Moonpreneur

Grade 8 Prep Guide

HomeGrade 2Grade 3Grade 4Grade 5Grade 6Grade 7Grade 9Grade 10Take Grade 8 Test

Adaptive Math Test — Grade 8 Prep Guide

A comprehensive, topic-by-topic study guide for the Grade 8 adaptive math assessment. None of the practice problems below are taken from the test bank — they are fresh examples designed to teach the same skills.

For each topic you'll find:

  • A short explanation of the key idea
  • Multiple worked examples (problems solved step by step)
  • Common mistakes to watch out for
  • Practice problems with answers at the end of each section

Table of Contents

Real Numbers

  1. Rational vs Irrational Numbers
  2. Exponent Rules
  3. Scientific Notation
  4. Square Roots and Cube Roots

Functions 5. Definition of a Function 6. Function Notation 7. Domain and Range 8. Increasing, Decreasing, and Constant

Linear Functions 9. Slope and Slope-Intercept Form 10. Writing Linear Equations 11. Comparing Linear Functions 12. Linear Inequalities

Systems 13. Solving Systems by Graphing 14. Substitution and Elimination

Geometry 15. Pythagorean Theorem 16. Distance Formula 17. Volume of 3D Solids (Cones, Cylinders, Spheres) 18. Transformations: Translation, Reflection, Rotation, Dilation

Statistics 19. Scatter Plots and Lines of Best Fit 20. Two-Way Tables

1. Rational vs Irrational Numbers

Key Concepts

  • Rational: any number that can be written as a fraction p/q with integers p, q (q ≠ 0). Includes all integers, terminating decimals (0.75), and repeating decimals (0.333...).
  • Irrational: cannot be written as a fraction. Decimals never repeat or terminate. Examples: π, √2, √7, e.

Worked Examples

Example 1. Is 0.25 rational? Yes — 1/4.

Example 2. Is √16 rational? Yes — equals 4.

Example 3. Is √20 rational? No — 20 isn't a perfect square.

Example 4. Classify: 7/3, π, −5, 0.1212..., √2. Solution. Rational: 7/3, −5, 0.1212... | Irrational: π, √2.

Common Mistakes

  • Assuming any square root is irrational (perfect squares give rationals).
  • Calling 0.333... irrational — it's rational (1/3).

Practice

  1. Rational or irrational: √36.
  2. Rational or irrational: 0.121221222... (non-repeating pattern).
  3. Rational or irrational: π/2.
  4. Rational or irrational: 22/7.

Answers: 1. Rational (6) | 2. Irrational | 3. Irrational | 4. Rational

2. Exponent Rules

Key Concepts

  • Product: aᵐ × aⁿ = aᵐ⁺ⁿ
  • Quotient: aᵐ / aⁿ = aᵐ⁻ⁿ
  • Power of a power: (aᵐ)ⁿ = aᵐⁿ
  • Power of a product: (ab)ⁿ = aⁿbⁿ
  • Zero exponent: a⁰ = 1 (a ≠ 0)
  • Negative exponent: a⁻ⁿ = 1/aⁿ

Worked Examples

Example 1. 2³ × 2⁵ = 2⁸ = 256.

Example 2. 7⁹ / 7⁶ = 7³ = 343.

Example 3. (3²)⁴ = 3⁸ = 6,561.

Example 4. 5⁻² = 1/25 = 0.04.

Example 5. (2x³)² = 4x⁶.

Common Mistakes

  • Adding exponents in a power of a power (3²)⁴ ≠ 3⁶.
  • Treating negative exponents as negative numbers (5⁻² is positive 1/25).

Practice

  1. 4² × 4⁵
  2. x⁸ / x³
  3. (2³)²
  4. 6⁻¹
  5. (xy)³

Answers: 1. 4⁷ | 2. x⁵ | 3. 64 | 4. 1/6 | 5. x³y³

3. Scientific Notation

Key Concepts

Numbers written as a × 10ⁿ where 1 ≤ a < 10. Positive n shifts decimal right; negative n shifts left.

  • 5,200 = 5.2 × 10³
  • 0.000045 = 4.5 × 10⁻⁵

For multiplication: multiply the a-values, add exponents. For division: divide the a-values, subtract exponents.

Worked Examples

Example 1. Write 67,500 in scientific notation. Solution. 6.75 × 10⁴.

Example 2. Convert 3.2 × 10⁻³ to standard. Solution. 0.0032.

Example 3. (2 × 10⁴)(3 × 10²) = 6 × 10⁶.

Example 4. (8 × 10⁵) ÷ (4 × 10²) = 2 × 10³.

Common Mistakes

  • Forgetting to keep a between 1 and 10 (e.g., writing 25 × 10³ instead of 2.5 × 10⁴).
  • Reversing direction of decimal shift for negative exponents.

Practice

  1. 0.00098 in scientific notation.
  2. 4.5 × 10⁶ to standard.
  3. (5 × 10³)(2 × 10⁴)
  4. (9 × 10⁷) ÷ (3 × 10⁴)

Answers: 1. 9.8 × 10⁻⁴ | 2. 4,500,000 | 3. 1 × 10⁸ | 4. 3 × 10³

4. Square Roots and Cube Roots

Key Concepts

  • √n = the non-negative number that, squared, gives n. √49 = 7.
  • ³√n = the number that, cubed, gives n. ³√27 = 3.
  • For non-perfect squares, leave in radical form or approximate.

Common perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Common perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.

Worked Examples

Example 1. √81 = 9.

Example 2. ³√125 = 5.

Example 3. Solve x² = 64. Solution. x = ±8 (both roots).

Example 4. Solve x³ = −27. Solution. x = −3 (cube roots keep sign).

Example 5. Approximate √20 between two integers. Solution. 16 < 20 < 25 → 4 < √20 < 5. (≈ 4.47)

Common Mistakes

  • Forgetting ± when solving x² = k.
  • Confusing √ with cube root.

Practice

  1. √169
  2. ³√216
  3. Solve x² = 100.
  4. Solve x³ = −64.
  5. Between which two integers is √50?

Answers: 1. 13 | 2. 6 | 3. ±10 | 4. −4 | 5. 7 and 8

5. Definition of a Function

Key Concepts

A function assigns exactly one output to each input.

Tests:

  • Vertical line test on a graph: any vertical line should cross the curve at most once.
  • In a table or mapping: each x maps to only one y.

Worked Examples

Example 1. Is {(1, 2), (3, 4), (5, 6)} a function? Yes — all x's are different.

Example 2. Is {(1, 2), (3, 4), (1, 5)} a function? No — x = 1 maps to both 2 and 5.

Example 3. Does a circle pass the vertical line test? No — a vertical through the middle hits two points.

Example 4. Is y = 2x + 1 a function? Yes — straight line, each x gives one y.

Common Mistakes

  • Mixing up "function" with "one-to-one" (functions don't require one-to-one, just that each x gives one y).
  • Forgetting that two points with the same y but different x is still a function.

Practice

  1. Is {(2, 4), (3, 9), (4, 16), (5, 25)} a function?
  2. Is x = y² a function? (y is the output)
  3. Is y = x² a function?

Answers: 1. Yes | 2. No (e.g. x = 4 gives y = ±2) | 3. Yes

6. Function Notation

Key Concepts

f(x) means "the function f evaluated at x". It's just a label — you can have g(x), h(t), etc.

f(3) = 7 means "when input is 3, output is 7".

Worked Examples

Example 1. f(x) = 2x + 5. Find f(3). Solution. 2(3) + 5 = 11.

Example 2. g(x) = x² − 4. Find g(−2). Solution. (−2)² − 4 = 0.

Example 3. f(x) = 3x − 1. Solve f(x) = 14. Solution. 3x − 1 = 14 → x = 5.

Example 4. From a graph, f(2) = 7 means the point (2, 7) is on the curve.

Common Mistakes

  • Reading f(x) as f times x.
  • Substituting only part of an expression for x.

Practice

  1. f(x) = 5x − 3. Find f(4).
  2. h(x) = x² + 2x. Find h(−1).
  3. f(x) = 4x + 7. Solve f(x) = 31.
  4. g(x) = (x − 3)². Find g(5).

Answers: 1. 17 | 2. −1 | 3. x = 6 | 4. 4

7. Domain and Range

Key Concepts

  • Domain: all valid inputs (x-values).
  • Range: all possible outputs (y-values).

Common restrictions:

  • Denominator can't be 0.
  • Square root needs non-negative input.

Worked Examples

Example 1. Domain of f(x) = 1/(x − 2). Solution. All real numbers except x ≠ 2.

Example 2. Domain of f(x) = √(x − 4). Solution. x − 4 ≥ 0 → x ≥ 4.

Example 3. Range of f(x) = x² (over all reals). Solution. y ≥ 0.

Example 4. Domain & range of {(1, 5), (2, 5), (3, 7)}. Solution. Domain {1, 2, 3}; range {5, 7}.

Common Mistakes

  • Forgetting to exclude values that make a denominator zero.
  • Mixing up domain (input) and range (output).

Practice

  1. Domain of f(x) = 2x + 1.
  2. Domain of f(x) = √(2x − 6).
  3. Range of f(x) = |x|.
  4. Domain of f(x) = 1 / (x² − 1).

Answers: 1. All real | 2. x ≥ 3 | 3. y ≥ 0 | 4. x ≠ ±1

8. Increasing, Decreasing, and Constant

Key Concepts

A function is:

  • Increasing on an interval if y goes up as x goes right.
  • Decreasing if y goes down as x goes right.
  • Constant if y stays the same.

Worked Examples

Example 1. f(x) = 3x + 2 is increasing everywhere (positive slope).

Example 2. f(x) = −x + 5 is decreasing (negative slope).

Example 3. f(x) = 7 is constant.

Example 4. f(x) = x² is decreasing for x < 0 and increasing for x > 0.

Common Mistakes

  • Mixing up positive slope with "starts high".

Practice

  1. f(x) = 5 − 2x. Increasing or decreasing?
  2. f(x) = 4. Behaviour?
  3. f(x) = (x − 3)². Where is it decreasing? Increasing?

Answers: 1. Decreasing | 2. Constant | 3. Decreasing for x < 3, increasing for x > 3

9. Slope and Slope-Intercept Form

Key Concepts

  • Slope m = (y₂ − y₁) / (x₂ − x₁).
  • Slope-intercept form: y = mx + b, where m = slope, b = y-intercept.
  • A horizontal line has slope 0. A vertical line has undefined slope.

Worked Examples

Example 1. Slope through (1, 4) and (5, 12). Solution. (12 − 4)/(5 − 1) = 8/4 = 2.

Example 2. y = 3x − 4. Slope and y-intercept? Solution. m = 3, b = −4.

Example 3. Write the equation of a line with slope −2 through (0, 5). Solution. y = −2x + 5.

Example 4. Slope of (3, 7) and (3, −2)? Solution. Same x → vertical line → undefined.

Common Mistakes

  • Computing rise / run with the wrong sign.
  • Confusing slope = 0 (horizontal) with undefined (vertical).

Practice

  1. Slope through (2, 3) and (6, 11).
  2. y = −x + 7. Slope?
  3. Equation: slope 4, passes through (0, −2).
  4. Slope through (5, −1) and (8, −1).

Answers: 1. 2 | 2. −1 | 3. y = 4x − 2 | 4. 0

10. Writing Linear Equations

Key Concepts

Given a slope m and a point (x₁, y₁): y − y₁ = m(x − x₁) (point-slope form). Convert to y = mx + b by simplifying.

From two points: find m first, then plug into point-slope.

Worked Examples

Example 1. Slope 3 through (2, 5). Solution. y − 5 = 3(x − 2) → y = 3x − 1.

Example 2. Line through (1, 4) and (3, 10). Solution. m = 3; y − 4 = 3(x − 1) → y = 3x + 1.

Example 3. Line through (0, 2) and (4, 0). Solution. m = (0 − 2)/(4 − 0) = −1/2 → y = −x/2 + 2.

Common Mistakes

  • Forgetting to distribute the slope through the parentheses.
  • Sign errors when subtracting coordinates.

Practice

  1. Slope 2 through (1, 3).
  2. Through (2, 5) and (4, 13).
  3. Through (0, −1) and (5, 14).

Answers: 1. y = 2x + 1 | 2. y = 4x − 3 | 3. y = 3x − 1

11. Comparing Linear Functions

Key Concepts

You can compare functions given in different forms (equation, table, graph, words) by computing slope and y-intercept and comparing those.

Steeper slope = faster rate of change.

Worked Examples

Example 1. Function A: y = 4x + 2. Function B: passes through (0, 1) and (2, 9). Which has greater slope? Solution. A's slope = 4. B's slope = (9 − 1)/2 = 4. Same.

Example 2. Function A starts at $50 and increases $5/month. Function B starts at $30 and increases $7/month. After 10 months who has more? Solution. A: 50 + 5(10) = 100. B: 30 + 7(10) = 100. Tied.

Example 3. Two cars. A: y = 60x (60 mph). B: graph through (0, 0) and (4, 200). Which is faster? Solution. B's slope = 50 mph. A is faster.

Common Mistakes

  • Forgetting to compare both slope and intercept when context matters.

Practice

  1. y = 2x + 5 vs y = 3x + 1. Which grows faster?
  2. Function A: $20 start, $4/week. Function B: $40 start, $2/week. Total after 12 weeks?
  3. Slope of {(0, 5), (4, 21)} vs y = 5x − 1. Greater slope?

Answers: 1. y = 3x + 1 (slope 3 > 2) | 2. A = 68, B = 64 | 3. Equal (both 4)

Wait — recompute #3: slope of points is (21−5)/4 = 4; y = 5x − 1 has slope 5. So y = 5x − 1 is greater.

12. Linear Inequalities

Key Concepts

Solve a linear inequality the same as an equation, but:

  • Flip the inequality sign when dividing/multiplying by a negative.
  • A solution set is graphed as a shaded region for two-variable inequalities; a number-line interval for one-variable.

Worked Examples

Example 1. Solve 3x − 5 ≤ 7. Solution. 3x ≤ 12 → x ≤ 4.

Example 2. Solve −2x + 1 > 9. Solution. −2x > 8 → divide by −2 (flip): x < −4.

Example 3. Graph y > 2x − 3. Solution. Dashed line for y = 2x − 3; shade above.

Example 4. Solve 5 − x ≥ 12. Solution. −x ≥ 7 → x ≤ −7.

Common Mistakes

  • Forgetting the sign flip with negative.
  • Using a solid line when it should be dashed (and vice versa).

Practice

  1. 4x + 3 < 19
  2. −3x + 1 ≤ −8
  3. 6 − 2x > 0

Answers: 1. x < 4 | 2. x ≥ 3 | 3. x < 3

13. Solving Systems by Graphing

Key Concepts

A system of equations is two or more equations sharing the same variables. The solution is the point(s) where graphs intersect.

  • One solution: lines cross at one point (most common).
  • No solution: lines are parallel.
  • Infinitely many: lines are the same.

Worked Examples

Example 1. Solve graphically: y = x + 1 and y = −x + 5. Solution. Intersection at (2, 3).

Example 2. y = 2x + 3 and y = 2x − 1. Parallel? Yes → no solution.

Example 3. y = x and 2y = 2x. Same line → infinitely many solutions.

Common Mistakes

  • Misreading the intersection point.
  • Assuming all systems have a unique solution.

Practice

  1. y = x − 2 and y = −x + 4. Solution?
  2. y = 3x + 1 and y = 3x − 5. Solution type?
  3. 2y = 4x and y = 2x. How many solutions?

Answers: 1. (3, 1) | 2. No solution (parallel) | 3. Infinitely many

14. Substitution and Elimination

Key Concepts

Substitution: solve one equation for a variable, substitute into the other.

Elimination: add or subtract the equations (scaled if needed) so one variable cancels.

Worked Examples

Example 1. Solve y = 2x + 1 and 3x + y = 11. Solution. Substitute: 3x + (2x + 1) = 11 → 5x = 10 → x = 2; y = 5.

Example 2. Solve x + y = 7 and x − y = 1. Solution. Add: 2x = 8 → x = 4; y = 3.

Example 3. Solve 3x + 2y = 12 and x − y = 1. Solution. From the second: x = y + 1. Sub: 3(y + 1) + 2y = 12 → 5y = 9 → y = 1.8; x = 2.8.

Example 4. Solve 2x + 3y = 16 and x − y = 2. Solution. Multiply 2nd by 3: 3x − 3y = 6. Add to 1st: 5x = 22 → x = 4.4; y = 2.4.

Common Mistakes

  • Forgetting to multiply both sides of an equation when scaling.
  • Substituting the answer back into only one equation (always check both).

Practice

  1. y = 3x − 4 and x + y = 8.
  2. 2x + y = 7 and x − y = 2.
  3. 4x + y = 10 and 2x + 3y = 12.

Answers: 1. (3, 5) | 2. (3, 1) | 3. (1.8, 2.8)

15. Pythagorean Theorem

Key Concepts

For a right triangle with legs a, b and hypotenuse c: a² + b² = c².

The hypotenuse is always the longest side, opposite the right angle.

Worked Examples

Example 1. Legs 3 and 4. Hypotenuse? Solution. √(9 + 16) = 5.

Example 2. Legs 5 and 12. Hypotenuse? Solution. √(25 + 144) = √169 = 13.

Example 3. Hypotenuse 25, one leg 7. Other leg? Solution. √(625 − 49) = √576 = 24.

Example 4. A ladder 10 ft long against a wall, base 6 ft from wall. How high? Solution. √(100 − 36) = 8 ft.

Common Mistakes

  • Using hypotenuse² + leg² = leg² (wrong).
  • Forgetting to take the square root.

Practice

  1. Legs 8 and 15. Hypotenuse?
  2. Legs 9 and 12. Hypotenuse?
  3. Hypotenuse 17, leg 8. Other leg?

Answers: 1. 17 | 2. 15 | 3. 15

16. Distance Formula

Key Concepts

Distance between (x₁, y₁) and (x₂, y₂): d = √[(x₂ − x₁)² + (y₂ − y₁)²].

Derived from the Pythagorean theorem.

Worked Examples

Example 1. Distance from (1, 2) to (4, 6). Solution. √(9 + 16) = 5.

Example 2. Distance from (−1, 3) to (2, −1). Solution. √(9 + 16) = 5.

Example 3. Distance from (0, 0) to (6, 8). Solution. √(36 + 64) = 10.

Example 4. A side of a polygon with vertices (3, 4) and (−2, 4). Length? Solution. Same y → just |Δx| = 5.

Common Mistakes

  • Forgetting to square the differences (using just Δx + Δy).
  • Mixing up signs in subtraction.

Practice

  1. (2, 3) to (5, 7).
  2. (−3, 1) to (4, 1).
  3. (0, 0) to (−6, −8).
  4. (1, 5) to (1, −2).

Answers: 1. 5 | 2. 7 | 3. 10 | 4. 7

17. Volume of 3D Solids (Cones, Cylinders, Spheres)

Key Concepts

  • Cylinder: V = πr²h.
  • Cone: V = (1/3)πr²h.
  • Sphere: V = (4/3)πr³.

A cone holds one-third of the cylinder it fits inside.

Worked Examples

Example 1. Cylinder r = 3, h = 8. Volume? Solution. π(9)(8) = 72π ≈ 226.08.

Example 2. Cone r = 4, h = 9. Volume? Solution. (1/3)π(16)(9) = 48π ≈ 150.72.

Example 3. Sphere r = 6. Solution. (4/3)π(216) = 288π ≈ 904.32.

Example 4. Cone with same radius and height as a cylinder of volume 90 — cone's volume? Solution. 90/3 = 30.

Common Mistakes

  • Forgetting the 1/3 factor for cone.
  • Cubing diameter instead of radius for sphere.

Practice

  1. Cylinder r = 5, h = 10.
  2. Cone r = 6, h = 12.
  3. Sphere r = 3.

Answers: 1. 250π ≈ 785 | 2. 144π ≈ 452 | 3. 36π ≈ 113

18. Transformations: Translation, Reflection, Rotation, Dilation

Key Concepts

  • Translation: slide. (x, y) → (x + a, y + b).
  • Reflection: flip. Across x-axis: (x, y) → (x, −y). Across y-axis: (x, y) → (−x, y).
  • Rotation (around origin): 90° ccw: (x, y) → (−y, x). 180°: (x, y) → (−x, −y).
  • Dilation (from origin) by factor k: (x, y) → (kx, ky). Lengths scale by k; areas by k².

The first three preserve shape and size (rigid). Dilation changes size.

Worked Examples

Example 1. Translate (3, 4) by (−2, 5). Solution. (1, 9).

Example 2. Reflect (−5, 2) across the x-axis. Solution. (−5, −2).

Example 3. Rotate (3, 1) by 180° around origin. Solution. (−3, −1).

Example 4. Dilate (4, 6) by scale factor 0.5. Solution. (2, 3).

Common Mistakes

  • Confusing axis reflections (x-axis flips y, not x).
  • Forgetting that dilation scales area by k².

Practice

  1. Translate (5, −1) by (3, 4).
  2. Reflect (2, 7) across y-axis.
  3. Rotate (4, 0) by 90° ccw.
  4. Dilate (6, 8) by factor 1/2.

Answers: 1. (8, 3) | 2. (−2, 7) | 3. (0, 4) | 4. (3, 4)

19. Scatter Plots and Lines of Best Fit

Key Concepts

A scatter plot shows pairs (x, y) as dots. Patterns indicate:

  • Positive association: y rises as x rises.
  • Negative association: y falls as x rises.
  • No association: scattered randomly.

A line of best fit approximates the trend. Use it to make predictions.

Worked Examples

Example 1. Study time vs test score: as study time goes up, score goes up. Association? Solution. Positive.

Example 2. Outside temperature vs hot chocolate sold. As temperature falls, sales rise. Solution. Negative association.

Example 3. Line of best fit y = 5x + 50. Predict score for 4 hours of study. Solution. 5(4) + 50 = 70.

Common Mistakes

  • Assuming correlation implies causation.
  • Extrapolating well beyond the data.

Practice

  1. Hours of sleep vs alertness — likely association?
  2. Best fit y = 2x + 10. Predict y when x = 7.
  3. A scatter plot shows no pattern. Association?

Answers: 1. Positive | 2. 24 | 3. None / no association

20. Two-Way Tables

Key Concepts

A two-way table groups data by two categorical variables (e.g., gender × sport). Useful for finding relative frequency, marginal totals, and conditional probabilities.

Worked Examples

Example 1. A table shows 30 boys / 20 girls; among boys, 18 play soccer; among girls, 12. Proportion of soccer players among boys? Solution. 18/30 = 0.6.

Example 2. What proportion of students play soccer overall? Solution. (18 + 12)/(30 + 20) = 30/50 = 0.6.

Example 3. Is there an association between gender and soccer? Same proportion both — no association.

Common Mistakes

  • Comparing raw counts when groups have different totals (use proportions).
  • Mixing up row and column when computing conditional proportions.

Practice

A two-way table: 40 cats / 60 dogs. Among cats, 25 like fish. Among dogs, 18 like fish.

  1. Proportion of cats that like fish?
  2. Proportion of dogs that like fish?
  3. Stronger preference among which group?

Answers: 1. 25/40 = 0.625 | 2. 18/60 = 0.30 | 3. Cats

That's Grade 8. The biggest tickets to a strong score: master slope-intercept form, Pythagorean theorem, and systems of equations — these compound into Grade 9 algebra. Practice mixed problems daily!