Algebra Honors (Periods 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 ... Jake and  Jordan in 6th period and Henri and Andy from 7th period.

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

The chart below was take from last year's examples

We set up a chart

Motion in Same Direction

Next we had a fictitious story about Ritika's Helicopter and Maya's plane ( or  Ryan's helicopter and Allison'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 Maya (or Allison) to over take  Ritika's ( or Ryan'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 Jenna ( or Brendan) up the slope at 6 km/h Jenna or Brendan 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

Algebra Honors (Periods 6 & 7)

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.

Math 6A ( Periods 1 & 2)

Exponents and Powers of Ten 3-1
When two or more numbers are multiplied together--each of the numbers is called a factor of the product.

A product in which each factor is the SAME is called a power of that factor.

2 X 2 X 2 X 2 = 16. 16 is called the fourth power of 2 and we can write this as
2⁴ = 16

The small numeral (in this case the 4) is called the exponent and represents the number of times 2 is a factor of 16.
The number two, in this case, is called the base.

When you are asked to evaluate... simplify... solve... find the answer
That is,
Evaluate
4³ = 4 X 4 X 4 = 16 X 4 = 64

The second and third powers of a numeral have special names.
The second power is called the square of the number and the third power is called the cube.

We read 12² as "twelve squared" and to evaluate it
12² = 12 X 12 = 144

Powers of TEN are important in our number system.
Make sure to check out the blue sheet and glue it into your spiral notebook
First Power: 10¹ but the exponent is invisible = 10
Second Power: 10² = 10 X 10 = 100
Third Power 10³ = 10 X 10 X 10 = 1000
Fourth Power 10⁴ =10 X 10 X 10 X 10 = 10,000
Fifth Power 10⁵ = 10 X 10 X 10 X 10 X 10 = 100,000

Take a look at this list carefully and you will probably see a pattern that we can turn into a general rule:

The exponent in a POWER of TEN is the same as the number of ZEROS when the number is written out.

The number of ZEROS in the product of POWERS OF TEN is the sum of the numbers of ZEROS in the factors.

For example Multiply.
100 X 1000
Since there are 2 Zeros in 100 and 3 zeros in 1000,
the product will have 2 + 3 , or 5 zeroes.
100 X 1000 = 100,000

When you need to multiply other bases:

first multiply each
For example

3⁴ X 2 ³ would be
(3 X 3 X 3X 3) X ( 2 X 2 X 2)
= 81 X 8 = 648

What happens when you multiply the same bases?
3⁴ ⋅ 3² = 3⋅3⋅3⋅3⋅3⋅3 or 3 ⁶
We just add the exponents if the bases are the same!!

Well then, what about (3⁴)² ?
Wait.. look carefully isn't that saying 3⁴ Squared?
That would be (3⁴)(3⁴), right?
.. and looking at the rule above all we have to do here is then add those bases or 4 + 4 = 8 so the answer would be 3⁸.
OR
we could have made each (3⁴) = (3⋅3⋅3⋅3)
so (3⁴)² would be 3⋅3⋅3⋅3⋅3⋅3⋅3⋅3 or still 3⁸
But wait... isn't that multiplying the two powers? So when raising a power to a power-- you multiply!!
(3⁴)² = 3⁸

1 to any more is still just 1
1⁵ = 1

0 to any power is still 0!!

Evaluate if a = 3 and b = 5
Just substitute in... but use hugs () we all love our hugs!!
a³ + b²
would be (3)³ + (5) ²
= 27 + 25 = 52




Check out this great Video on the Powers of Ten
POWERS OF TEN

Algebra Honors ( Period 6 & 7)

Multiplying Polynomials by Monomials 4-5
This is just the distributive property
x(x + 3) = x² + 3x

-2x(4x² - 3x + 5)

-8x³ + 6 x² -10x
The book shows you how to multiply using a vertical method but I think using the original method taught with the distributive property works just as well-- if not better.

n(2-5n) + 5(n² -2 ) = 0
2n - 5n² + 5n² - 10 = 0
2n - 10 = 0
2n = 10
n = 5
and in set notation {5}

1/2(6xc + 4) -2(c + 5/2) = 2/3 (9-3c)
3c + 2 - 2c - 5 = 6 - 2c
3c -3 = 6
3c = 9
c = 3
and in set notation {3}

Multiplying Polynomials  4-6
This is just double distributive property or triple distributive property so you really need to understand the DP

You have learned how to use the DP to multiply
2x(3x + 2)
but... what happens if you had instead
(2x +5)(3x + 2)
There are a number of different strategies to simplify this multiplication
(2x +5)(3x) + (2x + 5)(2)

6x² + 15x + 4x + 10

6x² + 19x + 10

You could also use Fireworks—as show in class or FOIL

Fà First
OàOuter
IàInner
Là Last

 (2x + 5)(3x + 2)

The F is the first terms   (2x)(3x)

The O is the Outer terms (2x)(2)

The I is the Inner terms (5)(3x)

the L is the Last Terms (5)(2)

or 6x² + 4x + 15x + 10

6x² + 19x + 10

Example:

(3x-2)(2x²- 5x-4)

The book shows you how to multiply in vertical form, similar to how you multiply multi-digit  numbers.

Read Page 161 if you are interested in reviewing that strategy

Step 1   2x² – 5x – 4                         Step 2                       2x² – 5x – 4                   

3x – 2                                                                         3x – 2             

6x³-15x² – 12x                                                    6x³-15x² – 12x

                                                                                                 -4x² +10x + 8     

Step 3        Add:

  2x² – 5x – 4                                                                        

   3x – 2             6x³-15x² – 12x                                                    

       -4x² +10x   + 8     

6x³ – 19x² -2x + 8

I showed Fireworks and Double Distributive Property with this example as well

but also showed my favorite… The BOX Method

Create a box as big as the polynomials

In this case it’s a 2 by 3

We talked about the order of the polynomials.

Make sure to place them in descending order.

The book terms it decreasing degree of x:

We discussed

x³ -3x² + xy² + 2y³

To see the advantage of rearranging terms, multiply the polynomial

(y +2x)(x³ – 2y³ + 3xy² + x²y)

We then rearranged both polynomials into decreasing degree in terms of x

(2x + y)(x³ + x²y+ 3xy² – 2y³)

Using the BOX Method

we could find the simplified form to be

2x⁴ + 3x³y + 7x²y² –xy³ -2y⁴