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Grade 5 Prep Guide

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Adaptive Math Test — Grade 5 Prep Guide

A comprehensive, topic-by-topic study guide for the Grade 5 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

Number & Operations

  1. Decimal Place Value to Thousandths
  2. Multi-Digit Multiplication
  3. Long Division
  4. Decimal Operations

Fractions 5. Equivalent and Comparing Fractions 6. Adding & Subtracting Unlike Denominators 7. Multiplying Fractions 8. Dividing Fractions by Whole Numbers 9. Mixed Numbers

Geometry & Measurement 10. Volume of Rectangular Prisms 11. Composite Volumes 12. Coordinate Plane (First Quadrant) 13. Distance and Polygons on a Coordinate Grid 14. Classifying 2D Figures 15. Measurement Conversion

Patterns 16. Numerical Patterns & Input-Output Tables

1. Decimal Place Value to Thousandths

Key Concepts

Decimals can extend several places to the right of the decimal point:

  • Tenths (0.1) — first place after the point.
  • Hundredths (0.01) — second place.
  • Thousandths (0.001) — third place.

Each place to the right is 10 times smaller than the one to its left. Each place to the left is 10 times larger.

You can read 4.275 as "four and two hundred seventy-five thousandths" (because 0.275 = 275/1000).

Worked Examples

Example 1. What is the value of the 6 in 3.764? Solution. The 6 is in the hundredths place. Value = 6/100 = 0.06.

Example 2. Write 0.085 as a fraction. Solution. 0.085 = 85/1000 (= 17/200 in lowest terms).

Example 3. Write "two and forty-three thousandths" in decimal form. Solution. 2 + 43/1000 = 2 + 0.043 = 2.043.

Example 4. Compare the value of the 7 in 0.74 vs. 7.4. Solution. In 0.74 the 7 is in tenths (0.7). In 7.4 the 7 is in ones (7). So the 7 in 7.4 is 10 times larger.

Example 5. Round 4.567 to the nearest hundredth. Solution. Hundredths place is 6; next digit is 7 (≥5) → round up. 4.57.

Common Mistakes

  • Confusing tenths (0.1) with hundredths (0.01) or thousandths (0.001).
  • Writing "five tenths" as 0.05 (it's 0.5).
  • Forgetting that 0.4 = 0.40 = 0.400 (trailing zeros don't change value).

Practice

  1. What is the value of the 3 in 8.234?
  2. Write 0.219 as a fraction (no need to simplify).
  3. Round 6.785 to the nearest tenth.
  4. How many thousandths in 0.5?
  5. Compare the value of the 5 in 0.025 vs. 5.2.

Answers: 1. 0.03 | 2. 219/1000 | 3. 6.8 | 4. 500 | 5. The 5 in 5.2 is much larger (5 vs 0.005)

2. Multi-Digit Multiplication

Key Concepts

Grade 5 multiplication includes 3-digit × 2-digit and 2-digit × 3-digit. Use:

  • Standard algorithm (vertical with placeholder zeros).
  • Partial products / area model (split each factor by place value).
  • Mental math for friendly factors (e.g., 25 × 16 → 25 × 4 × 4 = 100 × 4 = 400).

For multiplying by 10, 100, 1,000: shift digits left (or append zeros).

Worked Examples

Example 1. 24 × 36 = ? Solution. Partial products: 20 × 30 = 600; 20 × 6 = 120; 4 × 30 = 120; 4 × 6 = 24. Sum: 600 + 120 + 120 + 24 = 864.

Example 2. 47 × 23 = ? Solution. (40 + 7)(20 + 3) = 800 + 120 + 140 + 21 = 1,081.

Example 3. 125 × 16 = ? Solution. 125 × 16 = 125 × 8 × 2 = 1,000 × 2 = 2,000.

Example 4. A school has 24 classes of 32 students each. Total students? Solution. 24 × 32 = (20 + 4)(30 + 2) = 600 + 40 + 120 + 8 = 768.

Example 5. 215 × 12 = ? Solution. 215 × 10 + 215 × 2 = 2,150 + 430 = 2,580.

Common Mistakes

  • Forgetting to add the placeholder zero when multiplying by tens.
  • Mis-aligning partial products in the standard algorithm.
  • Skipping a partial product in the area model (especially the corner ones × ones).

Practice

  1. 36 × 24 = ?
  2. 52 × 47 = ?
  3. 145 × 13 = ?
  4. 125 × 32 = ?
  5. A library has 18 shelves with 145 books each. Total books?

Answers: 1. 864 | 2. 2,444 | 3. 1,885 | 4. 4,000 | 5. 2,610

3. Long Division

Key Concepts

Grade 5 long division extends to 2-digit divisors: e.g., 5,616 ÷ 16. The DMSB cycle still applies (Divide, Multiply, Subtract, Bring down).

Steps for 5,616 ÷ 16:

  1. 16 doesn't go into 5; into 56? Yes, 3 times (16 × 3 = 48). Write 3.
  2. Subtract 56 − 48 = 8.
  3. Bring down 1 → 81. 16 × 5 = 80. Write 5.
  4. Subtract 81 − 80 = 1.
  5. Bring down 6 → 16. 16 × 1 = 16. Write 1.
  6. Subtract 16 − 16 = 0.

Answer: 351.

When there's a remainder, you can express it as "R 3" or write the answer as a decimal.

Worked Examples

Example 1. 432 ÷ 12 = ? Solution. 12 × 36 = 432, so 36.

Example 2. 1,485 ÷ 15 = ? Solution. 15 × 99 = 1,485, so 99.

Example 3. 4,896 ÷ 24 = ? Solution. 24 × 200 = 4,800; remainder 96. 24 × 4 = 96. So 4,896 ÷ 24 = 204.

Example 4. 7,812 ÷ 21 = ? Solution. 21 × 300 = 6,300; remainder 1,512. 21 × 70 = 1,470; remainder 42. 42 ÷ 21 = 2. So 300 + 70 + 2 = 372.

Example 5. A factory packs 1,536 bottles into 12 cases equally. How many per case? Solution. 1,536 ÷ 12 = 128 bottles.

Common Mistakes

  • Mis-estimating the quotient digit (e.g., guessing 5 when 4 is right).
  • Forgetting to bring down a digit, especially when 0 belongs in the quotient.
  • Putting the quotient digit in the wrong place.

Practice

  1. 252 ÷ 14 = ?
  2. 858 ÷ 13 = ?
  3. 4,032 ÷ 18 = ?
  4. 6,720 ÷ 24 = ?
  5. 9,000 ÷ 25 = ?

Answers: 1. 18 | 2. 66 | 3. 224 | 4. 280 | 5. 360

4. Decimal Operations

Key Concepts

Adding/Subtracting decimals. Line up by the decimal point. Add or subtract column by column.

Multiplying decimals. Multiply as if they were whole numbers; place the decimal in the answer so the total decimal places in the answer equals the total in both factors.

  • 0.4 × 0.5 (1 + 1 = 2 decimal places) → 4 × 5 = 20 → 0.20.
  • 1.2 × 0.3 (1 + 1 = 2) → 12 × 3 = 36 → 0.36.

Dividing decimals by whole numbers. Divide normally; place the decimal point in the quotient directly above the decimal point in the dividend.

  • 4.56 ÷ 4 = 1.14.

Worked Examples

Example 1. 3.4 + 5.6 = ? Solution. Line up; 4 + 6 = 10 → carry. 3 + 5 + 1 = 9. Result: 9.0 (or 9).

Example 2. 0.6 × 0.7 = ? Solution. 6 × 7 = 42; total decimal places = 2. Result: 0.42.

Example 3. 2.4 × 5 = ? Solution. 24 × 5 = 120; one decimal place in factor → 12.0 (or 12).

Example 4. 8.4 ÷ 4 = ? Solution. 84 ÷ 4 = 21; place the decimal: 2.1.

Example 5. A pizza costs $7.85. How much for 3 pizzas? Solution. 7.85 × 3 = 23.55. So $23.55.

Common Mistakes

  • Forgetting to place the decimal point at all.
  • Counting decimal places in only one factor when multiplying.
  • Not lining up the decimal point when adding/subtracting (treating 0.5 + 0.05 as 0.10 instead of 0.55).

Practice

  1. 4.7 + 2.85 = ?
  2. 8.6 − 3.45 = ?
  3. 0.5 × 0.4 = ?
  4. 1.2 × 0.6 = ?
  5. 12.6 ÷ 6 = ?

Answers: 1. 7.55 | 2. 5.15 | 3. 0.20 | 4. 0.72 | 5. 2.1

5. Equivalent and Comparing Fractions

Key Concepts

Two fractions are equivalent if they represent the same value:

  • 1/2 = 2/4 = 3/6 = 5/10
  • Multiply or divide both numerator and denominator by the same number.

To compare fractions with different denominators, convert to a common denominator (often the LCM):

  • 2/3 vs 5/8: LCM of 3 and 8 is 24. So 16/24 vs 15/24. 2/3 > 5/8.

A fraction is in simplest form when GCF(numerator, denominator) = 1.

Worked Examples

Example 1. Are 4/6 and 6/9 equivalent? Solution. 4/6 = 2/3 (÷ 2); 6/9 = 2/3 (÷ 3). Yes, equivalent.

Example 2. Compare 5/8 and 7/12. Solution. LCM 24: 15/24 vs 14/24. So 5/8 > 7/12.

Example 3. Simplify 18/24 to lowest terms. Solution. GCF(18, 24) = 6. 18/24 = 3/4.

Example 4. Order from least to greatest: 3/5, 4/9, 1/2. Solution. LCM(5, 9, 2) = 90. So 54/90, 40/90, 45/90. Order: 4/9 < 1/2 < 3/5.

Example 5. Find a fraction equivalent to 5/6 with denominator 30. Solution. 6 × 5 = 30; numerator: 5 × 5 = 25/30.

Common Mistakes

  • Comparing only numerators when denominators differ.
  • Multiplying only the numerator (or only the denominator) to make equivalents.
  • Forgetting to reduce to simplest form when asked.

Practice

  1. Are 9/12 and 6/8 equivalent?
  2. Compare 4/5 and 7/10.
  3. Simplify 28/35.
  4. Order least to greatest: 5/8, 1/2, 3/4.
  5. Find an equivalent of 2/7 with denominator 28.

Answers: 1. Yes (both = 3/4) | 2. 4/5 > 7/10 | 3. 4/5 | 4. 1/2 < 5/8 < 3/4 | 5. 8/28

6. Adding & Subtracting Unlike Denominators

Key Concepts

To add or subtract fractions with different denominators:

  1. Find a common denominator (the LCM is the cleanest choice).
  2. Rewrite both fractions with that denominator.
  3. Add or subtract the numerators.
  4. Simplify if possible.

Worked Examples

Example 1. 1/3 + 1/4 = ? Solution. LCM(3, 4) = 12. 1/3 = 4/12; 1/4 = 3/12. Sum: 4/12 + 3/12 = 7/12.

Example 2. 2/3 + 1/6 = ? Solution. LCM = 6. 2/3 = 4/6. 4/6 + 1/6 = 5/6.

Example 3. 7/8 − 1/2 = ? Solution. LCM = 8. 1/2 = 4/8. 7/8 − 4/8 = 3/8.

Example 4. 5/6 − 1/4 = ? Solution. LCM = 12. 5/6 = 10/12; 1/4 = 3/12. 10/12 − 3/12 = 7/12.

Example 5. A jug had 3/4 L of juice. After pouring out 1/3 L, how much remains? Solution. 3/4 − 1/3. LCM = 12. 9/12 − 4/12 = 5/12 L.

Common Mistakes

  • Adding numerators AND denominators directly (1/2 + 1/3 ≠ 2/5).
  • Picking a common denominator that's too big and making arithmetic ugly.
  • Forgetting to simplify the result.

Practice

  1. 1/2 + 1/5 = ?
  2. 3/4 − 1/3 = ?
  3. 5/8 + 1/4 = ?
  4. 7/10 − 1/5 = ?
  5. 1/3 + 2/9 = ?

Answers: 1. 7/10 | 2. 5/12 | 3. 7/8 | 4. 5/10 (= 1/2) | 5. 5/9

7. Multiplying Fractions

Key Concepts

To multiply two fractions: multiply numerators together and denominators together.

  • a/b × c/d = (a × c) / (b × d).

Examples: 1/2 × 2/3 = 2/6 = 1/3.

To multiply a whole by a fraction, treat the whole as having denominator 1:

  • 4 × 2/3 = 4/1 × 2/3 = 8/3.

You can simplify before multiplying by canceling common factors across diagonals.

Worked Examples

Example 1. 1/2 × 1/3 = ? Solution. (1 × 1) / (2 × 3) = 1/6.

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

Example 3. 5 × 2/7 = ? Solution. 5/1 × 2/7 = 10/7 = 1 3/7.

Example 4. 3/4 × 8 = ? Solution. 3/4 × 8/1 = 24/4 = 6.

Example 5. A recipe needs 3/4 cup of flour. Half-batch needs? Solution. 1/2 × 3/4 = 3/8 cup.

Common Mistakes

  • Adding when you should multiply (1/2 × 1/2 = 1/4, not 2/4).
  • Forgetting to simplify the result.
  • Treating 4 × 2/3 as 4 × 2 / 4 × 3 — the whole only multiplies the numerator.

Practice

  1. 2/5 × 1/4 = ?
  2. 3/8 × 4/9 = ?
  3. 6 × 1/3 = ?
  4. 2/3 × 9 = ?
  5. A pan holds 3/4 L. How much do 2/3 of the pan hold?

Answers: 1. 2/20 (= 1/10) | 2. 12/72 (= 1/6) | 3. 2 | 4. 6 | 5. 1/2 L

8. Dividing Fractions by Whole Numbers

Key Concepts

To divide a fraction by a whole number, multiply by 1 over the whole number (the reciprocal):

  • 1/2 ÷ 3 = 1/2 × 1/3 = 1/6.
  • 4/5 ÷ 2 = 4/5 × 1/2 = 4/10 = 2/5.

Visual: dividing 4/5 into 2 equal parts → each part is 2/5.

Worked Examples

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

Example 2. 4/5 ÷ 4 = ? Solution. 4/5 × 1/4 = 4/20 = 1/5.

Example 3. 2/3 ÷ 3 = ? Solution. 2/3 × 1/3 = 2/9.

Example 4. A 3/4 L jug is shared equally among 3 friends. How much each? Solution. 3/4 ÷ 3 = 3/4 × 1/3 = 3/12 = 1/4 L.

Example 5. 5/6 ÷ 2 = ? Solution. 5/6 × 1/2 = 5/12.

Common Mistakes

  • Dividing just the numerator (1/3 ÷ 2 → "1/2 ÷ 3" — wrong).
  • Forgetting that "÷ 4" means "× 1/4".
  • Mixing up which number is the dividend.

Practice

  1. 1/2 ÷ 4 = ?
  2. 3/5 ÷ 3 = ?
  3. 2/7 ÷ 2 = ?
  4. A 6/7 cup of flour is split equally into 3 small bags. How much per bag?
  5. 4/9 ÷ 8 = ?

Answers: 1. 1/8 | 2. 1/5 (= 3/15) | 3. 1/7 | 4. 2/7 | 5. 1/18 (= 4/72)

9. Mixed Numbers

Key Concepts

A mixed number = whole + fraction. 2 3/4 = 2 + 3/4 = 11/4.

Conversion:

  • Mixed → improper: whole × denominator + numerator → over the same denominator. 2 3/4 → 2 × 4 + 3 = 11/4.
  • Improper → mixed: divide. 17/5 = 3 r 2 → 3 2/5.

To add or subtract mixed numbers:

  • Add wholes and fractions separately, then combine.
  • If fractions add to ≥ 1, regroup into the whole part.
  • If subtracting and the top fraction is smaller, borrow 1 from the whole as a fraction.

Worked Examples

Example 1. Convert 13/4 to a mixed number. Solution. 13 ÷ 4 = 3 r 1 → 3 1/4.

Example 2. Convert 4 2/5 to an improper fraction. Solution. 4 × 5 + 2 = 22/5.

Example 3. 2 1/4 + 3 1/2 = ? Solution. 1/2 = 2/4. Wholes: 2 + 3 = 5. Fractions: 1/4 + 2/4 = 3/4. Total: 5 3/4.

Example 4. 5 1/3 − 2 2/3 = ? Solution. Borrow from whole: 5 1/3 = 4 4/3. 4 4/3 − 2 2/3 = 2 2/3.

Example 5. 1 1/2 + 2 3/4 = ? Solution. 1/2 = 2/4. Wholes: 1 + 2 = 3. Fractions: 2/4 + 3/4 = 5/4 = 1 1/4. Total: 3 + 1 1/4 = 4 1/4.

Common Mistakes

  • Adding only the wholes and forgetting the fractions.
  • Forgetting to borrow when the top fraction is smaller.
  • Multiplying when converting (3 × 5 + 2, not 3 + 5 × 2).

Practice

  1. Convert 19/6 to a mixed number.
  2. Convert 3 5/8 to an improper fraction.
  3. 2 1/3 + 1 1/3 = ?
  4. 4 1/2 − 1 3/4 = ?
  5. 2 3/4 + 1 1/2 = ?

Answers: 1. 3 1/6 | 2. 29/8 | 3. 3 2/3 | 4. 2 3/4 | 5. 4 1/4

10. Volume of Rectangular Prisms

Key Concepts

Volume is the amount of space a 3-D shape occupies, measured in cubic units (cubic cm, cubic in, etc.).

For a rectangular prism (a "box"):

  • V = length × width × height (or V = base area × height).

You can also count unit cubes that fill the shape.

Worked Examples

Example 1. A box is 4 cm long, 3 cm wide, 2 cm tall. Volume? Solution. V = 4 × 3 × 2 = 24 cubic cm.

Example 2. A cube has side 5 cm. Volume? Solution. V = 5 × 5 × 5 = 125 cubic cm.

Example 3. A prism has base area 12 sq cm and height 5 cm. Volume? Solution. V = 12 × 5 = 60 cubic cm.

Example 4. A box of dimensions 6 × 5 × 4 in. Volume? Solution. V = 6 × 5 × 4 = 120 cubic in.

Example 5. A water tank shaped like a box is 10 cm × 8 cm × 6 cm. How many 1 cm³ cubes can fit? Solution. V = 10 × 8 × 6 = 480 unit cubes.

Common Mistakes

  • Confusing volume with surface area (surface area is the total area of all faces).
  • Forgetting the cubic units (volume answers should be in cubic cm, not just cm).
  • Multiplying only two dimensions (you need all three).

Practice

  1. A prism is 5 × 3 × 4. Volume?
  2. A cube has side 6. Volume?
  3. A box has length 8 cm, width 5 cm, height 2 cm. Volume?
  4. A prism has base area 20 sq cm, height 3 cm. Volume?
  5. A box of 2 × 2 × 2 cm. How many 1 cm³ unit cubes fit inside?

Answers: 1. 60 | 2. 216 | 3. 80 | 4. 60 | 5. 8

11. Composite Volumes

Key Concepts

A composite 3-D shape is made of two or more rectangular prisms joined together. To find the total volume:

  • Split the shape into rectangular prism pieces.
  • Compute each volume separately.
  • Add the volumes.

Sometimes you can also subtract: total enclosing box minus a missing piece.

Worked Examples

Example 1. Two boxes glued together: one is 4 × 3 × 2 (volume 24), the other is 2 × 2 × 2 (volume 8). Total volume? Solution. 24 + 8 = 32 cubic units.

Example 2. An L-shaped solid splits into a 5 × 4 × 3 piece and a 2 × 3 × 3 piece. Total? Solution. 60 + 18 = 78 cubic units.

Example 3. A 6 × 5 × 4 box has a 2 × 2 × 2 cube cut out of one corner. Remaining volume? Solution. 120 − 8 = 112 cubic units.

Example 4. A T-shape: top piece is 6 × 2 × 2 (volume 24); base is 4 × 4 × 2 (volume 32). Total? Solution. 24 + 32 = 56 cubic units.

Example 5. A box 8 × 6 × 4 with a hollow channel of 2 × 2 × 4 going through it. Solid volume? Solution. 192 − 16 = 176 cubic units.

Common Mistakes

  • Double-counting overlapping parts.
  • Splitting incorrectly so pieces overlap.
  • Forgetting to subtract a hollow region.

Practice

  1. Two prisms: 5 × 4 × 3 and 2 × 2 × 3. Total volume?
  2. An L-shape: 6 × 4 × 2 piece + 4 × 2 × 2 piece. Total?
  3. A 5 × 4 × 3 box with a 2 × 2 × 2 cube cut out. Remaining volume?
  4. T-shape: top 5 × 2 × 2 + base 6 × 6 × 2. Total?
  5. A 10 × 5 × 4 box with a 4 × 5 × 1 slab removed. Remaining?

Answers: 1. 60 + 12 = 72 | 2. 48 + 16 = 64 | 3. 60 − 8 = 52 | 4. 20 + 72 = 92 | 5. 200 − 20 = 180

12. Coordinate Plane (First Quadrant)

Key Concepts

The coordinate plane has two perpendicular axes:

  • x-axis — horizontal.
  • y-axis — vertical.

A point is given by an ordered pair (x, y):

  • x = how far right of the origin (0, 0).
  • y = how far up.

Grade 5 focuses on the first quadrant (x ≥ 0, y ≥ 0).

To plot (4, 3), move 4 right and 3 up from the origin.

Worked Examples

Example 1. What are the coordinates of a point 5 units right and 2 units up from the origin? Solution. (5, 2).

Example 2. Plot (3, 4). Where does it land? Solution. 3 right, 4 up from origin. The point sits in the first quadrant.

Example 3. What is the y-coordinate of (7, 0)? Solution. 0 (the point is on the x-axis).

Example 4. Two points: A(2, 5) and B(2, 8). What is the distance between them? Solution. Same x; y differs by 8 − 5 = 3 units.

Example 5. A treasure map says "go 4 right, then 3 up". The starting point is (0, 0). Where is the treasure? Solution. (4, 3).

Common Mistakes

  • Mixing up x and y (always: x first, y second).
  • Counting from the wrong axis (going up first when y comes second in the pair).
  • Plotting (3, 4) at (4, 3) by accident.

Practice

  1. Where is (6, 2)?
  2. Plot (0, 5). Where does it land?
  3. The x-coordinate of (8, 4) is what?
  4. Distance between (3, 1) and (7, 1)?
  5. Distance between (4, 2) and (4, 9)?

Answers: 1. 6 right, 2 up | 2. On the y-axis at height 5 | 3. 8 | 4. 4 | 5. 7

13. Distance and Polygons on a Coordinate Grid

Key Concepts

Horizontal distance between two points with the same y-coordinate = difference of x-coordinates. Vertical distance between two points with the same x-coordinate = difference of y-coordinates.

A polygon can be drawn by plotting its vertices and connecting them. To find the perimeter of a rectangle on the grid: add the side lengths.

For a rectangle with vertices at (1, 1), (5, 1), (5, 4), (1, 4):

  • Width = 5 − 1 = 4 (horizontal).
  • Height = 4 − 1 = 3 (vertical).
  • Perimeter = 2(4 + 3) = 14. Area = 4 × 3 = 12.

Worked Examples

Example 1. A rectangle has vertices (2, 1), (8, 1), (8, 5), (2, 5). What is its area? Solution. Width 6, height 4. Area = 24.

Example 2. Distance from (1, 3) to (1, 9)? Solution. Vertical: 9 − 3 = 6.

Example 3. Perimeter of a rectangle with corners (0, 0), (5, 0), (5, 3), (0, 3)? Solution. Width 5, height 3. P = 2(5 + 3) = 16.

Example 4. A square has corners (1, 1) and (4, 4) as opposite vertices. Side length? Solution. Width = 4 − 1 = 3. Side = 3.

Example 5. Plot a triangle with vertices (1, 1), (5, 1), (1, 4). Length of horizontal side? Solution. From (1, 1) to (5, 1): 4 units.

Common Mistakes

  • Subtracting in the wrong order (e.g., 1 − 5 instead of 5 − 1).
  • Confusing the two axes when computing distance.
  • Counting tick marks twice when they overlap with vertices.

Practice

  1. Distance from (3, 2) to (3, 8)?
  2. Rectangle with corners (0, 0), (6, 0), (6, 4), (0, 4). Perimeter? Area?
  3. Distance from (4, 5) to (10, 5)?
  4. A square of side 5 has one corner at (2, 1). What are the other corners (going right then up)?
  5. A rectangle has corners (1, 2), (1, 6), (5, 6), (5, 2). What is its area?

Answers: 1. 6 | 2. P = 20, A = 24 | 3. 6 | 4. (7, 1), (7, 6), (2, 6) | 5. 16

14. Classifying 2D Figures

Key Concepts

Polygons are categorized by sides and properties.

Polygon Sides Properties
Triangle 3 sum of angles = 180°
Quadrilateral 4 sum of angles = 360°
Pentagon 5
Hexagon 6
Octagon 8

Special quadrilaterals:

  • Square — 4 equal sides AND 4 right angles.
  • Rectangle — 4 right angles (opposite sides equal).
  • Rhombus — 4 equal sides (angles can be any).
  • Parallelogram — opposite sides parallel and equal.
  • Trapezoid — exactly one pair of parallel sides.

A square is a special rectangle, rhombus, and parallelogram. A rectangle and rhombus are both special parallelograms.

Worked Examples

Example 1. A 4-sided shape with 4 right angles but only opposite sides equal. What is it? Solution. Rectangle (it isn't a square because not all sides equal).

Example 2. Is a square always a rhombus? Solution. Yes — a square has 4 equal sides, satisfying the rhombus property.

Example 3. A polygon has 6 sides. Name? Solution. Hexagon.

Example 4. What is the sum of interior angles of any quadrilateral? Solution. 360°.

Example 5. A rhombus has angles 60°, 120°, 60°, 120°. Are opposite angles equal? Solution. Yes — opposite angles in a rhombus (or any parallelogram) are equal.

Common Mistakes

  • Calling all 4-sided shapes "rectangles".
  • Forgetting that a square is also a rectangle and rhombus.
  • Saying a trapezoid has "no parallel sides" — it has exactly one pair.

Practice

  1. A polygon with 5 sides is a what?
  2. Is every rhombus a parallelogram?
  3. The sum of interior angles of a triangle is what?
  4. A 4-sided shape with one pair of parallel sides is called a what?
  5. Is every rectangle a square?

Answers: 1. Pentagon | 2. Yes | 3. 180° | 4. Trapezoid | 5. No

15. Measurement Conversion

Key Concepts

You should comfortably convert within a system. Common conversions:

Metric:

  • 1 km = 1,000 m
  • 1 m = 100 cm = 1,000 mm
  • 1 kg = 1,000 g
  • 1 L = 1,000 mL

Customary (US):

  • 1 ft = 12 in
  • 1 yd = 3 ft = 36 in
  • 1 mi = 5,280 ft
  • 1 lb = 16 oz
  • 1 gal = 4 qt = 8 pt = 16 cups

To go from a larger unit to a smaller unit, multiply. To go from smaller to larger, divide.

Worked Examples

Example 1. How many m in 5 km? Solution. 5 × 1,000 = 5,000 m.

Example 2. How many g in 0.75 kg? Solution. 0.75 × 1,000 = 750 g.

Example 3. How many ft in 6 yd? Solution. 6 × 3 = 18 ft.

Example 4. How many oz in 3 lb? Solution. 3 × 16 = 48 oz.

Example 5. Convert 4,500 mL to L. Solution. 4,500 ÷ 1,000 = 4.5 L.

Common Mistakes

  • Using the wrong factor (1 km = 1,000 m, not 100 m).
  • Multiplying when you should divide (or vice versa).
  • Mixing metric and customary in the same problem.

Practice

  1. How many cm in 7.5 m?
  2. How many ft in 24 in?
  3. How many mL in 3.5 L?
  4. How many kg in 2,500 g?
  5. How many qt in 2 gallons?

Answers: 1. 750 | 2. 2 | 3. 3,500 | 4. 2.5 | 5. 8

16. Numerical Patterns & Input-Output Tables

Key Concepts

A numerical pattern has a rule that produces each next term:

  • Arithmetic — add the same constant. (5, 10, 15, 20 → +5)
  • Geometric — multiply by the same factor. (1, 2, 4, 8 → ×2)
  • Two-rule patterns — combine two rules (e.g., +2 then ×3).

An input-output table lists pairs (input, output) where each output is computed from the input by a rule:

Input Output
1 5
2 9
3 13
4 17

The rule here is Output = 4 × Input + 1.

Worked Examples

Example 1. Find the rule: input 2 → output 6; input 3 → output 9; input 4 → output 12. Solution. Output = 3 × Input.

Example 2. Continue: 7, 14, 21, 28, ___, ___. Solution. +7 each time → 35, 42.

Example 3. Find the missing term: 4, 12, 36, ___, 324. Solution. ×3 each step → 108.

Example 4. Find the rule: input 1 → 4; input 2 → 7; input 3 → 10. Solution. Each step input +1 makes output +3. Output = 3 × Input + 1.

Example 5. Two patterns: A: +2 starting at 0 (0, 2, 4, …) and B: +3 starting at 0 (0, 3, 6, …). Compare 5th terms. Solution. A's 5th term: 0 + 4 × 2 = 8. B's 5th term: 0 + 4 × 3 = 12. B is 4 more.

Common Mistakes

  • Stopping after one row to "find a rule" — verify with at least 2-3 rows.
  • Mixing up arithmetic (+) and geometric (×) when the pattern doubles or triples.
  • Forgetting to include the starting term when counting position.

Practice

  1. Continue: 8, 16, 24, 32, ___, ___.
  2. Find the rule: input 1 → 5; input 2 → 8; input 3 → 11.
  3. Continue: 3, 9, 27, 81, ___.
  4. Find the missing: 5, ___, 15, 20, 25.
  5. Two patterns: P: +4 (0, 4, 8, 12, …) and Q: +5 (0, 5, 10, 15, …). Compare the 6th terms.

Answers: 1. 40, 48 | 2. Output = 3 × Input + 2 | 3. 243 | 4. 10 | 5. P's 6th = 20, Q's 6th = 25; Q is 5 more

Final Tips Before the Test

  • Take a deep breath and read the question. Many traps come from rushing.
  • Estimate first. If 348 × 16 should be near 5,500, an answer of 558 means you missed a step.
  • Show your work for long division and decimal multiplication — alignment matters.
  • Check your units. Volume = cubic; area = square; perimeter = regular.
  • Re-read the question to make sure you answered exactly what was asked.
  • Watch the form. "In simplest form" means simplify; "as a mixed number" means convert; "in cubic centimeters" means cm³.
  • Skip and return if a question is hard. Don't burn time on one tough problem.
  • Pace yourself. Roughly 1 minute per question is a healthy rate.

You've got this! 🚀