MrE's Algebra Blog

Algebra: Chapter 3-1 and 3-2, pages 114 and 119.

The Addition and Multiplication Properties of Equality (I call these 1 STEP EQUATIONS)

You can add, subtract, multiply or divide the same number to both sides of an equation and get an equivalent equation. We call these the addition and multiplication property of equality.

If `a=b`, then `a+c=b+c` and if `a=b`, then `ac=bc`

Subtraction and division are opposites of addition and multiplication, so we have no problems there. There are lots of examples at my favorite site, Purplemath, give these a look! Remember to show ALL the STEPS and that you can do these either vertically or horizontally. DON’T TAKE SHORTCUTS!

Examples:

(Addition)

`-6 = y-8`, we add 8, the opposite of -8

`-6+8=y-8+8`, we use the addition property to add 8 to both sides of the equation and finally,

`2=y`

(and Multiplication)

`y/9=14`, we will multiply both sides by `9/1` or just `9`

`9* (y/9)=9*14`, remembering that `9/9 =1`, we have

`y=126`

Whatever you do to an equation, do the S A M E thing to B O T H sides of that equation! If its `x+7`, then subtract 7 from both sides. If its `x-6`, then add 6 to both sides. If its `5x`, then divide both sides by `5` and if its `x/3`, then multiply both sides by `3`! Always do the opposite operation in these 1 step equations.

Two of tonight’s homework problems (lesson 1) solved by MrE are here! Just click it!

Two of tonight’s homework problems (lesson 2) solved by MrE are here! Just click it!

======================================================================

Algebra 1a: Chapter 2, Lesson 1, page 54.

Integers and Rational Numbers

Integers consist of the whole numbers and their opposites. On the number line, the positive integers are to the right of `0` and the negative integers are to the left of `0`. Zero is neither positive or negative.

A mathematical sentence that contains an inequality symbol is called an inequality.

The symbol `>` means greater than.

The symbol `<` means less than.

On the number line, numbers increase as we move from left to right.

The absolute value of a number is its distance from `0` on the number line. We use the symbol `|n|` to represent “the absolute value of n”.

Algebra 1a: Chapter 2, Lesson 2, page 59.

Rational Numbers

“Any number that can be expressed as the ratio of two integers, `a/b`, is called a rational number.

There is a point on the number line for every rational number. The number is called the coordinate of the point and the point is the graph of the number. When we draw a point for a number on a number line, we say that we have graphed the number.“

Algebra: Chapter 2 BENCHMARK!

======================================================================

Algebra 1a: Chapter 1 BENCHMARK!

Algebra: Chapter 2 Review

Skills Practice 5 and 6 review, make sure you got it all down!

• Distribute
• Factor
• Combine Like Terms
• Add/Subtract/Multiply/Divide with ± numbers

======================================================================

Algebra 1a: Chapter 1 Review

Skills Practice 2 and 3 review, make sure you got it all down Wednesday, odd problems for classwork and Thursday EVEN problems for homework. Make sure your notes are ready for Friday’s Chapter 1 BENCHMARK TEST! Bring your calculator!!

Algebra: Chapter 2 Review

Quiz #4.

======================================================================

Algebra 1a: Chapter 1, Lesson 10, page 44

Reasoning Strategies – or Try, Test and Revise

1. UNDERSTAND the Problem
2. Develop and carry out a PLAN
3. Find the ANSWER and CHECK

Strategies look like:

• Draw a diagram
• Make an Organized list
• Look for a Pattern
• Try, Test and Revise
• Use Logical Reasoning
• Simplify the Problem
• Write an Equation
• Make a Table
• Work Backwards

Algebra: Chapter 2, Lesson 9, page 98.

Writing Equations

Plan to write and solve an equation by:

• Can I use a variable to represent an unknown number?
• Can I represent other conditions in terms of the variable?
• Can I find equivalent expressions?
• Can I write and solve an equation?

Remember too:

• UNDERSTAND the problem
• Develop and carry out a PLAN
• Find the ANSWER and CHECK

Two of tonight’s homework problems solved by MrE are here! Just click it!

======================================================================

Algebra 1a: Chapter 1, Lesson 10, page 44

Reasoning Strategies – or Try, Test and Revise

1. UNDERSTAND the Problem
2. Develop and carry out a PLAN
3. Find the ANSWER and CHECK

Strategies look like:

• Draw a diagram
• Make an Organized list
• Look for a Pattern
• Try, Test and Revise
• Use Logical Reasoning
• Simplify the Problem
• Write an Equation
• Make a Table
• Work Backwards

Algebra: Chapter 2, Lesson 8, page 93.

Inverse of a Sum and Simplifying

The inverse of a SUM Property: For any rational numbers, `−(a+b)=−a+(−b)`. The additive inverse of a sum is the sum of the additive inverses.

In other words, if you have a `−` in front of a paranthesis, then just change the sign of EVERYTHING inside.

For example: `−(2a−7b−6)` becomes the opposite of each term, `−2a+7b+6`.

Another example: `3y-2-(2y-4)=3y-2-2y+4`. Combining like terms, we see the answer `=2y-4`

Two of tonight’s homework problems solved by MrE are here! Just click it!

======================================================================

Algebra 1a: Chapter 1, Lesson 9, page 40.

Using Formulas

A formula is an equation that shows a relationship between 2 or more variables. The formula for the area of a rectangle is `a = l⋅w`. Given the length and width of an object, you can easily find the area. Be careful, however, and make sure that you are using consistent UNITS in the formula. Do not mix up seconds with minutes or hours or inches with feet or miles. The units must ALWAYS be consistent.

Don’t forget too to add the units to the answer!

Two of tonight’s homework problems solved by MrE are here! Just click it!

Algebra, Chapter 2, Lesson 7, page 89.

Using the Distributive Property

The distributive property of multiplication over addition and subtraction is:

`a(b+c)=ab+ac` and `a(b−c)=ab−ac`

You can also combine LIKE TERMS ONLY IF THE VARIABLE AND THE EXPONENT OF THE VARIABLE ARE EXACTLY THE SAME!

Examples, (remember to use the wacky arrows!):

`9(x – 5) = 9x – 9(5) = 9x – 45`

`-4(x – 2y + 3z) = -4x +(-4)(-2y) + (-4)(3z) = -4x -4(-2y) -4(3z)`

`= -4x + 8y – 12z`

(if it comes up, don’t forget the change-change or bling-bling, double negative thingy …)

Here is a link from purplemath.com about the distributive property with more examples.

Two of tonight’s homework problems solved by MrE are here! Just click it!

======================================================================

Algebra 1a: Chapter 1, Lesson 9, page 40.

Using Formulas

A formula is an equation that shows a relationship between 2 or more variables. The formula for the area of a rectangle is `a = l⋅w`. Given the length and width of an object, you can easily find the area. Be careful, however, and make sure that you are using consistent UNITS in the formula. Do not mix up seconds with minutes or hours or inches with feet or miles. The units must ALWAYS be consistent.

Don’t forget too to add the units to the answer!

Two of tonight’s homework problems solved by MrE are here! Just click it!

Algebra 1: Chapter 2, Lessons 5 and Lesson 6, pages 77 and 81

Multiplication and Division of Rational Numbers

Today is easy because the rules for multiplication and division are simple.

• When multiplying 2 numbers AND if the SIGNS are the same, the product is ALWAYS positive.
• If the signs are different, them the product is ALWAYS negative. This is pretty straightforward.

Division follows the same rules as multiplication.

2 rational numbers whose product is 1 are called multiplicative inverses or reciprocals of each other. Just flip the rational expression over and keep the same sign. For example, the reciprocal of `2/3` or `m/n` is `3/2 ` and `n/m` respectively.

Remember too, to divide rational numbers, sometimes its easier to express them as improper fractions, then convert the 2nd term to its reciprocal and change the `/` to a `⋅`.

Two of tonight’s homework problems solved by MrE are here! Just click it!

Purplemath.com has these tutorials about multiplying and dividing rational numbers, check it out!

======================================================================

Algebra 1a: Chapter 1, Lesson 8, page 38.

Reasoning Strategies

Reasoning Strategies to use:

Phase 1: UNDERSTAND the problem

• What am I trying to find out?
• What data am I given?
• Have I ever solved a similar problem?

Phase 2: Develop and carry out a PLAN

• What strategies might I use to solve the problem?
• How can I correctly carry out the strategies I selected?

Phase 3: Find the ANSWER and CHECK

• Does the proposed solution check?
• What is the answer to the problem?
• Does the answer seem reasonable?
• Have I stated the answer clearly?

One of the most useful strategies is to DRAW A DIAGRAM of the situation.

Algebra: Chapter 2, Lesson 4, page 71.

Subtraction of Rational Numbers

Today, we are working on subtraction of rational numbers. Remember, just do the inverse of subtraction and ADD the inverse (or opposite) of the number to the other number.

Examples:

`2-6 = 2 + (-6) = -4`

`-5 – 7` is really `-5 + (-7)`, remembering that the + sign is invisible. So this becomes just `-5 + (-7)` and remembering the adding rule of 2 negatives, the answer becomes `-12`.

`-4 – (-5)` is really `-4 + 5` because 2 negative are a positive (bling-bling or make change-change from Ms. Phillips and Ms. Craig) and the answer is `+5 – 4` and the final answer is `+1`!

Two of tonight’s homework problems solved by MrE are here! Just click it!

Hotmath.com has these tutorials for the addition and subtraction of rational numbers too!

======================================================================

Algebra 1a: Chapter 1, Lesson 7, page 33.

Solving Equations, An Introduction

An EQUATION is a mathematical sentence that uses the equal sign `=` to state that 2 expressions represent the same number or are equivalent. An equation that contains at least 1 variable is called an OPEN SENTENCE.

The set of numbers from which you can select replacements for the variable is called the REPLACEMENT SET, usually noted with squiggly brackets `{2, 5, 12}`. A replacement for a variable that makes an equation true is called the SOLUTION. To SOLVE and equation means to find all of its solutions. The collection of all the solutions is called the SOLUTION SET, usually contained in the { brackets}.

2 equations are EQUIVALENT if one can be obtained from the other by a sequence of the following steps. You can:

• add the same number to both sides of an equation
• subtract the same number from both sides of an equation
• multiply both sides of an equation by the same number
• divide both sides of an equation by the same number

Equivalent equations have the same solutions set.

Two of tonight’s homework problems solved by MrE are here! Just click it!!

Algebra: Chapter 2, Lesson 3, page 63.

Addition of Rational Numbers.

Adding 2 positive or negative numbers

• Add the absolute values. The sum has the same sign as the 2 numbers.

Adding a positive and a negative number

• Subtract the absolute values. The sum has the sign of the number with the bigger sign!

I have a podcast on iTunes that talks about this and tomorrow’s lesson. Just search on iTunes “MrE Algebra” and you’ll find it. CLick here and you can check it out here on your desktop!

Two of tonight’s homework problems solved by MrE are here! Just click it!

======================================================================

Algebra 1a: Chapter 1, Lesson 7, page 33.

Solving Equations, An Introduction

An EQUATION is a mathematical sentence that uses the equal sign `=` to state that 2 expressions represent the same number or are equivalent. An equation that contains at least 1 variable is called an OPEN SENTENCE.

The set of numbers from which you can select replacements for the variable is called the REPLACEMENT SET, usually noted with squiggly brackets `{2, 5, 12}`. A replacement for a variable that makes an equation true is called the SOLUTION. To SOLVE and equation means to find all of its solutions. The collection of all the solutions is called the SOLUTION SET, usually contained in the { brackets}.

2 equations are EQUIVALENT if one can be obtained from the other by a sequence of the following steps. You can:

• add the same number to both sides of an equation
• subtract the same number from both sides of an equation
• multiply both sides of an equation by the same number
• divide both sides of an equation by the same number

Equivalent equations have the same solutions set.

Two of tonight’s homework problems solved by MrE are here! Just click it!!