Algebra Honors (Period 6 & 7)

Cost Income, and Value Problems 3-7
Objective: To organize the facts of a problem in a chart & solve problems involving cost, income, and value
Using a chart to organize the facts of a problem can be a helpful problem solving strategy.
Cost = number of items X price per item
Income = hours worked X wage per hour
Total value = # of items X value per item

Example: Tickets for the senior class play cost $6 for adults and $3 for students. A total of 846 tickets worth $3846 were sold. How many student tickets were sold?
Let x = the number of student tickets sold
Then 846- x = the number of adult tickets sold

The only fact NOT recorded in this chart is that the total cost of the tickets is $3846.
The equation becomes
3x + 6(846 –x) = 3846.
3x + 5076 – 6x = 3826
-3x = -1230
x = 410
Check to make sure what x represented… in this case the number of student tickets so
410 student tickets were sold.


We then turned to Problem 4 on Page 128

A collection of 52 dimes and nickels is worth $4.50. How many nickels are there?
Let d = the number of dimes, so If there are 52 in the collection
52-d must = the number of nickels. So our Chart looks like:

This time we reread the given facts and with the second fact, we realize that we can add the two expressions and set them equal to $4.50
10d + 5(52 - d) = 450
solving this equation, we find that d = 38
Make sure to reread the question… it asked “how many nickels?”
So we need to use 52-d and substitute in 38… 52 -38 = 14
14 nickels is the correct answer


Next we complete # 6
Celia bought 12 apples, ate two and sold the rest at 20 cents more per apple than she paid. Her total profit was $1.00 How much did she sell each apple for?
Let b = the price she bought each of the apples for
We glued in the “yellow-colored” chart here and completed it to look like:

Now, her profit is the difference between what she sold them for and what she bought them for… Therefore the equation becomes
10(b + 20) – 12b = 100
10b + 200 -12b = 100
-2b = -100
b = 50
Re reading the question we realize we solved for what she bought the apples each for. We need to add 20cents to find out what she sold them for

She sold each apple for 70 cents.

The last problem we did from our textbook was # 14 on Page 129

Jo has 37 coins ( nickels, dimes & quarters) for $5.50 She has 4 more quarters than nickels. How many dimes does Jo have?
Let n = the number of nickels
n + 4 = the number of quarters
So if there was a total of 37 the rest must be dimes
37 – [ n + (n+4)] or 37 –(2n +4)
Let’s set up the chart

Now we know that all of them combined equal $5.50
so
5n + 10[37-(2n+4) + 25(n +4) = 550
5n + 370 – 20n – 40 + 25n + 100 = 550
combining all the n’s

10n + 330 +100 = 550
10n = 120
n = 12
Re reading the question, we find we need to see how many dimes so substitute in
37 -[2(12) +4] = 37 – [24 +4] = 37 -28 = 9
Jo had 9 dimes.

Algebra Honors (Period 6 & 7)

Problem Solving: Using Charts 3-6
Objective: To organize the facts of a problem in a chart.
Using a chart to organize the facts of a problem can be a helpful problem solving strategy.

Organize the given information in a chart
A swimming pool 25 m long is 13 m narrower than a pool 50 m long
There are two different charts you could create:

In the first chart we started with
Let w = the width of the 2nd pool; so to write the 1st pool in terms of the second we would place w-13 in place of the ?

In the 2nd chart we started with
Lt w = the width of the 1st pool; so we write the 2nd pool in terms of the first, so we would place w + 13 in place of that ?


We then solved the following:

Find the number of calories in an apple & a pear…

1) the pear contains 30 calories more than the apple.
2) Ten apples have as many calories as 7 pears.
Let a = the number of calories in an apple
Then a + 30 = the number of calories in a pear.
We glued in the “orange- colored” chart here and completed it to look like:

This time we reread the given facts and with the second fact, we realize that we can set
The last column expressions equal to each other
10a = 7(a + 30)
solving this equation, we find that a = 70
Therefore, an apple has 70 calories and a pear has 100 calories
Next we used the following two given facts to set up a chart and create an equation
1) An egg scrambled with butter & milk has 1 more gram of protein than an egg fried in butter.
2) Ten scrambled eggs have as much protein as a dozen fried eggs.
Let x = the number of protein in a fried egg.
Then x + 1 = the number of protein in a scrambled egg.
We glued in the “orange- colored” chart here and completed it to look like:

Our equation would be 10(x + 1) = 12x
We then turned to our textbook to Page 122-123 and completed problems 1 and 3 in our spiral notebook as follows:

Solve each problem using the two given facts. Complete a chart to help you solve each problem

1. Find the number of full 8 hour shifts that Maria worked last month
1) She worked twice as many 6 hour shifts as 8 hour shifts
2) She worked a total of 280 hours.

Math 6 Honors ( Periods 1, 2, & 3)

Writing Mathematical Expressions 2-1

Make sure to glue the 'purple 1/2 sheet' of math word phrases that we associate with each of the four basic operations -- into your spiral notebook (SN)

We can use the same mathematical expression to translate many different word phrases
Five less than a number n
The number n decreased by five
The difference when five is subtracted from a number n

All three of those phrases can be translated into the variable expression

n-5

The quotient of a number y divided by ten becomes y/10. It may look like only a fraction to you-- but if you read y/10 as always " y divided by 10" you have used the proper math language.

Twelve more than three times a number m

Wait-- where are you starting from... in this case you are adding 12 to 3m so you must write

3m + 12


Not all word phrases translate directly into mathematical expressions. Sometimes we need to interpret a situation.. we might need to use relationships between to help create our word phrase.

In writing a variable expression for the number of hours in w workdays, if each workday consists of 8 hours...

8w would be our expression


Some everyday words we use to so relationships with numbers:
consecutive whole numbers are whole numbers that increase by 1 for example 4, 5, 6
So the next consecutive whole number after w is w + 1.

A preceding whole number is the whole number that is 1 less and the next whole number is the whole number that is 1 greater.

So the number which precedes x would be x - 1.
The next number after n is n + 1


Thursday, September 15th Lesson:


What if I asked what is the next consecutive EVEN number after the even number "m"
It would be m + 2





What would it be if I asked what was the next consecutive odd number, after the odd number x?
x + 2
Let's look at those relationships like the workdays from Tuesday's lesson... We are going to find another strategy to use for some of the more complicated expressions...

First set up a T chart- as discussed in class
put the unknown on the left side of the T chart... The unknown is always the one that reads like " w workdays"

so in this case

w workdays on the left side and under it you put
1
2
3

On the right side put the other variable-- in this case hours
under hours put the corresponding facts you know-- the relationship between workdays and hours as given in this case
hours
8
16
24

all of those would be on the right side of the T chart.

Now look at the relationships and ask yourself--
What do you do to the left side to get the right side?

and in this case

What do you do to 1 to get 8?
What do you do to 2 to get 16?
What do you do to 3 to get 24?

Do you see the pattern?

For each of those the answer is "Multiply by 8" so
what do you do to w-- The answer is Multiply b 8
so the mathematical expression in this case is "8w."

What about writing an expression for
The number of feet in i inches

i inches is the unknown... so that goes on the left side of the T chart... with feet on the right

i inches ___feet
12...............1
24...............2
36...............3

I filled in three known relationships between inches and feet Now, ask your self those questions again...
What do you do to the left side to get the right side?

and in this case

What do you do to 12 to get 1?
What do you do to 24 to get 2?
What do you do to 36 to get 3?

In each of these, the answer is divide by 12
so What do you do to i? the answer is divide by 12
i inches ___feet
12...............1
24...............2
36...............3
i................i/12



and it is written i/12

Algebra Honors (Period 6 & 7)

Equations w/ The Variable on Both Sides 3-5

Partial Notes
2 unique types of equations:
Identity equation: You solve it and you get the same thing on both sides...if you solve until you cannot do anything more, you get 0 = 0.
What this means is that you can pick any number and the equation will work!
The Distributive Property is the simplest example of an Identity Equation:
3(x + 7) = 3x + 21  
Distribute on the left side and you'll get:
3x + 21 = 3x + 21  
At this point, you should already know this is an Identity!
If you keep solving, you would subtract 3x from each side and you'll get:
21 = 21
and you know again that this is an Identity.
If you now subtract 21 from each side:
0 = 0
BUT I WOULDN'T GO THIS FAR! AS SOON AS YOU HAVE THE SAME THING ON BOTH SIDES, YOU CAN STOP AND SAY IT'S AN IDENTITY EQUATION!!!

Null set equation:
You solve it and you get an impossible answer:
3(x + 7) = 3x + 10
3x + 21 = 3x + 10
21 = 10
WHEN WILL THAT HAPPEN??? NEVER!!! SO THERE IS NO POSSIBLE SOLUTION TO THIS! The answer is the null set.