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

Scale Drawing 7-9

Using the book’s drawing on page 237, find the length and width of the room shown, if the scale of the drawing is 1cm: 1.5 m

Measuring the drawing, we find that it has a length of 4 cm and a width of 3 cm
Method 1: write a proportion for the length
Let l = the actual length
1/1.5 = 4/ l
l= 4 (1.5)
l = 6
The room is 6 m long
Write a proportion for the width
Let w = the actual width
1/1.5 = 3/w
w = 3 (1.5)
w = 4.5 m
The room is 4.5 m wide

A scale drawing is a diagram of an object where its length and width aare proportional to the actual length and width.

The scale for the drawing gives the relationship between the drawing's measurements and the actual measurements.
For example 1 inch represents 2 feet

The Scale Factor is the ratio of the length in the drawing to the corresponding actual lenth.

A Scale Factor on a matchbox car of 1/64 would mean that every part of the matchbox car was 1/64 of the actual car-- the windshield, the door, the wheels, the roof... would all have the same relationship.

Scale Factor MUST HAVE the same units of measurements.



Method 2 : Use the scale ratio

1 cm/ 1.5 m = 1/cm/150 cm = 1/150
You need to change the units to the same and then set up a ratio

The actual length is 150 times the length in the drawing so
l =150 (4) = 600 cm = 6 m
w =150 (3) = 450 cm = 4.5 m

The scale on a map is 1 cm to 240 m


Open your books to Page 237, you will notice a drawing of a house. In this drawing of the house, the actual height of 9 meters is represented by a length of 3 centimeters, and the actual length of 21 meters is represented by a length of 7 centimeters.
This means that 1 cm in the drawing represents 3 m in the actual building. Such a drawing in which all lengths are in the same ratio to actual lengths is called a scale drawing.
The relationship of length in the drawing to actual length is called the scale. In the drawing of the house the scale is 1cm: 3m

We can express the scale as a ratio, called the scale ratio, if a common unit of measure is used. Since 3 m = 300 cm, the scale ratio is 1/300

The distance from Ryan’s house to his school is 10 cm on the map. What is the actual distance?
let d = the distance from Ryan’s house to school
1/240 = 10/d so d = 240(10)
d = 2400m or
The distance from Ryan's house to school is 2.4 km


A picture of an insect has a scale 7 to 1. The length of the insect in the picture is 5.6 cm. What is the actual length of the insect?
Let l = the actual length of the insect

7/1 = 5.6/ l
7l = 5.6
l = .8 cm
The actual length of the insect is 0.8 cm which is 8 mm

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

Problem Solving: Using Proportion 7-8

Proportions can be used to solve word problems. Use the following steps to help you in solving problems using proportions
~ Decide which quantity is to be found and represent it by a variable
~ Determine whether the quantities involved can be compared using ratios (rates)
~ Equate the ratios in a proportion
~ Solve the proportion

Dr Irshay bought 4 tires for her car at a cost of $264. How much would 5 tires cost at the same rate?
Let c = the cost of 5 tires. Set up a proportion

4/264 = 5/c
Solve the proportion
4c = 5(264)
Now divide both sides by 4
4c/4 = 5(264)/4
c = 330
Therefore, 5 tires would cost $330

Notice, the proportion in this example could also have been written as

2/264 = c/5
In fact--Any of the following proportions can be used to solve the problem

4/5 = 264/c 5/4 = c/264 4/264 = 5/c 264/4 = c/ 5
All of the above proportions result in the same equation

4c = 5(264)
But be careful you need to use a proportion that does relate.
4/5 DOES NOT EQUAL c/264. That is not an accurate proportion and would result in an inaccurate solution.

tires ___________ = _____________tires
cost _____________=_____________cost


If 4 bars of soap cost $1.50, how much would 6 bars of soap cost?

let x = the cost of 6 bars of soap

bars ___________ = _____________bars
cost _____________=_____________cost
4/1.50 = 6/x
simplify first up and down ( BUT NEVER DIAGONALLY!!)
x = 3 (.75)
x = 2.25
$2.25
6 bars would cost $2.25


If 9 kg of fertilizer will feed 300 m² of grass, how much fertilizer would be required to feed 500 m² of grass?

kg ___________ = _____________kg
m² _____________=_____________m²

9/300 = n/500
SIMPLIFY FIRST
9/3 = n/5
3/1 = n/5
n = 15
15kg of fertilzer







For every 5 sailboats in a harbor, there are 3 motorboats. If there are 30 sailboats in the harbor, how many motorboats are there?

Let m = the number of motorboats
5/3 = 30/m

5m = 3(30) divide both sides by 5 5m/5 = 3(30)/5


m=18
There are 18 motorboats in the harbor.

Some guidelines you can use to determine when it is appropriate to use a proportion to solve a word problem.

Ask the following questions

If one quantity increase does the other quantity also increase? (If one quantity decreases, does the other quantity decrease?) When the number of tires is increase, the cost is also increased.

Does the amount of change (increase or decrease) o one quantity depend upon the amount of change (increase or decrease) of the other quantity? The amount of increase in the cost depends upon the number of additional tires bought.

Does one quantity equal some constant times the other quantity? The total costs equals the cost of one tire times the number of tires. The cost of one tire is constant.

If the answers to all the questions above is YES, then it is appropriate to use a proportion.

Sometimes setting up a table can be useful




Although the problems in this lesson may be solved without using proportions, I must insist that you write a proportion for each problem and solve using this method. You may check your work using another other method you know.

Algebra Honors (Period 6 & 7)

Quadratic Equations 8-8

A QUADRATIC FUNCTION IS NOT y = mx + b
(which is a LINEAR function),
but instead is
y = ax² + bx + c
OR
f(x) = ax² + bx + c
where a, b, and c are all real numbers and
a cannot be equal to zero because
it must have a variable that is squared (degree of 2)

Quadratics have a squared term, so they have 2 possible solutions (roots)
You already saw this when you factored the trinomial and used zero products property.

If the domain is all real numbers, then you will have a PARABOLA which looks like a smile when the a coefficient is positive or
looks like a frown when the a coefficient is negative.

What happens as the "a" coefficient gets really big or really small (fraction/decimal)? We'll look at that together on my graphing calculator.

But think, what happened when the "m" (slope) coefficient got big?

The slope got steeper.

So now think that both sides of the U get steeper at the same time.

What's happening to the shape of the U???

Now think, what happened when the "m" (slope) coefficient got tiny?
The slope was a bunny slope.
So now think that both sides of the U are bunny slopes at the same time.
What's happening to the shape of the U???

Putting in standard form:
Standard form is:
y = ax² + bx + c

OR

f(x) = ax² + bx + c

You can't read the sign of a, b, or c until it's in standard form (just like y = mx + b!)

Graphing quadratics:

You can graph quadratics exactly the way you graphed lines
...by plugging in your choice of an x value and using the equation to find your y value.



Because it's a U shape, you should graph 5 points as follows:

First MAKE SURE THE EQUATION IS IN STANDARD FORM!

y must be isolated on one side and then you can read the a and b coefficients.

y = ax² + bx + c


Point 1) the vertex - the minimum value of the smile or the maximum value of the frown


The x value of the VERTEX = -b/2a


Plug that into the equation and then find the y value of the vertex



Next, draw the AXIS OF SYMMETRY :

x = -b/2a


a line through the vertex parallel to the y axis


Point 2) Pick an x value IMMEDIATELY to the right or left of the AXIS OF SYMMETRY and find its
y by plugging into the equation.



Point 3) Graph its mirror image on the other side of the AXIS OF SYMMETRY by counting from the axis of symmetry


Points 4 and 5) Repeat point 2 and 3 directions with another point ONE STEP FARTHER from the AXIS OF SYMMETRY.



JOIN YOUR 5 POINTS IN A SMOOTH "U" SHAPE ( not a V shape!)

AND 
EXTEND LINES WITH ARROWS ON END



Parabolas are functions whose domains are ALL REAL NUMBERS.


Their ranges depend on where the vertex is and also if the ‘a’ coefficient is positive or negative



EXAMPLE: f(x) = -3x²

(or y = -3x² )

the ‘a’ coefficient is negative so it is a frowny face


The vertex is called the maximum.


The x value of the vertex is -b/2a 


a = -3 and b = 0 (it's missing!)


The x value of the vertex = -b/2a = -0/2(-3) = 0


Plug the x value of 0 back into the function to find the y value of the
vertex:


y = -3(02) = 0
So the vertex is (0, 0)

The domain is all real numbers.


The range is y is less than or equal to zero (It's a frowny face)



To graph this function:


1) Graph vertex (0, 0)


2) Draw the AXIS OF SYMMETRY –
a dotted line at x = 0 (actually this is the y axis!)


3) Pick x value immediately to the right of axis of symmetry, x = 1


Plug it in the equation to find the y value: y = -3(1) = -3


Plot (1, -3)


4) Count the same 1 step from axis of symmetry on the other side of the axis
and place another point to the LEFT of axis at the same y value
(-1, -3)


5) Pick another x value to the right 2 steps away from the axis of symmetry, x = 2

Plug it in the equation to find y:
y = -3(22) = -12
Plot (2, -12)


6) Count 2 steps from axis of symmetry on the other side of it and
place another point to the LEFT of axis at the same y value
(-2, -12)



JOIN YOUR 5 POINTS IN A "U" SHAPE AND EXTEND LINES WITH ARROWS ON END

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

Ratio 7-6 Word Problems
Since all of the word problems from Ratios 7-6 are excellent examples of using Ratios, I thought I would create a blog posting with the solutions. Although I will list the actually word problems, you might need to turn to the textbook and page 229 for any charts included.
Read each problem carefully and set up the solutions based on the question asked
Page 229
1. What is the cost of grapes in dollars per kilogram if 4.5 kg of grapes costs $ 7.56?
Since they want $/kg you need to have 7.56/ 4.5 = 1.68; so the grapes cost $ 1.68/kg.

2. This one looked confusing, but just read it a few times: The index of refraction of a transparent substance is the ratio of the speed of light in space to the speed of light in the substance. (Read it again—it is just a ratio). Using the table on Page 229, find the index of refraction of
A. Glass
B. Water

According to the table Speed of light in space is 300,000 km/sec
Speed of light in glass is 200,000 km/ sec.
So
300,000
200,000
or 3/2

Speed of Light in Water is 225,000

so that ratio is
300,000
225,000
This takes a little more time dividing but you can simplify it to 4/3 Since it is a ratio you want to LEAVE it as an improper fraction.


3. The mechanical advantage of a simple machine is the ratio of the weight lifted by the machine to the force necessary to lift it. Now—just read that again—you don’t need to truly understand physics to get this problem. It is a RATIO again!! What is the mechanical advantage of a jack that lifts a 3200-pound car with a force of 120 pounds?

3200 =
120
80/3; so the mechanical advantage is 80/3

4. The C-String of a cello vibrates 654 times in 5 seconds, How many vibrations per second is that? (We are finding unit rate)
It’s a rate of time/sec.
654/ 5 = 130 4/5 vibrations per second

5. A four cubic foot volume of water at sea level weighs 250 lb. What is the density of water in pounds per cubic foot? (Do you need to understand density to do this problem?) No—just unit rates!! We want lbs/cubic foot. Look at the information and place it with lbs/ft and then divide to get the unit rate.
250lb/ 4 ft³. That equals 62 1/2 lb/ft³ .

6. A share of stock that costs $88 earned $16 last year. What was the price-to-earnings ratio of this stock? (Wish that was happening now!!)
Price/earnings so look carefully 88/16 = 11/2 so the Ratio is 11/2

In exercises 7 and 8 you will need to look at the diagrams on page 230. You are finding the ratio in lowest terms.
Look at both the small and large triangles. For A you are finding the ratio between segment AB of the larger triangle and DE of the smaller triangle.
4.8
3.2
Simplify ( I like to clear decimals to make it easier but in this case you could easily divide
= 3/2
B.
Perimeter of Triangle ABC
Perimeter of Triangle DEF

First you need to find the perimeter of the larger triangle by adding up the three sides
6.3 + 4.8 + 3 = 14.1
and then finding the perimeter of the smaller triangle by adding up those three sides
4.2 + 3.2 + 2 = 9.4

14.1
9.4
I like to clear decimals by multiplying by 10
141
94
= 3/2

What did you notice?

8. Again look at Page 230 for the figure
We are finding the following Ratios:
PQ:TU 2.6:6.5 or 26:65 = 2:5
QR:UV 1.2:3 = 12:30 = 2:5

Perimeter of PQRS: Perimeter of TUVW
You need to add up all the sides of each of the figures
2.2 + 2.6 + 1.2 + 3 = 9
and 5.5 + 6.5 + 3 + 7.5 = 22.5
9:22.5 = 90:225 = 2:5

What did you notice here?


For exercised 9-12 you need to use the Table on Page 230 as well

9. The population of Centerville in 1980 to its population in 1970?
44/36 = 11/9

10. The growth in the population of Easton to its 1980 population.
You need to subtract the 1970 population from the 1980 population to get the growth so
28-16 = 12 12/18 = 3/7

11. The total population of both towns in 1970 to their total population in 1980
36 + 16 = 52 and 44 + 28 = 72 52/72 = 13/ 18

12. The total growth in the population of both towns to their total 1980 population
Like number 10 you need to subtract to find the growth
(44-36) + (28-16) = 8 + 12 = 20
44 + 28 = 72 (from before)
20/72 = 5/18

13. During a season a baseball player hit safely in 135times at bat; the player struck out or was fielded out in 340 times at bat. What is the player’s ratio of hits to times at bat?

You take the 135 + 340 = 475 (the total times at bat)
135/475 = 37/95

14. A fruit drink recipe requires fruit juice and milk in the ration 3:5. What fraction of the drink is milk? What fraction is juice? Remember we need to find the total first. 3 + 5 = 8
so the fraction that is milk is 5/8 and the fraction that is fruit juice is 3/8