Math 7, Lesson 11, Fall 2021, 12/5/2021

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Section 1:

Chapter 5.3 Simple Fractional Equations

Chapter 5.4 Forming Linear Equations to Solve Problems

1. Fractional equation: when the variable of an equation is in the denominations of a term, the equation is called fractional equation

6 /(x – 2) = 3

1/(x + 3) = 2/x

Note: it is important to check the solutions, that they can’t be those values that make a denominator of the original equation zero.

2. Constructing/Forming Linear Equations to Solve Problems

Two pages handout of teaching material. The steps involved in problem solving with linear equations are:

  • Step 1. Read the question carefully and identify the unknown quantity
  • Step 2. Use a letter to represent the unknown quantity (e.g. x)
  • Step 3. Express other quantities in terms of x
  • Step 4. Construct/Form an equation based on the given information
  • Step 5. Solve the equation
  • Step 6. Write down the answer statement

3. Review basic concept of “open parenthesis”

  • 2(3x+y) – 5(0.2x-0.6y) =
  • -2(3x+y) + 5(0.2x-0.6y) =
  • -2(3x-y) + 5(-0.2x-0.6y) =
  • 2(3x-y) – 5(0.2x+0.6y) =

4.Homework:

  • Handout
    • Three pages
  • Workbook:
    • Page 30: 15, 16, 17, 18, 19, 20

Section 2:

1. Constructing/Forming Linear Equations to Solve Problems

The steps involved in problem solving with linear equations are:

  • Step 1. Read the question carefully and identify the unknown quantity
  • Step 2. Use a letter to represent the unknown quantity (e.g. x)
  • Step 3. Express other quantities in terms of x
  • Step 4. Construct/Form an equation based on the given information
  • Step 5. Solve the equation
  • Step 6. Write down the answer statement

2. Homework:

  • Handout
    • Two pages
  • Workbook:
    • Page 31: 21, 22, 23, 24, 25
    • Page 32: 26, 27, 28, 29

Math 7, Lesson 10, Fall 2021, 12/1/2021

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Make up class. Exam on Algebraic Manipulation.

Math 7, Lesson 9, Fall 2021, 11/21/2021

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Section 1: Chapter 5.1 Simple Linear Equations in One Variable

1. Starting chapter 5. Solve simple linear equations in one variable:

  • Concepts: equation, variable, solution/root, linear equation (ax+b=c, where a,b,c are constant and a != 0), LHS, RHS, balancing.
  • Methods: subtract, add, divide or multiply to both sides by the same number.

2. Introduce concepts and methods of “isolate”, “move items to other side and change of sign”, “plug the answer/solution back in the equation”.

Key word: isolate, isolate, isolate. The key to solving many equations is to get the variables alone on one side of the equation. To solve a linear equation with one variable, we isolate the variable by following a few simple steps:

  • simplify both sides of the equation by combining like terms on each side;
  • move all the terms with the variable to one side and all the constants to the other using addition and subtraction, or just moving them to other side with change of signs;
  • after simplify the equation that results from the previous step, multiply by the reciprocal of the variable’s coefficient to solve for the variable.
  • you can always check your answer by plug the solution back to the variable in the equation, both sides should be equal. if not, go check your calculation.

3. Home Work

  • handout:
    • two pages
  • Workbook:
    • page 27: 1, 2, 3
    • page 28: 6

Section 2: Chapter 5.2. Equations involving Parentheses

1. We apply the distributive law of multiplication over addition to help us solve equations involving parentheses.

Recall a(x + b) = ax + ab

Solve equation 9(x + 1) = 2(3x + 8)

2. When working with equations, always apply the same action to both sides of the equation.

Solve equation 5(2x – 9)/3 -8 = 2x

Solve equation (3x + 2)/5 = (4x – 7)/6

3. Rewriting equations

Giving the formula A = ½ * (a + b)h, find the value of a when b = 13, h = 9, and A = 90.

4. Homework:

  • Handout
    • Two pages
  • Workbook:
    • Page 27: 4
    • Page 28: 5
    • Page 29: 11, 12, 13, 14

Math 7, Lesson 8, Fall 2021, 11/14/2021

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Section 1: Simplification of Linear Algebraic Expressions

A 10 minutes quick quiz

1. The distributive law is applicable when removing parentheses in an algebraic expression. Examples in class:

  • 2(3x + 4y) =
  • -4(5a – 3b) + 7a
  • a(2x + 7y + 5z)
  • (2p + 3q -4r)(-6b)
  • 2[x- 5(3-x)]

2. Express each of the following as a single fraction in the simplest form. Recall LCM of denominators.

  • -p + p/3 +(3p)/5
  • (3p + 10)/4 -2
  • (3x – 4)/4 + (2x+5)/3
  • (1 – 2x)/3 + (3x + 1)/5 + (4x -3)/6

3. Home Work:

  • Handout:
    • two pages
  • Workbook: correct last week’s HW problem of
    • page 21: 3, 4, 5
    • Page 23: 11, 12, 13, 14, 15
    • Page 24: 21
    • Page 25: 22, 23

Section 2: Factorization by Extracting Common Factors, Factorization by grouping terms

1. The process of writing an algebraic expression as a product of its factors is called factorization or factoring:

  • 600 = 2x2x2x3x5x5
  • ax + ay = a(x+ y)
  • 15a + 20b = 5(3a) + 5(4b)=5(3a + 4b)
  • 24ax – 40ay + 8a = (8a)(3x) – (8a)(5y) + (8a)(1) = 8a(3x – 5y + 1)

2. Factorization by grouping

  • 12ax – 3ay + 8bx -2by

= (12ax -3ay) + (8bx -2by)

= 3a(4x-y) +2b(4x -y)

=(3a + 2b)(4x-y)

  • 49a + 42c -7ay -6cy

= (49a -7ay) + (42c – 6cy)

= 7a(7 – y) + 6c(7 – y)

= (7a + 6c)(7 – y)

3. Home Work:

  • Handout:
    • two pages
  • Workbook:
    • page 22: 7, 8
    • Page 24: 16, 17, 18, 20
    • Page 26: 26, 27, 28, 29

Math 7, Lesson 7, Fall 2021, 11/07/2021

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Section 1: Like Terms and Unlike Terms

1. Terms, coefficient, and constant terms

  • The expression 2x – 3y + 8 consists of three terms. They are 2x, -3y and 8. The numerical part, including the sign, of a term is called the coefficient of the variable.
  • Term 2x, the coefficient of x is 2
  • Term -3y, the coefficient of y is -3
  • Term 8, is called constant term

2. Classify the like terms and unlike terms: explained in class

3. Simplify the algebraic expression by combining (or collecting) like terms

  • 2x + 3x = 5x
  • 8y – 3y = 5y
  • 3a + 4b – 2a + 5b = (3a -2a) + (4b +5b) = a + 9b

4. Home Work:

  • Handout: two pages
  • Workbook:
    • page 21: 1, 2
    • Page 22: 6, 9, 10

Section 2: Distributive Law, Addition and Subtraction of Linear Algebraic Expressions

1. Use of parentheses and distributive law

  • a(x + y) = ax + ay
  • (x + y)a = a(x + y) = ax + ay = xa + xy
  • a(x – y) = a{x + (-y)] = ax + a(-y) = ax – ay
  • a(x + y + z) = ax + ay + az
  • x – (a – b) = a -a + b

2. Addition and subtraction of linear algebraic expressions: explained in class

  • (2a +3b) + (5a -4b) =
  • Find the sum of -2p + 3q – 4 and p + 5q – 3
  • (4x – 5) – (7x – 3)

3. Home Work:

  • Handout:
    • one pages
  • Workbook:
    • page 21: 3, 4, 5
    • Page 23: 11, 12, 13, 14, 15
    • Page 24: 21
    • Page 25: 22, 23

Math 7, Lesson 6, Fall 2021, 10/24/2021

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1. Exam on chapter “Introduction to Algebra”

2. Home Work: 

  • Redo all the problems you got wrong in last three homework assignment.

Math 7, Lesson 5, Fall 2021, 10/17/2021

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Section 1: 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

Section 2: Writing Algebraic Expressions to Represent Real-world Situation

1. We may use algebraic expressions and formulas to express the relationship between two or more quantities in our daily life

  • Lots of examples teaching in the classroom, and lots of exercise on whiteboard
  • Visualizing, drawing
  • Variables are representing quantities with similar units.

2. Home Work:

  • Handout: three pages
  • Workbook:
    • page 17: 15, 16, 17, 18, 19
    • Page 18: 20, 21, 22, 23, 24
    • Page 19: 25

Math 7, Lesson 4, Fall 2021, 10/3/2021

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The use of letters in algebra

Quiz-2 today.

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 7, Lesson 3, Fall 2021, 9/26/2021

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Section 1:

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

Section 2:

1. Quiz 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 7, Lesson 2, Fall 2021, 9/19/2021

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Section 1:

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

Section 2:

1. (Not today) 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