IB MYP Mathematics: Number sense, estimation, exponents and proportion across MYP (MYP Mathematics)

This guide teaches the Number rows of the numerical and abstract reasoning branch in the IB MYP Mathematics guide (for use from September 2020/January 2021, first assessment 2022). The guide gives topics and skills as examples in three columns: MYP 1-3, MYP 4-5 standard and MYP 4-5 extended, plus optional enrichment. Extended and enrichment items are labelled below. The guide notes that schools decide the exact split between standard and extended, and that enrichment topics are not included in eAssessments.

Related pages: revision notes, practice questions, the MYP Mathematics hub, the checklist and the free diagnostics. Algebra topics (equations, formulae, general rules for sequences) have their own pages.

What the branch asks you to understand

The guide lists ideas such as number sense, estimating and approximating, working with measurement and units, rounding to decimal places and significant figures, equivalence, proportional reasoning with ratios and percentages, accuracy and its limits, appreciation of errors, and money and financial literacy. Related concepts attached to this branch include approximation, equivalence, quantity, simplification and validity. In Criterion D tasks you are often asked to comment on whether a rounded or estimated answer is valid, so keep that question in mind throughout.

Topic	Level
Forms of numbers, recurring decimals, estimation, time zones, factors, primes, GCF, LCM, squares and roots, exponents, ratios	MYP 1-3
N, Z, Q, Q’, R; absolute values; compound and double inequalities; number sequences; surds; laws of exponents with negative integers; standard form; direct and inverse proportion	MYP 4-5 standard
Lower and upper bounds; logarithms and their laws; fractional exponents	MYP 4-5 extended
Rationalizing the denominator; number bases	Enrichment

Forms of numbers and number systems

Fractions, decimals and percentages are three forms of one value. A recurring decimal always equals a fraction. To convert, multiply by powers of 10 so the recurring tails line up, then subtract.

Example 1. Write 0.21818… (the 18 recurs) as a fraction. Let x = 0.21818… Then 10x = 2.1818… and 1000x = 218.1818… Subtract: 990x = 216, so x = 216/990 = 12/55.

The sets, in the guide’s notation: N is the positive integers and zero, Z the integers, Q the rational numbers (any a/b with b not 0), Q’ the irrational numbers (non-terminating, non-recurring decimals) and R the real numbers. Every member of N is in Z, every member of Z is in Q, and Q and Q’ together make R.

Example 2. Classify √49, 22/7, π, -3, 0.3 (recurring) and √20. √49 = 7 is in N. 22/7 is a fraction, so Q (it only approximates π). π is in Q’. -3 is in Z. 0.3 recurring = 1/3, so Q. √20 = 2√5 is in Q’. All are in R.

Absolute value |x| is the distance from x to 0, so it is never negative. |-7| = 7 and |3 - 9| = 6. For |x - 2| = 6, x is 6 away from 2: x = 8 or x = -4.

Factors, primes, GCF and LCM

Write each number as a product of primes, then compare.

Example 3. For 252 and 198: 252 = 2² × 3² × 7 and 198 = 2 × 3² × 11. GCF: take the lowest power of shared primes: 2 × 3² = 18. LCM: take the highest power of every prime: 2² × 3² × 7 × 11 = 2772. Check: 18 × 2772 = 252 × 198 = 49 896. GCF answers “largest equal pieces” (square tiles cutting a 252 cm by 198 cm floor with no waste: 18 cm). LCM answers “when do two cycles coincide”.

Squares, roots and surds (standard)

A surd is an irrational root left unevaluated, which keeps the answer exact. Use √(ab) = √a × √b and √(a/b) = √a / √b. Pull out the largest square factor.

Example 4. √72 = √36 × √2 = 6√2. 3√8 + √50 = 6√2 + 5√2 = 11√2 (only like surds combine). A difference of squares removes the surd: (2 + √3)(2 - √3) = 4 - 3 = 1.

Rationalizing the denominator (enrichment). Multiply top and bottom by the conjugate. 6/√3 = 6√3/3 = 2√3. 5/(3 - √2) = 5(3 + √2) / ((3 - √2)(3 + √2)) = 5(3 + √2)/7 = (15 + 5√2)/7.

Laws of exponents and standard form

Law	Example
aᵐ × aⁿ = aᵐ⁺ⁿ	2⁵ × 2⁻² = 2³
aᵐ ÷ aⁿ = aᵐ⁻ⁿ	7⁶ ÷ 7⁴ = 7²
(aᵐ)ⁿ = aᵐⁿ	(3²)³ = 3⁶
a⁰ = 1 (a not 0)	9⁰ = 1
a⁻ⁿ = 1/aⁿ	3⁻² = 1/9

Example 5. (2⁵ × 2⁻²) / 2⁴ = 2⁵⁻² ⁻ ⁴ = 2⁻¹ = 1/2. Algebraic form: (3a²b⁻¹)² / (a³b) = 9a⁴b⁻² / (a³b) = 9a/b³.

Standard form is a × 10ᵏ, with a between 1 (included) and 10 (excluded) and k a whole number. Large numbers have k positive, small numbers k negative.

Example 6. 0.000047 = 4.7 × 10⁻⁵. Multiply: (3.2 × 10⁵)(2.5 × 10⁻²) = 8 × 10³ (multiply the numbers, add the powers). Add: 6 × 10⁴ + 7.5 × 10³ = 60 000 + 7500 = 67 500 = 6.75 × 10⁴. Convert to the same power first.

Extended: fractional exponents and logarithms

a^(1/n) means the nth root of a, and a^(m/n) means that root raised to the power m. Do the root first. Example 7. 27^(2/3) = 3² = 9. 16^(-3/4) = 1 ÷ (fourth root of 16)³ = 1 ÷ 2³ = 1/8.

A logarithm answers “what power?”: log_a x = y means aʸ = x. Laws: adding logs multiplies the arguments, subtracting divides them, and a number in front becomes a power. So log(xy) = log x + log y, log(x/y) = log x - log y and log(xⁿ) = n log x. Example 8. log₂ 8 = 3 and log₁₀ 0.001 = -3. log 4 + log 25 = log 100 = 2. 2 log₃ 6 - log₃ 4 = log₃ (36/4) = log₃ 9 = 2. To solve 2ˣ = 50, take logs: x = log 50 / log 2 = 5.64 (3 s.f.), found with technology.

Number lines, inequalities and sequences

Open circle: the end value is not included (< or >). Closed circle: included (≤ or ≥).

• Double inequality -2 < x ≤ 5: open circle at -2, closed at 5, line between.
• Compound “x < -1 or x ≥ 3”: two separate pieces going outwards.
• Integers with -3 ≤ x < 2: -3, -2, -1, 0, 1.

(Solving inequalities is an algebra skill; here you represent them.)

Number sequences (describe, predict). For 5, 8, 13, 20, 29, the gaps are 3, 5, 7, 9, which grow by 2 each time. Continue the gaps (11, 13): next terms 40 and 53. Describe it in words: “the gap increases by 2 each time”.

Estimation, rounding and measurement

Round each number to 1 significant figure, then calculate. Example 9. 48.7 × 3.12 / 0.52 ≈ 50 × 3 / 0.5 = 300. The exact product gives 292.2, so the estimate is sensible.

Decimal places count digits after the point. Significant figures are counted from the first digit that is not zero. 0.0040763 is 0.0041 to 2 s.f. and 0.0041 to 4 d.p. (same number, different rules). 25 849 is 25 800 to 3 s.f.; the zeros are placeholders.

Units: convert in steps and keep the unit on every line. 3.6 km/h = 3600 m ÷ 3600 s = 1 m/s. 0.85 m³ = 850 litres. 2.5 g/cm³ = 2500 kg/m³.

Time zones. A flight leaves city A (UTC+5) at 21:15 and lasts 11 h 50 min. Arrival in A-time: 21:15 + 11:50 = 09:05 next day. City B is UTC-3, 8 hours behind: 01:05 next day, local time.

Accuracy, errors and bounds

Absolute error = |measured - true|. Percentage error = absolute error ÷ true value × 100. A tool’s precision limits how accurately you can report: a ruler marked in mm cannot justify four decimal places.

Extended: lower and upper bounds. A value rounded to the nearest unit u lies in [x - u/2, x + u/2). Lengths of 8.4 cm and 5.2 cm (1 d.p.) lie in 8.35 ≤ L < 8.45 and 5.15 ≤ W < 5.25. Area upper bound: 8.45 × 5.25 = 44.3625. Lower bound: 8.35 × 5.15 = 43.0025. For a quotient, divide the largest by the smallest for the upper bound. Distance 120 m (nearest 10 m, so 115 ≤ d < 125) in 9.6 s (1 d.p., so 9.55 ≤ t < 9.65): maximum speed = 125/9.55 = 13.1 m/s; minimum = 115/9.65 = 11.9 m/s (3 s.f.).

Ratio, percentage and proportion

Sharing. Share 3600 in the ratio 5 : 4 : 3. Parts = 12, one part = 300, shares 1500, 1200, 900. Percentage change. New = old × multiplier (increase 15%: ×1.15; decrease 15%: ×0.85). Reverse percentage. After a 15% discount the price is 340, so 0.85 × original = 340 and original = 400.

Direct proportion y = kx: y = 45 when x = 6, so k = 7.5 and at x = 14, y = 105. Inverse proportion y = k/x (xy constant): y = 12 when x = 5, so k = 60 and at x = 8, y = 7.5. Context: 6 workers take 15 days, so 9 workers take 6 × 15 ÷ 9 = 10 days.

Money and financial literacy

Simple interest: I = P × r × t. Compound interest: A = P(1 + r)ⁿ. On 2000 at 4% for 3 years, simple interest is 240; compound gives 2000 × 1.04³ = 2249.73, so interest 249.73. Depreciation uses a multiplier below 1: a car worth 18 000 losing 12% a year is worth 18 000 × 0.88⁴ = 10 794.52 after 4 years. Round money to 2 d.p. only at the end.

Number properties and bases

Addition and multiplication are commutative and associative, and multiplication distributes over addition. Use them for mental calculation: 25 × 17 × 4 = 25 × 4 × 17 = 1700, and 7 × 98 = 7 × 100 - 7 × 2 = 686. Subtraction and division are neither commutative nor associative: 8 - 3 ≠ 3 - 8.

Number bases (enrichment). In base 2, 1101₂ = 8 + 4 + 0 + 1 = 13. For 45 in base 2, divide by 2 repeatedly and read the remainders upwards: 101101₂. In base 5, 234₅ = 2 × 25 + 3 × 5 + 4 = 69.

Common errors

• Writing a recurring decimal as a rounded one and treating it as exact.
• Adding exponents when bases differ, or treating 3⁻² as -9.
• Giving 12.5 × 10⁴ as standard form; the first factor must be below 10.
• Rounding in the middle of a calculation, then reporting too many figures.
• Using the lower bound for both numbers when finding the upper bound of a quotient.
• Calling 22/7 irrational; it is rational.

Official syllabus

IB Mathematics guide, Middle Years Programme (for use from September 2020/January 2021, first assessment 2022), International Baccalaureate Organization: mathematics framework, numerical and abstract reasoning.