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Grades 6-8Algebra

Pre-Algebra Bridge

Build the foundation for algebra: integers and order of operations, evaluating and simplifying expressions, solving one-step equations, ratios, proportions, percents, and the coordinate plane.

1Integers, Opposites, and Absolute Value

Key idea: Integers are whole numbers and their opposites. The absolute value is a number's distance from 0, so it is never negative.
  1. Integers: …, -3, -2, -1, 0, 1, 2, 3, … (no fractions or decimals).
  2. The opposite of a number flips its sign: opposite of -5 is 5, opposite of 8 is -8.
  3. The absolute value |n| is the distance from 0: |-6| = 6 and |6| = 6.
  4. Comparing integers: further LEFT on the number line = smaller. So -9 < -4.
Worked example
-|-7| + 5
Absolute value first: |-7| = 7. Then -7 + 5 = -2.
Worked example
The opposite of -11
The opposite is the same distance from 0 on the other side: 11.
Watch out: Absolute value is always non-negative! |-9| = 9, never -9. And -|-9| = -9 because the minus sign applies after the absolute value.

2Order of Operations (PEMDAS)

Key idea: PEMDAS: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right). The order is never optional.
  1. Do whatever is inside Parentheses (innermost first).
  2. Evaluate Exponents.
  3. Do Multiplication and Division from LEFT to RIGHT (they are a tie).
  4. Do Addition and Subtraction from LEFT to RIGHT (also a tie).
Worked example
3 + 4 × 2
Multiply before adding: 3 + (4 × 2) = 3 + 8 = 11. Adding first (7 × 2 = 14) is wrong!
Worked example
23 + 10 ÷ 2 – 3
Exponents first: 8 + 10 ÷ 2 – 3. Then divide: 8 + 5 – 3. Then left to right: 13 – 3 = 10.
Watch out: Multiplication does NOT always come before division, and addition does NOT always come before subtraction! Each pair is done left to right: 10 – 4 + 2 = 6 + 2 = 8, not 10 – 6 = 4.

3Evaluating Algebraic Expressions

Key idea: Evaluating an expression means replacing each variable with its value and simplifying with the order of operations.
  1. Write the expression, then substitute each variable's value in parentheses.
  2. Follow PEMDAS to simplify.
  3. Watch the signs: a negative value inside an exponent base needs parentheses: (-3)2 = 9, but -32 = -9.
Worked example
Evaluate 3x + 5 when x = 4
Substitute 4 for x: 3(4) + 5 = 12 + 5 = 17.
Worked example
Evaluate x2 – 2x when x = -3
(-3)2 – 2(-3) = 9 + 6 = 15. The parentheses around -3 keep the sign with the number.
Watch out: Never drop the negative sign when substituting! If x = -2, then 5x means 5(-2) = -10, not 10.

4Combining Like Terms

Key idea: Like terms have the exact same variable part. Add or subtract their coefficients and keep the variable part unchanged.
  1. Identify like terms: same letter AND same exponent (3x and 5x are like; 3x and 3x2 are not).
  2. Add or subtract the coefficients.
  3. Keep the variable part exactly as it is.
  4. Leave unlike terms alone — 2x + 3y cannot be combined.
Worked example
5x + 3x – 2x
All three terms have the same variable x: 5 + 3 – 2 = 6, so 6x.
Worked example
4y – 2 + 3y + 7
Combine y-terms: 4y + 3y = 7y. Combine constants: -2 + 7 = 5. Result: 7y + 5.
Watch out: You can only combine terms with identical variable parts! 5x^2 + 3x = 8x^2 is a classic error — the answer is 5x^2 + 3x, fully simplified.

5One-Step Equations

Key idea: A one-step equation needs one inverse operation: undo what is done to the variable, and do it to BOTH sides.
  1. Identify what is happening to x: added, subtracted, multiplied, or divided.
  2. Apply the inverse (opposite) operation to both sides of the equation.
  3. Simplify to get x = value.
  4. Check by substituting the value back into the original equation.
Worked example
x + 9 = 15
Subtract 9 from both sides: x = 15 – 9 = 6. Check: 6 + 9 = 15.
Worked example
x/4 = 7
Multiply both sides by 4: x = 7 × 4 = 28. Check: 28 ÷ 4 = 7.
Watch out: Whatever you do to one side, do to the other! Solving x + 5 = 12 as x = 12 + 5 = 17 breaks the balance — the answer is 12 – 5 = 7.

6Ratios and Proportions

Key idea: A ratio compares two quantities. A proportion is two equal ratios — solve it by cross-multiplying (or simplifying first).
  1. Write the proportion with the unknown as x.
  2. Simplify each ratio if you can.
  3. Cross-multiply: multiply each numerator by the other denominator.
  4. Solve the one-step equation that remains.
Worked example
Simplify the ratio 18:24
Divide both parts by their GCF, 6: 18 ÷ 6 = 3 and 24 ÷ 6 = 4, so 3:4.
Worked example
Solve x/4 = 9/12
9/12 simplifies to 3/4, so x/4 = 3/4 and x = 3. (Or cross-multiply: 12x = 36, so x = 3.)
Watch out: Units matter! If a ratio is miles per hour, keep the same units on both sides of the proportion: miles/hours = miles/hours.

7Percent Basics

Key idea: Percent means 'out of 100'. Convert a percent to a decimal (drop the %, move the decimal point two places left) and multiply to find 'percent of' a number.
  1. Convert the percent: divide by 100 (42% = 0.42, 6% = 0.06).
  2. Multiply by the whole number.
  3. For a percent from a part: (part ÷ whole) × 100.
  4. Round money answers to the nearest cent.
Worked example
20% of 80
20% = 20/100 = 0.2. Then 0.2 × 80 = 16.
Worked example
What is 75% as a fraction?
75% = 75/100, which simplifies (divide by 25) to 34.
Watch out: Decimal placement trips everyone up: 5% = 0.05, NOT 0.5! And 150% = 1.5, which is MORE than the whole.

8Coordinate Plane Basics

Key idea: An ordered pair (x, y) is a set of directions: x moves left-right, y moves up-down. The signs of the coordinates tell you the quadrant.
  1. Start at the origin (0, 0).
  2. First number (x): right if positive, left if negative.
  3. Second number (y): up if positive, down if negative.
  4. Quadrants: I (+,+), II (-,+), III (-,-), IV (+,-) — counterclockwise from upper right.
Worked example
Plot (3, -2)
Start at the origin. The x-coordinate 3 moves RIGHT 3; the y-coordinate -2 moves DOWN 2. Mark the point.
Worked example
Which quadrant is (-3, 5)?
x is negative, y is positive: that is the upper-left region — Quadrant II.
Watch out: Order matters! (3, -2) and (-2, 3) are completely different points. Always read x first, then y.
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60-Second Challenge

How many pre-algebra bridge problems can you solve in 60 seconds?