GCSE Mathematics May 2024 Foundation Tier Paper 1 Flashcards

Examination Details and Instructions

• Subject: GCSE MATHEMATICS Foundation Tier, Paper 1 Non-Calculator.

• Examination Date: Thursday 16 May 2024 (Morning).

• Time Allowed: 1 hour 30 minutes.

• Total Marks: 80.

• Permitted Materials: Mathematical instruments, Formulae Sheet (enclosed).

• Prohibited Materials: Use of a calculator is strictly forbidden for this paper.

• Formatting Requirements: Use black ink or ball-point pen; draw diagrams strictly in pencil.

Basic Arithmetic and Unit Conversions

• Division:

  • $280 \div 7 = 40$

• Subtraction:

  • $1062 - 438 = 624$

• Metric Length Conversion:

  • $2\,m = 200\,cm$

• Metric Mass Conversion:

  • $8\,kg = 8000\,g$

• Distance Conversion (Metric to Imperial):

  • Conversion Standard: $8\,km = 5\,miles$

  • Process for $24\,km$:

    • $\frac{24\,km}{8\,km} = 3$

    • $3 \times 5\,miles = 15\,miles$

Fractions and Percentages

• Grid Shading Identification:

  • In a grid where $4$ out of $16$ small squares are shaded, the percentage is $25\%$.

• Target Shading Calculation:

  • Goal: Shade $\frac{3}{4}$ of a grid.

  • If total squares in grid are $16$, target shaded squares = $\frac{3}{4} \times 16 = 12$.

  • If $2$ squares are already shaded, remaining to shade = $12 - 2 = 10$ squares.

• Calculating Percentages of Amounts:

  • Requirement: Work out $35\%$ of $1200$.

  • $10\% = 120$

  • $30\% = 3 \times 120 = 360$

  • $5\% = \frac{120}{2} = 60$

  • Total ($35\%$) = $360 + 60 = 420$.

• Fractional Comparison:

  • Bag X contains $\frac{7}{20}$ red discs.

  • Bag Y contains $\frac{2}{5}$ red discs.

  • Normalization: $\frac{2}{5} = \frac{8}{20}$.

  • Conclusion: Bag Y has the greater proportion (\frac{8}{20} > \frac{7}{20}).

Number Properties and Logic

• Rounding and Proximity:

  • From the list: $6.92, 7.27, 7.18, 7.14$

  • Number closest to $7$: $6.92$ (Difference is $0.08$).

  • Number rounding to $7.2$ to $1$ decimal place: $7.18$.

• Operations with Integers:

  • From the list: $-10, -5, -2, 4, 6, 10$

  • Adding to make $-1$: $-5$ and $4$.

  • Multiplying to make $20$: $-2$ and $-10$ OR $4$ and $5$ (Note: Only $-2$ and $-10$ exist in this specific list for positive product $20$).

• Even and Odd Expressions:

  • Given: $c$ is a positive even number, $d$ is a positive odd number.

  • $c + d$: Always Odd (Even + Odd = Odd).

  • $4c$: Always Even (Any integer multiplied by $4$ is even).

  • $\frac{c}{2}$: Cannot tell. (Example: if $c = 2$, $\frac{2}{2} = 1$ [Odd]; if $c = 4$, $\frac{4}{2} = 2$ [Even]).

• Estimating Limits:

  • Rugby attendance is $8400$ (rounded to nearest $100$).

  • Minimum possible attendance: $8350$

  • Maximum possible attendance: $8449$ (or < 8450).

Geometry and Coordinate Geometry

• Parallel Lines: Lines are parallel if they maintain a constant distance and never meet, having the same gradient on the grid.

• Coordinates (Integer Solutions):

  • Requirement: $x + y = 3$ where both are whole numbers.

  • Potential plotted points: $(0,3), (1,2), (2,1), (3,0)$.

• Powers and Roots:

  • $3^2 = 9$

  • $\sqrt{144} = 12$

  • $2^4 = 2 \times 2 \times 2 \times 2 = 16$

• Congruency:

  • Two triangles are congruent if they are identical in shape and size. Corresponding sides must be equal.

• Triangle Area:

  • Formula: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$.

  • Example: Base $3\,cm$, Height $10\,cm$. Area = $\frac{1}{2} \times 3 \times 10 = 15\,cm^2$.

• Translations and Vectors:

  • If vector $\begin{pmatrix} 3 \\ -7 \end{pmatrix}$ translates point A to B, the inverse vector translates B to A.

  • Inverse Vector: $\begin{pmatrix} -3 \\ 7 \end{pmatrix}$.

• Curved Surface Area of a Cone:

  • Formula: $\text{CSA} = \pi r l$, where $r$ is radius and $l$ is slant height.

  • Error Analysis: Common mistake is using perpendicular height $h$ instead of slant height $l$ in the formula.

• Circle/Base Area Approximations:

  • Area of base = $\pi r^2$.

  • Adam using $\pi = 3$ vs Beth using $\pi = 3.14$. Beth's estimate will be higher because 3.14 > 3.

Problem Solving and Applied Mathematics

• Combinatorics (Toppings):

  • Vegan toppings: Sweetcorn (S), Mushrooms (M), Peppers (P).

  • Possible pairs: $(S, M), (S, P), (M, P)$.

• Inventory Optimization (Dough Balls):

  • Portion Sizes: Small ($6$ balls), Large ($10$ balls).

  • Aim: Exactly $44$ balls.

  • Solution: $4$ Small portions ($24$ balls) + $2$ Large portions ($20$ balls) = $44$ balls.

• Budgeting (Apples and Oranges):

  • Total Budget: $£10$.

  • Purchases: $9$ apples at $25p$ each = $225p = £2.25$.

  • Remaining Budget: $£10 - £2.25 = £7.75$.

  • Orange cost: $60p$.

  • calculation: $775 \div 60 = 12$ oranges remainder $55p$.

• Daily Consumption (Erik's Milk):

  • Drinking Schedule: $\frac{1}{4}$ pint (AM) + $\frac{1}{2}$ pint (PM) = $\frac{3}{4}$ pint per day.

  • Over $30$ days: $30 \times \frac{3}{4} = \frac{90}{4} = 22.5\,pints$.

Statistics and Data Handling

• Goal Records for 8 Games:

  • Alina: $12, 15, 17, 17, 21, 22, 24, 26$

  • Sue: $13, 13, 17, 20, 22, 23, 24, 31$

• Median and Range Analysis:

  • Alina Median: $\frac{17+21}{2} = 19$.

  • Sue Median: $\frac{20+22}{2} = 21$.

  • Alina Range: $26 - 12 = 14$.

  • Sue Range: $31 - 13 = 18$.

• Consistency: Alina is more consistent because her range ($14$) is smaller than Sue's range ($18$).

• Time-Series Analysis: Includes tracking worker numbers over consecutive years (2015\u20132022) to forecast future values (2023).

Algebra and Ratios

• Cost Formulae:

  • Equation: $C = 2W + 5$.

  • Determining invalidity: For a cost like $£24$, $24 - 5 = 19$. Since $19$ is not divisible by $2$, the cost cannot be correct for a whole number of windows $W$.

• Ratio Sharing:

  • Total: $£240$ in ratio $1:3$.

  • Total parts: $1 + 3 = 4$.

  • Value per part: $240 \div 4 = £60$.

  • Larger share ($3$ parts): $3 \times 60 = £180$.

• Win/Loss Ratio:

  • Ratio Win : Lose = $5 : 9$.

  • Fraction of wins: $\frac{\text{wins}}{\text{total matches}} = \frac{5}{5+9} = \frac{5}{14}$.

• Linear Sequences:

  • $1st\,term = 10$.

  • $1st + 2nd = 39 \rightarrow 10 + 2nd = 39 \rightarrow 2nd = 29$.

  • Common difference ($d$): $29 - 10 = 19$.

  • 5th term ($a_5$): $10 + (4 \times 19) = 10 + 76 = 86$.

• Linear Equations:

  • Solve: $7x - 22 = 4x + 29$

  • Subtract $4x$: $3x - 22 = 29$

  • Add $22$: $3x = 51$

  • Divide by $3$: $x = 17$.

• Simplifying Area Fractions:

  • Living room ($26\,m^2$) as fraction of Kitchen ($16.4\,m^2$).

  • $\frac{26}{16.4} = \frac{260}{164}$.

  • Simplify by $4$: $\frac{65}{41}$.

• Inequalities:

  • Represent -2 < x < 4: Empty circle at $-2$, line connecting to empty circle at $4$ on a number line.

  • Solve $5y + 14 \ge 11$:

    • $5y \ge 11 - 14$

    • $5y \ge -3$

    • $y \ge -0.6$ (or $\ge -\frac{3}{5}$).

Multiplication Table Data Extraction

• Reference Table for values involving 61, 63, 65, 67:

  • $61 \times 61 = 3721$

  • $61 \times 63 = 3843$

  • $67 \times 67 = 4489$

• Calculations using Table:

  • $3843 \div 63 = 61$.

  • $6.1 \times 6.7$: From table, $61 \times 67 = 4087$. Adjusting for decimals ($10^{-1} \times 10^{-1} = 10^{-2}$) results in $40.87$.

  • $63 \times 66$: Since $63 \times 65 = 4095$, then $4095 + 63 = 4158$.