GCSE Mathematics Foundation Tier Non-Calculator Study Guide

Arithmetic Fundamentals and Powers

• Calculation of square root:   $\sqrt{49} = 7$

• Calculation of cubing integers:   $3^3 = 3 \times 3 \times 3 = 27$

• Expressing powers of 10:   $10\,000 = 10^4$

• Imperial unit conversion between pounds and ounces:

  • Standard conversion factor: $1\,\text{pound} = 16\,\text{ounces}$
  • Calculation for $3\,\text{pounds}$:     $3 \times 16 = 48\,\text{ounces}$

Fractions, Decimals, and Ordering

• Mixed numbers and improper fractions:

  • Conversion of $\frac{3}{2}$ to a mixed number:     $\frac{3}{2} = 1\frac{1}{2}$

• Addition of fractions with common denominators:   $\frac{1}{5} + \frac{1}{5} = \frac{1 + 1}{5} = \frac{2}{5}$

• Subtraction of mixed numbers and fractions:

  • Expression: $1\frac{1}{5} - \frac{3}{10}$
  • Step 1: Convert $1\frac{1}{5}$ to an improper fraction:     $1\frac{1}{5} = \frac{6}{5}$
  • Step 2: Convert to a common denominator of $10$:     $\frac{6}{5} = \frac{12}{10}$
  • Step 3: Perform subtraction:     $\frac{12}{10} - \frac{3}{10} = \frac{9}{10}$

• Ordering fractions, decimals, and percentages:

  • Values to order: $80\%$, $0.7$, $\frac{3}{4}$
  • Convert all values to decimals for comparison:     $80\% = 0.80$$0.7 = 0.70$$\frac{3}{4} = 0.75$
  • Order from smallest to largest:     $0.7, \frac{3}{4}, 80\%$

Number Properties, Factors, and Proof by Counterexample

• Factors of an integer:

  • All positive factors of $20$: $1, 2, 4, 5, 10, 20$

• Counterexample in number theory:

  • Statement disproof: "When two multiples of 5 are added, the answer is always a multiple of 10."
  • Counterexample:     $5 + 10 = 15$     Both $5$ and $10$ are multiples of $5$, but their sum $15$ is not a multiple of $10$

Practical Financial Mathematics

• Calculating unit costs from total purchases:

  • Scenario: Buying $2\,\text{hats}$ and $3\,\text{scarves}$ for a total cost of $\text{£}28.50$
  • Cost of each hat: $\text{£}4.50$
  • Total cost of $2\,\text{hats}$:     $2 \times 4.50 = \text{£}9.00$
  • Remaining cost for $3\,\text{scarves}$:     $28.50 - 9.00 = \text{£}19.50$
  • Cost per scarf:     $\frac{19.50}{3} = \text{£}6.50$

• Percentage discounts on items:

  • Scenario: A multipack costs $10\%$ less than $6\,\text{single bags}$
  • Discount multiplier:     $1 - 0.10 = 0.90$
  • Cost formula for the multipack:     $\text{Multipack Cost} = 0.90 \times (6 \times \text{Cost of single bag})$

Data Representation and Frequency Trees

• Frequency tree analysis:

  • Total visitors to a maze: $120\,\text{people}$
  • Age category breakdown: $80\,\text{children}$, $120 - 80 = 40\,\text{adults}$
  • Directional decisions (left or right):
  • Total people turning left: $75\,\text{people}$
  • Children turning left: $45\,\text{children}$
  • Adults turning left: $75 - 45 = 30\,\text{adults}$
  • Children turning right: $80 - 45 = 35\,\text{children}$
  • Adults turning right: $40 - 30 = 10\,\text{adults}$
  • Calculating fraction of children turning left:     $\frac{45}{80}$
  • Simplifying the fraction by dividing numerator and denominator by $5$:     $\frac{45 \div 5}{80 \div 5} = \frac{9}{16}$

• Critiquing statistical bar chart representations:

  • Dataset: Library = $30$, Home = $68$, In class = $37$
  • Key potential structural errors in bar chart construction:
  • Incorrect or non-linear scale increments on the vertical frequency axis
  • Mislabeled or omitted horizontal/vertical axis categories
  • Inaccurate bar heights that do not correspond to dataset values
  • Non-uniform spacing or unequal bar widths across categories

Probability and Sample Spaces

• Combined events and sample space grid construction:
  • Selection set 1 (first three positive odd numbers): $\{1, 3, 5\}$
  • Selection set 2 (first four prime numbers): $\{2, 3, 5, 7\}$
  • Score definition: Product of one number from set 1 and one number from set 2
  • Multiplication outcomes ($3 \times 4 = 12\,\text{total outcomes}$):
  • For odd number $1$: $1 \times 2 = 2$, $1 \times 3 = 3$, $1 \times 5 = 5$, $1 \times 7 = 7$
  • For odd number $3$: $3 \times 2 = 6$, $3 \times 3 = 9$, $3 \times 5 = 15$, $3 \times 7 = 21$
  • For odd number $5$: $5 \times 2 = 10$, $5 \times 3 = 15$, $5 \times 5 = 25$, $5 \times 7 = 35$
  • Square number outcomes in the product grid:
  • $9 = 3^2$
  • $25 = 5^2$
  • Number of square number outcomes: $2$
  • Probability of obtaining a square number score:     $P(\text{square}) = \frac{2}{12} = \frac{1}{6}$

Algebraic Simplification and Functions

• Combining like terms:

  • Expression: $8m + 4 - 2m + 7$
  • Grouping variables and constants:     $(8m - 2m) + (4 + 7) = 6m + 11$

• Algebraic multiplication:

  • Expression: $\frac{1}{2}c \times 6d$
  • Multiplying coefficients and combining variables:     $\left(\frac{1}{2} \times 6\right) \times c \times d = 3cd$

• Truth evaluation of algebraic inequalities:

  • Given $x$ and $y$ are distinct positive numbers ($x > 0$, $y > 0$, $x \neq y$):
  • Statement $x + y$ is positive: Always true (sum of two positive quantities is strictly positive)
  • Statement $x - y$ is negative: Sometimes true (true if $x < y$, false if $x > y$)

• Functional mapping machines:

  • Function $y = 4x + 5$: Input $x$ $\rightarrow$ Multiply by $4$ $\rightarrow$ Add $5$ $\rightarrow$ Output $y$
  • Function $y = 3x - 24$: Input $x$ $\rightarrow$ Multiply by $3$ $\rightarrow$ Subtract $24$ $\rightarrow$ Output $y$
  • Identity function $y = x$: Input $x$ $\rightarrow$ Multiply by $1$ $\rightarrow$ Output $y$

Ratios and Proportions

• Simplifying ratios to $n : 1$ form:

  • Ratio: $6 : 2$
  • Divide both parts by $2$:     $\frac{6}{2} : \frac{2}{2} \implies 3 : 1$
  • Value of $n$: $n = 3$

• Division of quantities in ratios:

  • Total books: $35$
  • Ratio of adult books to children books: $6 : 1$
  • Sum of ratio parts:     $6 + 1 = 7\,\text{parts}$
  • Books per part:     $\frac{35}{7} = 5\,\text{books}$
  • Adult books quantity:     $6 \times 5 = 30\,\text{books}$
  • Children books quantity:     $1 \times 5 = 5\,\text{books}$
  • Difference between adult and children books:     $30 - 5 = 25\,\text{more adult books}$

Geometry, Transformations, and Speed

• Congruence and geometric transformations:

  • Congruence: Two shapes are congruent if all corresponding side lengths and angles are identical
  • Fractional enlargement: Enlarging a shape by scale factor $\frac{1}{3}$ scales every side length to $\frac{1}{3}$ of its original length

• Average speed calculations:

  • Car travels $4\,\text{miles}$ in $5\,\text{minutes}$
  • Time conversion factor to $1\,\text{hour}$ ($60\,\text{minutes}$):     $\frac{60}{5} = 12$
  • Average speed calculation:     $\text{Speed} = 4 \times 12 = 48\,\text{mph}$

• Circle area and concentric shapes:

  • Large circle radius $R = 12\,\text{cm}$
  • Ratio of large circle radius to small circle radius: $4 : 1$
  • Radius of small circle $r$:     $r = \frac{12}{4} = 3\,\text{cm}$
  • Area of large circle:     $\text{Area}_{\text{large}} = \pi R^2 = \pi \times 12^2 = 144\pi\,\text{cm}^2$
  • Area of small circle:     $\text{Area}_{\text{small}} = \pi r^2 = \pi \times 3^2 = 9\pi\,\text{cm}^2$
  • Shaded area between circles:     $\text{Shaded Area} = 144\pi - 9\pi = 135\pi\,\text{cm}^2$

Coordinate Geometry and Linear Vectors

• Line segment vector steps:
  • Points $J(0, 12)$ and $K(5, 10)$ lie on straight line $JKLM$
  • Segment equality: $JK = KL = LM$
  • Step displacement vector from $J$ to $K$:     $\Delta x = 5 - 0 = 5$$\Delta y = 10 - 12 = -2$
  • Coordinates of point $L$:     $(5 + 5, 10 - 2) = (10, 8)$
  • Coordinates of point $M$:     $(10 + 5, 8 - 2) = (15, 6)$

Statistical Data Shift Properties

• Effect of increasing every dataset value by $10$:
  • Mode: True (increases by $10$)
  • Median: True (increases by $10$)
  • Range: False (remains unchanged since $(\text{Max} + 10) - (\text{Min} + 10) = \text{Max} - \text{Min}$)

Sequences and Progressions

• Geometric progression:

  • Sequence: $1, 4, 16, \dots, 256$
  • Common ratio $r$:     $r = \frac{4}{1} = 4$
  • Missing term:     $16 \times 4 = 64$
  • Check term:     $64 \times 4 = 256$

• Fibonacci-type sequence:

  • Term generation rule: $T_n = T_{n-1} + T_{n-2}$
  • Sequence terms given: $\dots, 5, -9, \dots$
  • Next term ($T_4$):     $5 + (-9) = -4$
  • Subsequent term ($T_5$):     $-9 + (-4) = -13$

Solid Geometry, Trigonometry, and Proportion

• Geometric properties and cross-sectional area of prisms:

  • Total faces on an $n$-gonal prism: $n + 2\,\text{faces}$
  • Prism volume relationship:     $\text{Volume} = \text{Cross-Sectional Area} \times \text{Length}$
  • Given: $\text{Volume} = 3500\,\text{cm}^3$, $\text{Length} = 20\,\text{cm}$
  • Cross-sectional area calculation:     $\text{Cross-Sectional Area} = \frac{3500}{20} = 175\,\text{cm}^2$

• Exact trigonometric values:   $\sin(90^\circ) = 1$

• Inverse proportion and varied work rates:

  • Uniform worker rate scenario: $10\,\text{people}$ complete a job in $9\,\text{hours}$
  • Total workload:     $10 \times 9 = 90\,\text{worker-hours}$
  • Time required for $15\,\text{people}$ at standard rate:     $\frac{90}{15} = 6\,\text{hours}$
  • Non-uniform worker rate analysis ($6\,\text{slower}$, $9\,\text{faster}$):
  • Without specific quantitative performance ratios for the faster and slower workers, the net average rate cannot be determined
  • Logical conclusion: It is not possible to say