Area

#$&*

course PHY 121

6/9/13 7:55 pm

qa areas etc

001. Areas

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Question: `q001. There are 11 questions and 7 summary questions in this assignment.

What is the area of a rectangle whose dimensions are 4 m by 3 meters.

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Your solution:

Area is length times width and is measured in square units, so 4 m x 3 m = 12 square

meters.

confidence rating #$&*: 3

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Given Solution:

`aA 4 m by 3 m rectangle can be divided into 3 rows of 4 squares, each 1 meter on a

side. This makes 3 * 4 = 12 such squares. Each 1 meter square has an area of 1 square

meter, or 1 m^2. The total area of the rectangle is therefore 12 square meters, or 12

m^2.

The formula for the area of a rectangle is A = L * W, where L is the length and W the

width of the rectangle. Applying this formula to the present problem we obtain area A =

L * W = 4 m * 3 m = (4 * 3) ( m * m ) = 12 m^2.

Note the use of the unit m, standing for meters, in the entire calculation. Note that m

* m = m^2.

FREQUENT STUDENT ERRORS

The following are the most common erroneous responses to this question:

4 * 3 = 12

4 * 3 = 12 meters

INSTRUCTOR EXPLANATION OF ERRORS

Both of these solutions do indicate that we multiply 4 by 3, as is appropriate.

However consider the following:

4 * 3 = 12.

4 * 3 does not equal 12 meters.

4 * 3 meters would equal 12 meters, as would 4 meters * 3.

However the correct result is 4 meters * 3 meters, which is not 12 meters but 12

meters^2, as shown in the given solution.

To get the area you multiply the quantities 4 meters and 3 meters, not the numbers 4 and

3. And the result is 12 meters^2, not 12 meters, and not just the number 12.

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Self-critique (if necessary): OK

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Self-critique Rating: 3

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Question: `q002. What is the area of a right triangle whose legs are 4.0 meters and 3.0

meters?

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Your solution:

The formula for area of a triangle is 1/2 base times height. It is also measured in

square units. That means that this triangle would be half of 4 times 3, or 1/2 of 12,

which is 6 square meters.

confidence rating #$&*: 3

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Given Solution:

`aA right triangle can be joined along its hypotenuse with another identical right

triangle to form a rectangle. In this case the rectangle would have dimensions 4.0

meters by 3.0 meters, and would be divided by any diagonal into two identical right

triangles with legs of 4.0 meters and 3.0 meters.

The rectangle will have area A = L * W = 4.0 m * 3.0 m = 12 m^2, as explained in the

preceding problem. Each of the two right triangles, since they are identical, will

therefore have half this area, or 1/2 * 12 m^2 = 6.0 m^2.

The formula for the area of a right triangle with base b and altitude h is A = 1/2 * b *

h.

STUDENT QUESTION

Looking at your solution I think I am a bit rusty on finding the area of triangles.

Could you give me a little more details

on how you got your answer?

INSTRUCTOR RESPONSE

As explained, a right triangle is half of a rectangle.

There are two ways to put two right triangles together, joining them along the

hypotenuse. One of these ways gives you a rectangle. The common hypotenuse thus forms a

diagonal line across the rectangle.

The area of either triangle is half the area of this rectangle.

If this isn't clear, take a blade or a pair of scissors and cut a rectangle out of a

piece of paper. Make sure the length of the rectangle is clearly greater than its

width. Then cut your rectangle along a diagonal, to form two right triangles.

Now join the triangles together along the hypotenuse. They will either form a rectangle

or they won't. Either way, flip one of your triangles over and again join them along

the hypotenuse. You will have joined the triangles along a common hypotenuse, in two

different ways. If you got a rectangle the first time, you won't have one now. And if

you have a rectangle now, you didn't have one the first time.

It should be clear that the two triangles have equal areas (allowing for a little

difference because we can't really cut them with complete accuracy).

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Self-critique (if necessary): OK

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Self-critique Rating: 3

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Question: `q003. What is the area of a parallelogram whose base is 5.0 meters and whose

altitude is 2.0 meters?

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Your solution:

The area of a parallelogram is base times height and is measured in square units.

Therefore, the area of this parallelogram is 5 times 2, or 10 square meters.

confidence rating #$&*: 3

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Given Solution:

`aA parallelogram is easily rearranged into a rectangle by 'cutting off' the protruding

end, turning that portion upside down and joining it to the other end. Hopefully you are

familiar with this construction. In any case the resulting rectangle has sides equal to

the base and the altitude so its area is A = b * h.

The present rectangle has area A = 5.0 m * 2.0 m = 10 m^2.

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Self-critique (if necessary): OK

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Self-critique Rating: 3

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Question: `q004. What is the area of a triangle whose base is 5.0 cm and whose altitude

is 2.0 cm?

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Your solution:

One half of 5 times 2 or 1/2 of 10, which is 5 square cm.

confidence rating #$&*:

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Given Solution:

`aIt is possible to join any triangle with an identical copy of itself to construct a

parallelogram whose base and altitude are equal to the base and altitude of the

triangle. The area of the parallelogram is A = b * h, so the area of each of the two

identical triangles formed by 'cutting' the parallelogram about the approriate diagonal

is A = 1/2 * b * h. The area of the present triangle is therefore A = 1/2 * 5.0 cm * 2.0

cm = 1/2 * 10 cm^2 = 5.0 cm^2.

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Self-critique (if necessary): OK

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Self-critique Rating: 3

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Question: `q005. Sketch on a set of x-y axes the four-sided quadrilateral whose corners

are at the points (3, 0), (3, 7), (9, 11) and (9, 0) (just plot these points, then

connect them in order with straight lines).

What would you say is the width of this figure, as measured from left to right?

If the width is measured from left to right, why does it make sense to say that the

figure has 'altitudes' of 7 and 11?

Do you agree that the figure appears to be a quadrilateral 'sitting' on the x axis, with

'altitudes' of 7 and 11?

We will call this figure a 'graph trapezoid'. You might recall from geometry that a

trapezoid has two parallel sides, and that its altitude is the distance between those

sides. The parallel sides are its bases. There is a standard formula for the area of a

trapezoid, in terms of its altitude and its two bases. We are not going to apply this

formula to our 'graph trapezoid', for reasons you will understand later in the course.

The 'graph trapezoid' you have sketched appears to be 'sitting' on the x axis. An

object typically sits on its base. So we will think of its base as the side that runs

along the x axis, the side it is 'sitting' on.

The 'graph trapezoid' appears to be 'higher' on one side than on the other. We often

use the word 'altitude' for height. This 'graph trapezoid' therefore will be said to

have two 'graph altitudes', 7 and 11.

What therefore would you say is the 'average graph altitude' of this trapezoid?

If you constructed a rectangle whose width is the same as that of this trapezoid, and

whose length is the 'average graph altitude' of the trapezoid, what would be its area?

Do you think this area is more or less than the area of the 'graph trapezoid'?

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Your solution:

The width of this figure at the bottom is 6 units. It comes to a point at the top.

On one side the height of the line segment is 7 units. On the other side of the figure,

it is 11 units.

I would say that the average graph altitude of this trapezoid is 9 units.

If I chopped off the piece of the trapezoid above the 9 and made the trapezoid into a

triangle, the area of the trapezoid would be 6 times 9, or 54 square units.

I think that this is the same as the graph trapezoid because the tiny triangle that you

would chop off would come around and nestle into the top of the figure that is left to

make a rectangle.

confidence rating #$&*: 3

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Given Solution:

The line segment from (3, 0) to (3, 7) is 'vertical', i.e., parallel to the y axis. So

is the line segment from (6, 11) to (6, 0). These line segment form what we call here

the 'graph altitudes' of the trapezoid.

These line segments have lengths of 7 and 11, respectively. The 'graph altitudes' are

therefore 7 and 11.

The 'average graph altitude' is the average of 7 and 11, which you should easily see is

9. (In case you don't see it, this should be obvious in two ways: 9 is halfway between

7 and 11; also (7 + 11) / 2 = 18 / 2 = 9)

The 'base' of the 'graph trapezoid' runs along the x axis from (3, 0) to (9, 0). The

distance between these points is 6. So the 'graph trapezoid' has a 'graph width' of 6.

A rectangle whose base is equal to that of this 'graph trapezoid' and whose length is

equal to the 'average graph altitude' of our 'graph trapezoid' has width 6 and length 9,

so its area is 6 * 9 = 54.

If this rectangle is positioned on an above the x axis, with one of its widths running

along the x axis from (3, 0) to (9, 0), i.e., so that its width corresponds with the

'graph width' of the 'graph trapezoid', then the other width cuts the top of the

trapezoid in half. Most of the trapezoid will be inside the rectangle, but a small

triangle in the top right corner will be left out. Also the trapezoid will fill most of

the rectangle, except for a small triangle in the upper left-hand corner of the

rectangle. The area of this triangle is equal to that of the 'left-out' triangle.

It follows that the trapezoid and the rectangle have identical areas.

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Self-critique (if necessary): OK

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Self-critique Rating: 3

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Question: `q006. What is the area of a 'graph trapezoid' whose width is 4 cm in whose

altitudes are 3.0 cm and 8.0 cm?

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Your solution:

The average graph altitude of this trapezoid would be (3+8)/2, or 5.5. 5.5 times 4 (the

base measurement) is 22 square units.

confidence rating #$&*: 3

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Given Solution:

`aThe area is equal to the product of the 'graph width' and the average 'graph

altitude'. Average altitude is (3 cm + 8 cm) / 2 = 5.5 cm so the area of the trapezoid

is A = 4 cm * 5.5 cm = 22 cm^2.

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Self-critique (if necessary): OK

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Self-critique Rating: 3

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Question: `q007. What is the area of a circle whose radius is 3.00 cm?

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Your solution:

The formula for the area of a circle is pi times the radius squared. The radius is 3

cm. The radius squared is 9 square cm. 9 times 3.14 = 28.26 square cm.

confidence rating #$&*: 2

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Given Solution:

`aThe area of a circle is A = pi * r^2, where r is the radius. Thus

A = pi * (3 cm)^2 = 9 pi cm^2.

Note that the units are cm^2, since the cm unit is part r, which is squared.

The expression 9 pi cm^2 is exact. Any decimal equivalent is an approximation. Using the

3-significant-figure approximation pi = 3.14 we find that the approximate area is A = 9

pi cm^2 = 9 * 3.14 cm^2 = 28.26 cm^2, which we round to 28.3 cm^2 to match the number of

significant figures in the given radius.

Be careful not to confuse the formula A = pi r^2, which gives area in square units, with

the formula C = 2 pi r for the circumference. The latter gives a result which is in

units of radius, rather than square units. Area is measured in square units; if you get

an answer which is not in square units this tips you off to the fact that you've made an

error somewhere.

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Self-critique (if necessary):

I just didn't round to the nearest tenth, but I do understand the concept. I just need

to be more confident about remembering the formula.

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Self-critique Rating: 3

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Question: `q008. What is the circumference of a circle whose radius is exactly 3 cm?

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Your solution:

Circumference is 2r times pi. This means 6 times 3.14 or 18.84, which would probably be

18.8 square cm.

confidence rating #$&*: 3

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Given Solution:

`aThe circumference of this circle is

C = 2 pi r = 2 pi * 3 cm = 6 pi cm.

This is the exact area. An approximation to 3 significant figures is 6 * 3.14 cm = 18.8

cm.

Note that circumference is measured in the same units as radius, in this case cm, and

not in cm^2. If your calculation gives you cm^2 then you know you've done something

wrong.

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Self-critique (if necessary): OK

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Self-critique Rating: 3

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Question: `q009. What is the area of a circle whose diameter is exactly 12 meters?

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Your solution:

The formula for area of a circle is pi times r squared. In order to find the radius,

you have to cut the diameter in half. Therefore you have pi times 6 squared. This is

would be 113.04, which I imagine will round to 113 square meters.

confidence rating #$&*: 2

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Given Solution:

`aThe area of a circle is A = pi r^2, where r is the radius. The radius of this circle

is half the 12 m diameter, or 6 m. So the area is

A = pi ( 6 m )^2 = 36 pi m^2.

This result can be approximated to any desired accuracy by using a sufficient number of

significant figures in our approximation of pi. For example using the 5-significant-

figure approximation pi = 3.1416 we obtain A = 36 m^2 * 3.1416 = 113.09 m^2.

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Self-critique (if necessary):

I used 3.14 for my multiplier, so I got 113.04. When I multiply by 3.1416, my answer

was 113.0976. I am unclear why the 76 just got dropped off and it didn't round to

113.10.

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Self-critique Rating: 2

@&

If you multiply by 3.14 you can't get 113.04. You are multiplying by a 3-significant-figure approximation of pi, and you can get only 3 significant figures in your result.

*@

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Question: `q010. What is the area of a circle whose circumference is 14 `pi meters?

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Your solution:

Circumference is pi times diameter. If you put 14 times pi = pi times d, you will divide

both sides by pi, which will give you 14 times 1 = 1 times d, or the diameter is 14. To

find the radius, you divide the diameter by 2, so the radius is 7 meters. Now you can

use the area = pi times r squared formula. First you find the square of 7, which is 49.

Then you multiply that by pi, or 3.1416, which gives you 153.9384 square meters. I'm

guessing the answer will either be 153.94 or 153.93 because there are 5 significant

figures in pi, but I'm not sure about that part.

confidence rating #$&*: 2

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Given Solution:

`aWe know that A = pi r^2. We can find the area if we know the radius r. We therefore

attempt to use the given information to find r.

We know that circumference and radius are related by C = 2 pi r. Solving for r we obtain

r = C / (2 pi). In this case we find that

r = 14 pi m / (2 pi) = (14/2) * (pi/pi) m = 7 * 1 m = 7 m.

We use this to find the area

A = pi * (7 m)^2 = 49 pi m^2.

STUDENT QUESTION:

Is the answer not 153.86 because you have multiply 49 and pi????

INSTRUCTOR RESPONSE

49 pi is exact and easier to connect to radius 7 (i.e., 49 is clearly the square of 7)

than the number 153.86 (you can't look at that number and see any connection at all to

7).

You can't express the exact result with a decimal. If the radius is considered exact,

then only 49 pi is an acceptable solution.

If the radius is considered to be approximate to some degree, then it's perfectly valid

to express the result in decimal form, to an appropriate number of significant figures.

153.86 is a fairly accurate approximation.

However it's not as accurate as it might seem, since you used only 3 significant figures

in your approximation of pi (you used 3.14). The first three figures in your answer are

therefore significant (though you need to round); the .86 in your answer is pretty much

meaningless.

If you round the result to 154 then the figures in your answer are significant and

meaningful.

Note that a more accurate approximation (though still just an approximation) to 49 pi is

153.93804. An approximation to 5 significant figures is 153.94, not 153.86.

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Self-critique (if necessary):

So, because it started with pi as a part of the circumference, you don't need to

multiply it out. The answer will be more exact as 49pi. I understand and hope I

remember in the future.

@&

You can, of course, approximate the result to an appropriate number of significant figures, if the situation calls for it.

*@

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Self-critique Rating: 2

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Question: `q011. What is the radius of circle whose area is 78 square meters?

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Your solution:

Since area = pi times r squared, I first put the area into the formula: 78 = pi times r

squared. Then I divided both sides by 3.1416, so I got r squared = 24.8281. I found

the square root of 24.8281 to determine the radius. This was 4.98278 meters.

confidence rating #$&*: 2

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Given Solution:

`aKnowing that A = pi r^2 we solve for r. We first divide both sides by pi to obtain A /

pi = r^2. We then reverse the sides and take the square root of both sides, obtaining r

= sqrt( A / pi ).

Note that strictly speaking the solution to r^2 = A / pi is r = +-sqrt( A / pi ),

meaning + sqrt( A / pi) or - sqrt(A / pi). However knowing that r and A are both

positive quantities, we can reject the negative solution.

Now we substitute A = 78 m^2 to obtain

r = sqrt( 78 m^2 / pi) = sqrt(78 / pi) m.{}

Approximating this quantity to 2 significant figures we obtain r = 5.0 m.

STUDENT QUESTION

Why after all the squaring and dividing is the final product just meters and not meters

squared????

INSTRUCTOR RESPONSE

It's just the algebra of the units.

sqrt( 78 m^2 / pi) = sqrt(78) * sqrt(m^2) / sqrt(pi). The sqrt(78) / sqrt(pi) comes out

about 5.

The sqrt(m^2) comes out m.

This is a good thing, since radius is measured in meters and not square meters.

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Self-critique (if necessary):

I was guessing that the final answer might be about 5, but I am unsure when to round and

when to keep things as they are. I understood the basic process, though.

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Self-critique Rating: 2

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Question: `q012. Summary Question 1: How do we visualize the area of a rectangle?

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Your solution:

The area of a rectangle is the number of square units that makes up the rectangle. We

can find out how many square units there are by multiplying the length times the width

of the rectangle.

confidence rating #$&*: 3

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Given Solution:

`aWe visualize the rectangle being covered by rows of 1-unit squares. We multiply the

number of squares in a row by the number of rows. So the area is A = L * W.

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Self-critique (if necessary): OK

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Self-critique Rating: 3

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Question: `q013. Summary Question 2: How do we visualize the area of a right triangle?

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Your solution:

The area of a right triangle is the number of squares and partial squares it takes to

fill the triangle. You would make off the number of units by which you are measuring up

the side and across the bottom and extend the lines across the triangle. It turns out

that it will be exactly half of the base times the height.

confidence rating #$&*: 3

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Given Solution:

`aWe visualize two identical right triangles being joined along their common hypotenuse

to form a rectangle whose length is equal to the base of the triangle and whose width is

equal to the altitude of the triangle. The area of the rectangle is b * h, so the area

of each triangle is 1/2 * b * h.

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Self-critique (if necessary):

I did not say that you should imagine a second congruent right triangle that can be used

to create a rectangle, which you would find the area of and then cut in half.

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Self-critique Rating: 3

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Question: `q014. Summary Question 3: How do we calculate the area of a parallelogram?

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Your solution:

The area of a parallelogram is base times height. Because the parallelogram is made of

2 sets of parallel line segments, you can actually cut off the end of the parallelogram

to make the angles right angles and move the triangle you cut off to the other end of

the parallelogram. This fits perfectly into the other end, creating a rectangle. And

the base / height of the parallelogram are the same measurements as the length/width of

the rectangle.

confidence rating #$&*: 3

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Given Solution:

`aThe area of a parallelogram is equal to the product of its base and its altitude. The

altitude is measured perpendicular to the base.

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Self-critique (if necessary): OK

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Self-critique Rating: 3

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Question: `q015. Summary Question 4: How do we calculate the area of a trapezoid?

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Your solution:

To find the area of a trapezoid, you measure the lengths of the two parallel lines. You

add the lengths together and divide by two to get the average height. Then you multiply

the average height by the width, and come up with the number of square units.

confidence rating #$&*: 3

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Given Solution:

`aWe think of the trapezoid being oriented so that its two parallel sides are vertical,

and we multiply the average altitude by the width.

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Self-critique (if necessary):

I just didn't remember the term average altitude.

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Self-critique Rating: 3

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Question: `q016. Summary Question 5: How do we calculate the area of a circle?

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Your solution:

The area of a circle is calculated by taking the radius and squaring it then multiplying

that number by pi (or 3.1416).

confidence rating #$&*:

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Given Solution:

`aWe use the formula A = pi r^2, where r is the radius of the circle.

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Self-critique (if necessary): OK

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Self-critique Rating: 3

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Question: `q017. Summary Question 6: How do we calculate the circumference of a circle?

How can we easily avoid confusing this formula with that for the area of the circle?

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Your solution:

Circumference is pi times the diameter. I do not know a good way to avoid confusing

this formula with the formula for the area of a circle.

confidence rating #$&*: 2

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Given Solution:

`aWe use the formula C = 2 pi r. The formula for the area involves r^2, which will give

us squared units of the radius. Circumference is not measured in squared units.

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Self-critique (if necessary):

I like remembering that the r square gives us square units and circumferences aren't

measured in square units.

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Self-critique Rating: 3

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Question: `q018. Explain how you have organized your knowledge of the principles

illustrated by the exercises in this assignment.

I have taught 3rd and 4th grades for 23 years now. We have worked on area of rectangles

and squares, so those I own. I participated in a math conference where we created, over

the course of several months, a lesson on finding the area of a triangle, so I remember

that. The area of the parallelogram, for some reason, I remember because it makes sense

to me. I don't remember learning about the area of a trapezoid, but it makes sense

because I can picture manipulating the piece mentally. I remember pi r squared but

didn't remember whether it was for area or circumference. I have had to refer to notes

and the back of the Physics book to remember that it was for area and what the formula

was for circumference. Hopefully remembering that the r squared means area so 2 times

pi times the radius (or pi times the diameter) will mean circumference. This will just

take some time.

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Self-critique (if necessary): OK

Repetition is what I need on the circle things.

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Self-critique Rating: 2"

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#*&!

&#Your work looks good. See my notes. Let me know if you have any questions. &#