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IB Class 7 Mathematics★ IB MYP Year 2 Mathematics · Full Unit Coverage · Board: IB MYP

Algebra, Geometry and Statistical Reasoning — Patterns in Space and Data

ALGEBRA as the language of GENERALISATION. GEOMETRY as the study of SHAPE and SPACE. And STATISTICS as the science of making SENSE of data — because in the 21st century, data literacy is as important as reading literacy. IB MYP Year 2 Mathematics.

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By the end of this chapter you'll be able to…

1Distinguish expressions (no equals sign) from equations (has equals sign)
2Solve linear equations using the golden rule: do the same to both sides
3Find the nth term formula for a linear arithmetic sequence
4Apply angle relationships: complementary, supplementary, vertically opposite, parallel-transversal
5State properties of parallelogram, rhombus, rectangle, square, trapezium
6Calculate area of triangle, parallelogram, and trapezium
7Calculate mean, median, mode, and range; choose the appropriate measure for a dataset
8Choose the appropriate graph type for a dataset (bar/histogram/line/pie/scatter)
9Identify misleading graphs by examining axes and scales
10Distinguish theoretical from experimental probability; state the Law of Large Numbers

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Why this chapter matters

This IB MYP Year 2 unit builds the algebraic and statistical foundations central to all subsequent MYP mathematics. Solving linear equations (golden rule: do the same to both sides) and finding nth term formulas are Criterion A anchors tested in every subsequent unit. Statistics — mean/median/mode/range with choosing the right measure (median for skewed data) — is a Criterion D: Real-Life Context task that connects to science and I&S investigations. The 'deceptive graph' analysis (checking axes, identifying misleading scales) is a critical thinking skill assessed in Criterion D. Probability (classical vs experimental, Law of Large Numbers) connects to IB's global context 'Scientific Innovation.'

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Algebra, Geometry and Statistical Reasoning

MYP Unit Framework

Key Concept: RELATIONSHIPS Related Concepts: Generalisation. Space. Representation. Global Context: Scientific and Technical Innovation (How do we use mathematics to model, predict, and make decisions?) Statement of Inquiry: Algebra GENERALISES the relationships we observe in numbers and shapes, while statistics allows us to make REASONED JUDGMENTS from data — both are essential tools for navigating a complex world.

Inquiry Questions

Type	Question
Factual	What is a variable? How do you solve a linear equation? What is the mean and how is it calculated?
Conceptual	Why does algebra use LETTERS? How can the same data be presented to support DIFFERENT conclusions?
Debatable	Is algebra 'useful' for everyday life — or is it primarily a tool for advanced science? Can statistics prove ANYTHING if you choose the right graph?

1. Algebra — The Language of Generalisation

From Arithmetic to Algebra

Arithmetic: '3 + 5 = 8.' Algebra: 'x + y = z, where x = 3, y = 5.' 'Algebra uses LETTERS to stand for numbers — so we can express GENERAL relationships that are true for ALL numbers, not just specific cases.'

Expressions and Equations

Expression: 3x + 2y. A mathematical phrase. NO equals sign.
Equation: 3x + 5 = 14. A mathematical SENTENCE. HAS an equals sign. States that two expressions are EQUAL.

Solving Linear Equations

Golden Rule: Whatever you do to ONE side, you must do to the OTHER. Goal: ISOLATE the variable.

3x + 5 = 14 → 3x = 9 → x = 3.

From Patterns to Algebra

'The nth term of a sequence is an ALGEBRAIC expression. Pattern: 3, 7, 11, 15... Going up by 4 each time. nth term = 4n — 1. 'Algebra CAPTURES the pattern — so you can find the 100th term without writing out 100 numbers.'

2. Geometry — Shapes, Angles and Spatial Reasoning

Angle Relationships

Complementary: Sum = 90°. Supplementary: Sum = 180°.
Vertically Opposite: Equal.
Parallel Lines Cut by Transversal: Corresponding = equal. Alternate interior = equal. Co-interior = supplementary (sum = 180°).

Triangles — Deeper Properties

Angle Sum: Always 180°.
Exterior Angle = Sum of two interior OPPOSITE angles.
Triangle Inequality: Sum of ANY two sides > the third side.

Quadrilaterals

Shape	Properties
Parallelogram	Opposite sides ∥ & =. Opposite angles =. Diagonals BISECT.
Rhombus	All sides =. Diagonals ⟂.
Rectangle	All angles 90°. Diagonals =.
Square	ALL properties of rhombus AND rectangle.
Trapezium	ONE pair of parallel sides.

Area of 2D Shapes

Triangle: ½bh. Parallelogram: bh. Trapezium: ½(a+b)h.

3. Statistics — Making Sense of Data

Measures of Central Tendency

Measure	What It Is	Best Used When
Mean	Average (sum ÷ count)	Data is SYMMETRICAL, no outliers
Median	Middle value	Data is SKEWED (income, house prices)
Mode	Most frequent	Categorical data. Finding what's MOST COMMON.

Measures of Spread

Range = Max — Min. Crude but INSTANT. 'The mean tells you WHERE the centre is. The range tells you HOW SPREAD OUT the data are.'

Data Visualisation — Choosing the Right Graph

Graph Type	Best For
Bar chart	Comparing CATEGORIES
Histogram	Showing DISTRIBUTION of continuous data
Line graph	Showing CHANGE over TIME
Pie chart	Showing PROPORTIONS of a whole
Scatter plot	Showing RELATIONSHIP between two variables

The Deceptive Graph

'Changing the SCALE of a graph can make a small change look DRAMATIC — or a dramatic change look SMALL. Always CHECK the axes. Always ASK: "What is this graph trying to make me believe?"'

Real Data Investigation

'In this unit, you will COLLECT real data — from your classmates, your school, or public sources — and ANALYSE it. You'll calculate the mean, median, and range. You'll create graphs. And you'll WRITE about what the data SHOWS.'

4. Probability — The Mathematics of Chance

Classical Probability: P(E) = n(E)/n(S). 0 ≤ P(E) ≤ 1.

Experimental vs. Theoretical Probability

Theoretical: What SHOULD happen (based on equally likely outcomes). P(Head on fair coin) = ½.
Experimental: What ACTUALLY happens when you try it. 'Toss a coin 10 times. You might get 7 heads. Experimental probability = 7/10. Toss it 1000 times. It will be CLOSER to ½. The Law of Large Numbers: the more trials, the closer experimental probability approaches theoretical probability.'

Your Summative Assessment

Task: 'The Data Investigation Project'

Choose a RESEARCH QUESTION: 'Do students who sleep more get higher grades?' 'Is there a relationship between screen time and physical activity?'
COLLECT data (minimum n=30). 3. Calculate mean, median, mode, range. 4. Create at least TWO different types of graphs. 5. Write a CONCLUSION: What does your data show? What are the LIMITATIONS of your study?

ATL Skills

Skill	Focus
Critical Thinking	Interpreting data. Evaluating misleading graphs.
Information Literacy	Collecting and organising data. Creating appropriate visualisations.
Communication	Presenting mathematical findings clearly.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Algebra, Geometry, Statistics, and Probability

MYP FRAMEWORK: Key Concept = RELATIONSHIPS. Related Concepts = Generalisation, Space, Representation. Global Context = Scientific and Technical Innovation. Statement of Inquiry = 'Algebra GENERALISES relationships, while statistics allows REASONED JUDGMENTS from data.' ALGEBRA: EXPRESSION: mathematical phrase, NO equals sign (e.g. 3x+2y). EQUATION: has equals sign, states two expressions are EQUAL (e.g. 3x+5=14). SOLVING LINEAR EQUATIONS: Golden Rule = do the SAME to BOTH sides. Goal: ISOLATE the variable. Example: 3x+5=14 → subtract 5: 3x=9 → divide by 3: x=3. CHECK: substitute back: 3(3)+5=14 ✓. NTH TERM: For an arithmetic sequence, nth term = dn + c where d = common difference. Example: 3, 7, 11, 15 → difference=4, nth term = 4n−1. Verify: n=1 → 4(1)−1=3 ✓. n=100 → 4(100)−1=399. GEOMETRY — ANGLE RELATIONSHIPS: Complementary = sum 90°. Supplementary = sum 180°. Vertically Opposite = EQUAL. PARALLEL LINES + TRANSVERSAL: Corresponding = equal (F-shape). Alternate interior = equal (Z-shape). Co-interior/Same-side interior = supplementary, sum 180° (C-shape). TRIANGLES: Angle sum = 180°. Exterior angle = sum of two non-adjacent interior angles. Triangle inequality: sum of any two sides > third side. QUADRILATERAL PROPERTIES: Parallelogram (opp sides ∥ and =, opp angles =, diagonals bisect). Rhombus (all sides =, diagonals ⊥, bisect angles). Rectangle (all 90°, diagonals =). Square (all rhombus + rectangle properties). Trapezium (ONE pair ∥ sides). AREA FORMULAS: Triangle = ½bh. Parallelogram = bh (height = perpendicular). Trapezium = ½(a+b)h where a,b = parallel sides, h = perpendicular height. STATISTICS — MEASURES OF CENTRAL TENDENCY: MEAN = sum ÷ count. Best for: symmetrical data, no outliers. MEDIAN = middle value (odd n: middle; even n: average of two middle). Best for: SKEWED data (income, house prices), when outliers exist. MODE = most frequent value. Best for: categorical data, finding most common. RANGE = Max − Min. Simple measure of SPREAD. GRAPH TYPES: Bar chart (comparing categories). Histogram (distribution of continuous data — no gaps). Line graph (change over TIME). Pie chart (proportions of a whole). Scatter plot (relationship between two variables). DECEPTIVE GRAPHS: Changing the y-axis scale can make a small change look dramatic (truncated axis). Check: Does the axis start at 0? Is the scale consistent? 'Always ask: What is this graph trying to make me believe?' PROBABILITY: Classical: P(E) = n(E)/n(S). Range: 0 ≤ P(E) ≤ 1. P=0: impossible. P=1: certain. THEORETICAL probability: what SHOULD happen (equally likely outcomes). EXPERIMENTAL probability: what ACTUALLY happens in trials. LAW OF LARGE NUMBERS: as number of trials increases, experimental probability → theoretical probability.

IB MYP CRITERION CONNECTIONS: (1) Criterion A (Knowing): Solve equations, calculate statistics, identify angle relationships, apply area formulas. (2) Criterion B (Investigating): Collect real data (n≥30), calculate statistics, create graphs, evaluate limitations. (3) Criterion C (Communicating): Show equation-solving steps clearly. Present graphs with labels, titles, appropriate scale. (4) Criterion D (Applying in Real Contexts): Choose appropriate measure of central tendency (when to use median vs mean). Identify misleading graphs. Probability in real decisions. ATL: 'Can statistics prove ANYTHING if you choose the right graph?' — this is the critical question for criterion D. KEY TRAPS: (1) MEDIAN for even number of values: AVERAGE the two middle values. Don't just take the lower one. (2) NTH TERM: always verify by substituting n=1 and n=2. (3) PARALLELOGRAM AREA: use HEIGHT (perpendicular), not the slant side. (4) HISTOGRAM vs BAR CHART: histogram = continuous data = NO gaps between bars. Bar chart = categorical = gaps between bars.

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Using mean when the data has extreme outliers (should use median), or forgetting to average two middle values for even-numbered datasets

✓ TWO CORRECTIONS: (1) MEAN vs MEDIAN: If a dataset has OUTLIERS (extreme values), the MEAN is pulled toward them and gives a misleading 'average.' Example: Salaries at a company: £25k, £28k, £30k, £32k, £35k, £500k (CEO). Mean = (25+28+30+32+35+500)/6 = 650/6 ≈ £108k — this makes it sound like a high-paying company, but 5 out of 6 employees earn below £35k. Median = average of 3rd and 4th values = (30+32)/2 = £31k — much more representative. Rule: Use MEDIAN when data is SKEWED or has outliers. Use MEAN when data is SYMMETRICAL and no extreme values. (2) MEDIAN FOR EVEN n: When you have an EVEN number of values, there is no single 'middle' value. STEP 1: Sort the data in ascending order. STEP 2: Find the two MIDDLE values (positions n/2 and n/2+1). STEP 3: AVERAGE them (add and divide by 2). Example: 7 values ordered: 12, 15, 18, 21, 24, 27, 30. n=7 (odd) → median = 4th value = 21. If 8 values: 12, 15, 18, 21, 24, 27, 30, 33. n=8 (even) → median = average of 4th and 5th = (21+24)/2 = 22.5.

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1MEDIUM· nth-term-statistics

The sequence 4, 11, 18, 25, 32... (a) Find the nth term formula. (b) Is 200 a term in this sequence? (c) If this sequence represents weekly savings (in rupees), how much is saved after 52 weeks? Then: Explain why a newspaper headline saying 'Average Salary Rises to £50,000' can be misleading.

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(a) NTH TERM: Difference between consecutive terms = 7 (arithmetic sequence). First term = 4. nth term = first term + (n−1) × difference = 4 + (n−1) × 7 = 4 + 7n − 7 = 7n − 3. CHECK: n=1: 7(1)−3=4 ✓. n=2: 7(2)−3=11 ✓. n=5: 7(5)−3=32 ✓. (b) IS 200 A TERM? Set 7n−3=200 → 7n=203 → n=29. Since n=29 is a WHOLE POSITIVE NUMBER, YES, 200 is the 29th term. (If n were not a whole number, 200 would NOT be in the sequence.) (c) SAVINGS AFTER 52 WEEKS: 52nd term = 7(52)−3 = 364−3 = 361 rupees (savings in week 52). But the question asks total after 52 weeks. Sum of arithmetic series = n/2 × (first term + last term) = 52/2 × (4 + 361) = 26 × 365 = 9490 rupees total saved after 52 weeks. WHY 'AVERAGE SALARY = £50,000' CAN BE MISLEADING: If the word 'average' means MEAN, then a few very high earners (CEOs, directors earning £500k+) can PULL UP the mean dramatically. Most workers could earn far less than £50,000 while the MEAN is £50,000. Example: 10 employees earning: £20k, £22k, £25k, £28k, £30k, £32k, £35k, £38k, £40k, £230k (CEO). Mean = £500k / 10 = £50k. But 9 out of 10 employees earn LESS than £50k. The MEDIAN would be (30+32)/2 = £31k — a much more accurate reflection of what a 'typical' employee earns. The MEDIAN is the appropriate measure when data is skewed by outliers.

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•EXPRESSION vs EQUATION: Expression has NO equals sign (e.g., 3x+2). Equation HAS equals sign (e.g., 3x+2=11). Only equations can be 'solved'; expressions can only be 'simplified' or 'evaluated.'
•GOLDEN RULE OF EQUATIONS: Whatever you do to one side, you must do to the OTHER side. This preserves equality. Goal: isolate the variable. Example: 3x+5=14 → subtract 5: 3x=9 → divide by 3: x=3. Always CHECK by substitution.
•NTH TERM of arithmetic sequence: dn + c, where d = common difference, c = first term − d. Example: 3, 7, 11, 15 → d=4, c=−1, so nth term = 4n−1. Verify with n=1, n=2.
•ANGLE RELATIONSHIPS: Complementary = 90°. Supplementary = 180°. Vertically opposite = EQUAL. Parallel lines + transversal: CORRESPONDING (F-shape) = equal; ALTERNATE INTERIOR (Z-shape) = equal; CO-INTERIOR (C-shape) = supplementary (sum 180°).
•TRIANGLE FACTS: Angle sum = 180°. Exterior angle = sum of two non-adjacent interior angles. Triangle inequality: any two sides sum to more than third side.
•QUADRILATERAL HIERARCHY: Parallelogram (opp sides ∥ and equal) → Rhombus (all sides equal, diagonals perpendicular) and Rectangle (all 90°, diagonals equal) → Square (both rhombus and rectangle). Trapezium has ONE pair of parallel sides.
•AREA FORMULAS: Triangle = ½bh. Parallelogram = bh (h = perpendicular height). Trapezium = ½(a+b)h (a, b = parallel sides). Always use PERPENDICULAR height, not slant height.
•MEASURES OF CENTRAL TENDENCY: MEAN = sum/count (best for symmetric data). MEDIAN = middle value (best for SKEWED data / outliers). MODE = most frequent (best for categorical data). RANGE = max − min (measure of spread).
•MEDIAN FOR EVEN n: Average the two middle values. Example: 8 values → average of 4th and 5th values.
•GRAPH TYPES (5): Bar chart (categories, gaps between bars). Histogram (continuous data, NO gaps). Line graph (change over time). Pie chart (proportions of a whole). Scatter plot (relationship between two variables).
•MISLEADING GRAPHS: Check the y-axis — does it start at 0? Is the scale consistent? Truncated axes can make tiny differences look enormous. Always ask: 'What is this graph trying to make me believe?'
•PROBABILITY: P(E) = number favourable / total outcomes. 0 ≤ P(E) ≤ 1. THEORETICAL probability: calculated from equally likely outcomes. EXPERIMENTAL probability: measured from actual trials. LAW OF LARGE NUMBERS: as trials → ∞, experimental → theoretical.

IB marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Equations: ALWAYS show every step. Use the golden rule explicitly: 'Subtracting 5 from both sides...' Each step earns marks.
Statistics: when asked for 'an appropriate measure of central tendency,' state both your CHOICE and your REASON (e.g., 'I will use median because the data has outliers').
Nth term: write the formula, then ALWAYS verify with n=1 and n=2. Missing verification loses a mark in MYP rubrics.
Graphs (Criterion D): when interpreting a graph, ALWAYS check the y-axis scale. Comment on whether the graph could be misleading.
Probability: state both theoretical and experimental values when both can be calculated. Note that experimental → theoretical with more trials (Law of Large Numbers).

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Research Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, ...) — each term is the sum of the previous two. The ratio of consecutive Fibonacci numbers approaches the GOLDEN RATIO φ = (1+√5)/2 ≈ 1.618. Fibonacci patterns appear in nature (sunflower seeds, pine cones, nautilus shells). Investigate why these patterns appear.
Investigate the difference between PERMUTATIONS and COMBINATIONS — how many ways to arrange 5 books on a shelf (permutation = 5! = 120)? vs how many ways to choose 3 books from 5 for a holiday (combination = 5C3 = 10)? This distinction underlies all of probability and counting.
Explore SIMPSON'S PARADOX — a famous statistical phenomenon where a trend appears in different groups of data but reverses when the groups are combined. Real example: A hospital's mortality rate may LOOK higher than a competitor's, but when broken down by severity of cases, the hospital is actually BETTER at every severity level — they just treat sicker patients. Investigate the 1973 Berkeley gender bias case.
Research GEOMETRIC PROOFS using parallel lines and angles — proving the angle sum of a triangle = 180° using a line parallel to one side. The proof technique generalises to many geometry problems. Investigate the IMO (International Mathematical Olympiad) for the kind of geometry proofs expected at the olympiad level.

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

Use this DECISION RULE: (1) Is the data CATEGORICAL (colours, names, types)? → Use MODE (most frequent). Mean and median don't make sense for non-numerical data. (2) Is the data NUMERICAL and SYMMETRIC (no extreme values)? → Use MEAN (it captures all values). (3) Is the data NUMERICAL but SKEWED (has outliers)? → Use MEDIAN (it isn't pulled by extremes). EXAMPLE: Salary data: most workers earn $30-40k, but the CEO earns $5 million. The MEAN salary will be inflated by the CEO; the MEDIAN gives a much better picture of what a 'typical' worker earns. Real-world test: would you describe a typical family using the family with one billionaire's income, or the family in the middle? The median is usually the better representative for SKEWED data. Statisticians say: 'Mean is sensitive to outliers; median is robust.'

A MISLEADING GRAPH uses visual tricks to make differences look bigger or smaller than they actually are — often to support a particular interpretation. COMMON TRICKS: (1) TRUNCATED Y-AXIS: instead of starting at 0, the axis starts at (say) 50. A change from 51 to 52 looks dramatic visually, but is actually only 2% change. (2) INCONSISTENT SCALES: different intervals on the same axis. (3) 3D EFFECTS: distorts pie chart proportions. (4) CHERRY-PICKED DATA RANGE: showing only the time period that supports your point. (5) MISSING CONTEXT: not showing comparable data. CRITICAL READING: Always check (1) Does the y-axis start at 0? (2) Are the intervals equal? (3) What is the source of the data? (4) What time period is shown — does it cherry-pick? (5) Is the title biased? Newspapers, advertisers, and politicians routinely use misleading graphs. Being able to spot them is a key 21st-century literacy skill.

THEORETICAL PROBABILITY: calculated mathematically based on equally likely outcomes. For a fair coin, P(heads) = 1/2 = 0.5 (because there are 2 equally likely outcomes and 1 is heads). For a fair die, P(rolling 6) = 1/6 ≈ 0.167. EXPERIMENTAL PROBABILITY: measured from ACTUAL TRIALS. Toss a coin 10 times, get 6 heads → experimental P(heads) = 6/10 = 0.6. RELATIONSHIP — LAW OF LARGE NUMBERS: as the number of trials INCREASES, experimental probability APPROACHES theoretical probability. With 1,000 coin tosses, you'd likely get ~500 heads (P≈0.5). With 1,000,000 tosses, even closer to 0.5. With only 10 tosses, you might get 6 heads (0.6) or even 8 (0.8) just by chance — but this shouldn't make you think the coin is biased. This is why scientific studies use LARGE SAMPLE SIZES — to make experimental results approximate theoretical truths.

Imagine a parallelogram. If you 'cut' off a right triangle from one side and move it to the other side, you get a RECTANGLE with the SAME area. The rectangle's dimensions are the BASE (b) and the PERPENDICULAR HEIGHT (h) — not the slant side. So Area = b × h. The slant side is irrelevant to area — only the perpendicular distance between the parallel sides matters. EXAMPLE: A parallelogram with base 6 cm, slant side 5 cm, but tilted so the perpendicular height is only 4 cm. Area = 6 × 4 = 24 cm². NOT 6 × 5 = 30 cm². COMMON ERROR: students use the slant side. Always look for or compute the PERPENDICULAR HEIGHT (often drawn as a dashed line in textbook diagrams).

The NTH TERM is a FORMULA that lets you find ANY term of the sequence without listing them all. For an arithmetic sequence (constant difference between terms): formula is nth term = dn + c, where d = common difference, c = first term − d. STEPS: (1) Find d (subtract any two consecutive terms). (2) Find c (subtract d from the first term, or back-calculate). (3) Write the formula. (4) VERIFY using n=1, n=2. EXAMPLE: Sequence 5, 9, 13, 17, 21... Difference d = 4. First term = 5. So c = 5 − 4 = 1. Formula: nth term = 4n + 1. Verify: n=1 → 4(1)+1 = 5 ✓. n=2 → 4(2)+1 = 9 ✓. n=10 → 4(10)+1 = 41. POWER: instead of listing 100 terms to find the 100th, just compute 4(100)+1 = 401. The nth term is the algebraic 'compression' of an infinite sequence into one formula.