Adaptive Math Test — Grade 4 Prep Guide

A comprehensive, topic-by-topic study guide for the Grade 4 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 in Base Ten

1. Place Value & Multi-Digit Numbers
2. Rounding
3. Multi-Digit Addition & Subtraction
4. Multi-Digit Multiplication
5. Long Division

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

Decimals 10. Tenths, Hundredths, & Decimal Place Value 11. Comparing & Adding Decimals 12. Decimal-Fraction Equivalence

Measurement & Geometry 13. Measurement Conversion 14. Area and Perimeter 15. Lines, Angles, and Symmetry 16. Classifying 2D Shapes

Patterns 17. Patterns & Input-Output Tables

1. Place Value & Multi-Digit Numbers

Key Concepts

A multi-digit number can be read by place value. In a 6-digit number like 374,825:

• 3 is in the hundred-thousands place (300,000)
• 7 is in the ten-thousands place (70,000)
• 4 is in the thousands place (4,000)
• 8 is in the hundreds place (800)
• 2 is in the tens place (20)
• 5 is in the ones place (5)

A digit's value is digit × place value. Each place is 10 times the place to its right.

You can also write numbers in expanded form: 374,825 = 300,000 + 70,000 + 4,000 + 800 + 20 + 5.

To compare multi-digit numbers, compare from the leftmost digit; if those tie, move right.

Worked Examples

Example 1. What is the value of the 6 in 36,478? Solution. The 6 is in the thousands place, so its value is 6,000.

Example 2. Write 52,837 in expanded form. Solution. 50,000 + 2,000 + 800 + 30 + 7.

Example 3. How does the digit 4 in 4,392 compare to the 4 in 437? Solution. In 4,392 the 4 means 4,000; in 437 the 4 means 400. 4,000 is 10 times 400.

Example 4. Compare 56,123 and 56,213. Solution. Hundred-thousands tie (5). Ten-thousands tie (6). Thousands tie (1, but actually let me re-look: 56,123 has thousands digit ... wait, 56,123 → digits are 5,6,1,2,3 in places 10K, 1K, 100, 10, 1). Hundreds: 1 vs 2. So 56,213 is greater.

Example 5. What is 1,000 more than 38,475? Solution. Bump the thousands digit by 1: 39,475.

Common Mistakes

• Reading a digit's value as the digit itself (the 6 in 36,478 is "6,000", not "6").
• Forgetting commas when writing big numbers.
• Comparing only the last digits instead of starting from the left.

Practice

1. What is the value of the 7 in 27,893?
2. Write 64,051 in expanded form.
3. What is 10,000 less than 92,345?
4. Compare 38,742 and 38,724.
5. Write the number for "five hundred thousand, three hundred twelve."

Answers: 1. 7,000 | 2. 60,000 + 4,000 + 0 + 50 + 1 | 3. 82,345 | 4. 38,742 > 38,724 | 5. 500,312

2. Rounding

Key Concepts

To round a number to a particular place:

1. Identify the rounding place.
2. Look at the digit directly to its right (the "decision digit").
3. If the decision digit is 5 or more, round up; if less than 5, round down.
4. All digits to the right of the rounding place become 0.

Common Grade 4 rounds: nearest 10, 100, 1,000, 10,000.

Worked Examples

Example 1. Round 4,587 to the nearest 100. Solution. Hundreds is 5; next digit is 8 (≥ 5) → round up to 4,600.

Example 2. Round 23,481 to the nearest 1,000. Solution. Thousands is 3; next digit is 4 (< 5) → round down to 23,000.

Example 3. Round 76,549 to the nearest 10,000. Solution. Ten-thousands is 7; next digit is 6 (≥ 5) → round up to 80,000.

Example 4. Round 49,950 to the nearest 1,000. Solution. Thousands is 9; next digit is 9 (≥ 5) → round up. 49 + 1 = 50, so 50,000.

Example 5. Round 8,495 to the nearest 10. Solution. Tens is 9; next digit is 5 (≥ 5) → round up. 9 → 10, cascade → 8,500.

Common Mistakes

• Looking at the wrong digit (when rounding to the nearest 100, look at the tens digit, not ones).
• Forgetting the cascade when rounding up across a 9 (49,950 → 50,000, not 410,000).
• Always rounding 5 down — by convention 5 rounds up.
• Rounding 245 to 200 by looking at the ones place; should look at tens.

Practice

1. Round 3,876 to the nearest 100.
2. Round 27,500 to the nearest 1,000.
3. Round 89,234 to the nearest 10,000.
4. Round 4,995 to the nearest 100.
5. Round 12,345 to the nearest 100.

Answers: 1. 3,900 | 2. 28,000 | 3. 90,000 | 4. 5,000 | 5. 12,300

3. Multi-Digit Addition & Subtraction

Key Concepts

Use the standard algorithm: line up by place value, work right to left, carry when adding (column ≥ 10) or borrow when subtracting (column needs more than the top digit has).

For mental math, try:

• Round and adjust — 4,732 + 2,968 → 4,732 + 3,000 − 32 = 7,700.
• Decompose — 5,247 − 1,389 → 5,247 − 1,000 − 300 − 89.

When subtracting across zeros (e.g., 5,000 − 287), cascade the borrow: 4 thousands, 9 hundreds, 9 tens, 10 ones.

Worked Examples

Example 1. 4,587 + 2,946 = ? Solution. Standard: 7 + 6 = 13 (carry 1); 8 + 4 + 1 = 13 (carry 1); 5 + 9 + 1 = 15 (carry 1); 4 + 2 + 1 = 7. Result: 7,533.

Example 2. 8,000 − 3,547 = ? Solution. 8,000 → 7 thousands, 9 hundreds, 9 tens, 10 ones. 10 − 7 = 3; 9 − 4 = 5; 9 − 5 = 4; 7 − 3 = 4. Result: 4,453.

Example 3. 6,432 + 5,789 = ? Solution. Place value: 6,000 + 5,000 = 11,000; 432 + 789 = 1,221; total 12,221.

Example 4. A school had 12,486 books. 3,728 were lost. How many remain? Solution. 12,486 − 3,728 = 8,758.

Example 5. 23,567 + 18,492 = ? Solution. 23,567 + 18,000 = 41,567; + 500 = 42,067; − 8 = 42,059.

Common Mistakes

• Mis-aligning columns when problems are written sideways.
• Forgetting cascading borrows across zeros.
• Skipping a carry (e.g., 8 + 4 = "12" written without carrying the 1).

Practice

1. 5,432 + 2,789 = ?
2. 9,000 − 4,567 = ?
3. 14,532 + 7,498 = ?
4. 23,000 − 8,765 = ?
5. A truck carried 8,432 kg on Monday and 5,679 kg on Tuesday. Total?

Answers: 1. 8,221 | 2. 4,433 | 3. 22,030 | 4. 14,235 | 5. 14,111 kg

4. Multi-Digit Multiplication

Key Concepts

Grade 4 multiplication includes 1-digit × multi-digit and 2-digit × 2-digit. Useful strategies:

• Distributive property. 8 × 47 = 8 × 40 + 8 × 7 = 320 + 56 = 376.
• Area model. Split each factor by place value, fill a grid, sum the parts.
• Standard algorithm — vertical with carries.
• Multiplying by 10, 100, 1,000. Just append zeros: 32 × 10 = 320, 32 × 100 = 3,200.

For 2-digit × 2-digit, use the partial-products or area model:

• 23 × 45 = (20 × 45) + (3 × 45) = 900 + 135 = 1,035.

Worked Examples

Example 1. What is 6 × 78? Solution. 6 × 70 + 6 × 8 = 420 + 48 = 468.

Example 2. What is 25 × 40? Solution. 25 × 4 = 100; tack on a 0 → 1,000.

Example 3. What is 24 × 36 using the area model? Solution. Split: 24 = 20 + 4; 36 = 30 + 6.

Example 4. A box has 47 books. 9 boxes have how many books? Solution. 9 × 47 = 9 × 40 + 9 × 7 = 360 + 63 = 423.

Example 5. 18 × 25 = ? Solution. Doubles trick: 18 × 25 = 18 × 100 ÷ 4 = 1,800 ÷ 4 = 450.

Common Mistakes

• Forgetting the placeholder zero in the standard algorithm when multiplying by tens.
• Multiplying only the ones digits (e.g., 23 × 45 → 3 × 5 only).
• Mis-positioning partial products in the area model.

Practice

1. 7 × 86 = ?
2. 32 × 5 = ?
3. 43 × 27 = ?
4. 15 × 48 = ?
5. A school orders 34 boxes of pencils. Each box has 24 pencils. Total pencils?

Answers: 1. 602 | 2. 160 | 3. 1,161 | 4. 720 | 5. 816

5. Long Division

Key Concepts

Long division uses a "DMSB" cycle — Divide, Multiply, Subtract, Bring down — repeated until no digits remain.

Grade 4 long division divides a multi-digit number by a 1-digit divisor. Sometimes there's a remainder.

You can also relate division to multiplication: 84 ÷ 7 = 12 because 7 × 12 = 84.

Worked Examples

Example 1. 96 ÷ 4 = ? Solution. 9 ÷ 4 = 2 remainder 1; bring down 6 → 16; 16 ÷ 4 = 4. Answer: 24.

Example 2. 145 ÷ 5 = ? Solution. 14 ÷ 5 = 2 r 4; bring down 5 → 45; 45 ÷ 5 = 9. Answer: 29.

Example 3. 327 ÷ 6 = ? Solution. 32 ÷ 6 = 5 r 2; bring down 7 → 27; 27 ÷ 6 = 4 r 3. Answer: 54 r 3.

Example 4. A teacher has 138 stickers to share equally among 6 students. How many each, with leftovers? Solution. 138 ÷ 6 = 23 stickers each, 0 left over.

Example 5. 248 ÷ 8 = ? Solution. 24 ÷ 8 = 3; bring down 8 → 8; 8 ÷ 8 = 1. Answer: 31.

Common Mistakes

• Skipping the "bring down" step.
• Misplacing the quotient digits over the wrong place.
• Forgetting the remainder, or expressing 4 ÷ 6 as 0 (it's 0 r 4 in long-division language).
• In word problems, forgetting that a remainder might mean an extra group is needed (e.g., 27 students need 7-seat vans → 4 vans, but 27 ÷ 7 = 3 r 6, so you actually need a 4th van).

Practice

1. 84 ÷ 4 = ?
2. 156 ÷ 3 = ?
3. 295 ÷ 5 = ?
4. 423 ÷ 7 = ?
5. 28 cookies are split equally among 6 children. How many cookies each, with how many left over?

Answers: 1. 21 | 2. 52 | 3. 59 | 4. 60 r 3 | 5. 4 each, 4 left over

6. Equivalent and Comparing Fractions

Key Concepts

Two fractions are equivalent when they represent the same amount. To get an equivalent fraction, multiply (or divide) both numerator and denominator by the same non-zero number.

• 2/3 = 4/6 = 6/9 = 8/12 = …

To compare fractions:

• Same denominator — compare numerators.
• Same numerator — bigger denominator means smaller piece.
• Different both — find a common denominator (often the LCM) or convert to decimals.

A fraction is in simplest form when the numerator and denominator share only 1 as a common factor.

Worked Examples

Example 1. Are 4/8 and 6/12 equivalent? Solution. 4/8 = 1/2 (divide both by 4). 6/12 = 1/2 (divide both by 6). Yes, equivalent.

Example 2. Compare 5/8 and 3/4. Solution. Common denominator 8: 5/8 vs 6/8. 5/8 < 3/4.

Example 3. Write 12/18 in simplest form. Solution. GCF of 12 and 18 is 6. 12 ÷ 6 = 2, 18 ÷ 6 = 3. Result: 2/3.

Example 4. Find a fraction equivalent to 3/5 with denominator 20. Solution. 5 × 4 = 20, so multiply numerator by 4: 3 × 4 = 12/20.

Example 5. Order from least to greatest: 2/5, 1/2, 3/10. Solution. Common denominator 10: 4/10, 5/10, 3/10. Order: 3/10 < 2/5 < 1/2.

Common Mistakes

• Adding or multiplying only the numerator (or only the denominator) when scaling.
• Comparing 1/3 < 1/4 because 3 < 4 (denominator confusion).
• Saying 2/4 ≠ 1/2 because the fractions look different.

Practice

1. Find a fraction equivalent to 2/5 with denominator 25.
2. Simplify 16/24 to lowest terms.
3. Compare 3/4 and 5/6.
4. Order from least to greatest: 1/3, 1/4, 1/2.
5. Are 6/9 and 8/12 equivalent?

Answers: 1. 10/25 | 2. 2/3 | 3. 3/4 < 5/6 | 4. 1/4 < 1/3 < 1/2 | 5. Yes (both = 2/3)

7. Adding & Subtracting Fractions

Key Concepts

To add or subtract fractions:

• Same denominator — add or subtract the numerators; keep the denominator.
• Different denominators — find a common denominator first.

After computing, simplify the result to lowest terms when asked.

Worked Examples

Example 1. 3/8 + 2/8 = ? Solution. Same denominator: 3 + 2 = 5. Result: 5/8.

Example 2. 5/6 − 1/3 = ? Solution. Common denominator 6: 1/3 = 2/6. 5/6 − 2/6 = 3/6 (or 1/2 in simplest form).

Example 3. 1/4 + 3/8 = ? Solution. Common denominator 8: 1/4 = 2/8. 2/8 + 3/8 = 5/8.

Example 4. 7/10 − 1/2 = ? Solution. Common denominator 10: 1/2 = 5/10. 7/10 − 5/10 = 2/10 (or 1/5).

Example 5. Maya ate 1/4 of a pizza and Sam ate 2/4. How much did they eat together? Solution. 1/4 + 2/4 = 3/4 of the pizza.

Common Mistakes

• Adding numerators and denominators (e.g., 1/4 + 1/4 = 2/8 — wrong; should be 2/4 = 1/2).
• Forgetting to find a common denominator before adding/subtracting.
• Not simplifying when the answer demands "in simplest form".

Practice

1. 2/5 + 1/5 = ?
2. 3/4 + 1/8 = ?
3. 7/8 − 3/8 = ?
4. 5/6 − 1/2 = ?
5. A jug had 7/10 L of water. After pouring out 3/10 L, how much remains?

Answers: 1. 3/5 | 2. 7/8 | 3. 4/8 (= 1/2) | 4. 2/6 (= 1/3) | 5. 4/10 (= 2/5) L

8. Mixed Numbers

Key Concepts

A mixed number has a whole part plus a fraction part: 2 3/4 means 2 + 3/4 = 11/4.

• Mixed → improper: multiply whole × denominator, add numerator. 2 3/4 → 2 × 4 + 3 = 11; result is 11/4.
• Improper → mixed: divide. 17/5 = 17 ÷ 5 = 3 r 2 → 3 2/5.

To add or subtract mixed numbers, you can:

• Add wholes and fractions separately, then combine.
• Convert to improper fractions, compute, convert back.

Worked Examples

Example 1. Convert 9/4 to a mixed number. Solution. 9 ÷ 4 = 2 r 1. So 9/4 = 2 1/4.

Example 2. Convert 3 2/5 to an improper fraction. Solution. 3 × 5 + 2 = 17. So 3 2/5 = 17/5.

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

Example 4. 4 3/8 − 1 1/8 = ? Solution. Wholes: 4 − 1 = 3. Fractions: 3/8 − 1/8 = 2/8. Total: 3 2/8 (= 3 1/4).

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

Common Mistakes

• Converting 3 2/5 to 3 × 5 = 15 (forgetting to add the numerator 2).
• Treating "2 3/4" as 2 × 3/4 (multiplication).
• Forgetting to handle the carry when fraction parts add to ≥ 1.

Practice

1. Convert 15/4 to a mixed number.
2. Convert 5 1/3 to an improper fraction.
3. 2 1/6 + 3 2/6 = ?
4. 5 3/4 − 2 1/4 = ?
5. 3 1/3 + 1 2/3 = ?

Answers: 1. 3 3/4 | 2. 16/3 | 3. 5 3/6 (= 5 1/2) | 4. 3 2/4 (= 3 1/2) | 5. 5

9. Multiplying Fractions by Whole Numbers

Key Concepts

To multiply a fraction by a whole number, multiply the numerator by that whole number; keep the denominator.

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

You can also picture it: 5 × 2/3 means "five copies of two-thirds".

Worked Examples

Example 1. 4 × 1/5 = ? Solution. 4 × 1/5 = 4/5.

Example 2. 3 × 2/7 = ? Solution. 3 × 2/7 = 6/7.

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

Example 4. A recipe needs 2/3 cup of sugar per batch. How much sugar for 5 batches? Solution. 5 × 2/3 = 10/3 cups (= 3 1/3 cups).

Example 5. 8 × 3/4 = ? Solution. 8 × 3/4 = 24/4 = 6.

Common Mistakes

• Multiplying both numerator AND denominator by the whole number (e.g., 3 × 1/4 = 3/12 — wrong).
• Forgetting to convert improper results to mixed numbers when asked.

Practice

1. 2 × 3/8 = ?
2. 5 × 1/3 = ?
3. 4 × 2/5 = ?
4. 7 × 1/2 = ?
5. A pitcher holds 1/4 L. How much do 12 pitchers hold?

Answers: 1. 6/8 (= 3/4) | 2. 5/3 (= 1 2/3) | 3. 8/5 (= 1 3/5) | 4. 7/2 (= 3 1/2) | 5. 3 L

10. Tenths, Hundredths, & Decimal Place Value

Key Concepts

Decimals extend place value past the ones place to the right of the decimal point:

• Tenths (1/10) — first place after the point. 0.4 = 4/10.
• Hundredths (1/100) — second place. 0.07 = 7/100.

Each place is 10 times smaller than the one to its left:

• 1 = ten 0.1s
• 0.1 = ten 0.01s

You can read 0.62 as "six tenths and two hundredths" or "sixty-two hundredths" (because 0.62 = 60/100 + 2/100 = 62/100).

Worked Examples

Example 1. What is 4 tenths in decimal form? Solution. 4/10 = 0.4.

Example 2. What is 23 hundredths? Solution. 23/100 = 0.23.

Example 3. Write 0.5 as a fraction. Solution. 0.5 = 5/10 (or 1/2 in simplest form).

Example 4. What is 6 tenths plus 7 hundredths? Solution. 6/10 + 7/100 = 60/100 + 7/100 = 67/100 = 0.67.

Example 5. Which is larger: 0.3 or 0.30? Solution. They are equal — 0.30 just adds a zero in the hundredths place.

Common Mistakes

• Reading 0.4 as "zero point four" without recognizing it means "four tenths".
• Confusing 0.05 (5 hundredths) with 0.5 (5 tenths).
• Adding extra zeros in the wrong place changes the value: 0.40 = 0.4, but 0.04 ≠ 0.4.

Practice

1. Write 7 tenths as a decimal.
2. Write 14 hundredths as a decimal.
3. Convert 0.2 to a fraction in lowest terms.
4. What is 5 tenths plus 8 hundredths in decimal form?
5. Which is greater: 0.07 or 0.7?

Answers: 1. 0.7 | 2. 0.14 | 3. 1/5 | 4. 0.58 | 5. 0.7

11. Comparing & Adding Decimals

Key Concepts

To compare decimals: line them up by the decimal point, then compare digit by digit from the left. Add zeros if helpful: 0.6 vs 0.59 → 0.60 vs 0.59 → 0.6 is greater.

To add decimals: align the decimal points, then add as if they were whole numbers — putting the decimal point in the answer in the same column.

To subtract: same idea — align the decimals, subtract column by column.

Worked Examples

Example 1. Compare 0.7 and 0.65. Solution. 0.70 vs 0.65 → 0.70 > 0.65. So 0.7 > 0.65.

Example 2. 0.5 + 0.3 = ? Solution. Like adding 5 tenths + 3 tenths = 8 tenths = 0.8.

Example 3. 0.42 + 0.36 = ? Solution. Hundredths: 2 + 6 = 8. Tenths: 4 + 3 = 7. Result: 0.78.

Example 4. 1.25 + 0.75 = ? Solution. Hundredths: 5 + 5 = 10 (write 0, carry 1). Tenths: 2 + 7 + 1 = 10 (write 0, carry 1). Ones: 1 + 0 + 1 = 2. Result: 2.00 (or 2).

Example 5. 5.6 − 1.8 = ? Solution. 5.6 − 1.8: tenths 6 − 8 needs regroup → 16 − 8 = 8. Ones 4 − 1 = 3. Result: 3.8.

Common Mistakes

• Comparing decimals as if they were whole numbers (saying 0.65 > 0.7 because "65 > 7").
• Misaligning decimal points when adding.
• Treating 1.5 + 1.5 as 1.10 instead of carrying.

Practice

1. Compare 0.4 and 0.38.
2. 0.6 + 0.7 = ?
3. 1.25 + 0.5 = ?
4. 3.7 − 1.4 = ?
5. 2.05 + 1.95 = ?

Answers: 1. 0.4 > 0.38 | 2. 1.3 | 3. 1.75 | 4. 2.3 | 5. 4.00 (or 4)

12. Decimal-Fraction Equivalence

Key Concepts

Decimals and fractions describe the same numbers in different ways:

• 0.5 = 1/2
• 0.25 = 1/4
• 0.75 = 3/4
• 0.1 = 1/10
• 0.01 = 1/100

To convert a fraction with denominator 10 or 100 to a decimal, just write it out by place value:

• 7/10 = 0.7
• 23/100 = 0.23
• 305/100 = 3.05

To go the other way (decimal to fraction), use the place value:

• 0.6 = 6/10 (= 3/5)
• 0.45 = 45/100 (= 9/20)

Worked Examples

Example 1. Write 0.4 as a fraction. Solution. 0.4 = 4/10 = 2/5.

Example 2. Write 3/10 as a decimal. Solution. 3/10 = 0.3.

Example 3. Write 0.75 as a fraction. Solution. 0.75 = 75/100 = 3/4.

Example 4. Write 1/4 as a decimal. Solution. 1/4 = 25/100 = 0.25.

Example 5. Order from least to greatest: 0.5, 1/4, 0.7. Solution. 1/4 = 0.25. So 0.25 < 0.5 < 0.7. Order: 1/4, 0.5, 0.7.

Common Mistakes

• Writing 0.5 as 5/10 and forgetting to simplify (acceptable but often the question wants 1/2).
• Confusing 0.05 with 5/10 (it's 5/100).
• Not lining up decimal places when comparing 0.5 and 0.55.

Practice

1. Write 0.6 as a fraction in lowest terms.
2. Write 9/10 as a decimal.
3. Write 2/4 as a decimal.
4. Write 0.20 as a fraction in lowest terms.
5. Order: 0.3, 1/2, 0.45.

Answers: 1. 3/5 | 2. 0.9 | 3. 0.5 | 4. 1/5 | 5. 0.3 < 0.45 < 1/2

13. Measurement Conversion

Key Concepts

Convert within the metric or customary systems:

To convert from larger to smaller, multiply. From smaller to larger, divide.

Worked Examples

Example 1. How many cm in 4 m? Solution. 4 × 100 = 400 cm.

Example 2. How many g in 3 kg? Solution. 3 × 1,000 = 3,000 g.

Example 3. How many mL in 2.5 L? Solution. 2.5 × 1,000 = 2,500 mL.

Example 4. How many feet in 4 yards? Solution. 4 × 3 = 12 feet.

Example 5. A movie is 2 hours 15 minutes. How many minutes total? Solution. 2 × 60 + 15 = 135 minutes.

Common Mistakes

• Mixing up conversion factors (1 m = 100 cm, not 1,000 cm).
• Going the wrong direction (multiplying when you should divide, or vice versa).
• Forgetting that 1 L = 1,000 mL, not 100 mL.

Practice

1. How many cm in 7 m?
2. How many g in 0.5 kg?
3. How many mL in 1.75 L?
4. How many inches in 5 feet?
5. How many seconds in 4 minutes?

Answers: 1. 700 | 2. 500 | 3. 1,750 | 4. 60 | 5. 240

14. Area and Perimeter

Key Concepts

For a rectangle:

• Area = length × width (in square units).
• Perimeter = 2 × (length + width) (in length units).

For a square:

• Area = side × side.
• Perimeter = 4 × side.

You can also work backwards: given perimeter and one side, find the other side; given area and one side, find the other side.

Worked Examples

Example 1. A rectangle is 8 cm by 5 cm. Area? Perimeter? Solution. Area = 8 × 5 = 40 sq cm. Perimeter = 2(8 + 5) = 26 cm.

Example 2. A square has side 6 m. Area? Perimeter? Solution. Area = 6 × 6 = 36 sq m. Perimeter = 4 × 6 = 24 m.

Example 3. A rectangle has area 48 sq cm and length 8 cm. What is its width? Solution. Width = 48 ÷ 8 = 6 cm.

Example 4. A rectangle has perimeter 30 cm and length 10 cm. Width? Solution. L + W = 15. W = 15 − 10 = 5 cm.

Example 5. A garden is 12 m by 7 m. Fencing cost is $4 per meter. Total fencing cost? Solution. Perimeter = 2(12 + 7) = 38 m. Cost = 38 × $4 = $152.

Common Mistakes

• Mixing up area (square units) and perimeter (regular units).
• Forgetting to multiply by 2 in P = 2(L + W).
• Reading "perimeter" as "area" (or vice versa).

Practice

1. Area of a 6 × 9 rectangle?
2. Perimeter of a 6 × 9 rectangle?
3. A square has perimeter 28 cm. What is its area?
4. A rectangle has area 60 sq m and width 5 m. Length?
5. A 4 × 4 square's perimeter and area?

Answers: 1. 54 sq | 2. 30 | 3. 49 sq cm (side 7) | 4. 12 m | 5. P=16, A=16

15. Lines, Angles, and Symmetry

Key Concepts

Line types:

• Parallel lines — never meet (think railroad tracks).
• Perpendicular lines — meet at right angles (form a square corner).
• Intersecting lines — cross at any angle.

Angle types:

• Right angle — 90° (a square corner).
• Acute angle — less than 90° (sharp).
• Obtuse angle — between 90° and 180° (wide).
• Straight angle — 180° (a straight line).

A shape has a line of symmetry if it can be folded along the line so the two halves match perfectly.

Worked Examples

Example 1. What kind of angle is 45°? Solution. Less than 90° → acute.

Example 2. What kind of angle is 120°? Solution. Between 90° and 180° → obtuse.

Example 3. A rectangle has how many right angles? Solution. 4 right angles (one at each corner).

Example 4. How many lines of symmetry does a regular hexagon have? Solution. 6 — three through opposite vertices and three through opposite edge midpoints.

Example 5. Are the lines y = 5 and x = 5 parallel or perpendicular? Solution. y = 5 is horizontal; x = 5 is vertical. They are perpendicular.

Common Mistakes

• Calling all small angles "acute" without checking they're under 90°.
• Mistaking obtuse (90°-180°) for reflex (>180°).
• Counting only one line of symmetry on a square (it has 4).

Practice

1. What kind of angle is 90°?
2. What kind of angle is 30°?
3. How many lines of symmetry does an equilateral triangle have?
4. Are the lines y = 3 and y = 7 parallel or perpendicular?
5. A rhombus has how many lines of symmetry?

Answers: 1. Right | 2. Acute | 3. 3 | 4. Parallel (both horizontal) | 5. 2 (the diagonals)

16. Classifying 2D Shapes

Key Concepts

Polygons are named by number of sides:

• 3 — triangle
• 4 — quadrilateral
• 5 — pentagon
• 6 — hexagon
• 8 — octagon

Triangle types by side:

• Equilateral — 3 equal sides.
• Isosceles — 2 equal sides.
• Scalene — no equal sides.

Triangle types by angle:

• Right — has a 90° angle.
• Acute — all angles less than 90°.
• Obtuse — has one angle greater than 90°.

Quadrilaterals include square, rectangle, rhombus, parallelogram, trapezoid.

Worked Examples

Example 1. A polygon has 5 sides. What is it called? Solution. Pentagon.

Example 2. A triangle has all three sides equal. What type? Solution. Equilateral.

Example 3. A triangle has one angle measuring 95°. Type by angle? Solution. Obtuse (one angle > 90°).

Example 4. Is a square a special rectangle? Solution. Yes — a square has all properties of a rectangle (4 right angles, opposite sides equal) plus all sides equal.

Example 5. A 6-sided shape with all sides equal is called a: Solution. Regular hexagon.

Common Mistakes

• Calling any 3-sided shape "scalene" without checking sides.
• Confusing "isosceles" with "equilateral" (equilateral has all 3 equal; isosceles has at least 2 equal).
• Missing that a square is also a rectangle and a rhombus.

Practice

1. A polygon with 8 sides is called a what?
2. A triangle with sides 5, 5, 7 is what type?
3. A triangle has angles 60°, 60°, 60°. Type by sides?
4. A 4-sided shape with exactly one pair of parallel sides is called a what?
5. A right triangle has one ___ angle.

Answers: 1. Octagon | 2. Isosceles | 3. Equilateral | 4. Trapezoid | 5. right (90°)

17. Patterns & Input-Output Tables

Key Concepts

A pattern has a rule that tells you the next term. Common patterns:

• Arithmetic — add or subtract the same amount each step (4, 7, 10, 13, … rule: +3).
• Geometric — multiply (or divide) by the same factor each step (2, 6, 18, 54, … rule: ×3).
• Repeating — same group cycles (A, B, C, A, B, C, …).

An input-output table shows pairs of values where each output is computed from the input by a rule. Example:

The rule is "× 3" or "Output = 3 × Input".

Worked Examples

Example 1. What is the next term: 5, 10, 15, 20, ___? Solution. Rule: +5. Next: 25.

Example 2. Find the rule: input 4 → output 11; input 5 → output 13; input 6 → output 15. Solution. Each step input increases by 1 and output increases by 2. Rule: Output = 2 × Input + 3. So at input 1 → 5, etc.

Example 3. Pattern: 2, 6, 18, 54, ___. Solution. Rule: × 3. Next: 162.

Example 4. Repeating pattern: ▲, ●, ■, ▲, ●, ■, … What is the 10th term? Solution. Cycle length 3. 10 ÷ 3 = 3 r 1; the 10th matches position 1 → ▲.

Example 5. Input-output: input 7 → output 21. What is the rule? Solution. Output = Input × 3 (or "21 ÷ 7 = 3").

Common Mistakes

• Assuming all patterns are arithmetic (sometimes they multiply).
• Mis-counting the position in repeating patterns.
• Stopping at one example to "find the rule" — always check at least 2-3 to confirm.

Practice

1. Find the next term: 3, 7, 11, 15, ___.
2. Find the next term: 4, 12, 36, 108, ___.
3. Repeating ABCABCABC… What is the 25th letter?
4. Find the rule for: input 2 → output 5; input 4 → output 11; input 6 → output 17.
5. The 8th term in 1, 4, 7, 10, … is?

Answers: 1. 19 | 2. 324 | 3. A (cycle 3 → 25 ÷ 3 = 8 r 1) | 4. Output = 3 × Input − 1 | 5. 22 (8th term: 1 + 7 × 3 = 22)

Final Tips Before the Test

• Read the question twice — once to understand, once to spot key numbers.
• Estimate first — if 4,587 + 2,946 should be near 7,500, an answer of 1,500 is wrong.
• Show your work — column-line up addition, subtraction, and long division.
• Check units — square cm for area, regular cm for perimeter, mL for volume.
• Watch the question type — "in simplest form" means you simplify; "as a mixed number" means you convert.
• Skip and return — if a question takes too long, mark it and move on. You can come back.
• Pace yourself — about 1 minute per question is a healthy pace.

You've got this! 💪