Ratio, Rate, Unit Rate, and Proportions

This past week we have been discussing how to simplify a ratio, and to identify a rate, how to find a unit rate, and how to find the missing value in a proportion.

A ratio is a comparison of 2 like quantities.  Ratios need to be in simplest form and they can be written 3 different ways.

Example:


This ratio can be written as 1:3, 1 to 3, or as a fraction 1/3.

A rate is a ratio that compares quantities of different units.
A unit rate is a comparison of a quantity to 1.

Example:

Unit rates become very important when doing price comparison.  We always want to get the best buy.

Example:

    

Which flash drive is the best buy?

5.99/8 = $0.75 per GB                                                         24.95/64 = $0.39

The 64 GB hard drive, according to price, is the better flash drive.

A proportion states that two ratios that are equal to each other.  To solve a proportion you find the cross products and solve for the missing variable by dividing.

Example:

Images from Google.

Week of Oct. 27th

This week we are going to try using our digital textbook to review and learn about ratios, unit rates, and proportions.

Monday - We will review the definition of a ratio, find equivalent ratios, and simplify ratios with different units.  This information can be found in Chapter 6 Section 1 of the digital textbook.  If you can not remember how to log in to the book click here.

Tuesday - We will find unit rates and use them to find other equivalents.  We will be discussing unit cost, price comparison, better buy, etc.  Our lesson will be Chapter 6 Section 2 in our digital textbook

Wednesday - We will learn to solve proportions.  We will solve them like an equation.  We will also have our 3rd Quiz over our Fraction to Decimal Equivalents.

Thursday - We will complete a hallway activity in groups over unit rate and proportions.  This activity will test the students ability to determine items that are a better buy.

Friday - Parent/Teacher Conference Day - Holiday for Students
If you would like to come and discuss how your child is doing in class please send me an email with a time you would like to come visit and I will schedule you into a slot around that time.  If your child seems to be doing fine you do not have to come unless you just want to.

Also, remember Monday, Nov. 3rd is a Teacher Staff Development day so it is also a Student Holiday.

Simple Interest

Today we learned how to calculate Simple Interest.  Simple interest is easy to calculate.  You use the formula I=Prt where I stands for Interest, P = principal (the amount of money you start with), r = rate (the percentage of interest that is used on the money), and t = time (that amount of time the money is earning interest).

Example:  I = _____, P = $1500, r = 5%, t = 3 years

          I = Prt
          I = 1500 x 0.05 x 3
          I = $225

Example:  I = $500, P = $2100, r = _____ , t = 4 years

          I = Prt
          500 = 2100 x r x 4
          500 = 8400 r

           500  =   8400 r 
          8400       8400

             r = 0.059...
             r = 5.95%

Just plug the information in to the equation and solve.

Week of Oct. 20th

This week is the middle of the 2nd six weeks.  Schools seems to be going quickly.  This week might be a long week for some though, we will be having 2 assessments this week.

Monday - Simple Interest

- This lesson will be an application of 1-Step equations.  Simple Interest can be figured using I=PRT (Interest = Principle x Rate x Time).

- Also, any student that did not perform satisfactorily on Test 2-1 will be taking a retest. 

Tuesday - Common Assessment

- We will be taking our first Common Assessment which will cover equations, inequalities, percents, and simple interest.

Wednesday - Test 2-2 Review

- We will review our unit test over equations and inequalities.

Thursday - Test 2-2

- Students will take their second test of the six weeks.

Friday - Begin creating a Digital Portfolio

- Portfolio's are becoming more and more necessary to use as part of a college application.  It is a collection of works by the student that demonstrates student growth.

This week is Homecoming and each day there is a way to show your spirit.

Bonus Problem of the Week

Jenna invests $13,000 into separate bank accounts, one earning 6% simple interest and the other earning 3% simple interest. If at the end of one year she earns $682.50 in interest, how much did she invest in each account?

Copy the problem, solve, and turn in by Friday, Oct. 24th for 15 Bonus Points.

2-Step Equations

Thursday and Friday we learned how to solve 2-Step Equations.  We began by reviewing the two types of 1-step equations:  equations with addition and subtraction, and equations with multiplication and division.

Our main goal in solving equations is to get the variable by itself, or isolate the variable.  With addition problems you add the opposite, with subtraction you change the subtraction sign to addition and the next number to its opposite then you add the opposite.

              x + 6 = -4                                           x - 8 = 4
    x + 6 + (-6) = -4 + (-6)                            x + (-8) = 4
                    x = -10                              x + (-8) + 8 = 4 + 8
                                                                              x = 12

With equations involving multiplication and division you do the opposite operation to isolate the variable.

             4x = 24                                  x    =  -3
                                                           5
            4x  =  24 
             4        4                                 5 x   =  -3 (5)
                x = 6                                   5
                                                               x = -15

With 2-Step equations you have both of the steps in the same problem.  To solve a 2-step equation you reverse the order of operations.  The last thing you did to solve an order of operation problem is add/subtract to the first thing you undo in an equation is add/subtract.

              2x + 6 = -10                                                     m  - 5 = 3
    2x + 6 + (-6) = -10 + (-6)                                           3
                     2x = -16                                         m  + (-5) + 5 = 3 + 5
                                                                            3
                     2x  =  -16                                                      m   =  8
                      2         2                                                        3
                       x = -8
                                                                                     (3)  m  = 8 (3)
                                                                                            3

                                                                                             m = 24

Graphing Inequalities

Today we began inequalities.  An inequality says that two quantities are not equal. Click on inequality for more explanation.

We learned how to write an inequality to represent a situation such as:

Situation:  My classroom can hold no more than 30 students.
Inequality:   s < 30   The number of students can be less than or equal to 30.

Situation:  The waiting room has at least 15 people in it.
Inequality:   p > 15    The number of people in the waiting room is 15 or more people.

We also learned how to graph inequalities.  If the inequality is < or > then you would use an open circle because the solution does not include the value.  If the inequality is < or > then you would use a closed circle because the solution does include the value.

Here are some examples.

Tuesday we will solve 1-step inequalities and graph the solutions.

Images used from Google images.

Week of Oct. 13th

We have now completed 7 weeks of school, that is hard to believe.  This week we will learn about 1-step inequalities and 2-step equations.  Last week we began with 1-step equations with addition and subtraction then multiplication and division.  This week we will put both steps together.

Monday and Tuesday:  1-Step Inequalities
- We will begin by learning how to graph inequalities and then we will solve them.

Wednesday and Thursday:  Solve 2-Step Equations
- We will put our knowledge of 1-step equations with add/subtract and multiply/divide and put them together.

Friday:  Right now the plan is to create equations and inequalities from word problems and solve them.  This plan may change.


Bonus Question of the Week:
A physical therapist earns $87,000 annually. The therapist owes 6.2% of her earned wages for social security tax. She also owes 1.45% of her earned wages for Medicare tax.

What are the physical therapist's total payroll deductions for social security tax and for Medicare tax for the year?

Copy the problem, solve, and turn in by Friday, Oct. 17th for 10 extra bonus points.

Solving 1-Step Equations

In order understand 1-step equations you need to understand four vocabulary words:
Variable, Constant, Expression, and Equation.  Click on the word for its definition.

1-Step Equations are equations that require only one step to solve.  For all equations we have one goal in mind and that is to get the variable by itself.  For an equation involving addition you add the opposite of the number with the variable to get the variable by itself.  If the equation involves subtraction you LCO the problem and then add the opposite.  For equations involving multiplication and division you don't add the opposite but you do the opposite.

Examples:

Addition:
          x + 5 = 9
x + 5 + (-5) = 9 + (-5)    Add the opposite
                x = 4

Subtraction:
             m - 8 = -4
        m + (-8) = -4     LCO
  m + (-8) + 8 = -4 + 8     Add the opposite
                   m = 4

Multiplication:
       4g = 12

      4g  =  12     Since you have a multiply by 4 you do a divide by 4 to get the variable by itself.
       4        4
         g = 3

Division:
       h  = -7
       5

 (5)  h  = -7 (5)   Since you have a divide by 5 you do a multiply by 5 to get the variable by itself.
       5

        h = -35

October 3, 2014 - Sales Tax, Discount, and Tip

Thursday, we discussed how to find percent of a number, today we learned how to apply what we learned.

Sales Tax - a percentage of the total purchase that is added to your purchase.  In Huntsville our tax rate is 8.25% of your purchase.

Example:
Ms. Acton spent $205.60 at Target. If the sales tax is 6%, what was her final bill?

The first thing you do is determine the percent question being asked:

What is 6% of $205.60?

.06 x $205.60 = 12.336  We round this amount to the nearest cent, $12.34 and add that amount to the amount Ms. Acton spent.

$205.60 + $12.34 = $217.94

Her final bill was $217.94.

Discount - a percentage of the total purchase that is subtracted from your purchase.

Example:
Macy's is having a one-day 35% off special. If Sara bought $124.50 worth of items, what would the final bill total after applying the discount of 35%?

So what is the percent question:  What is 35% of $124.50?

0.35 x $124.50 = 43.575 = $43.58 is the amount of discount.  You subtract this amount from $124.50.

$124.50 - $43.58 = $80.92   Sara's final bill is $80.92.

Tip (gratuity) - a percentage of your bill at a restaurant that you leave for the wait staff.

Example:

The Kerwoods went out to eat at Chille's. If there bill was $58.65 and they gave their server a 15% tip, how much did they pay altogether?

The question is What is 15% of $58.65?

0.15 x $58.65 = 8.7975 = $8.80 is the amount of tip the Kerwoods should leave.  Now the question asks how much did they pay all together?  So you need to add the bill total and the tip together.

$58.65 + $8.80 = $67.45 is the total amount spent by the Kerwood family at Chille's.

October 2, 2014 - Percent of a Number

Today, in class we learned how to calculate percent of a number.  This topic, in my opinion, is one of the most important topics a student can learn.  Percent of a number is calculated for us all the time:  sales tax, discount, a tip at a restaurant, interest rates etc.

Percent of a number questions come in the form of questions like:   What is 40% of 80?

I discuss with the students that this is a Language Arts style question.  We need to translate this into a mathematical question.

  What   is 40% of 80?         translate into
______ = 40%  x  80

The What is a blank, is means =, and of means multiply.  We are using calculators in class so they multiply 40% by 80.  There is only one problem, our calculator, the TI-83, does not have a percent key.  TI assumes that if you are smart enough to use their calculator you are smart enough to know how to type a percent into the calculator.  All you do is change the percent into a decimal by moving the decimal 2 places to the left.

_ 32    = .4 x 80

Here are a couple more examples.

What is 25% of 75?   -   .25 x 75 = 18.75
What is 15% of $120?     -     .15 x 120 = $18
What is 0.5% of 12?     -     0.005 x 12 = 0.06
What is 140% of 8?     -     1.4 x 8 = 11.2