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

Quick recall for the Number rows of the numerical and abstract reasoning branch in the IB MYP Mathematics guide (from 2020, first assessment 2022). Full explanations and worked examples are in the study guide; test yourself with the practice questions. See also the MYP Mathematics hub, the checklist and the free diagnostics. Items marked ext are in the guide’s MYP 4-5 extended column; enr means optional enrichment.

Number sets and forms

Symbol	Set	Example members
N	positive integers and zero	0, 4, 250
Z	integers	-9, 0, 6
Q	rational numbers (a/b)	3/8, -0.25, 0.7 recurring
Q’	irrational numbers	√2, π, √20
R	real numbers (Q and Q’)	all of the above

• A decimal is rational if it stops or recurs. Irrational decimals neither stop nor recur.
• |x| is the distance from 0. |a - b| is the distance between a and b.
• Recurring decimal to fraction, in steps: let x equal the decimal; multiply by 10 or 100 or 1000 so the recurring tails match; subtract; divide.
• Worked reminder: x = 0.4545… gives 100x - x = 45, so x = 45/99 = 5/11.

Factors, primes, GCF, LCM

Task	Method
Prime factorization	divide by 2, 3, 5, 7 … until 1 is left; write with powers
GCF	shared primes, lowest powers
LCM	all primes, highest powers
Check	GCF × LCM = product of the two numbers

Worked reminder: 60 = 2² × 3 × 5 and 150 = 2 × 3 × 5². GCF = 2 × 3 × 5 = 30; LCM = 2² × 3 × 5² = 300; 30 × 300 = 9000 = 60 × 150.

Roots, surds and rationalizing

Rule	Example
√(ab) = √a × √b	√48 = √16 × √3 = 4√3
√(a/b) = √a / √b	√(5/9) = √5 / 3
like surds combine	2√3 + 5√3 = 7√3
(a + √b)(a - √b) = a² - b	(4 + √2)(4 - √2) = 14

Method for simplifying: find the largest square factor, take its root outside, leave the rest inside. enr Rationalizing: multiply top and bottom by √b (single surd) or by the conjugate (a binomial). 4/√2 = 4√2/2 = 2√2. Note: √(a + b) is not √a + √b.

Laws of exponents and standard form

Law	Note
aᵐ × aⁿ = aᵐ⁺ⁿ	the bases must match
aᵐ ÷ aⁿ = aᵐ⁻ⁿ	subtract, even if it goes negative
(aᵐ)ⁿ = aᵐⁿ	multiply powers
(ab)ⁿ = aⁿbⁿ	applies to every factor in the bracket
a⁰ = 1	a not 0
a⁻ⁿ = 1/aⁿ	negative power means reciprocal, not a negative number
a^(m/n) ext	root first (the denominator n), then the power (the numerator m)

Standard form is written a × 10ᵏ, with a between 1 (included) and 10 (excluded) and k a whole number, positive or negative.

• To multiply or divide, handle the numbers and the powers of 10 separately, then tidy.
• To add or subtract, write both with the same power of 10 first.
• Worked reminder: 5 × 10⁻³ ÷ (2 × 10²) = 2.5 × 10⁻⁵.

Logarithms (ext)

• log_a x = y means aʸ = x. So log₃ 81 = 4 and log_a 1 = 0.
• Adding logs multiplies the arguments, subtracting logs divides them, and a multiplier in front becomes a power: log(xy) = log x + log y, log(x/y) = log x - log y, log(xⁿ) = n log x.
• To solve aˣ = b: x = log b / log a, evaluated with technology.
• Worked reminder: 3ˣ = 20 gives x = log 20 / log 3 = 2.73 (3 s.f.).
• Logarithms and exponents are inverses; this is why logs appear in models of growth and decay.

Inequalities on a number line

Symbol	Circle
< or >	open (end value excluded)
≤ or ≥	closed (end value included)

• Double inequality: 1 ≤ x < 6 is one segment between two circles.
• Compound with “or”: x ≤ -2 or x > 4 is two rays pointing outwards.
• To list integers, check each end value against the circle type.

Number sequences: find the term-to-term difference; if the differences are not constant, look at the differences of the differences. Describe in words, then predict the next terms.

Estimation, rounding and units

• Estimate by rounding each number to 1 s.f. first.
• Decimal places are counted after the point. Significant figures are counted from the first digit that is not zero. Look only at the next digit; 5 or more rounds up.
• Do not round in the middle of working. Round once, at the end, to the accuracy the question asks for or the data supports.
• Conversions: length ×1000 (km to m); area factor is squared (1 m² = 10 000 cm²); volume factor is cubed (1 m³ = 1000 litres); speed km/h to m/s divide by 3.6.
• Time zones: convert to the departure city’s clock, add the duration, then shift by the difference in UTC offsets.

Accuracy, errors, bounds

Quantity	Formula
Absolute error	the size of (measured - true), always positive
Percentage error	absolute error ÷ true × 100
Rounded to nearest u	x - u/2 ≤ value < x + u/2

ext Bounds of calculations:

• a + b: add lowers, add uppers.
• a - b: lower = lower a - upper b; upper = upper a - lower b.
• a × b: multiply lowers, multiply uppers.
• a ÷ b: lower = lower a ÷ upper b; upper = upper a ÷ lower b.

Worked reminder (ext): a rectangle is 12 cm by 7 cm, both to the nearest cm. Area upper bound = 12.5 × 7.5 = 93.75 cm². Lower bound = 11.5 × 6.5 = 74.75 cm². The nominal area, 84 cm², sits between them. Report the answer with a sensible number of figures, such as “about 84 cm², to 2 s.f.”, and explain that the data limit the accuracy.

Commenting on validity: a result is only as accurate as the least accurate measurement used. Say which measurement limits it.

Ratio, percentage, proportion and money

Idea	Method
Share in ratio a : b	add parts, divide, multiply
Increase by p%	× (1 + p/100)
Decrease by p%	× (1 - p/100)
Reverse percentage	divide by the multiplier
Direct y = kx	y ÷ x constant; graph is a straight line through the origin
Inverse y = k/x	xy constant; graph is a curve
Simple interest	P × r × t
Compound interest	A = P(1 + r)ⁿ

Worked reminder: 300 after a 25% increase means original = 300 ÷ 1.25 = 240.

Worked reminder (compound interest): 800 at 5% for 3 years gives 800 × 1.05³ = 926.10, so the interest is 126.10. Simple interest would be 800 × 0.05 × 3 = 120.

Worked reminder (inverse proportion): 8 taps fill a tank in 15 minutes. Taps × time is constant at 120, so 10 taps take 120 ÷ 10 = 12 minutes.

Number properties and bases

• Commutative: a + b = b + a and ab = ba. Associative: grouping can change. Distributive: a(b + c) = ab + ac.
• Subtraction and division are not commutative or associative.
• enr Place values in base b are powers of b. 312₄ = 3 × 16 + 1 × 4 + 2 = 54. To convert from base 10, divide repeatedly by b and read remainders upwards.

Quick self-test

1. Is 0.1 recurring (0.111…) in N, Z, Q, Q’ or R? Name all that apply.
2. Write 0.2777… (7 recurring) as a fraction.
3. Find the GCF and LCM of 36 and 90.
4. Simplify √75 + √12.
5. Simplify (4x³y⁻²)² ÷ (2xy).
6. Write 3.6 × 10⁻³ × 5 × 10⁷ in standard form.
7. Round 0.0070468 to 3 s.f.
8. Share 840 in the ratio 2 : 3 : 7.
9. A price falls 20% to 360. What was it before?
10. y is inversely proportional to x and y = 9 when x = 4. Find y when x = 10.
11. ext Evaluate 81^(3/4) and log₂ 32.
12. ext A length is 20 cm to the nearest cm. Give its bounds.

Answers

1. It equals 1/9, so it is in Q and R only.
2. x = 0.2777…: 100x = 27.777…, 10x = 2.777…; 90x = 25; x = 25/90 = 5/18.
3. 36 = 2² × 3², 90 = 2 × 3² × 5. GCF = 2 × 9 = 18; LCM = 4 × 9 × 5 = 180.
4. 5√3 + 2√3 = 7√3.
5. 16x⁶y⁻⁴ ÷ (2xy) = 8x⁵y⁻⁵ = 8x⁵/y⁵.
6. 3.6 × 5 = 18; 10⁻³ × 10⁷ = 10⁴; 18 × 10⁴ = 1.8 × 10⁵.
7. 0.00705 (the digit after 0.00704 is 6, so round up).
8. 12 parts, 70 each: 140, 210, 490.
9. 360 ÷ 0.8 = 450.
10. k = 36; y = 36/10 = 3.6.
11. 81^(3/4) = 3³ = 27; log₂ 32 = 5.
12. 19.5 ≤ length < 20.5 cm.

Where marks are usually lost

• Stopping at 16.5 × 10⁷ without converting to standard form.
• Treating a negative exponent as a negative answer.
• Rounding a recurring decimal and calling it exact in a “show” or “justify” step.
• Forgetting that unit conversions for area and volume use the squared or cubed factor.
• Rounding before the final step, so the last digits are wrong.
• Mixing up the bounds in a division.
• Leaving out units, or giving too many significant figures for rounded data.
• In Criterion D tasks, giving an answer without commenting on whether it is reasonable in the context.

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.