## Math 6A, Lesson 11, Fall 2019, 11/17/2019

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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

## Math 6A, Lesson 10, Fall 2019, 11/10/2019

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1. Exam

2. No homework, enjoy a break.

## Math 6A, Lesson 9, Fall 2019, 11/03/2019

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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

## Math 6A, Lesson 8, Fall 2019, 10/27/2019

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

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

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

## Math 6A, Lesson 7, Fall 2019, 10/20/2019

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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

## Math 6A, Lesson 6, Fall 2019, 10/13/2019

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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.

• 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)

• 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

## Math 6A, Lesson 5, Fall 2019, 10/6/2019

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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

## Math 6A, Lesson 4, Fall 2019, 9/29/2019

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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

## Math 6A, Lesson 3, Fall 2019, 9/22/2019

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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

## Math 6A, Lesson 2, Fall 2019, 9/15/2019

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1. Greatest Common Factor (GCF): the largest common factor of a group of numbers is the largest positive integer that can divide all the numbers in the group.

The factors of 18: 1, 2, 3, 6, 9, 18

The factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

1, 2, 3, and 6 are called the common factors of 18 and 24. The largest of these common factors is 6. Thus, 6 is the greatest common factor of 18 and 24.

HCF: the highest common factor, same as GCF

GCD: the greatest common divisor, same as GCF

Relatively Prime or Mutually Prime: when two numbers, such as 15 and 16, have no common factors greater than 1, their GCF = 1 and the numbers are said to be relatively prime, or mutually prime.

2. Methods to find GCF

• Using prime factorization. GCF is obtained by multiplying the lowest power of each common 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 GCF?

• Reduce a fraction to its lowest terms: 18/48 = (18 / 6) / (48 /6) = ⅜
• Algebra, GCF of the coefficients: 18x*x + 48x = 6x(3x + 8)
• Word problems: Teaching on whiteboard in the class

4. Home Work:

• Handout: one page
• Workbook:
• page 2: 6, 10(a), 10(b), 11
• Page 3: 12, 14, 15(a), 15(b)(i), 17(a), 17(b)(i)
• Page 4: 21, 22
• Page 5: 26, 27, 28, 29(b)