Translations 9-8
You can move pattern blocks by sliding them, flipping them, or turning them. Each of these moves is a transformation.
Sometimes called a slide.
Moves a figure - Doesn't change the shape or size
A translation is a transformation that moves points in the same distance and in the same direction. A figure and its translated image are congruent. You can see examples of translations in wallpaper, fabric and wrapping paper.
The figure you get after a transformation is called the image.
To name the image of a point, you use the prime notation,
Point A and its image Point A’
After the figure is translated (moved), it is then called the image and you use ' (the prime notation)
So triangle ABC once it's moved up 2 spaces is now named ABC'
You can describe a transformation using arrow notation, which describes the mapping of a figure onto its image. A A’
You can also use arrow notation to write a general rule that describes a transformation. Choose corresponding points on a figure and its image. Subtract the coordinates of the pre image from the coordinates of the image.
An arrow pointing to the right tells what the translation is.
So if point A (3, 5) is moved to (5, 1), the notation is A(3, 5) arrow A'(5, 1)
To write the rule of translation:
Use (x, y) then arrow to the right then (x + or - translation number, y + or - translation number)
So for the translation of A above:
(x, y) arrow to the right (x + 2, y - 4)
Symmetry and Reflection 9-9
Symmetry means that if you divide a figure in half, both halves look exactly the same
Line of symmetry: where you can "fold" the figure and have it exactly the same
Figures can have 0 or more lines of symmetry
A random drawing would probably have no lines of symmetry
A butterfly has one line of symmetry
A rectangle has 2 lines of symmetry
A square has 4 lines of symmetry
A circle has infinite lines of symmetry (you can rotate it infinitely and divide it in half)
A figure has reflective symmetry when one half is a mirror image of the other half. A line of symmetry divides a figure with reflectional symmetry into two congruent halves.
Reflection
Reflection is sometimes called a flip of the figure
The figure flips over a LINE OF REFLECTION
For example, if a triangle is in Quadrant I, it may flip over the y axis into Quadrant II
(the right side of the triangle is now the left side)
Or it may flip over the x axis into Quadrant IV
(the top of the triangle is now the bottom)
Or it may flip over any other line like y = x. This is more complicated to understand!
A Reflection flips a figure over a line of reflection. The reflected figure, or image, is congruent to the original figure, called a pre image.
In any to these cases, the vertices of the triangle are not in the same place as before
(as they would be in a translation or slide)
Rotation
This transformation takes a figure and TURNS IT
about a point called the CENTER OF ROTATION.
The angle of turning is called the ANGLE OF ROTATION.
The two most common angles:
180 degrees: ends up in the opposite quadrant (I goes to III , II goes to IV)
The image is: (x, y) goes to (-x, -y) SAME NUMBERS...OPPOSITE SIGNS!
90 degrees: ends up in the next quadrant over counterclockwise
The image is (x, y) goes to (-y, x).
Tuesday, May 19, 2009
Math 6 H Periods 1, 6 & 7 (Tuesday)
Discount and Markup 9-5
A discount is a decrease in the price of an item. A markup is an increase in the price of an item. Both of these changes can be expressed as an amount of money or as a percent of the original price of the item. A store may announce a discount of $3 off the original price of $30 basketball, or a discount of 10%
A warm-up suit that sold for $42.50 is on sale at a 12% discount. What is the sale price?
Method 1: Use the formula
amount of change = percent of change X original amount
= 12% X $42.50
therefore the discount is 0.12 X 42.50 or 5.10
The amount of discount is $5.10
The sale price is 42.50 – 5.10 = $37.40
Method 2: Since the discount is 12%, the sale price is 100% - 12% = 88%.
The sale price is 0.88 X 42.50 = $ 37.40
When you know the amount of discount you subtract to find the new price. When dealing with a markup you add to find the new price.
The price of a new car model was marked up 6% over the previous year’s model. If the previous year’s model sold for $7800, what is the cost of the new car? {and what kind of a car could that be?}
Method 1: Use the formula
amount of change = percent of change X original amount
= 6% X 7800
Therefore the markup is 0.06 X7800= $468
The new price is 7800 + 468 = $8268
Method 2: Since the markup is 6% the new price is 100% + 6% or 106% of the original price. so the new price is 1.06 X7800 = $8268
This year a pair of ice skates sells for $46 after a 15% mark up over last year’s price. What was last year’s price?
This year’s price is 100 + 15 or 115% of last year’s price. Let n present last year’s price
46 = (115/100)n
46 = 1.15n
46/.15 = 1.15n/1.115
40 = n
So last year’s price was $40.
A department store advertised eclectic shavers at a sale price of $36.
If this is a 20% discount, what was the original price?
The sale price is 100 - 20 or 80% of the original price. Let n represent the original price.
36 = (80/100)n
36 = .8n
36/.8 = .8n/.8
45 = n
The original price was $45.
Check to see that your answers are logical and reasonable.
Try these: A service station (that’s gas station, now—they no longer provide service!!) give cash customers a 5% discount on the price of gasoline. If gasoline regularly sells for $3.00 a gallon, what is the discounted price?
A store marks up the price of a $5 item to $12. What is the percent of markup?
Commission and Profit 9-6
Some sales jobs pay an amount based on how much you sell. This amount is called a commission.
Like a discount, the commission can be expressed as a percent or as an amount of money.
amount of commission = percent of commission X total sales.
Using the examples from our textbook,
Maria sold $42,000 word of insurance in January. If her commission is 3% of the total sales, what was the amount of her commission in January?
amount of commission = percent X total sales
0.03 X 42,000 = 1260
Her commission was $1,260.
Profit is the difference between total income and total operating costs.
profit = total income – total costs
The percent of profit is the percent of total income that is profit
percent of profit = profit/total income
MATH AT WORK:
Caterer
A caterer provides food for parties, weddings, bar/bat mitzvahs, and other events. Caterers plan the menu, buy the ingredients, and cook the food. Often they provide seating and music as well. For each event, a caterer determines the cost per guest. The catering business requires a thorough knowledge of rations, proportions, and percents.
A discount is a decrease in the price of an item. A markup is an increase in the price of an item. Both of these changes can be expressed as an amount of money or as a percent of the original price of the item. A store may announce a discount of $3 off the original price of $30 basketball, or a discount of 10%
A warm-up suit that sold for $42.50 is on sale at a 12% discount. What is the sale price?
Method 1: Use the formula
amount of change = percent of change X original amount
= 12% X $42.50
therefore the discount is 0.12 X 42.50 or 5.10
The amount of discount is $5.10
The sale price is 42.50 – 5.10 = $37.40
Method 2: Since the discount is 12%, the sale price is 100% - 12% = 88%.
The sale price is 0.88 X 42.50 = $ 37.40
When you know the amount of discount you subtract to find the new price. When dealing with a markup you add to find the new price.
The price of a new car model was marked up 6% over the previous year’s model. If the previous year’s model sold for $7800, what is the cost of the new car? {and what kind of a car could that be?}
Method 1: Use the formula
amount of change = percent of change X original amount
= 6% X 7800
Therefore the markup is 0.06 X7800= $468
The new price is 7800 + 468 = $8268
Method 2: Since the markup is 6% the new price is 100% + 6% or 106% of the original price. so the new price is 1.06 X7800 = $8268
This year a pair of ice skates sells for $46 after a 15% mark up over last year’s price. What was last year’s price?
This year’s price is 100 + 15 or 115% of last year’s price. Let n present last year’s price
46 = (115/100)n
46 = 1.15n
46/.15 = 1.15n/1.115
40 = n
So last year’s price was $40.
A department store advertised eclectic shavers at a sale price of $36.
If this is a 20% discount, what was the original price?
The sale price is 100 - 20 or 80% of the original price. Let n represent the original price.
36 = (80/100)n
36 = .8n
36/.8 = .8n/.8
45 = n
The original price was $45.
Check to see that your answers are logical and reasonable.
Try these: A service station (that’s gas station, now—they no longer provide service!!) give cash customers a 5% discount on the price of gasoline. If gasoline regularly sells for $3.00 a gallon, what is the discounted price?
A store marks up the price of a $5 item to $12. What is the percent of markup?
Commission and Profit 9-6
Some sales jobs pay an amount based on how much you sell. This amount is called a commission.
Like a discount, the commission can be expressed as a percent or as an amount of money.
amount of commission = percent of commission X total sales.
Using the examples from our textbook,
Maria sold $42,000 word of insurance in January. If her commission is 3% of the total sales, what was the amount of her commission in January?
amount of commission = percent X total sales
0.03 X 42,000 = 1260
Her commission was $1,260.
Profit is the difference between total income and total operating costs.
profit = total income – total costs
The percent of profit is the percent of total income that is profit
percent of profit = profit/total income
MATH AT WORK:
Caterer
A caterer provides food for parties, weddings, bar/bat mitzvahs, and other events. Caterers plan the menu, buy the ingredients, and cook the food. Often they provide seating and music as well. For each event, a caterer determines the cost per guest. The catering business requires a thorough knowledge of rations, proportions, and percents.
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