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Broad General Education · S1–S3

BGE Maths Progression

Find a starting point, follow progression through Third and Fourth Level, and connect learning with National course preparation.

Follow a clear pathway through BGE maths, with practice for Third and Fourth Level and links into National courses. Choose the areas that match current learning and move on when pupils are ready. CfE levels are not tied to a particular school year.

Start or revisit

Earlier learning and readiness

Revisit Second Level skills where needed, or use Third Level Readiness to identify areas that may need more practice.

Choose a strand

Progress from Third to Fourth Level

Open a strand to see how skills build from Third into Fourth Level. Revisit earlier learning where needed, then choose the next appropriate step.

Some practice includes a range of difficulty. Teacher notes highlight where selecting suitable questions may be helpful.

Number and calculation

Calculate → choose a method → check accuracy

  1. Third Level

    Calculate and compare

    Connect whole numbers, decimals and numbers below zero.

    Next step: Use Negative numbers for ordering, all four integer operations and short mixed calculations. Explain sign choices using number lines and contexts before moving to signed algebra.

  2. Third → Fourth Level

    Choose and check

    Use estimates and the order of operations to check a method.

  3. Fourth Level

    Make accuracy decisions

    Choose suitable rounding and consider the effect of error.

What next? Move on when pupils can choose a calculation, explain their method and check whether the answer is sensible.

Powers, roots and prime factors

Prime factors → powers and roots → scientific notation

  1. Third Level

    Build prime-factor products

    Write a number as prime factors, using powers for repeated factors.

    Connect in class: Also explore common factors, common multiples and the reasons for each choice.

  2. Third → Fourth Level

    Connect powers and roots

    Explain how a power and a root undo one another.

    Connect in class: Teach root notation and simple power/root pairs before moving on.

  3. Fourth Level

    Represent large and small numbers

    Use powers of ten to write and compare quantities.

What next? Connect repeated multiplication and its inverse before introducing index laws or scientific notation.

Fractions, decimals and percentages

Equivalent forms → fraction calculations → mixed numbers and decisions

  1. Third Level

    Connect equivalent forms

    Use fractions of quantities and connect decimals and percentages.

  2. Third → Fourth Level

    Calculate with fractions

    Use common denominators and connect mixed and improper fractions.

  3. Fourth Level

    Compare and decide

    Extend mixed-number calculations and use percentages in decisions.

What next? Move from representations to calculations when pupils can explain equivalence and choose a useful form.

Ratio and proportion

Compare parts → share and scale → connect related quantities

  1. Third Level

    Compare and share

    Keep the order of quantities clear and explain each share.

  2. Third → Fourth Level

    Scale in context

    Connect recipes, quantities and unit rates.

  3. Fourth Level

    Reason about related quantities

    Use proportional reasoning to compare plans and explain changes.

    Connect in class: Use inverse relationships as a later extension once direct proportion is secure.

What next? Look for explanations of the scale factor or unit rate, as well as a correct answer.

Algebra, expressions and equations

Rules and unknowns → expressions and equations → brackets and inequalities

  1. Third Level

    Express a rule

    Connect patterns, unknown numbers and simple formulae.

    Connect in class: Start with positive values, then use signed values and squared terms. Ask learners to write and explain their methods as well as choosing answers. These short practice topics are not a complete Third Level course.

  2. Third → Fourth Level

    Connect a situation to a formula

    Describe a fixed cost and a cost that changes, then evaluate a rule.

    Connect in class: Use tables and simple equations to explain the relationship. State the allowance range for each phone-plan rule. Next, connect equal groups to a single bracket, then reverse expansion to explore common factors at Fourth Level.

  3. Fourth Level

    Extend expressions and equations

    Connect the distributive law, common factors and equation solving.

What next? Explain variables, collect like terms, substitute signed values and check simple equations before moving to Fourth Level brackets and common factors. National 5 topic pages contain a wider mix.

Geometry and angles

Shape and angle facts → scale and circles → right-angled triangles

  1. Third Level

    Explain angle relationships

    Use shape properties and intersecting or parallel lines.

    Connect in class: Include practical drawing with a ruler and protractor.

  2. Third → Fourth Level

    Connect shape and measure

    Apply scale, circle measures and right-triangle relationships.

    Connect in class: Introduce squares and roots before using Pythagoras.

  3. Fourth Level

    Choose a triangle method

    Distinguish side-length and angle problems in right-angled triangles.

What next? Ask pupils to name the property or relationship they used before increasing the number of steps.

Measurement, area and volume

Units and formulae → compound shapes → prisms and material choices

  1. Third Level

    Choose units and formulae

    Connect measurements to area and volume calculations.

  2. Third → Fourth Level

    Combine shapes

    Split a design into useful parts and check the total.

  3. Fourth Level

    Use prisms and materials

    Connect prism volume, capacity and choices about materials.

    Connect in class: Teach surface area using nets and faces alongside these volume resources.

What next? Check that pupils distinguish length, area, volume and surface area and can explain their units.

Coordinates, scale and transformations

Positions and scale → grids and movement → transformations and graphs

  1. Third Level

    Describe position and scale

    Read and plot points, then connect positions and lengths to a plan.

    Connect in class: Include zero, points on either axis and all four quadrants. Compare points with the same x- or y-coordinate before applying coordinates in a map. Continue symmetry in classroom work.

  2. Third → Fourth Level

    Work across a grid

    Use secure coordinate reading and plotting to describe how a shape moves.

    Connect in class: Use separate grid tasks for reflection, translation and rotational symmetry; navigation practice covers scale and bearings.

  3. Fourth Level

    Connect points and relationships

    Build a table of values, plot a line and interpret its gradient.

    Connect in class: Continue to identify and explain reflections and translations on a four-quadrant grid.

What next? Secure reading and plotting points before connecting movement on a grid to straight-line graphs.

Time, rates and travel

Timetables → speed, distance and time → compare journeys

  1. Third Level

    Read and plan time

    Find durations and interpret schedules.

  2. Third → Fourth Level

    Connect distance and time

    Use the relationship between speed, distance and time.

  3. Fourth Level

    Compare journey plans

    Combine travel calculations, stops and timing constraints.

What next? Move on when pupils can explain the units and choose which quantity they need to find.

Data and statistics

Read and collect → compare and question → choose a useful summary

  1. Third Level

    Read, collect and display

    Interpret graphs and gather data for a useful question.

  2. Third → Fourth Level

    Compare and question data

    Use summaries and consider how a display could mislead.

    Connect in class: Discuss sample choice, sample size and whether a conclusion is fair.

  3. Fourth Level

    Choose and explain a summary

    Compare averages, spread and relationships in context.

What next? Ask pupils to justify a graph or average and explain what the data can and cannot show.

Probability and risk

Describe chance → calculate and compare → predict and decide

  1. Third Level

    Describe and calculate chance

    Connect likelihood words, fractions and the probability scale.

  2. Third → Fourth Level

    Connect probability and frequency

    Compare outcomes, experiments and expected frequency.

  3. Fourth Level

    Use evidence to decide

    Use predictions and consider the consequences of a choice.

What next? Look for explanations of likelihood and consequences, not just a probability calculation.

Money and financial mathematics

Budget and compare → pay and currency → justify financial choices

  1. Third Level

    Budget and find value

    Use totals, change and best-value comparisons.

  2. Third → Fourth Level

    Connect income and spending

    Use budgets, pay, currency and savings in context.

  3. Fourth Level

    Justify a financial choice

    Compare costs over time and explain assumptions.

What next? Move on when pupils can compare options for a particular situation and explain their choice.

Problem solving and mathematical reasoning

Choose a method → connect ideas → justify and review

  1. Third Level

    Choose a useful method

    Identify what is known, plan a method and check it.

  2. Third → Fourth Level

    Connect several ideas

    Bring number, measure, money and data together.

  3. Fourth Level

    Justify and review a plan

    Compare constraints, explain a decision and consider alternatives.

    Connect in class: Include mathematical enquiry and discussion of how maths is used in work and everyday life.

What next? Revisit reasoning throughout every strand: ask what changed, why the method works and whether the answer is reasonable.

Next pathways

Prepare for National courses

Choose the next course with a teacher, using evidence across the relevant strands. Pupils do not need to complete every National course in order.

Planning for National 4 Mathematics? Use the algebra and geometry progression with your teacher. The National 4 course on this site is Applications of Mathematics.

Check a starting point or review learning

Use a short check alongside classroom evidence to plan next steps. Judgements about CfE levels and course choices draw on a wider picture of each pupil’s learning.