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Grade 6 Prep Guide

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Adaptive Math Test — Grade 6 Prep Guide

A comprehensive, topic-by-topic study guide for the Grade 6 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

Ratios, Rates & Proportional Reasoning

  1. Ratios and Equivalent Ratios
  2. Ratio Tables
  3. Unit Rates
  4. Proportional Reasoning

Number System 5. Dividing Fractions by Fractions 6. Operations with Fractions and Decimals 7. Positive and Negative Numbers 8. Absolute Value 9. Integers in the Coordinate Plane

Percent 10. Percent of a Number 11. Finding the Whole

Expressions and Equations 12. Writing Expressions 13. Evaluating Expressions 14. One-Step Equations 15. Inequalities (Introduction)

Geometry 16. Area of Triangles and Quadrilaterals 17. Surface Area and Volume

Statistics 18. Measures of Center: Mean, Median, Mode 19. Variability and Data Displays

1. Ratios and Equivalent Ratios

Key Concepts

A ratio compares two quantities. You can write 3 to 5 three ways: 3 : 5, 3 to 5, or 3/5.

Two ratios are equivalent when one can be scaled up or down to match the other (multiply or divide both parts by the same number).

Worked Examples

Example 1. Are 4 : 6 and 10 : 15 equivalent? Solution. Divide each by its GCF: 4 : 6 → 2 : 3; 10 : 15 → 2 : 3. Yes, both simplify to 2 : 3.

Example 2. A recipe uses 2 cups of flour for every 3 cups of milk. How much milk for 8 cups of flour? Solution. 2 : 3 scaled by 4 → 8 : 12. 12 cups of milk.

Example 3. Write the ratio of boys to girls if there are 12 boys and 18 girls, in simplest form. Solution. GCF(12, 18) = 6. 12 : 18 = 2 : 3.

Example 4. A paint mix is 5 red : 3 blue. How much blue for 25 parts red? Solution. 5 : 3 × 5 → 25 : 15. 15 parts blue.

Common Mistakes

  • Reversing the order — "3 boys to every 2 girls" is not the same as "3 girls to every 2 boys".
  • Adding instead of multiplying when scaling (e.g., 2 : 3 becoming 3 : 4 instead of 4 : 6).
  • Forgetting to simplify to lowest terms when asked.

Practice

  1. Simplify 18 : 24.
  2. If 4 pencils cost $1, how much do 20 pencils cost?
  3. A class has 8 girls and 12 boys. Ratio of girls to total students?
  4. Are 9 : 12 and 6 : 8 equivalent?

Answers: 1. 3 : 4 | 2. $5 | 3. 8 : 20 = 2 : 5 | 4. Yes (both = 3 : 4)

2. Ratio Tables

Key Concepts

A ratio table organises equivalent ratios in columns. Each column is the previous one multiplied (or divided) by the same factor. Tables help you find missing values and compare ratios.

Worked Examples

Example 1. Complete the table for the ratio 3 : 7. | Part A | 3 | 6 | ? | 15 | | Part B | 7 | ? | 21 | ? | Solution. ×2: 6, 14. ×3: 9, 21 (so missing A = 9). ×5: 15, 35. Table: 9 and 14 and 35.

Example 2. A car travels 60 miles in 1 hour. How far in 4 hours? In 2.5 hours? Solution. 60 × 4 = 240 miles; 60 × 2.5 = 150 miles.

Example 3. Find the missing value: 4 : 9 = ? : 36. Solution. 9 × 4 = 36, so 4 × 4 = 16.

Common Mistakes

  • Multiplying only one part of the ratio instead of both.
  • Mixing addition with multiplication (e.g., +3 in one column but ×2 in the next).

Practice

  1. If 5 books cost $20, how much do 12 books cost?
  2. Fill in: 7 : 12 = 21 : ?
  3. A bike travels 24 km in 2 hours. Speed for 5 hours?

Answers: 1. $48 | 2. 36 | 3. 60 km

3. Unit Rates

Key Concepts

A unit rate is a rate with denominator 1. To find it, divide the first quantity by the second.

Common unit rates: miles per hour, dollars per pound, words per minute.

Worked Examples

Example 1. 120 miles in 3 hours. What is the unit rate? Solution. 120 ÷ 3 = 40 miles per hour.

Example 2. 6 apples cost $4.50. Cost per apple? Solution. 4.50 ÷ 6 = $0.75 per apple.

Example 3. Which is the better buy: 12 oz for $3 or 16 oz for $3.60? Solution. $3/12 = $0.25 per oz; $3.60/16 = $0.225 per oz. 16 oz is better.

Example 4. Maria reads 90 pages in 3 hours. At this rate, how many pages in 8 hours? Solution. Unit rate = 30 pages/hour. 30 × 8 = 240 pages.

Common Mistakes

  • Dividing in the wrong order — unit rate of dollars-per-pound divides dollars by pounds, not the other way.
  • Comparing prices without converting to the same unit.

Practice

  1. 5 candy bars for $4. Cost per bar?
  2. A printer prints 240 pages in 8 minutes. Pages per minute?
  3. 30 students need 7.5 kg of clay. How much per student?
  4. Which is the better deal — 3 lbs for $4.20 or 5 lbs for $6.50?

Answers: 1. $0.80 | 2. 30 pages/min | 3. 0.25 kg | 4. 3 lbs ($1.40/lb beats $1.30/lb? recompute: $4.20/3 = $1.40 and $6.50/5 = $1.30; 5 lbs is cheaper)

4. Proportional Reasoning

Key Concepts

Two quantities are proportional when their ratio is always the same. You can solve proportions by:

  • Cross-multiplying (a/b = c/d ⟹ a·d = b·c)
  • Scaling (multiply both sides by the same factor)

Worked Examples

Example 1. Solve x/8 = 15/24. Solution. Cross-multiply: 24x = 120 → x = 5.

Example 2. Solve 6/9 = 14/x. Solution. 6x = 126 → x = 21.

Example 3. If 5 yards of fabric costs $35, how much does 12 yards cost? Solution. 5/35 = 12/x → 5x = 420 → x = $84.

Example 4. A map's scale is 1 inch = 25 miles. How many miles is 4.5 inches? Solution. 1/25 = 4.5/x → x = 112.5 miles.

Common Mistakes

  • Cross-multiplying when the equation isn't actually a proportion (e.g., 2/3 + 1/x = 4 — that's not a proportion).
  • Mixing up which side the unknown goes on.

Practice

  1. Solve: 4/x = 12/27.
  2. If 3 pizzas feed 8 people, how many pizzas for 24 people?
  3. Solve: x/14 = 5/7.
  4. A car uses 6 gallons to drive 180 miles. Gallons for 300 miles?

Answers: 1. x = 9 | 2. 9 pizzas | 3. x = 10 | 4. 10 gallons

5. Dividing Fractions by Fractions

Key Concepts

To divide by a fraction, multiply by its reciprocal (flip and multiply): a/b ÷ c/d = a/b × d/c.

Simplify before or after multiplying — both work; simplifying first is usually easier.

Worked Examples

Example 1. 3/4 ÷ 2/5 = ? Solution. 3/4 × 5/2 = 15/8 = 1 7/8.

Example 2. 2/3 ÷ 4 = ? Solution. 4 = 4/1. 2/3 × 1/4 = 2/12 = 1/6.

Example 3. A ribbon is 7/8 yard long. How many 1/4-yard pieces? Solution. 7/8 ÷ 1/4 = 7/8 × 4/1 = 28/8 = 3 1/2 pieces.

Example 4. 5/6 ÷ 5/12 = ? Solution. 5/6 × 12/5 = 60/30 = 2.

Common Mistakes

  • Flipping the wrong fraction (the divisor flips, not the dividend).
  • Multiplying instead of dividing for word problems with "how many groups of...".

Practice

  1. 3/4 ÷ 1/2
  2. 5/6 ÷ 2/3
  3. 9/10 ÷ 3/5
  4. A pitcher holds 3/4 gallon. How many 1/8-gallon cups can it fill?

Answers: 1. 1 1/2 | 2. 5/4 = 1 1/4 | 3. 3/2 = 1 1/2 | 4. 6 cups

6. Operations with Fractions and Decimals

Key Concepts

For decimals: line up the decimal points when adding/subtracting; for multiplication, multiply ignoring the points then count total decimal places; for division, shift the decimal in both numbers so the divisor becomes whole.

For fractions: get common denominators for + and −; multiply numerators and denominators for ×; flip and multiply for ÷.

Worked Examples

Example 1. 3.45 + 2.7 = ? Solution. Align: 3.45 + 2.70 = 6.15.

Example 2. 0.6 × 0.4 = ? Solution. 6 × 4 = 24; total of 2 decimal places → 0.24.

Example 3. 1/2 + 2/3 = ? Solution. LCD = 6: 3/6 + 4/6 = 7/6 (or 1 1/6).

Example 4. 0.75 ÷ 0.25 = ? Solution. Shift both two places: 75 ÷ 25 = 3.

Common Mistakes

  • Forgetting to line up decimals when adding 5.2 + 0.45 (must be 5.20 + 0.45).
  • Mis-counting decimal places in multiplication.

Practice

  1. 4.7 − 2.85
  2. 0.5 × 0.08
  3. 1/4 + 5/8
  4. 0.6 ÷ 1.5

Answers: 1. 1.85 | 2. 0.04 | 3. 7/8 | 4. 0.4

7. Positive and Negative Numbers

Key Concepts

Integers extend the number line in both directions. Negative numbers are to the left of zero.

  • Adding a positive moves right; adding a negative moves left.
  • Subtracting is the same as adding the opposite: 5 − (−3) = 5 + 3.
  • Multiplying two same signs gives positive; different signs gives negative.

Worked Examples

Example 1. −7 + 4 = ? Solution. Start at −7, move right 4. −3.

Example 2. −3 − 8 = ? Solution. −3 + (−8) = −11.

Example 3. (−4) × (−6) = ? Solution. Same sign → positive. 24.

Example 4. −20 ÷ 5 = ? Solution. Different signs → negative. −4.

Example 5. A submarine descends from −50 ft to −120 ft. How far did it move? Solution. −120 − (−50) = −70 ft (70 ft deeper).

Common Mistakes

  • Treating subtraction as "always make smaller" — sometimes it makes larger (subtracting a negative).
  • Forgetting the sign rules for multiplication.

Practice

  1. −9 + 12
  2. 6 − (−4)
  3. (−5) × 7
  4. (−18) ÷ (−3)
  5. Temperature dropped from 4°C to −9°C. Change?

Answers: 1. 3 | 2. 10 | 3. −35 | 4. 6 | 5. −13°C

8. Absolute Value

Key Concepts

Absolute value is the distance from zero on a number line — always non-negative. Written |x|.

|6| = 6, |−6| = 6, |0| = 0.

Worked Examples

Example 1. |−12| = 12.

Example 2. |8 − 15| = ? Solution. 8 − 15 = −7; |−7| = 7.

Example 3. Two friends are at positions −4 and 7 on a number line. Distance between? Solution. |7 − (−4)| = |11| = 11.

Example 4. Which is larger: |−9| or |3|? Solution. 9 vs 3 → |−9| is larger.

Common Mistakes

  • Thinking |−5| = −5. Absolute value is never negative.
  • Computing |a − b| as a − b without absolute value bars when distance is asked.

Practice

  1. |−15|
  2. |0|
  3. |3 − 10|
  4. Distance between −6 and 9 on a number line.

Answers: 1. 15 | 2. 0 | 3. 7 | 4. 15

9. Integers in the Coordinate Plane

Key Concepts

The coordinate plane is split into four quadrants by the x- and y-axes:

  • Q1 (top right): x > 0, y > 0.
  • Q2 (top left): x < 0, y > 0.
  • Q3 (bottom left): x < 0, y < 0.
  • Q4 (bottom right): x > 0, y < 0.

To plot (x, y): start at origin, move x right (or left if negative), then y up (or down if negative).

Worked Examples

Example 1. Plot (−3, 4). Which quadrant? Solution. Left 3, up 4 — Quadrant II.

Example 2. Reflect (5, −2) across the x-axis. Solution. Flip the y-coordinate: (5, 2).

Example 3. Distance between (2, 3) and (2, −5). Solution. Same x-coordinate; |3 − (−5)| = 8 units.

Example 4. A rectangle has vertices (−2, 1), (4, 1), (4, −3), (−2, −3). Find its area. Solution. Width |4 − (−2)| = 6; height |1 − (−3)| = 4. Area = 6 × 4 = 24 sq units.

Common Mistakes

  • Confusing x-axis and y-axis reflections (x-axis flips y; y-axis flips x).
  • Mixing up the order of (x, y) when plotting.

Practice

  1. Which quadrant is (−4, −7) in?
  2. Reflect (3, 5) across the y-axis.
  3. Distance from (−1, 4) to (−1, −2)?
  4. Plot (0, −3). Which axis is it on?

Answers: 1. Q3 | 2. (−3, 5) | 3. 6 | 4. The y-axis

10. Percent of a Number

Key Concepts

A percent is a ratio out of 100. p% of n = (p/100) × n.

Quick conversions:

  • 50% = 1/2 → halve it.
  • 25% = 1/4 → divide by 4.
  • 10% = 1/10 → divide by 10.
  • 1% = 1/100 → divide by 100.

Worked Examples

Example 1. What is 30% of 80? Solution. 0.30 × 80 = 24.

Example 2. Find 15% of 60. Solution. 10% = 6; 5% = 3; 15% = 6 + 3 = 9.

Example 3. A jacket costs $48. With 25% off, what's the discount? Solution. 0.25 × 48 = $12.

Example 4. What is 120% of 50? Solution. 1.20 × 50 = 60.

Common Mistakes

  • Using 30 instead of 0.30 in calculations (multiplying by 30 gives 30× too much).
  • Forgetting that more than 100% means more than the original.

Practice

  1. 40% of 200
  2. 75% of 32
  3. 12% of 50
  4. 150% of 80

Answers: 1. 80 | 2. 24 | 3. 6 | 4. 120

11. Finding the Whole

Key Concepts

When you know a part and what percent of the whole it represents, divide to find the whole.

If p% of W = part, then W = part ÷ (p/100).

Worked Examples

Example 1. 20% of a number is 14. What's the number? Solution. 14 ÷ 0.20 = 70.

Example 2. 30 is 60% of what number? Solution. 30 ÷ 0.60 = 50.

Example 3. A student got 18 questions right, which was 75% of the test. How many total? Solution. 18 ÷ 0.75 = 24 questions.

Example 4. 5% sales tax on a purchase was $1.40. What was the price? Solution. 1.40 ÷ 0.05 = $28.

Common Mistakes

  • Multiplying by the percent instead of dividing.
  • Confusing "20% of" with "20% more than".

Practice

  1. 25% of x is 15. Find x.
  2. 9 is 30% of what?
  3. 80% of a class voted yes — that's 24 students. Class size?
  4. A 6% tax is $3. Original price?

Answers: 1. 60 | 2. 30 | 3. 30 students | 4. $50

12. Writing Expressions

Key Concepts

Translate words into algebra:

  • "plus / more than / increased by" → +
  • "minus / less than / decreased by" → −
  • "times / product of" → ×
  • "divided by / quotient of" → ÷
  • "a number" → use a variable like n or x

Order matters with subtraction and division: "5 less than n" = n − 5 (not 5 − n).

Worked Examples

Example 1. "Three more than twice a number" → 2n + 3.

Example 2. "5 less than the quotient of x and 4" → x/4 − 5.

Example 3. "The sum of a and b, divided by 2" → (a + b)/2.

Example 4. A taxi costs $2 plus $0.50 per mile. Cost for m miles? Solution. 2 + 0.5m.

Common Mistakes

  • Reversing "less than" (3 less than x is x − 3, not 3 − x).
  • Missing parentheses around sums or differences that are then divided/multiplied.

Practice

  1. "7 more than 4 times a number"
  2. "12 minus half of x"
  3. "The product of 3 and the sum of x and 5"
  4. A movie costs $8 plus $1.50 for popcorn. Total for p people?

Answers: 1. 4n + 7 | 2. 12 − x/2 | 3. 3(x + 5) | 4. 8p + 1.5p or 9.5p

13. Evaluating Expressions

Key Concepts

To evaluate an expression, substitute the values for the variables and follow the order of operations (PEMDAS):

  1. Parentheses
  2. Exponents
  3. Multiplication / Division (left to right)
  4. Addition / Subtraction (left to right)

Worked Examples

Example 1. Evaluate 3x + 5 when x = 4. Solution. 3(4) + 5 = 12 + 5 = 17.

Example 2. Evaluate x² − 2y when x = 5, y = 7. Solution. 25 − 14 = 11.

Example 3. Evaluate (a + b) ÷ 2 when a = 3, b = 11. Solution. 14 ÷ 2 = 7.

Example 4. Evaluate 4(2x − 3) when x = 5. Solution. 4(10 − 3) = 4 × 7 = 28.

Common Mistakes

  • Skipping parentheses (e.g., −3² should be −9, but (−3)² is 9 — different).
  • Adding before multiplying when the order says otherwise.

Practice

  1. 5x − 2 when x = 6.
  2. x² + y when x = 3, y = 4.
  3. (a − b)² when a = 7, b = 3.
  4. 2(x + 5) − 3 when x = 4.

Answers: 1. 28 | 2. 13 | 3. 16 | 4. 15

14. One-Step Equations

Key Concepts

To solve, do the inverse operation on both sides:

  • x + 5 = 12 → subtract 5 from both sides.
  • x − 7 = 2 → add 7.
  • 4x = 20 → divide by 4.
  • x / 3 = 6 → multiply by 3.

Always check by substituting back.

Worked Examples

Example 1. Solve x + 8 = 15. Solution. x = 15 − 8 = 7.

Example 2. Solve x − 12 = 5. Solution. x = 5 + 12 = 17.

Example 3. Solve 6x = 42. Solution. x = 42 ÷ 6 = 7.

Example 4. Solve x/5 = 9. Solution. x = 9 × 5 = 45.

Example 5. Solve x + 2.5 = 7. Solution. x = 4.5.

Common Mistakes

  • Doing the operation to only one side.
  • Choosing the same operation instead of the inverse (e.g., adding 5 to "x + 5 = 12").

Practice

  1. y − 11 = 4
  2. 9x = 81
  3. m/7 = 3
  4. p + 1.2 = 5

Answers: 1. y = 15 | 2. x = 9 | 3. m = 21 | 4. p = 3.8

15. Inequalities (Introduction)

Key Concepts

An inequality uses one of: < (less than), > (greater than), ≤ (≤), ≥ (≥).

Solving is similar to equations: do the same operation to both sides. The big exception comes in Grade 7+: dividing or multiplying by a negative flips the inequality. In Grade 6 we mostly stick with positive values.

A solution set is often a range — draw it on a number line with open circles (< or >) or closed circles (≤ or ≥).

Worked Examples

Example 1. Solve x + 3 < 10. Solution. x < 7.

Example 2. Solve 2x ≥ 14. Solution. x ≥ 7.

Example 3. Graph x > 4 on a number line. Solution. Open circle at 4, arrow pointing right.

Example 4. Sarah needs at least $50 for the trip. She has $32. How much more does she need (use an inequality)? Solution. 32 + x ≥ 50 → x ≥ $18.

Common Mistakes

  • Confusing < and > (think of the symbol as an "alligator's mouth" eating the larger number).
  • Using open vs closed circle incorrectly.

Practice

  1. Solve x − 4 ≤ 10.
  2. Solve 5x > 25.
  3. Graph x ≥ −2 (describe).
  4. Tom needs more than 15 stickers. He has 8. Inequality?

Answers: 1. x ≤ 14 | 2. x > 5 | 3. Closed circle at −2, arrow right | 4. 8 + x > 15 → x > 7

16. Area of Triangles and Quadrilaterals

Key Concepts

  • Rectangle: A = length × width.
  • Parallelogram: A = base × height (perpendicular height).
  • Triangle: A = (1/2) × base × height.
  • Trapezoid: A = (1/2)(b₁ + b₂) × h.

Worked Examples

Example 1. Area of a triangle with base 10 and height 6. Solution. (1/2)(10)(6) = 30.

Example 2. A parallelogram with base 8 cm, height 5 cm. Solution. 8 × 5 = 40 cm².

Example 3. Trapezoid with parallel sides 6 and 10, height 4. Solution. (1/2)(6 + 10)(4) = 32.

Example 4. L-shape: 5×4 rectangle joined with a 3×2 rectangle. Area? Solution. 20 + 6 = 26.

Common Mistakes

  • Using a slanted side instead of perpendicular height.
  • Forgetting the 1/2 factor for triangles.

Practice

  1. Triangle base 14, height 9.
  2. Rectangle 12 by 7.
  3. Trapezoid b₁ = 5, b₂ = 9, h = 6.
  4. Parallelogram base 11, height 4.

Answers: 1. 63 | 2. 84 | 3. 42 | 4. 44

17. Surface Area and Volume

Key Concepts

  • Rectangular prism volume: V = l × w × h.
  • Surface area: add areas of all 6 faces. For a rectangular prism: 2(lw + lh + wh).
  • Cube (side s): V = s³, SA = 6s².

Worked Examples

Example 1. Volume of a 3 × 4 × 5 box. Solution. 60 cubic units.

Example 2. SA of a 2 × 3 × 4 prism. Solution. 2(6 + 8 + 12) = 52 sq units.

Example 3. Volume of a cube with side 5. Solution. 5³ = 125.

Example 4. A swimming pool is 25 m × 10 m × 2 m. How much water? Solution. 500 m³.

Common Mistakes

  • Confusing volume (cubic units) and surface area (square units).
  • Missing a face in surface area.

Practice

  1. V of 6 × 6 × 6 cube.
  2. SA of 4 × 5 × 6 prism.
  3. V of 8 × 3 × 5 box.
  4. SA of cube with side 4.

Answers: 1. 216 | 2. 148 | 3. 120 | 4. 96

18. Measures of Center: Mean, Median, Mode

Key Concepts

  • Mean = sum ÷ count (the average).
  • Median = middle value when sorted (if even count, mean of the two middle).
  • Mode = most frequent value (can be more than one, or none).

Worked Examples

Example 1. Mean of 4, 8, 10, 6, 12. Solution. Sum = 40; 40 ÷ 5 = 8.

Example 2. Median of 3, 7, 9, 12, 15. Solution. Middle is the 3rd value → 9.

Example 3. Median of 2, 4, 6, 8. Solution. (4 + 6)/2 = 5.

Example 4. Mode of 1, 3, 3, 5, 7, 3, 9. Solution. 3 (appears 3 times).

Example 5. Scores: 70, 80, 80, 90, 100. Mean / median / mode? Solution. Mean = 84; median = 80; mode = 80.

Common Mistakes

  • Forgetting to sort the data before finding the median.
  • Reporting frequency instead of the value for the mode.

Practice

  1. Mean of 5, 9, 11, 7.
  2. Median of 12, 4, 18, 9, 7.
  3. Mode of 2, 5, 2, 8, 9, 2, 5.
  4. Mean of 100, 90, 95, 85, 80, 100.

Answers: 1. 8 | 2. 9 | 3. 2 | 4. 91.67 (≈ 91.7)

19. Variability and Data Displays

Key Concepts

  • Range = max − min.
  • Interquartile range (IQR) = Q3 − Q1.
  • Common displays: dot plot, box plot (5-number summary), histogram (groups data into bins).

A box plot shows: minimum, Q1, median, Q3, maximum. The "box" stretches from Q1 to Q3 with a line at the median.

Worked Examples

Example 1. Range of 4, 7, 12, 9, 15, 6. Solution. 15 − 4 = 11.

Example 2. Find Q1, median, Q3 of 2, 4, 6, 8, 10, 12, 14. Solution. Median = 8; lower half {2, 4, 6} → Q1 = 4; upper half {10, 12, 14} → Q3 = 12. IQR = 8.

Example 3. A histogram bin "10–20" with frequency 8 means how many data points? Solution. 8 values fall in that range.

Common Mistakes

  • Mixing up range (spread of all data) with IQR (spread of middle 50%).
  • Not sorting before computing quartiles.

Practice

  1. Range of 22, 30, 15, 28, 19.
  2. Find median of 11, 14, 9, 17, 20, 8.
  3. What does Q1 represent?
  4. Five-number summary of 3, 5, 7, 9, 11, 13, 15.

Answers: 1. 15 | 2. 12.5 | 3. 25th percentile (lower quartile) | 4. min 3, Q1 5, med 9, Q3 13, max 15

You've covered every major Grade 6 topic. The best way to lock these in is to practise mixed problems on the adaptive test, then come back to any topic where your accuracy was below 70%. Good luck!