Prealgebra Reference

Unit 1 30 topics

Ratios & Percentages

Comparing quantities, rates, and parts of 100.

1.1 Ratios

Tape diagram. A ratio is a count of equal shares; find one share and everything follows.

1.1.1
Introduction to Ratios
A ratio compares two quantities in a set order, read “a to b.” Order matters: 3 : 2 is not 2 : 3.
e.g. 3 cats and 2 dogs → cats to dogs is $3:2$
$\displaystyle a : b$
1.1.2
Equivalent Ratios
Multiply or divide both parts by the same nonzero number and the comparison stays the same.
e.g. $2:3 = 4:6 = 10:15$
$\displaystyle a : b = ka : kb$
1.1.3
Writing Ratios Using Fractions
The ratio $a:b$ can be written $\tfrac{a}{b}$ . If $a:b$ is part to part, the whole has $a+b$ parts.
e.g. boys : girls $= 3:2$ → boys are $\tfrac{3}{5}$ of the class
$\displaystyle \frac{a}{a+b} = \text{fraction of the whole that is } a$
1.1.4
Reasoning With Equivalent Ratios
Find the scale factor between the parts you know, then apply it to the missing part.
e.g. $2:5 = 6:\,?$ → scale ×3 → $15$
$\displaystyle \frac{a}{b} = \frac{c}{d}$
1.1.5
Further Reasoning With Equivalent Ratios
Given a total, split it into $a+b$ equal shares and hand out $a$ shares and $b$ shares.
e.g. Split 40 in the ratio $3:5$ → share $=5$ → $15$ and $25$
$\displaystyle \text{one share} = \frac{\text{total}}{a+b}$
1.1.6
Ratio Tables
Each column is an equivalent ratio. Move between columns by multiplying, or add two columns together.
e.g. 2 : 3 → 4 : 6 → 6 : 9 → 8 : 12
1.1.7
Graphing Ratios
Plot equivalent ratios as points; they fall on a straight line through the origin.
e.g. $(2,3),(4,6),(6,9)$ all lie on $y=1.5x$
$\displaystyle y = \tfrac{b}{a}\,x$

1.2 Unit Rates

1.2.1
Unit Rates
A rate compares different units. A unit rate is the amount “per 1.” Divide by the second quantity.
e.g. $6 for 3 lb → $2 per lb
$\displaystyle \text{unit rate} = \frac{\text{amount}}{\text{number of units}}$
1.2.2
Solving Problems Using Unit Rates
Find the unit rate, then multiply. For best buy, the lower price per unit wins.
e.g. $2/lb × 7 lb = $14
$\displaystyle \text{total} = \text{unit rate} \times \text{units}$
1.2.3
Speed as a Unit Rate
Speed is distance per unit of time. Cover the one you want to find.
e.g. 150 miles in 3 hours → 50 mph
$\displaystyle s = \frac{d}{t}$
$\displaystyle d = s\,t$
$\displaystyle t = \frac{d}{s}$
1.2.4
Unit Rates Associated With Ratios of Fractions
Divide the fractions: multiply by the reciprocal of the bottom one.
e.g. $\tfrac12$ mile in $\tfrac14$ hour → $\tfrac12 \cdot \tfrac41 = 2$ mph
$\displaystyle \frac{a/b}{c/d} = \frac{a}{b} \cdot \frac{d}{c}$
1.2.5
Solving Unit Rate Problems Using Ratios of Fractions
Same move: get the rate “per 1 whole unit,” then scale up or down.
e.g. $\tfrac23$ cup per $\tfrac12$ batch → $\tfrac43$ cups per batch

1.3 Proportional Relationships

Solid: proportional, a straight line through the origin. Dashed: a starting amount lifts the line off the origin, so it is not proportional.

1.3.1
Understanding Proportional Relationships Using Tables
Proportional when $\tfrac{y}{x}$ is the same constant $k$ in every row.
e.g. $x$ : 2, 4, 5 / $y$ : 6, 12, 15 → $k = 3$ every time
$\displaystyle k = \frac{y}{x}$
1.3.2
Understanding Proportional Relationships Using Graphs
A straight line through the origin $(0,0)$ . The point $(1, k)$ shows the constant.
$\displaystyle y = kx$
1.3.3
Understanding Proportional Relationships From Descriptions
Look for a constant rate (“per,” “each”) with no starting amount. A fixed fee breaks proportionality.
e.g. $4 per ticket: yes. $4 per ticket + $5 fee: no.
$\displaystyle y = kx \;\text{(proportional)}$
$\displaystyle y = kx + b,\ b \neq 0 \;\text{(not proportional)}$

1.4 Percentages

Percent bar. The whole is always 100%. Line the percents up with the amounts: 25% of 80 is 20.

1.4.1
Understanding Percentages Using Models
Percent means “per hundred.” A 10 × 10 grid or a bar split into 100 shows it.
$\displaystyle p\% = \frac{p}{100}$
1.4.2
Converting Between Percentages and Fractions
Percent to fraction: put it over 100 and simplify. Fraction to percent: multiply by 100%.
$\displaystyle 35\% = \frac{35}{100} = \frac{7}{20}$
$\displaystyle \frac{3}{8} \times 100\% = 37.5\%$
1.4.3
Converting Between Percentages and Decimals
Move the decimal point two places: left to go from percent to decimal, right to go back.
$\displaystyle 45\% = 0.45$
$\displaystyle 0.075 = 7.5\%$
1.4.4
Comparing Fractions, Percents, and Decimals
Convert everything to one form (decimals are easiest), then compare.
e.g. $\tfrac{2}{5} = 0.4$ , $38\% = 0.38$ → $\tfrac25$ is larger
1.4.5
Finding Part of a Number Given a Whole and a Percentage
“Of” means multiply.
e.g. 20% of 60 $= 0.2 \times 60 = 12$
$\displaystyle \text{part} = \frac{p}{100} \times \text{whole}$
1.4.6
Finding Part of a Number: Word Problems
Identify the whole first; it usually follows the word “of.”
e.g. 15% tip on $40 → $0.15 \times 40 =$ $6
1.4.7
Finding a Percentage Given Two Numbers
Divide the part by the whole, then convert to a percent.
e.g. 12 out of 48 → $0.25$ → 25%
$\displaystyle \text{percent} = \frac{\text{part}}{\text{whole}} \times 100\%$
1.4.8
Finding a Percentage Given Two Numbers: Word Problems
The whole is the total, the original, or what follows “out of.”
e.g. 18 correct out of 24 → 75%
1.4.9
Finding a Total Given a Part and a Percentage
Divide the part by the percent as a decimal.
e.g. 30 is 40% of ? → $30 \div 0.4 = 75$
$\displaystyle \text{whole} = \frac{\text{part}}{p/100}$
1.4.10
Applying Percentages in Succession
Multiply the multipliers one after another. Never just add the percents.
e.g. +10% then −10% → $1.1 \times 0.9 = 0.99$ → 1% less than you started
$\displaystyle \text{new} = \text{original} \times m_1 \times m_2$

1.5 Percentage Change

Percent change is a multiplier. Up 25% means × 1.25; down 25% means × 0.75. To find the original, divide.

1.5.1
Applying Percentage Increases
Multiply by one plus the percent.
e.g. $80 up 25% → $80 \times 1.25 = 100$
$\displaystyle \text{new} = \text{old} \times \left(1 + \frac{p}{100}\right)$
1.5.2
Applying Percentage Decreases
Multiply by one minus the percent.
e.g. $80 down 25% → $80 \times 0.75 = 60$
$\displaystyle \text{new} = \text{old} \times \left(1 - \frac{p}{100}\right)$
1.5.3
Calculating Percentage Change
Change divided by the original. Always divide by the old value.
e.g. 50 → 65: $\tfrac{15}{50} = 30\%$ increase
$\displaystyle \%\text{ change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%$
1.5.4
Finding Original Values From Percentage Increases
Divide the new value by the multiplier.
e.g. 60 after a 20% rise → $60 \div 1.2 = 50$
$\displaystyle \text{old} = \frac{\text{new}}{1 + p/100}$
1.5.5
Finding Original Values From Percentage Decreases
Divide the new value by the multiplier.
e.g. 45 after a 25% drop → $45 \div 0.75 = 60$
$\displaystyle \text{old} = \frac{\text{new}}{1 - p/100}$

Unit 2 29 topics

The Number System

Negatives, fractions, and how whole numbers break into primes.

2.6 Addition and Subtraction of Rational Numbers

Adding a negative moves left. Subtracting a negative moves right, the same as adding.

2.6.1
Subtracting a Positive Number From a Smaller Positive Number
The answer is negative. Find the gap, then make it negative.
e.g. $3 - 8 = -5$
$\displaystyle a - b = -(b - a) \quad \text{when } b > a$
2.6.2
Subtracting a Positive Number From a Negative Number
Move further left. Add the sizes, keep the negative sign.
e.g. $-4 - 6 = -10$
$\displaystyle -a - b = -(a + b)$
2.6.3
Subtracting a Positive Fraction From a Smaller Fraction
Get a common denominator, then subtract the same way.
$\displaystyle \frac{1}{4} - \frac{2}{3} = \frac{3}{12} - \frac{8}{12} = -\frac{5}{12}$
2.6.4
Subtracting a Positive Decimal From a Smaller Decimal
Line up the decimal points, subtract larger minus smaller, attach the negative.
$\displaystyle 1.2 - 3.75 = -2.55$
2.6.5
Adding a Positive Number to a Negative Number
Signs differ: subtract the sizes and keep the sign of the bigger size.
$\displaystyle -7 + 3 = -4$
$\displaystyle -2 + 9 = 7$
2.6.6
Adding a Negative Number
Adding a negative is the same as subtracting.
e.g. $5 + (-8) = -3$
$\displaystyle a + (-b) = a - b$
2.6.7
Subtracting a Negative Number
Subtracting a negative is the same as adding.
e.g. $-2 - (-6) = 4$
$\displaystyle a - (-b) = a + b$
2.6.8
Additive Inverses of Numbers
A number and its opposite add to zero. They sit the same distance from 0 on opposite sides.
$\displaystyle a + (-a) = 0$
2.6.9
Interpreting Additive Opposites and Addition Using Absolute Value
Absolute value is distance from zero, so it is never negative. $a + b$ lands $\lvert b \rvert$ units from $a$ .
$\displaystyle \lvert a \rvert = \text{distance from } 0$
$\displaystyle \lvert -5 \rvert = \lvert 5 \rvert = 5$
2.6.10
Solving Problems Using Addition and Subtraction of Rational Numbers
Model it on a number line: right or up is +, left or down is −. Temperatures, elevation, bank balances.
e.g. 3°F falls 10° → $3 - 10 = -7$ °F

2.7 Multiplication and Division of Rational Numbers

Sign rules for multiplying and dividing.

2.7.1
Multiplying Positive Numbers With Negative Numbers
Different signs give a negative.
e.g. $4 \times (-3) = -12$
$\displaystyle (+)(-) = (-)$
2.7.2
Multiplying Negative Numbers
Same signs give a positive. An even count of negatives is positive; an odd count is negative.
e.g. $(-2)(-3)(-1) = -6$
$\displaystyle (-)(-) = (+)$
2.7.3
Dividing With Negative Numbers
Same sign rules as multiplication.
$\displaystyle \frac{-12}{4} = -3$
$\displaystyle \frac{-12}{-4} = 3$
2.7.4
Equivalent Fractions With Negative Numbers
One negative can sit in any of three places. Two negatives cancel.
$\displaystyle -\frac{a}{b} = \frac{-a}{b} = \frac{a}{-b}$
$\displaystyle \frac{-a}{-b} = \frac{a}{b}$
2.7.5
Dividing With Negative Decimals and Fractions
Settle the sign first, then divide the sizes. For fractions, multiply by the reciprocal.
e.g. $-\tfrac34 \div \tfrac12 = -\tfrac34 \cdot 2 = -\tfrac32$
2.7.6
Reciprocals of Rational Numbers
Flip the fraction. A number times its reciprocal is 1. The sign stays; zero has no reciprocal.
e.g. reciprocal of $-\tfrac25$ is $-\tfrac52$
$\displaystyle \frac{a}{b} \cdot \frac{b}{a} = 1$
2.7.7
Multiplying and Dividing With Zero
Anything times zero is zero. Zero divided by a number is zero. Dividing by zero is undefined.
$\displaystyle a \times 0 = 0$
$\displaystyle \frac{0}{a} = 0$
$\displaystyle \frac{a}{0} \text{ is undefined}$
2.7.8
Fractions of Fractions
“Of” means multiply. Multiply across.
e.g. $\tfrac23$ of $\tfrac34$ $= \tfrac{6}{12} = \tfrac12$
$\displaystyle \frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$
2.7.9
Solving Problems Using Multiplication and Division of Rational Numbers
Decide the operation, settle the sign, then compute.
e.g. Drops 1.5 m per hour for 4 hours → $-1.5 \times 4 = -6$ m
2.7.10
Solving Problems Using the Four Operations on Rational Numbers
Use order of operations and the sign rules together.
e.g. $-3 + 4 \times (-2) = -3 - 8 = -11$

2.8 Number Theory

Each set sits inside the next. Every natural number is an integer; every integer is rational (5 = 5⁄1).

2.8.1
Divisibility of Integers by 2, 5, 10, 4, and 8
Check the last digits.
$\displaystyle \begin{array}{ll} 2: & \text{last digit even} \\ 5: & \text{ends in 0 or 5} \\ 10: & \text{ends in 0} \\ 4: & \text{last two digits divisible by 4} \\ 8: & \text{last three digits divisible by 8} \end{array}$
2.8.2
Divisibility of Integers by 3, 9, 6, and 7
Check the digit sum, or combine rules.
e.g. 203: $20 - 2\cdot3 = 14$ → divisible by 7
$\displaystyle \begin{array}{ll} 3: & \text{digit sum divisible by 3} \\ 9: & \text{digit sum divisible by 9} \\ 6: & \text{divisible by 2 and by 3} \\ 7: & \text{double last digit, subtract from rest; repeat} \end{array}$
2.8.3
Natural Numbers, Integers, and Rational Numbers
Each set sits inside the next. Rational means it can be written as a fraction of integers.
$\displaystyle \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q}$
$\displaystyle \text{rational} = \frac{a}{b},\ a, b \text{ integers},\ b \ne 0$
2.8.4
Rational Numbers as Finite or Repeating Decimals
Every rational number either ends or repeats. It ends when the simplified denominator has only 2s and 5s as prime factors.
$\displaystyle \frac{3}{8} = 0.375$
$\displaystyle \frac{1}{3} = 0.\overline{3}$
$\displaystyle \frac{2}{11} = 0.\overline{18}$
2.8.5
Prime and Composite Numbers
Prime: exactly two factors, 1 and itself. Composite: more than two. 1 is neither. 2 is the only even prime.
e.g. Primes to 30: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
2.8.6
Prime Factors
Break a number down with a factor tree until every branch ends in a prime. Write repeats as powers.
$\displaystyle 60 = 2^2 \cdot 3 \cdot 5$
2.8.7
The Fundamental Theorem of Arithmetic
Every whole number greater than 1 has exactly one prime factorization, apart from the order of the factors.
e.g. However you split 60, you always end at $2 \cdot 2 \cdot 3 \cdot 5$
2.8.8
Finding Greatest Common Factors Using Prime Factorization
Take only the primes both share, each at its lowest power.
$\displaystyle 12 = 2^2 \cdot 3,\quad 18 = 2 \cdot 3^2$
$\displaystyle \text{GCF} = 2 \cdot 3 = 6$
2.8.9
Finding Lowest Common Multiples Using Prime Factorization
Take every prime that appears, each at its highest power.
$\displaystyle \text{LCM}(12, 18) = 2^2 \cdot 3^2 = 36$
$\displaystyle \text{GCF} \times \text{LCM} = a \times b$

Unit 3 35 topics

Exponents & Radicals

Repeated multiplication, very big and very small numbers, and roots.

3.9 Exponents

3.9.1
Squaring Rational Numbers
Multiply the number by itself. A negative squared is positive.
$\displaystyle x^2 = x \cdot x$
$\displaystyle \left(-\tfrac{2}{3}\right)^2 = \tfrac{4}{9}$
3.9.2
Cubing Rational Numbers
Three copies multiplied. A cube keeps the sign.
$\displaystyle x^3 = x \cdot x \cdot x$
$\displaystyle (-2)^3 = -8$
3.9.3
Exponents With Rational Bases
Raise the top and the bottom. Watch the parentheses: the exponent only touches what it is attached to.
$\displaystyle \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$
$\displaystyle -3^2 = -9 \quad (-3)^2 = 9$
3.9.4
The Zeroth Power
Any nonzero number to the zero power is 1.
e.g. $(-7)^0 = 1$
$\displaystyle a^0 = 1 \quad (a \ne 0)$
3.9.5
Powers of Negative One
Even power gives 1, odd power gives −1.
$\displaystyle (-1)^n = \begin{cases} 1 & n \text{ even} \\ -1 & n \text{ odd} \end{cases}$
3.9.6
Negative Exponents
A negative exponent means reciprocal. It does not make the number negative.
e.g. $2^{-3} = \tfrac18$
$\displaystyle a^{-n} = \frac{1}{a^n}$
$\displaystyle \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}$

3.10 The Rules of Exponents

Why the product rule works. Every exponent rule comes from counting copies.

3.10.1
The Product Rule for Exponents
Same base, multiplying: add the exponents.
e.g. $x^3 \cdot x^4 = x^7$
$\displaystyle a^m \cdot a^n = a^{m+n}$
3.10.2
The Quotient Rule for Exponents
Same base, dividing: subtract the exponents.
e.g. $\tfrac{x^7}{x^2} = x^5$
$\displaystyle \frac{a^m}{a^n} = a^{m-n}$
3.10.3
The Power Rule for Exponents
Power of a power: multiply the exponents.
e.g. $(x^2)^5 = x^{10}$
$\displaystyle (a^m)^n = a^{mn}$
3.10.4
The Power of Product Rule for Exponents
The exponent goes to every factor inside.
e.g. $(2x)^3 = 8x^3$
$\displaystyle (ab)^n = a^n b^n$
3.10.5
The Power of Quotient Rule for Exponents
The exponent goes to the top and the bottom.
$\displaystyle \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$
3.10.6
Combining the Rules of Exponents
Work inside parentheses, apply outside powers, combine like bases, finish with positive exponents.
e.g. $\dfrac{(x^2)^3 \cdot x}{x^4} = \dfrac{x^7}{x^4} = x^3$

3.11 Scientific Notation

Scientific notation: a number from 1 up to 10, times a power of ten that says how far the decimal moved.

3.11.1
Writing Numbers in Scientific Notation
A number from 1 up to (not including) 10, times a power of ten. Big numbers get positive powers; tiny numbers get negative powers.
$\displaystyle a \times 10^n,\quad 1 \le \lvert a \rvert < 10$
$\displaystyle 45{,}000 = 4.5 \times 10^4$
$\displaystyle 0.0032 = 3.2 \times 10^{-3}$
3.11.2
Converting Scientific Notation to Standard Notation
Positive power: move the decimal right $n$ places. Negative power: move it left.
e.g. $6.1 \times 10^{-4} = 0.00061$
3.11.3
Comparing Numbers in Scientific Notation
Compare the powers of ten first. Only if they match, compare the front numbers.
e.g. $2 \times 10^5 > 9 \times 10^4$
3.11.4
Addition and Subtraction in Scientific Notation
With the same power, add or subtract the front numbers and keep the power.
$\displaystyle (a \times 10^n) \pm (b \times 10^n) = (a \pm b) \times 10^n$
3.11.5
Addition in Scientific Notation With Different Powers of Ten
Rewrite one number so the powers match, add, then put it back in proper form.
$\displaystyle 3 \times 10^5 + 4 \times 10^4$
$\displaystyle = 3 \times 10^5 + 0.4 \times 10^5 = 3.4 \times 10^5$
3.11.6
Subtraction in Scientific Notation With Different Powers of Ten
Match the powers first, then subtract.
$\displaystyle 5 \times 10^3 - 2 \times 10^2$
$\displaystyle = 5 \times 10^3 - 0.2 \times 10^3 = 4.8 \times 10^3$
3.11.7
Further Addition in Scientific Notation With Different Powers of Ten
If the front number reaches 10 or more, shift the decimal and raise the power by one.
$\displaystyle 12 \times 10^4 = 1.2 \times 10^5$
3.11.8
Multiplying Numbers in Scientific Notation
Multiply the front numbers, add the powers, then fix the form.
e.g. $(3\times10^4)(5\times10^2) = 15\times10^6 = 1.5\times10^7$
$\displaystyle (a \times 10^m)(b \times 10^n) = ab \times 10^{m+n}$
3.11.9
Dividing Numbers in Scientific Notation
Divide the front numbers, subtract the powers.
$\displaystyle \frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}$

3.12 Order of Operations

Order of operations. M and D share a step, as do A and S: whichever comes first, reading left to right, goes first.

3.12.1
Evaluating Rational Number Expressions Without Exponents
Parentheses first. Then multiply and divide left to right. Then add and subtract left to right.
e.g. $8 - 6 \div 3 \times 2 = 8 - 4 = 4$
3.12.2
Evaluating Expressions With Positive Integer Exponents
PEMDAS: exponents come right after parentheses.
e.g. $2 + 3 \cdot 4^2 = 2 + 48 = 50$
$\displaystyle \text{P} \to \text{E} \to \text{MD} \to \text{AS}$
3.12.3
Evaluating Expressions With Integer Exponents
Handle negative exponents at the exponent step by turning them into reciprocals.
e.g. $2 + 3 \cdot 2^{-1} = 2 + \tfrac32 = \tfrac72$

3.13 Radicals

3.13.1
The Square Root of a Perfect Square
$\sqrt{\ }$ asks “what nonnegative number squared gives this?” Know squares through 15² = 225.
e.g. $\sqrt{49} = 7$ , $\sqrt{\tfrac{9}{16}} = \tfrac34$
$\displaystyle \sqrt{a^2} = a \quad (a \ge 0)$
3.13.2
Squaring a Square Root
Squaring undoes the square root.
e.g. $(\sqrt{7})^2 = 7$
$\displaystyle \left(\sqrt{a}\right)^2 = a \quad (a \ge 0)$
3.13.3
The Cube Root of a Perfect Cube
Asks “what cubed gives this?” Negatives are allowed and stay negative.
$\displaystyle \sqrt[3]{a^3} = a$
$\displaystyle \sqrt[3]{-27} = -3$
3.13.4
Surds
A root that never comes out to a whole number or fraction, like $\sqrt2$ . It is irrational, so leave it in exact form.
e.g. $\sqrt2, \sqrt3, \sqrt5$ are surds; $\sqrt9$ is not
3.13.5
Simplifying Expressions With Surds
Pull out the largest perfect-square factor.
$\displaystyle \sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$
3.13.6
Evaluating Numerical Expressions Containing Radicals
A radical acts like parentheses: simplify everything under it first.
e.g. $3 + \sqrt{25 - 9} = 3 + 4 = 7$
3.13.7
Adding and Subtracting Radicals
Only like radicals (same number under the root) combine. Simplify each one first.
$\displaystyle 3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$
$\displaystyle \sqrt{8} + \sqrt{2} = 2\sqrt2 + \sqrt2 = 3\sqrt{2}$
3.13.8
Rationalizing Denominators
Clear a root from the bottom by multiplying top and bottom by that root.
e.g. $\tfrac{6}{\sqrt3} = \tfrac{6\sqrt3}{3} = 2\sqrt3$
$\displaystyle \frac{a}{\sqrt{b}} = \frac{a\sqrt{b}}{b}$

3.14 The Rules of Radicals

3.14.1
Radicals as Fractional Exponents
A root is a fractional power. The bottom of the fraction is the root.
e.g. $8^{2/3} = (\sqrt[3]{8})^2 = 4$
$\displaystyle \sqrt[n]{a} = a^{1/n}$
$\displaystyle \sqrt[n]{a^m} = a^{m/n}$
3.14.2
The Product Rule for Radicals
The root of a product is the product of the roots.
e.g. $\sqrt{3}\sqrt{12} = \sqrt{36} = 6$
$\displaystyle \sqrt{ab} = \sqrt{a}\,\sqrt{b} \quad (a, b \ge 0)$
3.14.3
The Quotient Rule for Radicals
The root of a quotient is the quotient of the roots.
e.g. $\sqrt{\tfrac{25}{4}} = \tfrac52$
$\displaystyle \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

Unit 4 17 topics

Algebraic Expressions

Letters standing in for numbers, and how to tidy them up.

4.15 Algebraic Expressions

Parts of an expression.

4.15.1
Introduction to Algebraic Expressions
Numbers, variables, and operations with no equals sign. A variable stands for a number you don't know yet.
$\displaystyle 3x + 5$
4.15.2
Constructing Algebraic Expressions
Translate the words. Sum +, difference −, product ×, quotient ÷. “Less than” and “more than” flip the order.
e.g. “5 less than twice $n$ ” → $2n - 5$
4.15.3
Evaluating Linear Expressions
Substitute the value in parentheses, then follow order of operations.
$\displaystyle 3x - 4,\ x = -2$
$\displaystyle 3(-2) - 4 = -10$
4.15.4
Evaluating Rational Expressions
Work out the top and the bottom separately, then divide. The bottom can never be 0.
e.g. $\dfrac{x+4}{x-1}$ at $x=3$ → $\tfrac72$
4.15.5
Identifying Terms in Linear Expressions
Terms are the pieces separated by + or −. The sign in front belongs to the term.
e.g. $4x - 7y + 2$ has terms $4x$ , $-7y$ , $2$
4.15.6
Identifying Coefficients and Constants
The coefficient is the number multiplying a variable. The constant has no variable. A lone $x$ has coefficient 1.
e.g. In $-5x + 8$ : coefficient $-5$ , constant $8$
4.15.7
The Greatest Common Factor of Two Linear Expressions
Factor each fully, then keep what they share.
$\displaystyle 6x + 9 = 3(2x + 3)$
$\displaystyle 12x + 18 = 6(2x + 3)$
$\displaystyle \text{GCF} = 3(2x + 3)$

4.16 Simplifying Linear Expressions

Area model. The outside number multiplies every piece inside. Factoring reads the picture right to left.

4.16.1
Identifying Like Terms in Linear Expressions
Like terms have the same variable to the same power. Constants are like each other.
e.g. $3x$ and $-5x$ : like. $3x$ and $3y$ : not.
4.16.2
Collecting Like Terms in Linear Expressions
Add the coefficients of like terms; the variable part stays.
$\displaystyle 3x + 5 - x + 2 = 2x + 7$
4.16.3
Applying the Distributive Law With Linear Expressions
Multiply the outside number by every term inside.
e.g. $4(x + 3) = 4x + 12$
$\displaystyle a(b + c) = ab + ac$
4.16.4
The Distributive Law With Three Terms
Same rule, one more term.
e.g. $2(x - y + 5) = 2x - 2y + 10$
$\displaystyle a(b + c + d) = ab + ac + ad$
4.16.5
Distributing the Negative Sign in a Linear Expression
A minus in front flips the sign of every term inside.
$\displaystyle -(a - b + c) = -a + b - c$
4.16.6
Simplifying Linear Expressions Using the Distributive Law
Distribute first, then collect like terms.
$\displaystyle 2(x + 3) + 4x = 6x + 6$
4.16.7
Simplifying Linear Expressions When a Group of Terms Is Subtracted
Distribute the −1 (or −k) to every term in the group.
$\displaystyle 5x - (2x - 3) = 5x - 2x + 3 = 3x + 3$
4.16.8
Simplifying Linear Expressions Containing Fractions
Distribute, then combine with a common denominator.
$\displaystyle \tfrac{1}{2}(4x - 6) = 2x - 3$
$\displaystyle \tfrac{x}{2} + \tfrac{x}{3} = \tfrac{5x}{6}$
4.16.9
Factoring Numerical Expressions
Pull out the GCF. Factoring is distributing in reverse.
$\displaystyle 12 + 18 = 6(2 + 3)$
4.16.10
Factoring Linear Expressions
Find the GCF of the terms and write it outside.
$\displaystyle ab + ac = a(b + c)$
$\displaystyle 8x - 12 = 4(2x - 3)$

Unit 5 29 topics

Equations & Inequalities

Finding the unknown, and graphing the lines equations make.

5.17 Solving One-Step Linear Equations

An equation is a balance. Whatever you do to one side, do to the other.

5.17.1
Solving Linear Equations by Trial and Error
Guess a value, substitute, check whether both sides match, adjust the guess.
e.g. $3x + 1 = 13$ : try 4 → $13$ ✓
5.17.2
Solving One-Step Addition and Subtraction Equations
Do the opposite operation to both sides.
$\displaystyle x + a = b \Rightarrow x = b - a$
$\displaystyle x - a = b \Rightarrow x = b + a$
5.17.3
Solving One-Step Multiplication and Division Equations
Divide to undo multiplying, multiply to undo dividing.
$\displaystyle ax = b \Rightarrow x = \frac{b}{a}$
$\displaystyle \frac{x}{a} = b \Rightarrow x = ab$

5.18 Solving Multi-Step Linear Equations

5.18.1
Solving Two-Step Equations
Undo adding or subtracting first, then undo multiplying or dividing.
e.g. $3x + 4 = 19 \Rightarrow 3x = 15 \Rightarrow x = 5$
$\displaystyle ax + b = c \Rightarrow x = \frac{c - b}{a}$
5.18.2
Solving Linear Equations With Fractional Coefficients
Multiply both sides by the reciprocal of the coefficient.
$\displaystyle \tfrac{2}{3}x = 8 \Rightarrow x = 8 \cdot \tfrac{3}{2} = 12$
5.18.3
Solving Linear Equations With Decimal Coefficients
Multiply both sides by 10, 100, … to clear the decimals, then solve.
$\displaystyle 0.3x + 1.2 = 3 \xrightarrow{\times 10} 3x + 12 = 30$
5.18.4
Solving Linear Equations With Variables on Both Sides
Gather the variable terms on one side and the numbers on the other.
$\displaystyle 5x + 2 = 3x + 10$
$\displaystyle 2x = 8 \Rightarrow x = 4$
5.18.5
Solving Linear Equations by Clearing a Rational Expression
Multiply every term by the LCD of all the denominators.
$\displaystyle \frac{x}{3} + \frac{x}{4} = 7 \xrightarrow{\times 12} 4x + 3x = 84$
5.18.6
Solving Linear Equations Using Cross-Multiplication
When one fraction equals another, multiply diagonally.
e.g. $\tfrac{x}{4} = \tfrac{9}{6} \Rightarrow 6x = 36 \Rightarrow x = 6$
$\displaystyle \frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc$
5.18.7
Solving Two-Variable Equations
Solve for one variable in terms of the other, or plug in a given value.
$\displaystyle 2x + y = 10 \Rightarrow y = 10 - 2x$
5.18.8
Linear Equations With Infinitely Many Solutions
The variables cancel and leave something always true. Every number works.
$\displaystyle 2(x + 1) = 2x + 2 \;\to\; 2 = 2$
5.18.9
Linear Equations With No Solutions
The variables cancel and leave something false. No number works.
$\displaystyle 3x + 1 = 3x + 5 \;\to\; 1 = 5$

5.19 Solving Linear Inequalities

Open circle for < and >. Closed circle for ≤ and ≥. Multiply or divide by a negative and the sign flips.

5.19.1
Inequalities
A statement that one side is bigger or smaller. Its solution is a range of numbers.
$\displaystyle < \;\; > \;\; \le \;\; \ge$
5.19.2
Compound Inequalities
“And” means both are true, so $x$ is between. “Or” means at least one is true.
$\displaystyle -2 < x \le 5$
$\displaystyle x < -1 \;\text{or}\; x > 4$
5.19.3
Verifying Solutions of Linear Inequalities
Substitute the value and check whether the statement is true.
e.g. Is 3 a solution of $2x - 1 > 4$ ? $5 > 4$ ✓
5.19.4
Solving One-Step Linear Inequalities
Solve like an equation, with one rule: multiplying or dividing by a negative flips the sign.
$\displaystyle -3x < 12 \Rightarrow x > -4$
5.19.5
Solving Two-Step Inequalities
Same order as a two-step equation. Watch for the flip.
$\displaystyle 5 - 2x \ge 11 \Rightarrow -2x \ge 6 \Rightarrow x \le -3$
5.19.6
Representing Solutions to Inequalities on Number Lines
Open circle for < or >. Closed circle for ≤ or ≥. Shade toward the solutions.
5.19.7
Further Solving Linear Inequalities
Distribute, collect, move variables to one side; flip if you divide by a negative.
$\displaystyle 3(x - 2) > 5x + 4$
$\displaystyle -2x > 10 \Rightarrow x < -5$

5.20 Graphs of Linear Equations

Slope-intercept form. Start at b on the y-axis, then step by the slope: up the rise, over the run.

5.20.1
The Cartesian Coordinate System
A point is $(x, y)$ : across first, then up. Four quadrants, numbered counterclockwise from upper right.
e.g. Quadrant I: $(+,+)$ , II: $(-,+)$ , III: $(-,-)$ , IV: $(+,-)$
$\displaystyle (x, y)$
5.20.2
Distances Between Points in the Coordinate Plane
Points that share an $x$ or a $y$ : subtract the other coordinates and take the absolute value.
e.g. $(3, -2)$ to $(3, 5)$ → $7$
$\displaystyle \text{same } x: \lvert y_2 - y_1 \rvert$
$\displaystyle \text{same } y: \lvert x_2 - x_1 \rvert$
5.20.3
Two-Variable Linear Equations and Their Solutions
A solution is an $(x, y)$ pair that makes the equation true. All solutions together form a line.
e.g. $(1, 8)$ solves $y = 2x + 6$
5.20.4
Graphing Linear Equations
Pick two or three $x$ values, find each $y$ , plot the points, draw the line through them.
5.20.5
Horizontal and Vertical Lines
$y = b$ is horizontal with slope 0. $x = a$ is vertical with undefined slope.
$\displaystyle y = b$
$\displaystyle x = a$
5.20.6
Calculating Slopes of Straight Lines
Slope is rise over run between any two points.
e.g. $(1, 2)$ to $(4, 8)$ → $m = \tfrac63 = 2$
$\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}$
5.20.7
Equations of Lines in Slope-Intercept Form
$m$ is the slope; $b$ is where the line crosses the $y$ -axis.
$\displaystyle y = mx + b$
5.20.8
Properties of Lines Given in Slope-Intercept Form
Positive $m$ rises, negative $m$ falls, bigger $\lvert m \rvert$ is steeper. Parallel lines have equal slopes.
e.g. $y = 3x - 1$ and $y = 3x + 4$ are parallel

5.21 Systems of Equations

A system’s solution is where the lines meet. Two lines can meet once, never, or everywhere.

5.21.1
Introduction to Systems of Linear Equations
Two equations sharing variables. The solution is where the lines cross: one point, none (parallel), or infinitely many (same line).
5.21.2
Solving Systems of Equations by Substitution
Solve one equation for a variable, substitute into the other, solve, then plug back in.
$\displaystyle y = 2x,\quad x + y = 9$
$\displaystyle x + 2x = 9 \Rightarrow x = 3,\ y = 6$

Unit 6 48 topics

Geometry

Points, lines, angles, shapes, and the space they cover.

6.22 Introduction to Geometry

Line, ray, segment. The arrowheads say which way it keeps going.

6.22.1
Units of Area and Volume
Area is in square units, volume in cubic units. To convert, square or cube the length conversion.
$\displaystyle 1\text{ ft}^2 = 12^2 = 144\text{ in}^2$
$\displaystyle 1\text{ yd}^3 = 3^3 = 27\text{ ft}^3$
6.22.2
Points, Lines, Rays, and Segments
A line runs forever both ways. A ray starts at a point and runs one way. A segment has two endpoints.
$\displaystyle \overleftrightarrow{AB} \qquad \overrightarrow{AB} \qquad \overline{AB}$
6.22.3
Parallel and Perpendicular Lines
Parallel lines never meet. Perpendicular lines meet at a right angle.
$\displaystyle \ell \parallel m \qquad \ell \perp m$
6.22.4
Measures of Segments
$AB$ with no bar means the length of $\overline{AB}$ , a number.
$\displaystyle AB = \text{length of } \overline{AB}$
6.22.5
Congruent Segments
Same length. Marked with matching tick marks.
$\displaystyle \overline{AB} \cong \overline{CD} \iff AB = CD$
6.22.6
Midpoints
The point that splits a segment into two equal halves.
$\displaystyle AM = MB = \tfrac{1}{2}AB$
6.22.7
Collinear Points
Points that all lie on one line. Any two points are always collinear.
6.22.8
The Segment Addition Postulate
If $B$ is on the segment between $A$ and $C$ , the pieces add to the whole.
e.g. $AB = 4$ , $BC = 7$ → $AC = 11$
$\displaystyle AB + BC = AC$

6.23 Angles

Angle types by size. A full turn is 360°; a null angle is 0°.

6.23.1
Angles and Measures of Angles
Two rays from one point (the vertex). Measured in degrees. The vertex goes in the middle of the name.
$\displaystyle m\angle ABC$
6.23.2
Connecting Angles and Circles
A full turn is 360°. A degree is $\tfrac{1}{360}$ of a full turn.
e.g. Quarter turn $= 90^\circ$
$\displaystyle \text{full turn} = 360^\circ$
6.23.3
Measuring Angles Using a Protractor
Center on the vertex, line up 0 with one ray, read the scale that starts at 0 on that ray.
6.23.4
Sums of Angles
Angles side by side add up.
$\displaystyle m\angle ABD = m\angle ABC + m\angle CBD$
6.23.5
Right, Straight, Full, and Null Angles
Right 90°, straight 180°, full 360°, null 0°.
$\displaystyle 0^\circ \quad 90^\circ \quad 180^\circ \quad 360^\circ$
6.23.6
Acute, Obtuse, and Reflex Angles
Acute is less than 90°. Obtuse is between 90° and 180°. Reflex is between 180° and 360°.
6.23.7
Congruent Angles
Same measure. Marked with matching arcs.
$\displaystyle \angle A \cong \angle B \iff m\angle A = m\angle B$
6.23.8
Vertical Angles
When two lines cross, the opposite angles are equal.
6.23.9
Solving Problems With Angles
Write an equation from a known relationship (sum, equal, straight line), then solve.
e.g. $2x + 40 = 180 \Rightarrow x = 70$

6.24 Advanced Angles

Parallel lines cut by a transversal. Highlighted: 3 and 5, a consecutive pair that adds to 180°.

6.24.1
Complementary Angles
Two angles that add to 90°.
$\displaystyle \alpha + \beta = 90^\circ$
6.24.2
Non-Adjacent Complementary Angles
They still add to 90° even when they are not side by side.
6.24.3
Supplementary Angles
Two angles that add to 180°. Side by side on a line, they form a linear pair.
$\displaystyle \alpha + \beta = 180^\circ$
6.24.4
Corresponding Angles
Same position at each crossing of the transversal. Equal when the lines are parallel.
6.24.5
Alternate Angles
On opposite sides of the transversal, both inside (or both outside) the lines. Equal when the lines are parallel.
6.24.6
Consecutive Angles
Same side of the transversal, both inside the lines. Supplementary when the lines are parallel.
$\displaystyle \alpha + \beta = 180^\circ$
6.24.7
Angle Criteria for Parallel Lines
Run it backward: equal corresponding angles, equal alternate angles, or supplementary consecutive angles prove the lines are parallel.
6.24.8
Angle Bisectors
A ray that splits an angle into two equal angles.
e.g. Bisect 70° → two 35° angles
6.24.9
Segment Bisectors and the Perpendicular Bisector Theorem
A segment bisector passes through the midpoint. Any point on the perpendicular bisector is the same distance from both endpoints.
$\displaystyle PA = PB$

6.25 Triangles

The two triangle angle facts.

6.25.1
Interior Angles of Triangles
The three inside angles always add to 180°.
$\displaystyle a + b + c = 180^\circ$
6.25.2
Exterior Angles of Triangles
An outside angle equals the two inside angles not touching it.
$\displaystyle d = a + b$
6.25.3
Problem Solving With Angles of Triangles
Combine the 180° sum, the exterior angle rule, and straight-line facts.
6.25.4
Classifying Triangles
By sides: equilateral (3 equal), isosceles (2 equal), scalene (none). By angles: acute, right, obtuse.
6.25.5
The Triangle Inequality
Any two sides together are longer than the third.
e.g. 3, 4, 8 fails: $3 + 4 < 8$
$\displaystyle a + b > c$
6.25.6
The Isosceles Triangle Theorem
Two equal sides means the two base angles are equal, and the reverse is true too.
6.25.7
Heights of Triangles
A height runs from a vertex perpendicular to the line holding the opposite side. In an obtuse triangle it can land outside.

6.26 Polygons

6.26.1
Polygons
A closed shape made of straight sides. Named by side count: 3 triangle, 4 quadrilateral, 5 pentagon, 6 hexagon, 8 octagon, 10 decagon.
6.26.2
Interior Angles of Polygons
Split into $n - 2$ triangles from one vertex.
e.g. Hexagon: $4 \times 180 = 720^\circ$
$\displaystyle \text{sum} = (n - 2) \times 180^\circ$
6.26.3
Exterior Angles of Polygons
One at each vertex, and they always add to 360°, whatever the number of sides.
$\displaystyle \text{sum} = 360^\circ$
6.26.4
Congruent Polygons
Same shape and size: every matching side and angle is equal. List vertices in matching order.
6.26.5
Regular Polygons
All sides equal and all angles equal.
e.g. Regular octagon: $135^\circ$ inside
$\displaystyle \text{each interior} = \frac{(n - 2) \times 180^\circ}{n}$
$\displaystyle \text{each exterior} = \frac{360^\circ}{n}$
6.26.6
The Perimeter of a Polygon
Add up the sides. For a regular polygon, multiply.
$\displaystyle P = n \cdot s \quad \text{(regular)}$

6.27 Quadrilaterals

Quadrilateral family tree. Each shape has every property of the shapes above it: a square is a rectangle and a rhombus.

6.27.1
Classifying Quadrilaterals
A parallelogram has two pairs of parallel sides. A rectangle adds right angles, a rhombus adds equal sides, and a square has both. A trapezoid has one pair of parallel sides.
6.27.2
Properties of Rectangles and Squares
Opposite sides equal and parallel, four right angles, equal diagonals that cut each other in half. A square’s diagonals also meet at right angles.
6.27.3
Properties of Trapezoids
The parallel sides are the bases; the others are legs. An isosceles trapezoid has equal legs, equal base angles, and equal diagonals.

6.28 Area and Perimeter

Height is always measured straight up from the base, at a right angle. Never use the slanted side.

6.28.1
Areas of Rectangles and Squares
Length times width.
$\displaystyle A = lw$
$\displaystyle A = s^2$
6.28.2
Areas of Parallelograms
Base times the perpendicular height, not the slanted side.
$\displaystyle A = bh$
6.28.3
Areas of Triangles
Half of a parallelogram.
$\displaystyle A = \tfrac{1}{2}bh$
6.28.4
Areas of Trapezoids
Average the two bases, times the height.
$\displaystyle A = \tfrac{1}{2}(b_1 + b_2)h$
6.28.5
Areas and Perimeters of Composite Shapes
Split into simple shapes and add the areas. For perimeter, count only the outside edges.
6.28.6
The Area Between Two Shapes
Big area minus small area.
$\displaystyle A = A_{\text{outer}} - A_{\text{inner}}$

Unit 7 22 topics

Statistics

Summarizing a pile of numbers with a few that tell the story.

7.29 Summarizing Data

7.29.1
Introduction to Statistics
Statistics is collecting, summarizing, and interpreting data. A statistical question expects answers that vary.
e.g. “How tall are 8th graders?” is statistical; “How tall is Sam?” is not
7.29.2
The Mean of a Data Set
Add the values, divide by how many there are.
e.g. $\{2, 4, 9\}$ → $\tfrac{15}{3} = 5$
$\displaystyle \bar{x} = \frac{\sum x}{n}$
7.29.3
The Median of a Data Set
Sort, then take the middle value. With an even count, average the middle two.
e.g. $\{1, 3, 4, 8\}$ → $\tfrac{3+4}{2} = 3.5$
7.29.4
The Mode of a Data Set
The value that shows up most. There can be several, or none.
e.g. $\{2, 3, 3, 7\}$ → mode 3
7.29.5
Range, Quartiles, and IQR
Range is max − min. Q1 is the median of the lower half, Q3 of the upper half. IQR is the middle 50%.
$\displaystyle \text{Range} = \max - \min$
$\displaystyle \text{IQR} = Q_3 - Q_1$
7.29.6
Mean Absolute Deviation
The average distance of each value from the mean.
e.g. $\{2, 4, 9\}$ , mean 5 → $\tfrac{3+1+4}{3} = \tfrac83$
$\displaystyle \text{MAD} = \frac{\sum \lvert x - \bar{x} \rvert}{n}$
7.29.7
Comparing Data Sets Using Measures of Centrality
Compare means or medians to say which set is typically higher. Use medians when there are outliers.
7.29.8
Comparing Data Sets Using Range and Quartiles
A larger range or IQR means the data is more spread out. IQR ignores extremes; range doesn't.
7.29.9
Comparing Data Set Visualizations Using Standard Deviation and MAD
The plot whose values sit farther from the center has the larger SD and MAD.
7.29.10
Comparing Data Sets Using Standard Deviation and MAD
Both measure typical distance from the mean; bigger means more spread. SD squares the distances, so it reacts more to far-off values.
$\displaystyle \sigma = \sqrt{\frac{\sum (x - \bar{x})^2}{n}}$
7.29.11
Outliers
A value far from the rest. The usual test is the 1.5 × IQR fence.
$\displaystyle \text{below } Q_1 - 1.5 \cdot \text{IQR}$
$\displaystyle \text{above } Q_3 + 1.5 \cdot \text{IQR}$
7.29.12
Removing Outliers
Dropping an outlier moves the mean, range, and SD a lot; the median and IQR barely move.

7.30 Representing Data

Box plot. Each of the four pieces (whisker, half-box, half-box, whisker) holds about a quarter of the data.

7.30.1
Frequency Tables
Each value (or interval) with a count of how often it occurs. The counts add up to $n$ .
7.30.2
Dot Plots
One dot per data point, stacked above a number line.
7.30.3
Measuring Centrality From Dot Plots
Count the dots to get $n$ . The tallest stack is the mode; count in from either end to find the median.
7.30.4
Measuring Range and MAD From Dot Plots
Range is the distance between the outermost dots. For MAD, find each dot’s distance from the mean.
7.30.5
Measuring Quartiles and IQR From Dot Plots
Find the median dot, then the middle of each half.
7.30.6
Symmetry, Skew, and Outliers
Symmetric: mean ≈ median. Skewed right (long tail right): mean > median. Skewed left: mean < median.
7.30.7
Box Plots
Draw the five-number summary. The box spans Q1 to Q3 with a line at the median; whiskers reach min and max.
$\displaystyle \min,\ Q_1,\ \text{median},\ Q_3,\ \max$
7.30.8
Histograms
Bars over equal-width intervals (bins). Height is the count. Bars touch because the intervals are continuous.
7.30.9
Comparing Measures of Center
Symmetric with no outliers: use the mean. Skewed or with outliers: use the median.
7.30.10
Comparing Measures of Spread
Symmetric: use SD or MAD (they pair with the mean). Skewed: use IQR (it pairs with the median).

No topic matches that search. Try a single word, like “area” or “exponent.”

Topic numbers match the course outline, so a topic here lines up with the same number in the lesson list.

1.1 Ratios

Introduction to Ratios

Equivalent Ratios

Writing Ratios Using Fractions

Reasoning With Equivalent Ratios

Further Reasoning With Equivalent Ratios

Ratio Tables

Graphing Ratios

1.2 Unit Rates

Unit Rates

Solving Problems Using Unit Rates

Speed as a Unit Rate

Unit Rates Associated With Ratios of Fractions

Solving Unit Rate Problems Using Ratios of Fractions

1.3 Proportional Relationships

Understanding Proportional Relationships Using Tables

Understanding Proportional Relationships Using Graphs

Understanding Proportional Relationships From Descriptions

1.4 Percentages

Understanding Percentages Using Models

Converting Between Percentages and Fractions

Converting Between Percentages and Decimals

Comparing Fractions, Percents, and Decimals

Finding Part of a Number Given a Whole and a Percentage

Finding Part of a Number: Word Problems

Finding a Percentage Given Two Numbers

Finding a Percentage Given Two Numbers: Word Problems

Finding a Total Given a Part and a Percentage

Applying Percentages in Succession

1.5 Percentage Change

Applying Percentage Increases

Applying Percentage Decreases

Calculating Percentage Change

Finding Original Values From Percentage Increases

Finding Original Values From Percentage Decreases

2.6 Addition and Subtraction of Rational Numbers

Subtracting a Positive Number From a Smaller Positive Number

Subtracting a Positive Number From a Negative Number

Subtracting a Positive Fraction From a Smaller Fraction

Subtracting a Positive Decimal From a Smaller Decimal

Adding a Positive Number to a Negative Number

Adding a Negative Number

Subtracting a Negative Number

Additive Inverses of Numbers

Interpreting Additive Opposites and Addition Using Absolute Value

Solving Problems Using Addition and Subtraction of Rational Numbers

2.7 Multiplication and Division of Rational Numbers

Multiplying Positive Numbers With Negative Numbers

Multiplying Negative Numbers

Dividing With Negative Numbers

Equivalent Fractions With Negative Numbers

Dividing With Negative Decimals and Fractions

Reciprocals of Rational Numbers

Multiplying and Dividing With Zero

Fractions of Fractions

Solving Problems Using Multiplication and Division of Rational Numbers

Solving Problems Using the Four Operations on Rational Numbers

2.8 Number Theory

Divisibility of Integers by 2, 5, 10, 4, and 8

Divisibility of Integers by 3, 9, 6, and 7

Natural Numbers, Integers, and Rational Numbers

Rational Numbers as Finite or Repeating Decimals

Prime and Composite Numbers

Prime Factors

The Fundamental Theorem of Arithmetic

Finding Greatest Common Factors Using Prime Factorization

Finding Lowest Common Multiples Using Prime Factorization

3.9 Exponents

Squaring Rational Numbers

Cubing Rational Numbers

Exponents With Rational Bases

The Zeroth Power

Powers of Negative One

Negative Exponents

3.10 The Rules of Exponents

The Product Rule for Exponents

The Quotient Rule for Exponents

The Power Rule for Exponents

The Power of Product Rule for Exponents

The Power of Quotient Rule for Exponents