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Grade 3 Prep Guide

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Adaptive Math Test — Grade 3 Prep Guide

A comprehensive, topic-by-topic study guide for the Grade 3 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

Multiplication and Division

  1. Multiplication within 100
  2. Division within 100
  3. Arrays
  4. Word Problems

Fractions 5. Unit Fractions 6. Fractions on a Number Line 7. Comparing Fractions

Place Value 8. Rounding 9. Multi-Digit Addition 10. Multi-Digit Subtraction

Geometry 11. Area 12. Perimeter 13. Perimeter and Area 14. Quadrilaterals 15. Partitioning Shapes

Measurement 16. Liquid Volume 17. Mass 18. Elapsed Time

Data 19. Bar Graphs 20. Pictographs 21. Scaled Graphs

1. Multiplication within 100

Key Concepts

Multiplication is repeated addition. 4 × 6 means "four groups of six" or "six added together four times" = 6 + 6 + 6 + 6 = 24.

Useful properties:

  • Commutative. 7 × 3 = 3 × 7 (the order doesn't matter).
  • Associative. (2 × 3) × 4 = 2 × (3 × 4).
  • Distributive. 6 × 12 = 6 × 10 + 6 × 2 = 72.
  • Identity. Any number × 1 = itself.
  • Zero. Any number × 0 = 0.

Master your multiplication facts up to 10 × 10 — this is the backbone of Grade 3.

Worked Examples

Example 1. What is 8 × 7? Solution. From the times table: 56. (Tip: 8 × 7 is the same as 7 × 8.)

Example 2. What is 9 × 6? Solution. Use distributive: 10 × 6 − 1 × 6 = 60 − 6 = 54.

Example 3. What is 12 × 4? Solution. Distributive: 10 × 4 + 2 × 4 = 40 + 8 = 48.

Example 4. What is 5 × 7 × 2? Solution. Use associative: (5 × 2) × 7 = 10 × 7 = 70.

Example 5. A shelf has 6 rows of 8 books. Total books? Solution. 6 × 8 = 48 books.

Common Mistakes

  • Confusing the times tables (e.g., 7 × 8 = 56, not 54).
  • Forgetting that × 1 leaves a number unchanged and × 0 makes it 0.
  • Adding when the problem asks to multiply ("groups of" = multiply, not add).
  • Slowing down by counting on fingers when the fact is memorizable.

Practice

  1. 8 × 6 = ?
  2. 9 × 7 = ?
  3. 11 × 4 = ?
  4. 5 × 5 = ?
  5. A box has 9 rows of 8 cookies. How many cookies in total?

Answers: 1. 48 | 2. 63 | 3. 44 | 4. 25 | 5. 72

2. Division within 100

Key Concepts

Division splits into equal groups. 24 ÷ 4 means "24 split into 4 equal groups" → each group has 6.

Division is the inverse of multiplication:

  • If 6 × 4 = 24, then 24 ÷ 4 = 6 and 24 ÷ 6 = 4.
  • To divide, ask: "What number times the divisor gives the dividend?"

Sometimes division has a remainder when it doesn't divide evenly: 17 ÷ 5 = 3 remainder 2 (because 5 × 3 = 15, leaving 2 left).

Worked Examples

Example 1. What is 56 ÷ 7? Solution. 7 × ? = 56. From facts: 7 × 8 = 56, so 56 ÷ 7 = 8.

Example 2. What is 36 ÷ 6? Solution. 6 × 6 = 36, so 36 ÷ 6 = 6.

Example 3. What is 40 ÷ 8? Solution. 8 × 5 = 40, so 40 ÷ 8 = 5.

Example 4. 23 ÷ 4 = ? Solution. 4 × 5 = 20, leaving 3 left over. So 5 remainder 3.

Example 5. A box of 84 markers is shared equally among 6 students. How many does each get? Solution. 84 ÷ 6 = 14 markers each.

Common Mistakes

  • Mixing up dividend and divisor (24 ÷ 4 ≠ 4 ÷ 24).
  • Forgetting that any number ÷ 1 = itself, and any number ÷ itself = 1.
  • Trying to "divide by zero" — undefined; impossible.
  • Ignoring the remainder when it matters.

Practice

  1. 63 ÷ 9 = ?
  2. 28 ÷ 4 = ?
  3. 50 ÷ 5 = ?
  4. 25 ÷ 4 = ? (with remainder)
  5. 72 marbles are shared equally among 8 children. How many marbles each?

Answers: 1. 7 | 2. 7 | 3. 10 | 4. 6 r 1 | 5. 9

3. Arrays

Key Concepts

An array is a rectangular arrangement of objects in equal rows and columns. Arrays are the visual model for multiplication: rows × columns = total.

Arrays show multiplication is commutative — 3 rows of 4 (3 × 4) has the same total as 4 rows of 3 (4 × 3).

If you know the total and one dimension, divide to find the other. 36 chairs in 4 equal rows → 36 ÷ 4 = 9 chairs per row.

Worked Examples

Example 1. An array has 4 rows and 5 columns. How many objects? Solution. 4 × 5 = 20.

Example 2. A box has 6 rows of 7 chocolates. How many in total? Solution. 6 × 7 = 42.

Example 3. 32 students stand in 4 equal rows. How many in each row? Solution. 32 ÷ 4 = 8 per row.

Example 4. Which is larger: a 3 × 9 array or a 5 × 6 array? Solution. 3 × 9 = 27; 5 × 6 = 30. 5 × 6 is larger by 3.

Example 5. An array of 48 has rows of 8. How many rows? Solution. 48 ÷ 8 = 6 rows.

Common Mistakes

  • Switching rows and columns when reading a problem.
  • Counting only one row instead of multiplying.
  • Forgetting that an array's total stays the same when rows and columns swap.

Practice

  1. An array has 7 rows of 6 objects. Total?
  2. 45 stickers in 5 equal rows. How many per row?
  3. An array of 24 has 8 rows. How many columns?
  4. Which is bigger: 4 × 7 or 5 × 6?
  5. A garden bed has 9 rows of 9 plants. How many plants in total?

Answers: 1. 42 | 2. 9 | 3. 3 | 4. 5 × 6 = 30 (larger than 28) | 5. 81

4. Word Problems

Key Concepts

Reading a word problem well is half the work. Look for:

  • Total / sum / altogether → addition
  • Difference / how many more / fewer / left → subtraction
  • Groups of / each / per / times as many → multiplication
  • Share / split / divide equally → division

A useful method: CUBES:

  • Circle the numbers
  • Underline the question
  • Box the keywords
  • Eliminate extra information
  • Show the work

Worked Examples

Example 1. A farmer has 4 baskets with 9 apples in each. Total apples? Solution. "Each" → multiply. 4 × 9 = 36 apples.

Example 2. Lucy had 75 stickers. She gave away 28. How many remain? Solution. "Gave away / remain" → subtract. 75 − 28 = 47.

Example 3. 56 cookies are shared equally among 7 friends. How many each? Solution. "Equally" → divide. 56 ÷ 7 = 8 each.

Example 4. A class has 8 boys and twice as many girls. How many students? Solution. Girls: 8 × 2 = 16. Total: 8 + 16 = 24.

Example 5. Jose buys 5 packs of 8 pens. He gives 12 away. How many pens does he keep? Solution. Bought: 5 × 8 = 40. Kept: 40 − 12 = 28.

Common Mistakes

  • Picking up numbers without reading the operation.
  • Adding when the problem says "each" (which suggests multiplication).
  • Forgetting to do the second step in multi-step problems.
  • Not checking if the answer makes sense ("Each child got 80 candies" — too many?).

Practice

  1. A teacher has 6 boxes of 12 crayons. Total crayons?
  2. A book has 96 pages. Tom read 38. How many pages remain?
  3. 64 students split into 8 equal teams. How many per team?
  4. A jar has 5 red marbles and three times as many blue marbles. Total marbles?
  5. Maya buys 4 packs of 7 stickers, then loses 9. How many stickers does she have?

Answers: 1. 72 | 2. 58 | 3. 8 | 4. 20 | 5. 19

5. Unit Fractions

Key Concepts

A unit fraction has a numerator of 1. Examples: 1/2, 1/3, 1/4, 1/5, …

Unit fractions represent one piece of a whole that has been split into equal parts. The bigger the denominator, the smaller each piece (because the whole is split into more parts).

So 1/8 < 1/4 < 1/2.

Sums of unit fractions can equal a whole: 1/2 + 1/2 = 1 (whole), 1/4 + 1/4 + 1/4 + 1/4 = 1.

Worked Examples

Example 1. Which is greater: 1/3 or 1/5? Solution. Bigger denominator = smaller piece. 1/3 > 1/5. So 1/3 is greater.

Example 2. A pizza is cut into 6 equal slices. What fraction is one slice? Solution. 1/6 (a unit fraction).

Example 3. Which of these is a unit fraction? 2/3, 1/8, 3/4, 5/6. Solution. 1/8 (numerator = 1).

Example 4. Order from smallest to largest: 1/2, 1/8, 1/4. Solution. Larger denominator → smaller piece: 1/8 < 1/4 < 1/2.

Example 5. How many one-thirds make a whole? Solution. 1/3 + 1/3 + 1/3 = 1. So 3 thirds make a whole.

Common Mistakes

  • Thinking 1/10 is bigger than 1/2 because 10 > 2.
  • Confusing "1/3" with "3" or "one-third" with "thirty-three".
  • Calling 2/3 a unit fraction (numerator must be exactly 1).

Practice

  1. Which is bigger: 1/4 or 1/9?
  2. Of these, which is a unit fraction? 3/5, 1/7, 2/4, 5/6.
  3. A cake is cut into 8 equal pieces. What fraction is one piece?
  4. Order from largest to smallest: 1/3, 1/2, 1/6.
  5. How many fifths make a whole?

Answers: 1. 1/4 | 2. 1/7 | 3. 1/8 | 4. 1/2 > 1/3 > 1/6 | 5. 5

6. Fractions on a Number Line

Key Concepts

A number line from 0 to 1 can be divided into equal parts. Each tick mark shows a fraction.

If the line has 4 equal parts, each part is 1/4, and the tick marks are at 0, 1/4, 2/4, 3/4, 1.

To find where a fraction sits, count the marks from 0:

  • 2/4 of the way from 0 to 1 is the 2nd tick (out of 4).
  • 3/8 of the way is the 3rd tick (out of 8).

A number line from 0 to 2 with 8 equal parts has marks every 1/4 (since 2 ÷ 8 = 1/4).

Worked Examples

Example 1. A number line from 0 to 1 is divided into 4 parts. What is the 3rd mark? Solution. 3/4.

Example 2. A line from 0 to 1 with 6 parts. What is the 5th mark? Solution. 5/6.

Example 3. Where on a 0-to-1 line is 1/2? Solution. Exactly halfway → the midpoint.

Example 4. A number line from 0 to 2 with 4 equal parts. What is at the 1st mark? Solution. Each section is 2 ÷ 4 = 1/2. The 1st mark is at 1/2.

Example 5. Which is closer to 1 on the number line: 5/8 or 7/8? Solution. 7/8 is closer (smaller gap to 1: 1 − 7/8 = 1/8; vs. 1 − 5/8 = 3/8).

Common Mistakes

  • Counting the 0 mark as the first instead of starting count at the first tick after 0.
  • Reading 3/4 as the third tick from the right (instead of from the left).
  • Forgetting to divide the total range by the number of parts when the line goes beyond 1.

Practice

  1. A 0-to-1 line is in 8 equal parts. What is at the 5th mark?
  2. A 0-to-1 line is in 3 equal parts. What is at the 2nd mark?
  3. Where on a 0-to-1 line is the midpoint?
  4. A 0-to-2 line has 4 equal parts. What is at the 3rd mark?
  5. Which is greater: 4/5 or 5/6?

Answers: 1. 5/8 | 2. 2/3 | 3. 1/2 | 4. 3/2 (or 1 1/2) | 5. 5/6 (since 24/30 < 25/30)

7. Comparing Fractions

Key Concepts

To compare two fractions:

  • Same denominator? Bigger numerator = bigger fraction. (3/8 vs. 5/8 → 5/8 is bigger.)
  • Same numerator? Bigger denominator = SMALLER fraction. (1/3 vs. 1/5 → 1/3 is bigger.)
  • Different both? Use a common denominator (find a number both denominators divide into).

Helpful benchmarks: 1/2 (halfway) and 1 (the whole).

Worked Examples

Example 1. Compare 3/7 and 5/7. Solution. Same denominator. 5 > 3, so 5/7 > 3/7.

Example 2. Compare 1/4 and 1/6. Solution. Same numerator (1). Bigger denominator means smaller piece. 1/4 > 1/6.

Example 3. Compare 2/3 and 3/5. Solution. Common denominator 15: 2/3 = 10/15; 3/5 = 9/15. 2/3 > 3/5.

Example 4. Order least to greatest: 3/4, 2/3, 1/2. Solution. Common denominator 12: 9/12, 8/12, 6/12. So 1/2 < 2/3 < 3/4.

Example 5. Is 7/10 greater or less than 1/2? Solution. 1/2 = 5/10. 7/10 > 5/10, so 7/10 > 1/2.

Common Mistakes

  • Comparing only numerators when denominators are different.
  • Assuming bigger denominators always mean bigger fractions.
  • Forgetting to use the same "size" when comparing (use a common denominator).

Practice

  1. Compare 4/9 and 5/9.
  2. Compare 1/5 and 1/8.
  3. Compare 3/4 and 5/8.
  4. Order least to greatest: 1/2, 1/3, 2/3.
  5. Is 4/10 greater or less than 1/2?

Answers: 1. 5/9 > 4/9 | 2. 1/5 > 1/8 | 3. 3/4 > 5/8 | 4. 1/3 < 1/2 < 2/3 | 5. less than (4/10 < 5/10)

8. Rounding

Key Concepts

Rounding simplifies a number to a friendlier value. The rule:

  • Find the place you're rounding to.
  • Look at the digit just to its right.
    • If it's 5 or more, round up (add 1 to the rounding place).
    • If it's less than 5, round down (keep the rounding place the same).
  • Replace all digits to the right with zeros.

Worked Examples

Example 1. Round 47 to the nearest 10. Solution. Tens digit is 4; ones is 7 (≥5), round up → 50.

Example 2. Round 234 to the nearest 100. Solution. Hundreds digit is 2; tens is 3 (<5), round down → 200.

Example 3. Round 2,765 to the nearest 1,000. Solution. Thousands digit is 2; hundreds is 7 (≥5), round up → 3,000.

Example 4. Round 845 to the nearest 100. Solution. Hundreds is 8; tens is 4 (<5), round down → 800.

Example 5. Round 95 to the nearest 10. Solution. Ones is 5, round up → 100. Yes, rounding up across a place boundary is normal.

Common Mistakes

  • Looking at the wrong digit (e.g., looking at the tens when rounding to the nearest 100 — should look at tens? Wait — you should: the digit one place to the right. So when rounding to nearest 100, look at the tens digit.)
  • Forgetting to replace digits after the rounding spot with 0s.
  • Always rounding 5 down — by convention 5 rounds up.

Practice

  1. Round 78 to the nearest 10.
  2. Round 432 to the nearest 100.
  3. Round 1,256 to the nearest 1,000.
  4. Round 89 to the nearest 10.
  5. Round 6,549 to the nearest 100.

Answers: 1. 80 | 2. 400 | 3. 1,000 | 4. 90 | 5. 6,500

9. Multi-Digit Addition

Key Concepts

For Grade 3, multi-digit addition usually means 3-digit + 3-digit (and sometimes 4-digit). Use the standard algorithm:

  1. Line up the digits by place value (ones under ones, tens under tens, …).
  2. Add column by column from right to left.
  3. Carry when a column sums to 10 or more.

You can also use mental math: add by place, or round-and-adjust.

Worked Examples

Example 1. 248 + 365 = ? Solution. Ones: 8 + 5 = 13 (write 3, carry 1). Tens: 4 + 6 + 1 = 11 (write 1, carry 1). Hundreds: 2 + 3 + 1 = 6. Total: 613.

Example 2. 472 + 188 = ? Solution. By place: 400 + 100 = 500; 70 + 80 = 150; 2 + 8 = 10. Total: 500 + 150 + 10 = 660.

Example 3. 1,234 + 567 = ? Solution. Standard: 1,234 + 500 = 1,734; 1,734 + 67 = 1,801.

Example 4. 549 + 273 = ? Solution. 549 + 300 − 27 = 849 − 27 = 822.

Example 5. A school bought 467 pencils on Monday and 285 on Tuesday. How many in total? Solution. 467 + 285 = 752.

Common Mistakes

  • Mis-aligning columns (writing 84 + 245 lined up with the 8 under the 4).
  • Forgetting cascading carries (e.g., 999 + 1 = 1000).
  • Skipping a column.

Practice

  1. 367 + 158 = ?
  2. 489 + 273 = ?
  3. 612 + 388 = ?
  4. 1,256 + 374 = ?
  5. A library has 478 fiction and 365 non-fiction books. How many books in total?

Answers: 1. 525 | 2. 762 | 3. 1,000 | 4. 1,630 | 5. 843

10. Multi-Digit Subtraction

Key Concepts

For 3-digit subtraction, line up by place value and subtract from right to left. Regroup (borrow) from the next-larger place when a column has a smaller top digit.

If the top number has a 0 where you need to borrow, the borrow cascades: 700 → 6 hundreds, 9 tens, 10 ones.

Worked Examples

Example 1. 624 − 287 = ? Solution. Ones: 4 − 7 → borrow: 14 − 7 = 7. Tens: 1 − 8 → borrow: 11 − 8 = 3 (hundreds drop to 5). Hundreds: 5 − 2 = 3. Total: 337.

Example 2. 800 − 459 = ? Solution. 800 → 7 hundreds, 9 tens, 10 ones. Ones: 10 − 9 = 1. Tens: 9 − 5 = 4. Hundreds: 7 − 4 = 3. Total: 341.

Example 3. 502 − 247 = ? Solution. 502 → 4 hundreds, 9 tens, 12 ones. Ones: 12 − 7 = 5. Tens: 9 − 4 = 5. Hundreds: 4 − 2 = 2. Total: 255.

Example 4. 1,000 − 478 = ? Solution. Count up: 478 → 500 = 22; 500 → 1,000 = 500. Total: 522.

Example 5. A factory had 945 boxes; 367 were shipped. How many remain? Solution. 945 − 367 = 578.

Common Mistakes

  • Borrowing without dropping the top digit by 1.
  • Trying to subtract upward (5 − 8 = "3" instead of borrowing).
  • Cascading borrows incorrectly across zeros.
  • Forgetting that subtraction is not commutative — 800 − 200 ≠ 200 − 800.

Practice

  1. 524 − 168 = ?
  2. 700 − 283 = ?
  3. 1,000 − 367 = ?
  4. 905 − 478 = ?
  5. A shop had 832 cans; 245 were sold. How many remain?

Answers: 1. 356 | 2. 417 | 3. 633 | 4. 427 | 5. 587

11. Area

Key Concepts

Area is the amount of space inside a shape. It's measured in square units: square cm, square in, etc.

For a rectangle (or square): Area = length × width.

  • A 4 × 3 rectangle has area 12 sq units.
  • A square with side 5 has area 5 × 5 = 25 sq units.

You can also count the unit squares inside a shape on grid paper.

Worked Examples

Example 1. A rectangle is 6 cm long and 4 cm wide. What is its area? Solution. 6 × 4 = 24 sq cm.

Example 2. A square has side 7 cm. What is its area? Solution. 7 × 7 = 49 sq cm.

Example 3. A rectangle has area 30 sq cm and length 6 cm. What is the width? Solution. 30 ÷ 6 = 5 cm.

Example 4. A square has area 36 sq cm. What is the side length? Solution. Side × side = 36, and 6 × 6 = 36. Side = 6 cm.

Example 5. A rectangle is 8 cm by 9 cm. Find its area. Solution. 8 × 9 = 72 sq cm.

Common Mistakes

  • Confusing area with perimeter (perimeter is the outline distance).
  • Forgetting to use square units (area is always squared).
  • Multiplying length + width instead of length × width.
  • Mixing up the formulas for square (side²) and rectangle (length × width).

Practice

  1. A 5 × 8 rectangle has area = ?
  2. A square with side 9 has area = ?
  3. A rectangle has area 42 sq m, width 6 m. Length = ?
  4. A 12 × 4 floor has area = ?
  5. A square has area 81 sq cm. Side = ?

Answers: 1. 40 sq | 2. 81 sq | 3. 7 m | 4. 48 sq | 5. 9 cm

12. Perimeter

Key Concepts

Perimeter is the distance around a shape — the sum of all sides.

  • Rectangle: P = 2(length + width).
  • Square: P = 4 × side.
  • Triangle: P = side1 + side2 + side3.
  • Any polygon: just add all the side lengths.

Perimeter is in regular length units (cm, m, in), not squared.

Worked Examples

Example 1. A rectangle is 7 cm by 3 cm. What is its perimeter? Solution. 2(7 + 3) = 2 × 10 = 20 cm.

Example 2. A square has side 5 cm. What is its perimeter? Solution. 4 × 5 = 20 cm.

Example 3. A triangle has sides 4, 6, 8 cm. What is its perimeter? Solution. 4 + 6 + 8 = 18 cm.

Example 4. A regular pentagon has 5 equal sides of 6 cm. Perimeter? Solution. 5 × 6 = 30 cm.

Example 5. A rectangle has perimeter 24 cm and width 4 cm. What is its length? Solution. 2(L + 4) = 24 → L + 4 = 12 → L = 8 cm.

Common Mistakes

  • Confusing perimeter with area.
  • Adding only two sides of a rectangle (forgetting that you have two of each).
  • Using square units for perimeter (perimeter is in regular length units).

Practice

  1. A rectangle 8 × 5. Perimeter?
  2. A square with side 10. Perimeter?
  3. A triangle with sides 5, 7, 9. Perimeter?
  4. A regular hexagon with side 4. Perimeter?
  5. A rectangle has perimeter 30 cm and length 9 cm. Width?

Answers: 1. 26 | 2. 40 | 3. 21 | 4. 24 | 5. 6 cm

13. Perimeter and Area

Key Concepts

These two ideas often appear in the same problem. Don't confuse them:

  • Perimeter — distance around (regular units).
  • Area — space inside (square units).

A common type of problem: given one (e.g., perimeter), find the other.

  • Two shapes can have the same perimeter but different areas.
  • Two shapes can have the same area but different perimeters.

Worked Examples

Example 1. A square has perimeter 20 cm. What is its area? Solution. Side = 20 ÷ 4 = 5; area = 5 × 5 = 25 sq cm.

Example 2. A square has area 36 sq cm. What is its perimeter? Solution. Side = 6 cm; perimeter = 4 × 6 = 24 cm.

Example 3. A rectangle has length 8 and area 24. Find its perimeter. Solution. Width = 24 ÷ 8 = 3. Perimeter = 2(8 + 3) = 22.

Example 4. Two rectangles have perimeter 12 cm. One is 4 × 2 and one is 5 × 1. Which has the bigger area? Solution. 4 × 2 = 8; 5 × 1 = 5. The 4 × 2 has the bigger area (8 sq cm).

Example 5. A path is laid around a 10 × 6 rectangle. How long is the path? Solution. Perimeter = 2(10 + 6) = 32 m of path.

Common Mistakes

  • Reading the problem as asking for one when it's asking for the other.
  • Using P = L + W (instead of 2(L + W)).
  • Forgetting to convert units consistently.

Practice

  1. A square with perimeter 24 — area?
  2. A rectangle with area 40 and length 10 — perimeter?
  3. A square with area 49 — perimeter?
  4. A rectangle 6 × 4: perimeter and area both?
  5. Two rectangles with the same perimeter: 4 × 5 and 3 × 6. Which has bigger area?

Answers: 1. 36 sq | 2. 28 | 3. 28 | 4. P = 20, A = 24 | 5. 3 × 6 (= 18) > 4 × 5 (= 20)? Wait: 4 × 5 = 20 and 3 × 6 = 18. So 4 × 5 has the bigger area.

14. Quadrilaterals

Key Concepts

A quadrilateral is a 4-sided polygon. Common types:

  • Square — 4 equal sides, 4 right angles.
  • Rectangle — 4 right angles, opposite sides equal.
  • Rhombus — 4 equal sides (angles do not have to be right).
  • Parallelogram — opposite sides parallel and equal.
  • Trapezoid — exactly one pair of parallel sides.

Every square is also a rectangle (because it has 4 right angles). Every square is also a rhombus (because it has 4 equal sides).

Worked Examples

Example 1. A polygon with 4 vertices is called a what? Solution. A quadrilateral (any 4-sided polygon — square, rectangle, trapezoid, etc.).

Example 2. A square with 4 sides of 6 cm has what perimeter? Solution. 4 × 6 = 24 cm.

Example 3. A shape has 4 right angles and opposite sides equal but not all equal. What is it? Solution. A rectangle (it's not a square because not all sides equal).

Example 4. How many right angles does a square have? Solution. 4 right angles.

Example 5. Is a square a rectangle? Solution. Yes — a square is a special rectangle (with all sides equal).

Common Mistakes

  • Calling any 4-sided shape a "rectangle" — only those with right angles count.
  • Forgetting trapezoids exist (they're often missed).
  • Confusing rhombus and rectangle (rhombus has equal sides but not right angles by default).

Practice

  1. How many sides does any quadrilateral have?
  2. A 5-by-5 square has perimeter = ?
  3. A shape with 4 equal sides but no right angles is called a __?
  4. Is every rectangle a square? (Yes/No)
  5. How many vertices does a trapezoid have?

Answers: 1. 4 | 2. 20 | 3. rhombus | 4. No | 5. 4

15. Partitioning Shapes

Key Concepts

To partition is to cut into equal pieces. Each piece becomes a fraction of the whole:

  • 2 equal parts → halves (1/2 each)
  • 3 equal parts → thirds (1/3 each)
  • 4 equal parts → fourths (1/4 each)
  • 6 equal parts → sixths (1/6 each)
  • 8 equal parts → eighths (1/8 each)

The pieces must have equal area (not just equal in count). A square cut into 4 long thin rectangles vs. 4 small squares — both partition into 1/4 each, as long as areas are equal.

Worked Examples

Example 1. A square is divided into 4 equal triangles. Each is what fraction? Solution. 1/4.

Example 2. A circle is split into 6 equal sectors. Each is what fraction? Solution. 1/6.

Example 3. A rectangle is cut into 8 equal strips. If 3 are colored, what fraction is colored? Solution. 3/8.

Example 4. A square is divided into 4 equal squares. If 2 are shaded, what fraction is unshaded? Solution. 2 of 4 are shaded → 2/4 unshaded. So 2/4 (or 1/2) is unshaded.

Example 5. A pizza has 12 equal slices; you eat 5. What fraction remains? Solution. 7/12.

Common Mistakes

  • Counting unequal pieces as fractions (each piece must be the same size).
  • Mixing up the colored fraction with the uncolored fraction.
  • Confusing the number of pieces with the fraction one piece represents.

Practice

  1. A bar is in 8 equal parts; 5 are filled. What fraction is filled?
  2. A circle in 3 equal sectors. Each sector is what fraction?
  3. A square is cut into 6 equal strips; 4 are colored. Fraction colored?
  4. A rectangle in 5 equal parts; 3 are shaded. Fraction unshaded?
  5. A pizza in 10 equal slices; 7 are eaten. Fraction remaining?

Answers: 1. 5/8 | 2. 1/3 | 3. 4/6 (= 2/3) | 4. 2/5 | 5. 3/10

16. Liquid Volume

Key Concepts

Liquid volume measures how much a container holds. Common units:

  • Milliliter (mL) — small (a teaspoon is about 5 mL).
  • Liter (L) — bigger (a soda bottle is often 2 L).
  • 1 L = 1,000 mL.

To convert: liters × 1,000 = milliliters; milliliters ÷ 1,000 = liters.

Worked Examples

Example 1. How many mL in 3 L? Solution. 3 × 1,000 = 3,000 mL.

Example 2. A bottle holds 750 mL. Two bottles together? Solution. 2 × 750 = 1,500 mL (or 1.5 L).

Example 3. A pitcher has 2 L of juice. After pouring 800 mL, how much remains? Solution. 2 L = 2,000 mL. 2,000 − 800 = 1,200 mL (or 1 L 200 mL).

Example 4. A water jug holds 6 L. How many 250-mL cups can it fill? Solution. 6,000 ÷ 250 = 24 cups.

Example 5. A recipe needs 500 mL of water for one batch. How much for 3 batches? Solution. 3 × 500 = 1,500 mL (or 1.5 L).

Common Mistakes

  • Mixing up mL and L (treating 1 L as 100 mL or 10,000 mL).
  • Adding numbers with different units without converting.
  • Reading "milli" as "mega" or "kilo".

Practice

  1. How many mL in 5 L?
  2. A bottle is 1.5 L. In mL?
  3. A tank has 4 L; you pour out 1,200 mL. How much remains?
  4. A jug holds 9 L. How many 300-mL cups can it fill?
  5. 750 mL + 250 mL = ? (in L)

Answers: 1. 5,000 | 2. 1,500 | 3. 2.8 L (or 2,800 mL) | 4. 30 | 5. 1 L

17. Mass

Key Concepts

Mass measures how heavy something is. Common units:

  • Gram (g) — small (a paperclip is ~1 g).
  • Kilogram (kg) — bigger (a textbook is ~1 kg).
  • 1 kg = 1,000 g.

To convert: kilograms × 1,000 = grams; grams ÷ 1,000 = kilograms.

Worked Examples

Example 1. How many grams in 4 kg? Solution. 4 × 1,000 = 4,000 g.

Example 2. A bag weighs 2.5 kg. In grams? Solution. 2,500 g.

Example 3. A cake weighs 750 g. Three cakes total? Solution. 3 × 750 = 2,250 g (or 2 kg 250 g).

Example 4. A box weighs 1 kg 200 g. In grams? Solution. 1,000 + 200 = 1,200 g.

Example 5. A truck carries 8 boxes each weighing 12 kg. Total? Solution. 8 × 12 = 96 kg (or 96,000 g).

Common Mistakes

  • Confusing g with kg in word problems.
  • Forgetting that 1 kg = 1,000 g (not 100 g).
  • Adding kg and g without converting first.

Practice

  1. How many grams in 7 kg?
  2. A pencil weighs 6 g. 12 pencils together?
  3. A box of 5 kg of flour. Half of it weighs how many kg?
  4. A bag has 3 kg 500 g. In grams?
  5. Two packages: 2.4 kg and 950 g. Total in grams?

Answers: 1. 7,000 | 2. 72 g | 3. 2.5 kg | 4. 3,500 | 5. 3,350 g

18. Elapsed Time

Key Concepts

Elapsed time is the amount of time between a start and an end:

  • End time − Start time = elapsed time.
  • Be careful with hours that cross past 12 or AM/PM boundaries.
  • Often easiest to count up in chunks: hours first, then minutes.

Useful conversions: 1 hour = 60 minutes.

Worked Examples

Example 1. A movie starts at 3:00 and ends at 5:30. How long? Solution. 3:00 → 5:00 = 2 hours; 5:00 → 5:30 = 30 min. Total: 2 h 30 min.

Example 2. A class starts at 9:15 and ends at 10:00. How long? Solution. 9:15 → 10:00 = 45 minutes.

Example 3. A train left at 10:45 and arrived at 1:20. How long was the trip? Solution. 10:45 → 12:45 = 2 hours; 12:45 → 1:20 = 35 min. Total: 2 h 35 min.

Example 4. It is 11:50. What time will it be in 25 minutes? Solution. 11:50 + 10 min = 12:00; + 15 min = 12:15.

Example 5. A meeting at 1:30 lasts 1 hour 45 minutes. End time? Solution. 1:30 + 1 h = 2:30; + 45 min = 3:15.

Common Mistakes

  • Mixing AM and PM without crossing 12:00 properly.
  • Forgetting that an hour has 60 (not 100) minutes.
  • Subtracting end − start when crossing noon (e.g., 11:00 to 1:00 isn't "10 hours").

Practice

  1. Start 4:00, end 6:30. Elapsed?
  2. Start 8:45, end 9:15. Elapsed?
  3. Start 11:50 AM, end 1:20 PM. Elapsed?
  4. It is 2:40. What time will it be in 35 minutes?
  5. A workout at 6:15 lasts 1 hour 50 minutes. End time?

Answers: 1. 2 h 30 min | 2. 30 min | 3. 1 h 30 min | 4. 3:15 | 5. 8:05

19. Bar Graphs

Key Concepts

A bar graph uses bars to compare values. Each bar's height (or length) shows how many.

To read:

  • Pick the category on the labelled axis.
  • Read across or up to the bar's top.
  • The scale tells you what each grid line represents (1, 2, 5, 10, etc.).

Worked Examples

Example 1. A bar graph shows: cats = 7, dogs = 4, fish = 3. Total pets? Solution. 7 + 4 + 3 = 14.

Example 2. Same data: how many more cats than fish? Solution. 7 − 3 = 4 more.

Example 3. A graph: red = 25, blue = 30, green = 18. Which has the fewest? Solution. Green (18 is the smallest).

Example 4. Class A scored 18, Class B scored 22 on a test. How much higher did B score? Solution. 22 − 18 = 4 points higher.

Example 5. Three bars: 12, 18, 20. What is the average? Solution. (12 + 18 + 20) ÷ 3 = 50 ÷ 3 ≈ 16.7 (around 17).

Common Mistakes

  • Reading the wrong axis.
  • Using the bar width instead of height.
  • Skipping the title or the labels (and missing the scale).

Practice

  1. A graph: dogs = 12, cats = 8, birds = 5. Total pets?
  2. From above: how many more dogs than birds?
  3. A graph shows test scores: 15, 22, 18, 27. Highest score?
  4. Two bars: 28 and 35. Difference?
  5. Three classes scored 24, 19, 28. Total?

Answers: 1. 25 | 2. 7 | 3. 27 | 4. 7 | 5. 71

20. Pictographs

Key Concepts

A pictograph uses small pictures (icons) to represent counts. Each pictograph has a key that says how much each icon represents (e.g., "1 ★ = 5 votes").

To find the count: number of icons × value per icon.

Half-icons are sometimes used to represent half the value.

Worked Examples

Example 1. Each ★ = 1 book. A row has 4 stars. Books? Solution. 4 × 1 = 4 books.

Example 2. Each ⛳ = 4 students. A row has 5 flags. Students? Solution. 5 × 4 = 20 students.

Example 3. Each ⚪ = 10 votes. Pizza got 6 circles; burger got 3.5 circles. Difference? Solution. (6 − 3.5) × 10 = 2.5 × 10 = 25 votes.

Example 4. Each ★ = 2. Two rows: 5 stars and 8 stars. Total? Solution. (5 + 8) × 2 = 13 × 2 = 26.

Example 5. A row of 4 stars represents 36 votes. What is each star worth? Solution. 36 ÷ 4 = 9 votes per star.

Common Mistakes

  • Forgetting the key (treating each icon as 1 when key says 5).
  • Counting half-icons as full ones.
  • Skipping the multiplication step.

Practice

  1. Each ★ = 3 cars. A row of 7 stars. Cars?
  2. Each ⛳ = 5 votes. Two rows: 4 and 6 flags. Total votes?
  3. Each ⚪ = 1 student. Three rows: 8, 6, 9. Total students?
  4. Each ★ = 2 cookies. Sara: 4 stars. Tom: 6 stars. Difference in cookies?
  5. A row of 5 icons stands for 40 birds. Each icon = how many birds?

Answers: 1. 21 | 2. 50 | 3. 23 | 4. 4 cookies | 5. 8

21. Scaled Graphs

Key Concepts

A scaled graph is a bar or pictograph where each grid line (or icon) represents more than 1. The graph has a scale: 1 unit = 2, 5, 10, 100, etc.

To read: bar height × scale = actual count. To plot: actual count ÷ scale = bar height.

Scaled graphs let you show big numbers compactly without huge bars.

Worked Examples

Example 1. A scaled graph (1 unit = 5). Bar reaches 4 units. Actual count? Solution. 4 × 5 = 20.

Example 2. Scale 10. Bar of 7. Count? Solution. 7 × 10 = 70.

Example 3. Scale 2. Three bars: 4, 6, 9 units. Total actual? Solution. (4 + 6 + 9) × 2 = 19 × 2 = 38.

Example 4. Scale 100. Bar reaches 3.5 units. Count? Solution. 3.5 × 100 = 350.

Example 5. Actual count is 240. Scale = 20. Bar height? Solution. 240 ÷ 20 = 12 units.

Common Mistakes

  • Ignoring the scale and reading bar height as the count.
  • Using the wrong scale (e.g., reading "1 unit = 10" but treating it as 5).
  • Misreading half-units.

Practice

  1. Scale 5. Bar of 6. Actual count?
  2. Scale 4. Two bars: 3 and 7. Actual difference?
  3. Scale 50. Bar of 3. Actual count?
  4. Actual count 80. Scale = 8. Bar height?
  5. Scale 2. A bar at the 9 mark. Count?

Answers: 1. 30 | 2. 16 (4 × 4) | 3. 150 | 4. 10 | 5. 18

Final Tips Before the Test

  • Read every word. Many word problems hinge on a single word — "each", "share", "fewer", "remaining".
  • Estimate first. If 645 + 287 looks like ≈ 950, an answer of 832 is in the right neighborhood; an answer of 1,832 is wrong.
  • Show your work. A messy mental calculation often hides a small error. On scratch paper, line up your columns.
  • Watch units. "How many cm in 3 m?" — make sure you convert (300 cm).
  • Check your answer. If a question says "5 children share 35 stickers", and your answer is 175, you went the wrong way (5 × 7 instead of 35 ÷ 5).
  • Don't dwell. If a question is hard, mark it for review and come back later.
  • Manage your pace. Roughly 1 minute per question is a healthy rate.

You've got this! 🚀