Evaluation of algebraic expressions and formulas

1. Evaluation of algebraic expressions

• The process of replacing each variable with its value to find the actual value of an algebraic expression is called substitution.

2. Formulas

• The area of a rectangle is given by
  • Area = Length x Width
  • A = lw

• This equality of connecting two or more variables is called a formula. When the values of l and w are known, we can find the value of A in the formula by substitution.

3. Home Work:

• Handout: two pages
• Workbook:
  • page 15: 3, 4, 5
  • Page 16: 11, 12, 13
  • Page 17: 14
  • Page 19: 26, 27
  • Page 20: 28

The use of letters in algebra

1. The use of letters

• In general, an algebraic expression involves numbers and letters that are connected with operation symbols such as “+”, “-”, “x” and “/”
• In 10 + 8n, we call n a variable and 10 + 8n an algebraic expression

2. Basic notation in algebra

• In algebra, there are rules for writing algebraic expressions. The operation symbols “+”, “-”, “x”, “/”  and “=” have the same meanings in both algebra and arithmetic.
• Add a to b: sum = a + b = b + a
• Subtract c from d: difference = d – c != c -d
• Multiply g by h: product = g x h = h x g = gh
• Divide x by y where y != 0: quotient = x/y

3. Exponential notation

• Teach in the class on white board

4. Simplify algebraic expressions

• Teach in the class on white board

5. Home Work:

• Handout: two pages
• Workbook:
  • page 15: 1, 2
  • page 16: 6, 7, 8, 9, 10

1. Exam

2. No homework, no class next week. Happy Thanksgiving break.

1. No quiz today, exam next week.

2. Real numbers and use of calculators:.

• Rational number: can be expressed as a decimal by dividing the numerator of the rational number by its denominator
• Terminating decimal: has a finite number of digits: ⅜, 5/16
• Repeating decimal: 9/11, 7/11
• Irrational number: non-terminating and non-repeating decimals
• Use of calculators

3. Rounding numbers to decimal places

• Mark the cut-off point after the desired place value
• Rule of rounding up: the first digit after the cut-off point,  is >=5
• Rule of rounding don: the first digit after the cut-off point, is  < 5
• Nearest 10, 100, 1000, one million, …
• Nearest tenth, hundredth, or thousandth, …
• 1 decimal place, two decimal places, 3 decimal places, …

4. Home Work:

• Handout: two pages
• Workbook:
  • page 11: 24, 25, 26, 27, 28, 29
  • page 12-13: 30, 31, 32, 33, 34, 35, 36

1. Quiz on negative number, addition and subtraction of integers

2. Multiplication, division and combined operations of integers:.

• If two numbers have the same sign, their product/quotient is positive
• If two numbers have opposite signs, their product/quotient is negative
• Any integer multiplies 0, the product is 0: a x 0 = 0
• 0/a = 0, given a !=0
• a/0 is undefined
• How about 0/0? https://www.khanacademy.org/math/algebra/introduction-to-algebra/division-by-zero/v/undefined-and-indeterminate
• Combined operation: PEMDAS

3. Rational numbers: rational numbers are numbers that can be expressed in the form of a/b, where a and b are integers, and b != 0

4. Addition, subtraction, multiplication and division of rational numbers

5. Home Work:

• Handout: three pages
• Workbook:
  • page 8: 6, 8, 9, 10, 11
  • page 10: 18, 19, 20, 21, 22, 23

1. Quiz on negative number and addition of integers

2. Subtraction of integers:.subtraction is the reverse process of addition. Using a number line:

• Subtraction of a positive number is taken as a movement to the LEFT of the number line
• Subtraction of a negative number is taken as a movement to the RIGHT of the number line

(-3)  – 4

2 –  3

(-1) –  (-3)

2 –  (-5)

0 – (-4)

3. Rule for subtraction: to subtract integers, we change the sign of the integer being subtracted and add them together according to the rule for addition of integers.

For any integers a, and,b,

a – b = a + (-b)

4. Absolute value of the difference:

For any integers a, and,b,

|a – b| = |b – a|

5. Home Work:

• Handout: two pages
• Workbook:
  • page 8: 7
  • page 9: 17
  • page 10: 20

1. Opposite Numbers (additive inverse): is a number that when added to a given number yields 0. The opposite number for any number x is -x. Note that x may be positive or negative.

(-5) + 5 = 0

x + (-x) = 0

On the number line, the numbers 5 and -5 are located at the same distance from zero. We say that the numbers 5 and -5 are opposites. We can also say that -5 is opposite to 5, and 5 is opposite to -5 on the number line.

2. Addition of integers:

• Addition of a positive number is taken as a movement to the RIGHT of the number line
• Addition of a negative number is taken as a movement to the LEFT of the number line

(-3) + 4

2 + 3

(-1) + (-3)

2 + (-5)

0 + (-4)

3. Rule for addition:

• If the signs of the integers being added are the same, the sum has the same sign as the integers and we add the absolute values of the integers

For any a > 0, and b > 0,

a + b = a + b;        3 + 5 = 8

(-a) + (-b) = -(a+b);         (-3) + (-5) = – (3 + 5) = -8

• If the signs of the integers being added are different,, the sum takes the sign of the integer with the greater absolute value and we find the difference of the absolute values of the integers.

For any a > 0, and b > 0,

a + (-b) = +(a – b) if a >=b;        9 + (-6) = +(9 – 6) = +3 = 3

a + (-b) = – (b -a) if b > a;         15 + (-20) = -(20 -15) = -5

-a + b = -(a – b) if a >= b;         -18 + 12 = -(18 -12) = -6

-a + b = +(b -a) if b > a;          -23 + 27 = +(27 – 23) = +4 = 4

4. Home Work:

• Handout: one page
• Workbook: page 7: 4

1. Negative Numbers: we often across quantities that have opposite directions or meanings. For example,  traveling due east and west, the rise and fall in price, profit and loss, and being above or below sea level. When we assign a certain meaning of quantity to be positive, a value of the quantity that has the  opposite meaning may be considered as negative and is represented with a “-” sign.

Positive numbers: 1, 2, 3, 0.8, ⅜, …

Negative numbers: -1, -2, -3, -⅔, -0.0025, …

Meanings of positive and negative numbers: teaching on whiteboard in the class.

2. The (horizontal) number line:

How to draw a number line:

• Draw a line, and mark the zero point on it;
• Choose a unit length, e.g., 1 cm, to mark the points 1, 2, 3, … at equal unit intervals on the right of 0, and the points -1, -2, -3, … on the left of 0;
• Draw an arrow at each end.

Inequality signs:

• > greater than
• < less than
• >= greater or equal
• <= less than or equal

On a horizontal number line:

• All the positive numbers are to the right of 0
• All the negative numbers are to the left of 0
• Numbers are arranged in ascending (increasing) order from left to right
• Every number is smaller than any number on its right and greater than any number on its left

3. Absolute Value: the absolute value of a number is the distance that number is from 0 on the number line. Both 3 and -3 are the same distance from 0. The absolute value of a number is never negative.

|3| = 3, or |x| = x if x >= 0

|-3| = 3, or |x| = -x if x < 0

4. Home Work:

• Handout: two pages
• Workbook: page 7: 1, 2, 3, 5

1. Quiz

2. Square Roots

3*3 = 9, 9 is called the square of 3, we also say that 3 is the positive square root of 9.

3. Perfect Squares: the number 1, 4, 9, 16, 25, … whose square roots are whole numbers are called perfect squares. We can find the square root of a perfect square by using prime factorization. Teaching on whiteboard in the class.

4. Cube Roots

2*2*2 = 8, 8 is called the cube of 2, we also say that 2 is the cube root of 8.

5. Perfect Cubes: the number 1, 8, 27, 64, … whose cube roots are whole numbers are called perfect cubes. We can find the cube root of a perfect cube by using prime factorization. Teaching on whiteboard in the class.

6. Home Work:

• Handout: three pages
• Workbook:
  • page 2: 8, 9
  • Page 4: 18, 19, 20

1. Least Common Multiple (LCM): the least common multiple of a group of numbers is the smallest positive integer which is divisible by all the numbers in the group.

The multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, …

The multiple of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, …

24 and 48 are the first two common multiples of 6 and 8. Since 24 is the least of all common multiples, we say the least common multiple (LCM) of 6 and 8 is 24.

2. Methods to find LCM

• Using prime factorization. LCM is obtained by multiplying the highest power of each prime factor of the given numbers. Teaching on whiteboard in the class.
• Continuous division. Teaching on whiteboard in the class.
• Venn diagram. Teaching on whiteboard in the class.

3. Why LCM?

• adding, subtracting, or comparing vulgar fractions, it is useful to find the least common multiple of the denominators, often called the lowest common denominator, because each of the fractions can be expressed as a fraction with this denominator. Ex. 1/15 + 1/18 + 1/21
• Word problems: Teaching on whiteboard in the class

4. Home Work:

• Handout: two pages
• Workbook:
  • page 2: 7
  • Page 3: 13, 15(a), 15(b)(ii), 17(a), 17(b)(ii)
  • Page 4: 23
  • Page 5: 24