Flashcards · IB Middle Years Programme Mathematics · Topic 1

Branches of mathematical study (Section I framework overview) flashcards

37 cards on Topic 1, subtopics 1 to 4. Pick a subtopic or study them all.

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1 Numerical and abstract reasoning

Rational compared with irrational numbers: Rational (Q): can be written as a/b; its decimal stops or recurs (3/8, −0.25). Irrational (Q′): the decimal neither stops nor recurs (√2, π). Together they make the real numbers (R).
Write 0.4545… as a fraction: x = 0.4545…, 100x = 45.4545… 100x − x = 99x = 45 x = 45/99 = 5/11
How do you simplify a surd such as √48?: Find the largest square factor, take its root outside and leave the rest inside: √48 = √16 × √3 = 4√3. Note √(a + b) is not √a + √b.
Laws of exponents: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; (ab)ⁿ = aⁿbⁿ; a⁰ = 1 (a ≠ 0); a⁻ⁿ = 1/aⁿ (a reciprocal, not a negative number). The bases must match.
Work out 5 × 10⁻³ ÷ (2 × 10²) in standard form: Numbers: 5 ÷ 2 = 2.5. Powers: 10⁻³ ÷ 10² = 10⁻⁵. Answer: 2.5 × 10⁻⁵ (a × 10ᵏ with 1 ≤ a < 10).
Absolute error and percentage error: Absolute error = the size of (measured − true), always positive. Percentage error = absolute error ÷ true value × 100.
Fractional exponents: a^(m/n)Extended: Take the nth root first (the denominator), then raise to the power m (the numerator).
Logarithms: what does log_a x = y mean, and how do you solve 3ˣ = 20?Extended: log_a x = y means aʸ = x, so log₃ 81 = 4 and log_a 1 = 0. 3ˣ = 20 gives x = log 20 ÷ log 3 ≈ 2.73 (3 s.f.).
Bounds of a − b and a ÷ bExtended: a − b: lower = lower a − upper b; upper = upper a − lower b. a ÷ b: lower = lower a ÷ upper b; upper = upper a ÷ lower b.
nth term of arithmetic and geometric sequencesExtended: Arithmetic: aₙ = a₁ + (n − 1)d. Geometric: aₙ = a₁ × rⁿ⁻¹. Example: the 20th term of 8, 13, 18, 23, … is 8 + 19 × 5 = 103.

2 Thinking with models

Function: A mapping where each input has exactly one output. f(x) is the output when the rule f is applied to x. A line or parabola is a function; a circle is a relation, not a function.
Three forms of a quadratic and what each shows: General ax² + bx + c: y-intercept c, axis x = −b/(2a). Factorised a(x − p)(x − q): x-intercepts p and q. Vertex a(x − h)² + k: vertex (h, k). a > 0 opens up (minimum); a < 0 opens down (maximum).
Write x² + 6x + 2 in vertex form: Complete the square: (x + 3)² − 9 + 2 = (x + 3)² − 7, so the vertex is (−3, −7).
Exponential function f(x) = a × bˣ: b > 0, b ≠ 1. Start value f(0) = a. Growth when b > 1, decay when 0 < b < 1. Horizontal asymptote y = 0. A 12% rise each year is a factor of 1.12; a 12% fall is 0.88.
Flowchart shapes and trace tables: Oval: start or stop. Parallelogram: input or output. Rectangle: process step. Diamond: yes/no decision. A trace table has one row per pass and one column per variable; record values after each step.
Linear search compared with halving search: A linear search of n items needs up to n checks. A halving search needs k checks where 2ᵏ − 1 ≥ n, but only works on a sorted list. For 1000 items: up to 1000 checks against 10.
Domain and range of 2ˣ and of x² − 4x + 1Extended: Domain: allowed inputs. Range: outputs that occur. 2ˣ: domain all reals, range f(x) > 0. x² − 4x + 1 = (x − 2)² − 3: domain all reals, range f(x) ≥ −3.
Asymptotes of f(x) = (ax + b)/(cx + d)Extended: Vertical: cx + d = 0, so x = −d/c. Horizontal: y = a/c. Domain x ≠ −d/c; range f(x) ≠ a/c.
How do you solve a linear programming problem?Extended: 1. Define the variables and write every constraint. 2. Draw the feasible region (solid lines for ≤ or ≥, dashed for < or >). 3. Find every corner. 4. Evaluate the objective function at each corner and pick the best.
Networks: degree of a vertex, and the degree checkExtended: The degree is the number of edges that end at a vertex. The sum of all the degrees is twice the number of edges.

3 Spatial reasoning

Angle sums of polygons: Interior angles of an n-sided polygon sum to (n − 2) × 180°. A regular polygon has exterior angle 360°/n.
Cone: volume, surface area and slant height: Volume = ⅓πr²h. Total surface area = πr² + πrl. Slant height l = √(r² + h²); r = 15, h = 8 gives l = 17. Slant height is not vertical height.
Circle theorems: Angle at the centre = 2 × angle at the circumference (same arc). Angles in the same segment are equal. Angle in a semicircle = 90°. Opposite angles of a cyclic quadrilateral add to 180°. Tangent and radius meet at 90°.
Distance and midpoint of (−2, 1) and (4, 9): Distance = √[(x₂ − x₁)² + (y₂ − y₁)²] = √(6² + 8²) = 10. Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2) = (1, 5).
Rotations about the origin: 90° anticlockwise: (x, y) → (−y, x). 180°: (x, y) → (−x, −y). 90° clockwise: (x, y) → (y, −x). For another centre: subtract the centre, rotate, add it back.
Congruence and similarity conditions for triangles: Congruent: SSS, SAS, ASA or AAS, RHS. Similar: AA (two pairs of equal angles) or all three pairs of sides in the same ratio.
Right-angled trigonometry: find the side opposite 52° when the adjacent side is 15 m: tan = opp/adj, unknown on top, so multiply: 15 × tan 52° ≈ 19.2 m.
Converse of PythagorasExtended: If a² + b² = c² for the longest side c, the triangle is right-angled. 15, 20, 25: 225 + 400 = 625, so yes. 5, 6, 8: 61 ≠ 64, so no.
Sine rule and cosine ruleExtended: Sine rule: a/sin A = b/sin B = c/sin C, when you know a side and its opposite angle. Cosine rule: a² = b² + c² − 2bc cos A, for two sides and the included angle, or three sides.
Line perpendicular to y = −3x + 2 through (6, 5)Extended: m₁ × m₂ = −1, so the new gradient is 1/3. Then 5 = (1/3)(6) + c, so c = 3: y = x/3 + 3.

4 Reasoning with data

What does the Reasoning with data branch cover?: Collecting and organising data, measures of central tendency and spread, graphical representation (bar charts, histograms, scatter plots, box plots), basic probability, and drawing and justifying conclusions from data.
Which measure of centre suits which data?: Roughly symmetric: mean (uses every value). Skewed or with outliers: median (not distorted by extreme values). Categorical: mode.
A few very high values pull a mean upwards. What should you do?: Recognise the data are skewed and consider the median as a more representative measure, and say why.
Theoretical compared with experimental probability: Theoretical: calculated from the possible equally likely outcomes. Experimental: calculated from the results of repeated trials.
Choosing a representation for spread, outliers or correlation: A box plot shows spread and outliers clearly; a scatter plot shows correlation between two variables. Label axes, give units and use sensible scales.
Correlation compared with causation: A correlation between two variables does not show that one causes the other.
Drawing a conclusion from data for criterion D: State what the data suggest in the real context, and reflect on the limitations, such as a small or unrepresentative sample that may not generalise.