Math 6A (Periods 2 & 4)

Solving Other Equations & Inequalities 2-5 cont'd
Parent: "What did you do today in Math?"
Student: "Today Mrs. Nelson taught us how to wrap and unwrap a present!!!"
Parent: "Huh..."
Student: "Well, it all has to do with PEMDAS... and undoing equations to solve them!!

Someone was paying attention today... :)


In order to solve two step or multi-step equations you must UNDO in the reverse order of PEMDAS...

UNDO by doing the reverse of PEMDAS

4x + 172 = 248

I will review the Formal Check on Monday but if you want to see-- read on...
CHECK:

Step 1: Rewrite the problem
Step 2: Substitute the solutions for the variable
Step 3: DO THE MATH

4x + 172 = 248
We found that x = 19 above-- > in the first problem
so for the check
4x + 172 = 248
4(19) + 172 ?=? 248
76 + 172 ?=? 248
248 = 248 CHECK!!!

For the above, you actually pace a “?” above the equal sign….rather than to each side…

Math 6A ( Periods 2 & 4)

Solving Equations & Inequalities 2-4 continued

GOAL: You use the INVERSE operation to ISOLATE the variable on one side of the equation


Here are the steps and justifications (reasons)

1. focus on the side where the variable is and focus specifically on what is in the way of the variable being by itself ( isolated)

2. What is the operation the variable is doing with that number in its way?

3. Get rid of that number by using the opposite (inverse) operation

*Use + if there is a subtraction problem
*Use - if there is an addition problem
*Use x if there is a division problem
*Use ÷ if there is a multiplication problem

GOLDEN RULE OF EQUATIONS; DO UNTO ONE SIDE OF THE EQUATION WHATEVER YOU DO TO THE OTHER!!



We solve by "undoing" the operations in each equation.

We use inverse operation & undo Aunt Sally
It's the reverse of PEMDAS!!


GOAL: You use the INVERSE operation to ISOLATE the variable on one side of the equation


GOLDEN RULE OF MATHEMATICS
What you do to one side of the equation you MUST do to the other side!!





4. Justification: You have just used one of the PROPERTIES OF EQUALITY
which one?

that's easy-- Whatever operation YOU USED to balance both sides that's the property of equality
We used:

" +prop= " to represent Addition Property of Equality
" -prop= " to represent Subtraction Property of Equality
" xprop= " to represent Multiplication Property of Equality
" ÷prop= " to represent Division Property of Equality

5. You should now have the variable all alone (isolated) on one side of the equal sign.

6. Justification: Why is the variable alone?
For + and - equations you used the Identity Property of Addition (ID+) which simply means that you don't bring down the ZERO because you add zero to anything-- it doesn't change anything... [Note: there is no ID of subtraction]

For x and ÷ equations, you used the Identity Property of Multiplication (IDx) which simply means that you don't bring down the ONE because when you multiply by one it doesn't change anything [NOTE: there is no ID of division]

7. Put answer in the final form of x = ____and box this in.

Algebra Honors (Periods 5 & 6)

Rate-Time- Distance Problems 4-8

D = rt

Uniform Motion

Three types of problems:

• Motion in opposite direction
• Motion in same direction
• Round Trip

Motion in opposite direction
For this we used different students bicycling ... Jeff K and Jeff B in 5th period and Rebecca and Cayla  from 6th period.

Last year's examples follow:
They start at noon --60 km apart riding toward each other. They meet at 1:30 PM. If Josh K speed is 4 km/h faster than Josh P ( Maddie is greater by the same from Jamie's rate) What are their speeds?

We set up a chart

Motion in Same Direction

Next we had a fictitious story about Stan's Helicopter and Brett's plane ( or David's helicopter and Noah's Smiling Plane) taking off from Camarillo Airport flying north. The helicopter flies at a speed of 180 mi/hr. 20 minutes later the plane takes off in the same direction going 330 mi/hr. How long will it take Brett (or Noah) to over take  Stan's ( or David's) helicopter?

Let t = plane's flying time

Make sure to convert the 20 minutes ---> 1/3 hours.
We set up a chart .. Here is the chart with last year's names:

When the plane over takes the helicopter they have traveled the exact same distance so set them equal
180(t + 1/3) = 330t

180 t + 60 = 330t
60 = 150t

t = 2/5
which means 2/5 hour. or 24 minutes.

Round Trip

A ski lift carries Sammy ( or Jessica) up the slope at 6 km/h Sammy or Jessica snowboard down 34 km/h. The round trip takes 30 minutes.

Did you see the picture?

Let t = time down
then set up a chart

6(.5 -t) = 34 t
3 - 6t = 34 t
3/40 = t
Now, what's that?

0.75 hr or 4.5 minutes

How far did they snowboard... plug it in
34(0.075) = 2.55 km

Math 6High ( Period 3)

 Greatest Common Factor 2-2
A whole number that is a factor of two or more non zero whole numbers is a common factor of the two numbers.
Find all the common factors of 14 and 42
List all the factors of each number-- making a T chart of the factors as shown in class.
Factors of 14: 1, 2, 7, 14
Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
You notice that the common factors as 1, 2, 7, and 14
The Greatest of these is 14
GCF(14, 42) = 14

Any two numbers have 1 as a common factor. the largest number that is a factor of two or more nonzero whole numbers is their greatest common factor ( GCF) or greatest common divisor (GCD)
Find the GCF of 28 and 36
Factors of 28: 1, 2, 4, 7, 14, 28
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
GCF( 28, 36) = 4

Find the GCF of 16 and 37
Factors of 16: 1, 2, 4, 8, 16
Factors of 37: 1, 37  (
GCF( 16, 37) = 1
If the GCF of 2 numbers is 1 the two numbers are said to be relatively prime. Notice the numbers do NOT need to be prime themselves. In this case, 37 is prime BUT 16 is not.

the GCF(8, 9)= 1
because
Factors of 8 : 1, 2, 4, 8
Factors of 9: 1, 3, 9
The only common factor they share is 1. 8 and 9 are said to be relatively prime-- even thought BOTH numbers are composite... neither is actually prime.

GCF ( 72, 84)
Making the T chart for these two numbers you discover that
Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84

The greatest common factor is 12
We write
GCF(72, 84) = 12

What about GCF(66, 96)
Factors of 66: 1, 2, 3, 6, 11, 22, 33, 66
Factors of 96: 1, 2, 3, 4, 6, 8, 12, 18, 24, 32, 48, 96

WE discover that GCF(66, 96) = 6
The BOX method was shown in class. It is another way to discover what the GCF is for two numbers.
If you aren't sure, come in and I will review the BOX method for finding the GCF. You will want to know how to use it because we will use it to find the LCM as well-- later in this chapter.

Math 6A (Periods2 &4)

Solving Equations & Inequalities 2-4

Before we began today's lesson, we reviewed some of the difficult mathematical expressions and equations found on Page 448. We discussed the importance of a well placed comma... in MATH as well as in Language Arts!!

Our math example was
the sum of three and a number b, times seven
(3 + b)7

and
the sum of three and a number b times seven
3 + 7b

Our Language Arts example... is one of my favorites.

Where does the comma belong in the following:
A woman without her man is nothing.

I insist it is...
A woman, without her, man is nothing.
However, I acknowledge that some men would differ and insist it is
A woman, without her man, is nothing.
So you see, the comma makes all the difference... in math expressions and equations as well as in language arts!!

We solve by "undoing" the operations in each equation.

We use inverse operation & undo Aunt Sally
It's the reverse of PEMDAS!!


GOAL: You use the INVERSE operation to ISOLATE the variable on one side of the equation

GOLDEN RULE OF MATHEMATICS
What you do to one side of the equation you MUST do to the other side!!


w + 18 = 64 we must undo addition using the inverse of + (that is, subtraction)
- 18 -18
w + 0 = 46
and then we write
w = 46


What property allows us to subtract 18 from both sides of the equation?
The SUBTRACTION property of equality which we abbreviate with
-prop=

What property allows us to write w instead of w + 0
w + 0 = w
That is the Identity Property of Addition. or  ID +

How about
p - 84 = 102 we must undo subtraction using the inverse of - (which is ADDITION)

p - 84 = 102
+84 +84
p + 0 = 186
and then we write
p = 186

What property allows us to add 84 to both sides of the equation?
the addition property of equality, which we abbreviate as
+prop=

What property allows us to write p instead of p + 0
The Identity property of Addition. ID+
Why is it called the Identity Property of Addition?
The number never changes its identity
a + 0 = a for all numbers!!

What happens if we have
b + 7 > 8
This is an inequality but we solve this as we would an equation

b + 7 > 8
- 7 -7
b + 0 > 1
and then we write b > 1

h - 7 > 7
+ 7 +7
h + 0 > 14
and then we write
h > 14

Algebra Honors (Periods 5 & 6)

Transforming Formulas 4-7

Formulas are used throughout real life applications-- the book gives an example of the formula for the total piston displacement of an auto engine... we discussed a number of formulas that students recalled such as the following:
A =lw
d = rt
I = Prt
A = ∏r²
A = P(1 + rt)
C = 5/9(F - 32)
y = mx + b
A = bh
C = ∏d
F = Ma
A = ½(b₁b₂h
E= mc<sup>2
A2 + B2 = C2
and even the quadratic formula-- which we will study later this year..



b = ax ; x
just divide both sides by a
b/a = x

Solve P = 2L + 2W for the width, w
P-2L = w
2

C = 5/9(F - 32) Solve for F
We need to multiply both sides by the reciprocal of 5/9
(9/5)C = F - 32
now add 32 to both sides
(9/5)C + 32 = F


Next we tackled</sup> 

We also discussed the restrictions and found that the denominator could not equal zero.


S = v/r ( solving for r) became one of the homework problems that caused some discussion-- until students realized that they had actually found the reciprocal of r or 1/r instead of solving for r!!
We discussed how to solve that dilemma.