Equations and Problem Solving Part 2 (Uniform Motion)

Topics for Today:

We continued our discussion about problem solving by investigating uniform motion problems.   Most students are familiar with the basic formula for motion: 
distance = rate * time, or in algebraic terms, d = rt.

Uniform motion problems fall into three main categories: same-direction travel, round-trip travel, or opposite-direction travel.  Depending upon what the problem is asking, we'll combine our problem data in different ways to find our answer, but in each case, we will still apply the general formula (d=rt) to set up our problem.

Drawing diagrams to help picture what is going on in the problem is another helpful strategy.  Using a table to solve problems helps organize all the supporting data, and provides a systematic way to solve more complex problems.  Students are encouraged to use a table and to draw a picture of what is going on in the question to help better understand what is being asked.

Sections Covered in Textbook:

2-5: Equations and Problem-Solving Part 2 (pages 103-110)

Resources & Tutorials:

1)  How to Solve Opposite-Directions problem.
2)  How to Solve Same-Directions problem.
3)  How to Solve Round-Trip Travel Problems.

Equations and Problem Solving Part 1

Topics for Today:

Now that we know how to solve all types of equations, we will use that knowledge to solve story problems.  There are many different types of story problems, but most of them can be categorized into one of several categories.  Today's lesson will focus on problem solving involving defining variables in terms of each other,  consecutive integers (which also involves defining variables in terms of another), and break-even problems.

Vocabulary: consecutive, break-even

Sections Covered in Textbook:

2-5: Equations and Problem Solving (pages 103 - 110)
We will continue working on this topic tomorrow with more examples.

Resources & Tutorials:

1) Solving Break-Even Problems
2) How to find the Break-Even Point
3) Solving Word Problems with Consecutive Integers

Solving Equations with Variables on Both Sides

Topics for Today:

Our discussion about solving equations moved on to situations where there are variables on both sides of an equation.  The basic properties of equality still apply when dealing with variable terms.  We used the addition, subtraction, multiplication, and division properties of equality to get variables on one side, and constants on the other side of the equation.  For equations with variables on both sides, it is possible for the equation to have infinitely many solutions (identity) or no solution at all.  Students will need to be on the lookout for these special cases, which show themselves is interesting mathematical ways.

Sections Covered in Textbook:

2-4: Equations with Variables on Both Sides (pages 96-100)

Resources & Tutorials:

1) Solving Equations with Variables on Both Sides
2) Solving Equations with Variables on Both Sides and Grouping Symbols

Solving Multi-Step Equations

Topics for Today:

We moved on from two-step to multi-step equation solving.  Today's lesson focused on simplifying with grouping symbols, and multiplying through to eliminate fractions and decimals to make the equation easier to solve.

Sections Covered in Textbook:

2-3: Solving Multi-Step Equations (pages 88 - 93)

Resources & Tutorials:

1) Solving Multi-Step Equations using Distributive Property
2) Solving Multi-Step Equations by Clearing Fractions

3) Solving Multi-Step Equations Class Notes

Solving Two-Step Equations

Topics for Today:

Our discussion about solving equations moved on to solving 2-step equations.  The order of operations still figures in to this process, although since we are undoing operations, we go in reverse.

Sections Covered in Textbook:

2-2: Solving Two-Step Equations (pages 81- 86)

Resources & Tutorials:

1) How do you solve a 2-step equation?
2) Math Antics - How to solve 2-step equations.

Solving One-Step Equations

Topics for Today:

Today we reviewed solving one-step algebraic equations.  We discussed what inverse operations are, and also defined the term solution.  Students are reminded that all they are accountable for all the mathematics that came before this class, as well as any new learned material.  Mathematics is a cumulative subject, and the skills built in the past will continue to be used to solve new problems.  That means that we will continue to integrate fraction and decimal operations into our problem solving.  Equation operations will include all types of real numbers.

Sections Covered in Textbook:

2-1: Solving One-Step Equations (pages 74-79)

Resources & Tutorials:

1) Math Antics - How to Solve a 1-step Equation.
2) How to solve an equation using addition.
3) How to solve an equation using subtraction.
4) How to solve an equation using multiplication.
5) How to solve an equation using division.
6) How to solve a decimal equation using subtraction.
7) How to solve an equation when multiplying fractions. 

Chapter 1 Review

Topics for Today:

Today we played a game of Money Grab to finish up our review of Chapter 1.  Students should make sure their binders are fully organized prior to joining class on Tuesday.  Binders will be collected at the beginning of class.  For a list of topics that need to be in the notebook, open the Chapter 1 Table of Contents Google Docs that I have shared with you from the Chapter 1 folder.  

Topics covered in Chapter 1:  Using variables, order of operations, classifications of real numbers, real number operations (adding, subtracting, multiplying, and dividing), as well as the properties of equality.  

Sections Covered in Textbook:

Chapter 1:  Tools of Algebra (pages 4-58) - Omit section 1-9

Resources & Tutorials:

See Blog entries for August 24th through September 11th.

Properties of Real Numbers Parts 1 & 2

Topics for Today:

Today we laid the groundwork for all the properties of real numbers, and we'll be taking two days to discuss and master them.  The properties are a set of truths that act as rules to allow us to rewrite numbers and equations to make them easier to solve.  We also use the properties to justify the different steps we take to simplify expressions and solve equations.  Students also made a set of notecards that can be used during any quiz or test this year.

This is the last section of this chapter.  I have posted a Chapter 1 Review on Delta Math for those who want more practice.  This assignment is optional but highly recommended.  

Sections Covered in Textbook:

1-8: Properties of Real Numbers (pages 54 - 57)

Resources & Tutorials:

1)  Video page for the addition, subtraction, multiplication and division properties.
2)  What is the substitution property of equality?
3)  Summary sheet of Properties from Math-Aids.

4) Properties of Equality Class Notes

The Distributive Property

Topics for Today:

Today we discussed the distributive property  of multiplication over addition and subtraction.  We also used the distributive property to multiply large numbers using mental math.  We also discussed how to combine algebraic like terms, and expanded our algebra vocabulary.  We revisited translating phrases into algebraic expressions with the addition of the distributive property.

Vocabulary: constant, like terms, distributive property, coefficient, term

Sections Covered in Textbook:

1-7: The Distributive Property (pages 47 - 52)

Resources & Tutorials:

1) What is the distributive property?

Multiplying and Dividing Real Numbers

Topics for Today:

Earlier this week we took our first quiz of the year.  How did it go?  Did you properly prepare?  If not, what improvements can you make for the future?  Students will be allowed and expected to complete quiz corrections for partial credit.  Quizzes are meant as assessment and learning tools.  Students should always be asking, "How can I do better?".

We moved on to multiplying and dividing real numbers today.

Some basic sign rules for multiplication and division:

+ · + = +      + / + = +
+ · - = -      + / - = -
- · + = -      - / + = -
- · - = +      - / - = +

Vocabulary: Identity property of Multiplication, Multiplication Property of Zero, Multiplication Property of -1, Inverse Property of Multiplication, multiplicative inverse, reciprocal

Sections Covered in Textbook:

1-6: Multiplying and Dividing Real Numbers (pages 37 - 44)

Resources & Tutorials:

1) How do you multiply and divide numbers with different signs?
2) What are Multiplicative Inverses?
3) What is a reciprocal?

Subtracting Real Numbers

Topics for Today:

We continued our discussion of adding real numbers (both positive and negative) using a number line and integer chips.  Our rules for addition are:

• If the signs are the same, add and keep the sign.
• If the signs are different, subtract and keep the sign of the number with the larger absolute value.  

Stated another way...

Subtraction is actually the addition of opposites!  

For subtraction problems, first change to addition by adding the opposite, then follow your addition rules!  

                  6 – 10 =

              6 + (-10) =  

Subtract 6 from 10 and keep the negative sign because |-10| > |6|

Therefore, the answer is – 4. 

Sections Covered in Textbook:

1-4: Adding Real Numbers (pages 24-31)
1-5: Subtracting Real Numbers (pages 32-36)

Resources & Tutorials:

1) How do you add two negative numbers?
2) Rules for Adding Integers
3)  How to rewrite a subtraction problem as addition