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Pre-Algebra · Grades 5–8 · Foundational Math

Pre-Algebra in 6 Concepts: Whole Numbers, Operations, Properties, Factors, Integers & Equivalent Fractions

The whole pre-algebra foundation on one page. Six SOMATH posters walk through the six concepts every middle-school student needs to master before algebra. Under each poster you’ll find the full theory expanded from the poster and 10 practice questions with click-to-reveal solutions — 60 questions total, plus a free class-pack PDF and a teacher answer key.

· By SOMATH · 226 W 79th St, UWS · (646) 668-6151

↓ Class Pack PDF (posters + theory + 60 questions) ↓ Teacher Answer Key PDF

What’s in this class

  1. Whole Numbers & Place Value
  2. Operations with Whole Numbers
  3. Number Properties
  4. Factors and Multiples
  5. Integers
  6. Equivalent Fractions
How to use this page. Study each poster first, then read the theory notes below it, then work the 10 practice questions. Click Show solution only after you’ve tried a problem. Bring questions to your SOMATH teacher — or book a free 30-minute evaluation at 226 W 79th St to see where your pre-algebra foundation stands.

1. Whole Numbers & Place Value

Place value tells us what a digit is worth based on its position. In 34,582 the 5 means 500, not 5.

SOMATH poster — Whole Numbers and Place Value

What are whole numbers?

The whole numbers are the counting numbers starting at zero: 0, 1, 2, 3, 4, 5, …. Whole numbers have no fractions, decimals, or negatives.

Place value

The place value of a digit depends on where it sits in the number. Each place to the left is 10 times greater than the place to its right. In 522, the two 2s are not the same — one is 2 tens (20), the other is 2 ones (2).

Place value chart for 34,582

Ten ThousandsThousandsHundredsTensOnes
34582

Standard, expanded, and word form

  • Standard form: 34,582
  • Expanded form: 30,000 + 4,000 + 500 + 80 + 2
  • Word form: thirty-four thousand, five hundred eighty-two

Comparing and ordering

Compare from the leftmost place first. If those digits match, move one place right. Example: 4,321 > 4,213 because both have 4 thousands, but 3 hundreds > 2 hundreds. Ordering least → greatest: 245 < 254 < 425.

10 Practice Questions — Whole Numbers & Place Value

Q1. What is the value of the digit 7 in 27,304?
Solution: The 7 is in the thousands place. Value = 7 × 1,000 = 7,000.
Q2. Write 58,206 in expanded form.
Solution: 50,000 + 8,000 + 200 + 6.
Q3. Write forty thousand, three hundred nine in standard form.
Solution: 40,309.
Q4. Which is greater: 63,481 or 63,418?
Solution: Compare place by place. Tens digit: 8 vs 1, so 63,481 > 63,418.
Q5. Order from least to greatest: 5,082, 5,802, 5,028, 5,820.
Solution: 5,028 < 5,082 < 5,802 < 5,820.
Q6. How many hundreds are in the hundreds place of 12,347?
Solution: The hundreds digit is 3, so 3 hundreds.
Q7. Place value of the underlined digit in 45,678?
Solution: Thousands; value 5,000.
Q8. Write 6,000 + 300 + 40 + 2 in standard form.
Solution: 6,342.
Q9. Greatest whole number less than 10,000?
Solution: 9,999.
Q10. A digit’s value moves one place to the LEFT. What happens to its value?
Solution: It becomes 10 times greater.

2. Operations with Whole Numbers

Every whole-number algorithm is the same idea: line up place values and operate one column at a time.

SOMATH poster — Operations with Whole Numbers

The four operations, one place-value system

Every whole-number algorithm is really place value + the properties: line up ones under ones, tens under tens, and operate one column at a time.

Addition — find the sum

Add each column right to left. If a column total is 10 or more, carry 1. Example: 2,847 + 1,596 = 4,443.

Subtraction — find the difference

Subtract right to left. If the top digit is smaller, borrow 1 from the next place. Example: 5,208 − 1,764 = 3,444.

Multiplication — find the product

Multiplying by n is adding n times. Multi-digit: multiply by each digit, shift each partial product one place left, then add. Example: 328 × 24 = 7,872 (328×4 = 1,312; 328×20 = 6,560; sum 7,872).

Division — find the quotient

How many times does the divisor fit? Long division: divide, multiply, subtract, bring down. Example: 1,472 ÷ 23 = 64.

10 Practice Questions — Operations with Whole Numbers

Q1. Compute 4,687 + 2,548.
Solution: 7,235.
Q2. Compute 8,003 − 2,547.
Solution: 5,456.
Q3. Compute 236 × 47.
Solution: 236×7 = 1,652; 236×40 = 9,440; sum 11,092.
Q4. Compute 1,296 ÷ 8.
Solution: 162 (8 × 162 = 1,296).
Q5. Estimate 4,872 + 3,158 by rounding each to the nearest thousand.
Solution: 5,000 + 3,000 = 8,000 (actual 8,030).
Q6. A theater has 24 rows of 38 seats. Total seats?
Solution: 24 × 38 = 912.
Q7. Divide 4,725 by 15.
Solution: 315.
Q8. Write 305,072 in words.
Solution: Three hundred five thousand, seventy-two.
Q9. Compare: 9,876 ? 9,867.
Solution: Tens 7 vs 6, so 9,876 > 9,867.
Q10. A truck holds 1,250 kg. Six identical crates weigh 7,320 kg total — does one crate fit?
Solution: One crate = 7,320 ÷ 6 = 1,220 kg. 1,220 < 1,250, so yes.

3. Number Properties

Commutative, associative, distributive, identity, inverse, and zero properties are the shortcuts you already use — named.

SOMATH poster — Number Properties

Closure

Add, subtract, multiply, or divide numbers in a set (when allowed) and the result stays in that set.

Commutative Property

Order doesn’t matter for + and ×: a + b = b + a, a × b = b × a. It does matter for subtraction and division.

Associative Property

Grouping doesn’t matter for + and ×: (3 + 5) + 2 = 3 + (5 + 2), (2 × 7) × 5 = 2 × (7 × 5).

Distributive Property

a × (b + c) = a×b + a×c. Example: 3(4 + 2) = 12 + 6 = 18. This is the property that makes mental math and algebra work.

Identity Properties

Additive identity: a + 0 = a. Multiplicative identity: a × 1 = a.

Inverse Properties

Additive inverse (opposite): a + (−a) = 0. Multiplicative inverse (reciprocal): a × (1/a) = 1 for a ≠ 0.

Zero Property of Multiplication

Any number times 0 equals 0: a × 0 = 0.

10 Practice Questions — Number Properties

Q1. Name the property: 6 + 9 = 9 + 6.
Solution: Commutative Property of Addition.
Q2. Name the property: (2 × 5) × 4 = 2 × (5 × 4).
Solution: Associative Property of Multiplication.
Q3. Fill in using distributive: 7(3 + 5) = 7·3 + ___.
Solution: 7 × 5 = 35. Full: 21 + 35 = 56.
Q4. What is 143 × 1?
Solution: 143 (multiplicative identity).
Q5. What is 89 + 0?
Solution: 89 (additive identity).
Q6. Additive inverse of −12?
Solution: 12.
Q7. Multiplicative inverse of 4?
Solution: 1/4.
Q8. Use distributive to compute 6 × 27 mentally.
Solution: 6(20 + 7) = 120 + 42 = 162.
Q9. True or false: subtraction is commutative.
Solution: False. 7 − 4 = 3 but 4 − 7 = −3.
Q10. Simplify using zero property: 15 × 0 + 8.
Solution: 0 + 8 = 8.

4. Factors and Multiples

Factors divide a number exactly; multiples are products of it. GCF simplifies fractions; LCM adds them.

SOMATH poster — Factors and Multiples

Factors

Factors of a number divide it exactly (no remainder). Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.

Multiples

Multiples of a number are what you get by multiplying it by 0, 1, 2, 3, .... First 10 multiples of 6: 0, 6, 12, 18, 24, 30, 36, 42, 48, 54.

Factor pairs

Two whole numbers whose product is the original number. Factor pairs of 24: (1, 24), (2, 12), (3, 8), (4, 6).

Prime vs composite

  • Prime: exactly two factors — 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19.
  • Composite: more than two factors. Examples: 4, 6, 8, 9, 10, 12.
  • 1 is not prime (only one factor).

GCF — Greatest Common Factor

The largest factor two or more numbers share. GCF(12, 18) = 6. Use it to simplify fractions.

LCM — Least Common Multiple

The smallest positive multiple two or more numbers share. LCM(4, 6) = 12. Use it to add fractions with unlike denominators.

10 Practice Questions — Factors and Multiples

Q1. List all factors of 36.
Solution: 1, 2, 3, 4, 6, 9, 12, 18, 36.
Q2. First 8 multiples of 7 (starting at 0).
Solution: 0, 7, 14, 21, 28, 35, 42, 49.
Q3. All factor pairs of 30.
Solution: (1, 30), (2, 15), (3, 10), (5, 6).
Q4. Is 29 prime or composite?
Solution: Prime — only factors are 1 and 29.
Q5. Is 51 prime or composite?
Solution: Composite: 51 = 3 × 17.
Q6. Find GCF(24, 36).
Solution: GCF = 12.
Q7. Find LCM(8, 12).
Solution: LCM = 24.
Q8. Common factors of 20 and 30.
Solution: 1, 2, 5, 10.
Q9. Find GCF(45, 60) and LCM(45, 60).
Solution: GCF = 15; LCM = 180.
Q10. Two lights blink every 6 and 9 seconds. Next time they blink together after 0?
Solution: LCM(6, 9) = 18 seconds.

5. Integers

Integers are whole numbers and their opposites. Same signs behave one way; different signs the other.

SOMATH poster — Integers

The set of integers

ℤ = {..., −3, −2, −1, 0, 1, 2, 3, ...}. Positive integers, zero, and negative integers. No fractions or decimals.

Number line and comparing

Numbers to the right are greater; numbers to the left are smaller. Any positive is greater than any negative. A “bigger-looking” negative is actually smaller: −6 < −2.

Opposites & absolute value

Every integer has an opposite (the same distance from 0 on the other side): opposite of 5 is −5. Absolute value is distance from zero, always non-negative: |5| = 5, |−8| = 8.

Adding integers

  • Same signs: add absolute values, keep the sign. (−3) + (−5) = −8.
  • Different signs: subtract the smaller absolute value from the larger, keep the sign of the larger. 4 + (−7) = −3.

Subtracting integers

a − b = a + (−b). Example: 5 − (−3) = 5 + 3 = 8.

Multiplying and dividing integers

  • Same signs → positive: (−6) × (−7) = 42, (−56) ÷ (−7) = 8.
  • Different signs → negative: 3 × (−4) = −12, (−20) ÷ 5 = −4.

10 Practice Questions — Integers

Q1. Compute −7 + 12.
Solution: 5.
Q2. Compute −8 + (−5).
Solution: −13.
Q3. Compute 4 − 9.
Solution: −5.
Q4. Compute −6 − (−10).
Solution: 4.
Q5. Compute (−3) × 8.
Solution: −24.
Q6. Compute (−6) × (−7).
Solution: 42.
Q7. Compute (−36) ÷ 4.
Solution: −9.
Q8. Compute (−56) ÷ (−7).
Solution: 8.
Q9. Order least → greatest: −3, 4, 0, −7, 2.
Solution: −7, −3, 0, 2, 4.
Q10. Evaluate |−15| + |4| − |−9|.
Solution: 15 + 4 − 9 = 10.

6. Equivalent Fractions

Multiply or divide numerator and denominator by the same number and the fraction's value doesn't change.

SOMATH poster — Equivalent Fractions

What are equivalent fractions?

Fractions that represent the same amount even if they look different: 1/2 = 2/4 = 4/8. They all shade the same portion of a whole.

Finding equivalent fractions — multiply

Multiply the numerator and denominator by the same non-zero number. Example: 3/5 × 2/2 = 6/10, so 3/5 and 6/10 are equivalent.

Finding equivalent fractions — divide (simplify)

Divide numerator and denominator by the same non-zero number (a common factor). Example: 8/12 ÷ 4/4 = 2/3. To reduce to lowest terms, divide both by their GCF.

Not equivalent

Different amounts mean not equivalent: 1/3 ≠ 1/4. Cross-multiplication check: a/b = c/d iff a×d = b×c.

Why we care

  • Simplifying: reduce a fraction to lowest terms using the GCF.
  • Comparing: rewrite fractions with a common denominator, then compare numerators.
  • Adding & subtracting fractions: requires a common denominator — equivalent fractions get you there.

10 Practice Questions — Equivalent Fractions

Q1. Write two fractions equivalent to 2/3.
Solution: 4/6, 6/9 (also 8/12, 10/15, ...).
Q2. Simplify 18/24.
Solution: GCF = 6, so 3/4.
Q3. Fill in: 5/8 = ?/40.
Solution: 25/40.
Q4. Are 6/9 and 8/12 equivalent?
Solution: Yes, both simplify to 2/3.
Q5. Are 3/5 and 4/7 equivalent?
Solution: No: 3·7 = 21, 5·4 = 20.
Q6. Simplify 45/60.
Solution: 3/4 (GCF = 15).
Q7. Which is larger: 2/3 or 5/8?
Solution: Common denominator 24: 16/24 vs 15/24. 2/3 > 5/8.
Q8. Fraction with denominator 100 equivalent to 7/20.
Solution: 35/100.
Q9. Simplify 24/36 in two steps.
Solution: ÷2 → 12/18; ÷6 → 2/3.
Q10. A recipe uses 6/8 cup of flour. Simplest form and two equivalents.
Solution: 3/4; equivalents: 9/12, 12/16.

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