Reveal Math Grade 8 General Mathematics End of Term Review Booklet

General Course Overview

• Subject: Mathematics / Reveal Math

• Level: Grade 8

• Stream: General

• Academic Year: 2025/2026

• Exam Duration: 120 Minutes

• Assessment Mode: SwiftAssess

• Structure: 25 Multiple-Choice Questions (MCQ)

• Max Grade: 100

• Calculators: Not allowed

• Prepared By: Ms Sumaiya - Al Foaa School

Transformations on the Coordinate Plane

1. Translations

• Definition: Sliding a figure without turning or flipping it. The figure remains the same shape but moves to a new position.

• Horizontal and Vertical Shifts Rule: $(x, y) \rightarrow (x + a, y + b)$

  • $a$: Horizontal shift (Positive is Right, Negative is Left).

  • $b$: Vertical shift (Positive is Up, Negative is Down).

• Worked Example:

  • Translate $\Delta A$ with vertices $A(-3, 1)$, $B(-4, -2)$, $C(-1, -1)$ by $(4, 2)$.

  • Apply rule: $(x+4, y+2)$.

  • $A'(-3+4, 1+2) = A'(1, 3)$.

  • $B'(-4+4, -2+2) = B'(0, 0)$.

  • $C'(-1+4, -1+2) = C'(3, 1)$.

2. Reflections

• Definition: Flipping a figure over a line (the axis of reflection).

• Reflection across the x-axis:

  • Rule: $(x, y) \rightarrow (x, -y)$.

  • Example: $P(2, 3) \rightarrow P'(2, -3)$, $Q(0, 1) \rightarrow Q'(0, -1)$, $R(4, 1) \rightarrow R'(4, -1)$.\n* Reflection across the y-axis:

  • Rule: $(x, y) \rightarrow (-x, y)$.

  • Example: $P(2, 3) \rightarrow P'(-2, 3)$, $Q(0, 1) \rightarrow Q'(0, 1)$, $R(4, 1) \rightarrow R'(-4, 1)$.

3. Rotations

• Definition: Turning a figure about a point (usually the origin). Counterclockwise direction is considered positive.

• 90° Counterclockwise Rotation:

  • Rule: $(x, y) \rightarrow (-y, x)$.

  • Example: $P(1, 2) \rightarrow P'(-2, 1)$.

• 180° Rotation:

  • Rule: $(x, y) \rightarrow (-x, -y)$.

  • Example: $P(1, 2) \rightarrow P'(-1, -2)$.

• 90° Clockwise Rotation:

  • Rule: $(x, y) \rightarrow (y, -x)$.

  • Example: $P(1, 2) \rightarrow P'(2, -1)$.

• 270° Clockwise Rotation:

  • Rule: $(x, y) \rightarrow (-y, -x)$.

  • Example: $P(1, 2) \rightarrow P'(-2, -1)$.

• Key Principle: Always maintain the same distance from the center of rotation.

4. Dilations

• Definition: Resizing a figure from a center point (usually the origin).

• Rule: $(x, y) \rightarrow (kx, ky)$, where $k$ is the scale factor.

• Scale Factor ($k$) Behaviors:

  • $k > 1$: Enlargement.

  • $0 < k < 1$: Reduction.

  • $k < 0$: Enlargement and flip through the origin.

• Worked Example (k = 2):

  • Vertices: $P(1, 2)$, $Q(3, 1)$, $R(1, -1)$.

  • $P'(2 \times 1, 2 \times 2) = P'(2, 4)$.

  • $Q'(2 \times 3, 2 \times 1) = Q'(6, 2)$.

  • $R'(2 \times 1, 2 \times -1) = R'(2, -2)$.

Congruence, Similarity, and Indirect Measurement

1. Congruence

• Definition: Two figures are congruent if they are exactly the same size and shape.

• Principle: Corresponding sides and corresponding angles are equal.

• Example 1 (Missing Side): Triangle with sides $6\text{ cm}$, $8\text{ cm}$, and $x\text{ cm}$. Since the triangles are congruent, if the corresponding side is $8\text{ cm}$, then $x = 8\text{ cm}$.

• Example 2 (Missing Angle): If triangles are congruent and the corresponding angle is $62^\circ$, then $y = 62^\circ$.

2. Similarity

• Definition: Two figures are similar if they have the same shape but may be different sizes.

• Principle: Corresponding sides are proportional.

• Example (Missing Side): Triangles have corresponding sides of $6$ and $8$, and $9$ and $x$.

  • Set up proportion: $\frac{6}{8} = \frac{9}{x}$.

  • Cross multiply: $6x = 72$.

  • Solve: $x = 12$.

3. Indirect Measurement

• Definition: Using similar triangles to calculate distances/heights that are difficult to measure directly.

• Example: A stick $1.5\,m$ tall casts a shadow of $2\,m$. At the same time, a tree casts a shadow of $16\,m$.

  • Set up proportion: $\frac{1.5}{2} = \frac{H}{16}$.

  • Cross multiply: $2H = 1.5 \times 16$.

  • Calculate: $2H = 24 \rightarrow H = 12\,m$.

Volume of Solids

1. Cylinder

• Formula: $V = \pi r^2 h$

  • $r$ = radius

  • $h$ = height

• Example: $r = 5\text{ cm}$, $h = 12\text{ cm}$.

  • $V = \pi (5)^2 (12) = 300\pi \text{ cm}^3$.

  • Decimal: $942.5\text{ cm}^3$.

2. Cone

• Formula: $V = \frac{1}{3} \pi r^2 h$

• Example: $r = 4\text{ cm}$, $h = 9\text{ cm}$.

  • $V = \frac{1}{3} \pi (4)^2 (9) = 48\pi \text{ cm}^3$.

  • Decimal: $150.8\text{ cm}^3$.

3. Sphere

• Formula: $V = \frac{4}{3} \pi r^3$

• Example: $r = 6\text{ cm}$.

  • $V = \frac{4}{3} \pi (6)^3 = 288\pi \text{ cm}^3$.

  • Decimal: $904.8\text{ cm}^3$.

4. Hemisphere

• Formula: $V = \frac{2}{3} \pi r^3$

• Example: $r = 7\text{ cm}$.

  • $V = \frac{2}{3} \pi (7)^3 = \frac{686}{3} \pi \text{ cm}^3$.

  • Decimal: $718.4\text{ cm}^3$.

5. Composite Solids

• Procedure: Find the volume of each individual part, then add or subtract as necessary.

• Example: A cone on top of a cylinder. Radius ($r$) = $5\text{ cm}$ for both. Cylinder height = $10\text{ cm}$, Cone height = $8\text{ cm}$.

  • Cylinder Part: $V = \pi (5)^2 (10) = 250\pi \approx 785.4\text{ cm}^3$.

  • Cone Part: $V = \frac{1}{3} \pi (5)^2 (8) = \frac{200}{3} \pi \approx 209.4\text{ cm}^3$.

  • Total Volume: $250\pi + \frac{200}{3} \pi = \frac{950}{3} \pi \approx 994.8\text{ cm}^3$.

Data Analysis and Tables

1. Scatter Plots

• Definition: Shows the relationship between two quantitative variables.

• Line of Fit (Trend Line): A line that models the trend. It should have roughly an equal number of data points above and below it.

• Finding the Equation ($y = mx + b$):

  1. Find the slope ($m$): Use "rise over run" with two points on the line.

  2. Find the y-intercept ($b$): Locate where the line crosses the y-axis (predicted value of $y$ when $x = 0$).

  3. Write the equation: Substitute $m$ and $b$ into $y = mx + b$.

• Slope Interpretation:

  • $m > 0$: Positive slope (line goes up).

  • $m < 0$: Negative slope (line goes down).

  • Larger $|m|$ indicates a steeper line and stronger relationship.

2. Two-Way Tables

• Definition: Shows the relationship between two categorical variables.

• Structure:

  • Rows (Horizontal): Represent categories of the first variable (e.g., Grade level).

  • Columns (Vertical): Represent categories of the second variable (e.g., Favorite subject).

  • Totals: Peripheral counts for rows and columns, including the grand total.

• Example Reading: If the cell for "7th Grade" and "Math" is "18", then 18 students in 7th grade chose Math.

Lesson and Exercise Inventory

• Q1 & Q2 (Lesson 8-1): Translations (p. 443, ex 1–6).

• Q3 & Q4 (Lesson 8-2): Reflections (p. 445 learn; p. 453 practice, ex 1–5).

• Q5 (Lesson 8-3): Rotations (p. 463, ex 1–5).

• Q6 (Lesson 8-4): Dilations (p. 473, ex 2–6).

• Q7 (Lesson 9-1): Composition of Transformations (p. 491, ex 4 & 5).

• Q8 & Q9 (Lesson 9-2): Congruence & Corresponding Parts (p. 499–500, ex 1–8).

• Q10 & Q11 (Lesson 9-3): Similarity & Transformations (p. 505 learn; p. 511 practice, ex 4 & 5).

• Q12 (Lesson 9-4): Similar Figures (p. 522, ex 8, 9 & 13).

• Q13 (Lesson 9-5): Indirect Measurement (p. 528, ex 7, 8 & 11).

• Q14 & Q15 (Lesson 10-1): Volume of Cylinders (p. 535 learn; p. 541 practice, ex 1–5).

• Q16 (Lesson 10-2): Volume of Cones (p. 549, ex 1–4).

• Q17 & Q18 (Lesson 10-3): Volume of Spheres (p. 557, ex 1–8).

• Q19 (Lesson 10-5): Volume of Composite Solids (p. 573, ex 1–6).

• Q20 (Lesson 11-1): Scatter Plots (p. 583 learn; p. 589 practice, ex 1 & 2).

• Q21 & Q22 (Lesson 11-2): Draw Lines of Fit (p. 597, ex 1–4).

• Q23 (Lesson 11-3): Equations for Lines of Fit (p. 607, ex 1–4).

• Q24 & Q25 (Lesson 11-4): Two-Way Tables (p. 609 learn; p. 617 practice, ex 1–3).

Exam Affirmations

• Trust your preparation.

• You are capable.

• Believe in yourself.

• Take it one step at a time.