Lynn Marecek; MaryAnne Anthony-Smith

Key Concepts

1.1 Introduction to Whole Numbers

Place Value as in Figure 1.3.
Name a Whole Number in Words
1. Step 1. Start at the left and name the number in each period, followed by the period name.
2. Step 2. Put commas in the number to separate the periods.
3. Step 3. Do not name the ones period.
Write a Whole Number Using Digits
1. Step 1. Identify the words that indicate periods. (Remember the ones period is never named.)
2. Step 2. Draw 3 blanks to indicate the number of places needed in each period. Separate the periods by commas.
3. Step 3. Name the number in each period and place the digits in the correct place value position.
Round Whole Numbers
1. Step 1. Locate the given place value and mark it with an arrow. All digits to the left of the arrow do not change.
2. Step 2. Underline the digit to the right of the given place value.
3. Step 3.
  Is this digit greater than or equal to 5?
  - Yes—add 1 to the digit in the given place value.
  - No—do not change the digit in the given place value.
4. Step 4. Replace all digits to the right of the given place value with zeros.
Divisibility Tests: A number is divisible by:
- 2 if the last digit is 0, 2, 4, 6, or 8.
- 3 if the sum of the digits is divisible by 3.
- 5 if the last digit is 5 or 0.
- 6 if it is divisible by both 2 and 3.
- 10 if it ends with 0.
Find the Prime Factorization of a Composite Number
1. Step 1. Find two factors whose product is the given number, and use these numbers to create two branches.
2. Step 2. If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree.
3. Step 3. If a factor is not prime, write it as the product of two factors and continue the process.
4. Step 4. Write the composite number as the product of all the circled primes.
Find the Least Common Multiple by Listing Multiples
1. Step 1. List several multiples of each number.
2. Step 2. Look for the smallest number that appears on both lists.
3. Step 3. This number is the LCM.
Find the Least Common Multiple Using the Prime Factors Method
1. Step 1. Write each number as a product of primes.
2. Step 2. List the primes of each number. Match primes vertically when possible.
3. Step 3. Bring down the columns.
4. Step 4. Multiply the factors.

1.2 Use the Language of Algebra

Notation The result is…

$\begin{matrix} \circ a + b & the sum of a and b \\ \circ a - b & the difference of a and b \\ \circ a \cdot b, a b, (a) (b) (a) b, a (b) & the product of a and b \\ \circ a \div b, a / b, \frac{a}{b}, b a & the quotient of a and b \end{matrix}$
Inequality

$\begin{matrix} \circ a < b is read “ a is less than b ” & a is to the left of b on the number line \\ \circ a > b is read “ a is greater than b ” & a is to the right of b on the number line \end{matrix}$
Inequality Symbols Words

$\begin{matrix} \circ a \neq b & a is not equal to b \\ \circ a < b & a is less than b \\ \circ a \leq b & a is less than or equal to b \\ \circ a > b & a is greater than b \\ \circ a \geq b & a is greater than or equal to b \end{matrix}$
Grouping Symbols
- Parentheses $()$
- Brackets $[]$
- Braces ${}$
Exponential Notation
- $a^{n}$ means multiply $a$ by itself, $n$ times. The expression $a^{n}$ is read $a$ to the $n^{t h}$ power.
Order of Operations: When simplifying mathematical expressions perform the operations in the following order:
1. Step 1. Parentheses and other Grouping Symbols: Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
2. Step 2. Exponents: Simplify all expressions with exponents.
3. Step 3. Multiplication and Division: Perform all multiplication and division in order from left to right. These operations have equal priority.
4. Step 4. Addition and Subtraction: Perform all addition and subtraction in order from left to right. These operations have equal priority.
Combine Like Terms
1. Step 1. Identify like terms.
2. Step 2. Rearrange the expression so like terms are together.
3. Step 3. Add or subtract the coefficients and keep the same variable for each group of like terms.

1.3 Add and Subtract Integers

Addition of Positive and Negative Integers
$\begin{matrix} 5 + 3 & - 5 + (−3) \\ 8 & −8 \\ both positive, & both negative, \\ sum positive & sum negative \\ −5 + 3 & 5 + (−3) \\ −2 & 2 \\ different signs, & different signs, \\ more negatives & more positives \\ sum negative & sum positive \end{matrix}$
Property of Absolute Value: $| n | \geq 0$ for all numbers. Absolute values are always greater than or equal to zero!
Subtraction of Integers
$\begin{matrix} 5 - 3 & −5 - (−3) \\ 2 & −2 \\ 5 positives & 5 negatives \\ take away 3 positives & take away 3 negatives \\ 2 positives & 2 negatives \\ −5 - 3 & 5 - (−3) \\ −8 & 8 \\ 5 negatives, want to & 5 positives, want to \\ subtract 3 positives & subtract 3 negatives \\ need neutral pairs & need neutral pairs \end{matrix}$
Subtraction Property: Subtracting a number is the same as adding its opposite.

1.4 Multiply and Divide Integers

Multiplication and Division of Two Signed Numbers
- Same signs—Product is positive
- Different signs—Product is negative
Strategy for Applications
1. Step 1. Identify what you are asked to find.
2. Step 2. Write a phrase that gives the information to find it.
3. Step 3. Translate the phrase to an expression.
4. Step 4. Simplify the expression.
5. Step 5. Answer the question with a complete sentence.

1.5 Visualize Fractions

Equivalent Fractions Property: If $a, b, c$ are numbers where $b \neq 0, c \neq 0,$ then
$\frac{a}{b} = \frac{a \cdot c}{b \cdot c}$ and $\frac{a \cdot c}{b \cdot c} = \frac{a}{b} .$
Fraction Division: If $a, b, c and d$ are numbers where $b \neq 0, c \neq 0, and d \neq 0,$ then $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} .$ To divide fractions, multiply the first fraction by the reciprocal of the second.
Fraction Multiplication: If $a, b, c and d$ are numbers where $b \neq 0, and d \neq 0,$ then $\frac{a}{b} \cdot \frac{c}{d} = \frac{a c}{b d} .$ To multiply fractions, multiply the numerators and multiply the denominators.
Placement of Negative Sign in a Fraction: For any positive numbers $a and b,$ $\frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} .$
Property of One: $\frac{a}{a} = 1;$ Any number, except zero, divided by itself is one.
Simplify a Fraction
1. Step 1. Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers first.
2. Step 2. Simplify using the equivalent fractions property by dividing out common factors.
3. Step 3. Multiply any remaining factors.
Simplify an Expression with a Fraction Bar
1. Step 1. Simplify the expression in the numerator. Simplify the expression in the denominator.
2. Step 2. Simplify the fraction.

1.6 Add and Subtract Fractions

Fraction Addition and Subtraction: If $a, b, and c$ are numbers where $c \neq 0,$ then
$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$ and $\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c} .$
To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.
Strategy for Adding or Subtracting Fractions
1. Step 1. Do they have a common denominator?
  Yes—go to step 2.
  No—Rewrite each fraction with the LCD (Least Common Denominator). Find the LCD. Change each fraction into an equivalent fraction with the LCD as its denominator.
2. Step 2. Add or subtract the fractions.
3. Step 3. Simplify, if possible. To multiply or divide fractions, an LCD IS NOT needed. To add or subtract fractions, an LCD IS needed.
Simplify Complex Fractions
1. Step 1. Simplify the numerator.
2. Step 2. Simplify the denominator.
3. Step 3. Divide the numerator by the denominator. Simplify if possible.

1.7 Decimals

Name a Decimal
1. Step 1. Name the number to the left of the decimal point.
2. Step 2. Write ”and” for the decimal point.
3. Step 3. Name the “number” part to the right of the decimal point as if it were a whole number.
4. Step 4. Name the decimal place of the last digit.
Write a Decimal
1. Step 1. Look for the word ‘and’—it locates the decimal point. Place a decimal point under the word ‘and.’ Translate the words before ‘and’ into the whole number and place it to the left of the decimal point. If there is no “and,” write a “0” with a decimal point to its right.
2. Step 2. Mark the number of decimal places needed to the right of the decimal point by noting the place value indicated by the last word.
3. Step 3. Translate the words after ‘and’ into the number to the right of the decimal point. Write the number in the spaces—putting the final digit in the last place.
4. Step 4. Fill in zeros for place holders as needed.
Round a Decimal
1. Step 1. Locate the given place value and mark it with an arrow.
2. Step 2. Underline the digit to the right of the place value.
3. Step 3. Is this digit greater than or equal to 5? Yes—add 1 to the digit in the given place value. No—do not change the digit in the given place value.
4. Step 4. Rewrite the number, deleting all digits to the right of the rounding digit.
Add or Subtract Decimals
1. Step 1. Write the numbers so the decimal points line up vertically.
2. Step 2. Use zeros as place holders, as needed.
3. Step 3. Add or subtract the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers.
Multiply Decimals
1. Step 1. Determine the sign of the product.
2. Step 2. Write in vertical format, lining up the numbers on the right. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.
3. Step 3. Place the decimal point. The number of decimal places in the product is the sum of the decimal places in the factors.
4. Step 4. Write the product with the appropriate sign.
Multiply a Decimal by a Power of Ten
1. Step 1. Move the decimal point to the right the same number of places as the number of zeros in the power of 10.
2. Step 2. Add zeros at the end of the number as needed.
Divide Decimals
1. Step 1. Determine the sign of the quotient.
2. Step 2. Make the divisor a whole number by “moving” the decimal point all the way to the right. “Move” the decimal point in the dividend the same number of places - adding zeros as needed.
3. Step 3. Divide. Place the decimal point in the quotient above the decimal point in the dividend.
4. Step 4. Write the quotient with the appropriate sign.
Convert a Decimal to a Proper Fraction
1. Step 1. Determine the place value of the final digit.
2. Step 2. Write the fraction: numerator—the ‘numbers’ to the right of the decimal point; denominator—the place value corresponding to the final digit.
Convert a Fraction to a Decimal Divide the numerator of the fraction by the denominator.

1.8 The Real Numbers

Square Root Notation
$\sqrt{m}$ is read ‘the square root of m.’ If $m = n^{2},$ then $\sqrt{m} = n,$ for $n \geq 0 .$
Order Decimals
1. Step 1. Write the numbers one under the other, lining up the decimal points.
2. Step 2. Check to see if both numbers have the same number of digits. If not, write zeros at the end of the one with fewer digits to make them match.
3. Step 3. Compare the numbers as if they were whole numbers.
4. Step 4. Order the numbers using the appropriate inequality sign.

1.9 Properties of Real Numbers

Commutative Property of
- Addition: If $a, b$ are real numbers, then $a + b = b + a .$
- Multiplication: If $a, b$ are real numbers, then $a \cdot b = b \cdot a .$ When adding or multiplying, changing the order gives the same result.
Associative Property of
- Addition: If $a, b, c$ are real numbers, then $(a + b) + c = a + (b + c) .$
- Multiplication: If $a, b, c$ are real numbers, then $(a \cdot b) \cdot c = a \cdot (b \cdot c) .$
  When adding or multiplying, changing the grouping gives the same result.
Distributive Property: If a,b,ca,b,c are real numbers, then
- $a (b + c) = a b + a c$
- $(b + c) a = b a + c a$
- $a (b - c) = a b - a c$
- $(b - c) a = b a - c a$
Identity Property
- of Addition: For any real number $a : a + 0 = a 0 + a = a$
  0 is the additive identity
- of Multiplication: For any real number $a : a \cdot 1 = a 1 \cdot a = a$
  $1$ is the multiplicative identity
Inverse Property
- of Addition: For any real number $a, a + (− a) = 0 .$ A number and its opposite add to zero. $− a$ is the additive inverse of $a .$
- of Multiplication: For any real number $a, (a \neq 0) a \cdot \frac{1}{a} = 1 .$ A number and its reciprocal multiply to one. $\frac{1}{a}$ is the multiplicative inverse of $a .$
Properties of Zero
- For any real number $a,$
  $a \cdot 0 = 0 0 \cdot a = 0$ – The product of any real number and 0 is 0.
- $\frac{0}{a} = 0$ for $a \neq 0$ – Zero divided by any real number except zero is zero.
- $\frac{a}{0}$ is undefined – Division by zero is undefined.

1.10 Systems of Measurement

Metric System of Measurement
- Length
  $\begin{matrix} 1 kilometer (km) & = & 1,000 m \\ 1 hectometer (hm) & = & 100 m \\ 1 dekameter (dam) & = & 10 m \\ 1 meter (m) & = & 1 m \\ 1 decimeter (dm) & = & 0.1 m \\ 1 centimeter (cm) & = & 0.01 m \\ 1 millimeter (mm) & = & 0.001 m \\ 1 meter & = & 100 centimeters \\ 1 meter & = & 1,000 millimeters \end{matrix}$
- Mass
  $\begin{matrix} 1 kilogram (kg) & = & 1,000 g \\ 1 hectogram (hg) & = & 100 g \\ 1 dekagram (dag) & = & 10 g \\ 1 gram (g) & = & 1 g \\ 1 decigram (dg) & = & 0.1 g \\ 1 centigram (cg) & = & 0.01 g \\ 1 milligram (mg) & = & 0.001 g \\ 1 gram & = & 100 centigrams \\ 1 gram & = & 1,000 milligrams \end{matrix}$
- Capacity
  $\begin{matrix} 1 kiloliter (kL) & = & 1,000 L \\ 1 hectoliter (hL) & = & 100 L \\ 1 dekaliter (daL) & = & 10 L \\ 1 liter (L) & = & 1 L \\ 1 deciliter (dL) & = & 0.1 L \\ 1 centiliter (cL) & = & 0.01 L \\ 1 milliliter (mL) & = & 0.001 L \\ 1 liter & = & 100 centiliters \\ 1 liter & = & 1,000 milliliters \end{matrix}$
Temperature Conversion
- To convert from Fahrenheit temperature, F, to Celsius temperature, C, use the formula $C = \frac{5}{9} (F - 32)$
- To convert from Celsius temperature, C, to Fahrenheit temperature, F, use the formula $F = \frac{9}{5} C + 32$