estimating by rounding

  Use approximation through rounding to estimate answers and calculate possible resulting errors expressed using inequality notation a < x ≤ b; understand upper and lower bounds of rounded values

  • Estimate the answer to a calculation by rounding all values appropriately
  • Calculate upper and lower bounds of a rounded measurement
  • Express error intervals using inequality notation

• Square and cube numbers

  Use integer powers and associated real roots (square, cube, and higher); recognise powers of 2, 3, 4, and 5; distinguish between exact representations of roots and their decimal approximations

  • Calculate squares, cubes, and higher integer powers of whole numbers
  • Find square roots and cube roots of perfect squares and perfect cubes
  • Recognise key powers (powers of 2 up to 2¹⁰, powers of 3 up to 3⁵, etc.) and distinguish exact roots from approximations

• Powers of Ten Notation

  Interpret and compare numbers in standard form A × 10ⁿ where 1 ≤ A < 10 and n is an integer; convert between ordinary numbers and standard form

  • Convert large and small numbers into standard form A × 10ⁿ
  • Convert numbers from standard form back to ordinary notation
  • Compare and order numbers given in standard form

Data & Statistics

Your child is learning to interpret and create different types of graphs and charts, and beginning to calculate averages to understand what data tells us about real-world situations.

• Scatter Graphs

  Plot bivariate data on a scatter graph with correctly labelled axes and appropriate scales; describe the correlation (positive, negative, none) and draw an estimated line of best fit where appropriate

  • Plot 10 data points on a scatter graph with correct axis labels and a consistent scale
  • Draw a line of best fit by eye, ensuring roughly equal numbers of points above and below
  • Use the line of best fit to estimate a y-value for a given x-value within the data range

• Scatter Graphs & Correlation

  Describe simple mathematical relationships between two variables using scatter graphs, identify positive, negative, or no correlation, and use a line of best fit to make predictions

  • Plot bivariate data on a scatter graph and describe the type of correlation observed
  • Draw a line of best fit by eye and use it to estimate a value within the data range
  • Explain what positive, negative, and no correlation mean in the context of real data (e.g., temperature vs. ice-cream sales)

• Comparing measurements

  Describe, interpret, and compare distributions of a single variable using appropriate measures of central tendency (mean, median, mode) and spread (range), including the effect of outliers

  • Calculate mean, median, and mode for a data set and explain when each is most appropriate
  • Find the range of a data set and explain how an outlier affects the mean versus the median
  • Compare two data sets using their averages and ranges to draw conclusions

• Pictograms and tally charts (age 11+)

  Construct and interpret frequency tables, bar charts, pie charts, pictograms, and vertical line charts for both categorical and grouped numerical data, choosing appropriate representations for the data type

  • Construct a grouped frequency table from raw continuous data, choosing appropriate class intervals
  • Draw a pie chart by calculating the angle for each category
  • Interpret a bar chart comparing two data sets and draw a conclusion about the difference

Fractions

Your child is mastering advanced fraction operations — adding, subtracting, and multiplying fractions with different denominators, dividing with fractions, and solving real-world problems using visual models and mathematical reasoning.

• Mixed & Improper Fractions

  Use the four operations with formal written methods applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative

  • Add, subtract, multiply, and divide with positive and negative integers
  • Apply formal written methods to calculations with decimals of any size
  • Perform all four operations with proper fractions, improper fractions, and mixed numbers

• Decimals and fractions (age 11+)

  Work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and 7/2 or 0.375 and 3/8); convert fluently between the two forms

  • Convert any terminating decimal to a fraction in simplest form
  • Convert any fraction with a denominator whose prime factors are only 2 and 5 to a terminating decimal
  • Explain why some fractions produce terminating decimals and others do not

Addition & Subtraction

Your child is mastering complex addition and subtraction — solving multi-step problems with large numbers and decimals, and choosing the best strategies to work out challenging calculations.

• Positive and Negative Numbers

  Understand positive and negative numbers as describing quantities with opposite directions or values; use them in context such as temperature, floors in a building, and bank balances

  • Represent positive and negative numbers on a number line and explain what zero means in context
  • Add a positive or negative number to any integer using number line reasoning
  • Subtract a positive or negative number from any integer, understanding that subtracting a negative is equivalent to adding

Multiplication & Division

Your child is advancing to sophisticated multiplication and division — using formal written methods for complex calculations, working with decimals, and applying the order of operations to solve multi-step problems.

• Sign Rules for Multiplication

  Multiply and divide with positive and negative integers and rational numbers, understanding the rules for the sign of the product or quotient

  • Apply the sign rules when multiplying two integers (positive × negative, negative × negative)
  • Apply the sign rules when dividing two integers
  • Solve multi-step real-world problems involving all four operations with positive and negative rational numbers