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Monday, January 25, 2010
Saturday, January 23, 2010
Daily Scribe:)
Today in class it was parent visitation day! We learned about the sun and it's shadows. We also learned how to tell the height of something by it's shadows. Here's some of the problems we did in class:
There's a clock tower in a local neighborhood that's height is unknown. There is also a stick that stands 3 meters high a little bit down the road. The shadow of the clock tower is 8 meters. The stick's shadow is 1.5 meters. How tall is the clock tower?
To find out how tall the clock tower is, the first thing you need to do is find two similar triangles. The first triangle consists of the clock tower, the shadow, and the imaginary line to close-off the shape. The second triangle is the stick, the shadow of the stick, and the imaginary line.There are two ways to complete the next step. You can find ratios such as: x/8 and 3/1.5. Or you can use Scale Factor. If you use ratios this is what you'll do: Divide 8 by 1.5 to get 5 1/3. Then multiply 3 by 5 1/3 to get about 16. (The fractions and decimals will make the numbers funny so try to round up if your answer still seems accurate.) Therefore, the length of the clock tower is 16 feet!
If you do it by scale factor this is what you can do: divide 3 by 1.5 to get 2 .Then, multiply 8 by 2 to get 16. Therefore, the length of the clock tower is 16 meters!
No matter which way you do it you will always get the same answer!
Another problem we did was using a mirror.
Jim is standing in a street with a mirror laid on the ground 100 cm. in front of him. 450 cm. past the mirror is a traffic light. From the ground to Jim's eyes is 150 cm. How tall is the traffic light? I will solve this problem using ratios. x/450 and 150/100. 450 divided by 100 equals 4.5. 150 times 4.5 equals 675. Therefore, the traffic light is 675 cm.
These problems are really simple if you break them down. Find the two similar triangles, find the ratios or scale factor and you have your missing length!
Somethings to remember when you're doing these problems are:
The suns rays are always parallel.
Always look at the unit because sometimes not all of the lenghts will be in the same unit. And lastly,
remember to check your work!
There will be a chapter test on Wednesday, so study up!
Tuesday, January 19, 2010
Daily Scribe
Today in class we wrapped up on a few different subjects, including ratios. There is to be a quiz on Thursday.
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We also began Labsheet 4.3, on page 64.
Here are the answers to the problems.
A) The value of X is 10 centimeters. This is because the scale factor happens to be (from the smaller shape to the larger shape) multiply by 2. Since X's similar side on the smaller shape is 5 centimeters, and 5 * 2 = 10 centimeters.
This is one example of using a scale factor to solve a problem.
B) The value of X is 13.75, because the scale factor of the smaller shape to the larger shape is multiply by 2.5. Using the same method as before, we learn that the side similar to X is 5.5 centimeters. 5.5 * 2.5 = 13.75.
C) The value of X is 2.5, because the scale factor from the larger shape to the smaller shape is divide by4. X's similar side on the larger shape is 10 centimeters in length, so 10 divided by 4 = 2.5.
D) Finding the measurement of angle B is easy. It is 68 degrees. This is because B and the other angle measuring 68 degrees are congruent. This means that they both have the same measurements.
Finding angles A and C is more challenging. They must be found using a number of clues.
First of all, since they are congruent, you know that they will have the same measurements.
Next, since an average parallelogram has angles adding up to 360 degrees, you must take the sum of the two angles whose measurements have already been determined (68 and 68, which adds up to 136) and subtract that sum from 360. 360-136 = 224.
Finally, since you now know that the sum of the two angles is 224, and that both angles must have the same measurements (because they are congruent) it only makes sense to divide 224 by 2.
224 divided by 2 = 112. Therefore, angles A and C are both 112 degrees.
In order to check that this is correct, all you have to do is add every angle together and see if the sum is 360. 68+68+112+112= 360 degrees. This proves that the angles have been measured correctly.
E) 1: The value of X can be found by using the scale factor. The scale factor from the larger shape to the smaller shape is divide by 1.75. So, since X's similar side on the larger shape is 1.75, the value of X is 1.
2: Again, the value can be found by using scale factors. Since Y's similar side on the smaller shape is 4, and 4 * 1.75.= 7, the value of Y is 7.
3: The area of the larger shape is 147. This is because the height adds up to 10.5, and the length is 14. 10.5 * 14 = 147.
4: The area of the smaller shape can be found by using the same strategy used during problem 3. The base of the smaller shape is 8, and the height is 6. 8 * 6 = 48, so the area of the smaller shape is 48.
**********************************************************************
Those are the answers to Labsheet 4.3, done in class today.
As previously mentioned, there will be a quiz on Thursday, so study hard!
Written by Grace T
*************************************************************
We also began Labsheet 4.3, on page 64.
Here are the answers to the problems.
A) The value of X is 10 centimeters. This is because the scale factor happens to be (from the smaller shape to the larger shape) multiply by 2. Since X's similar side on the smaller shape is 5 centimeters, and 5 * 2 = 10 centimeters.
This is one example of using a scale factor to solve a problem.
B) The value of X is 13.75, because the scale factor of the smaller shape to the larger shape is multiply by 2.5. Using the same method as before, we learn that the side similar to X is 5.5 centimeters. 5.5 * 2.5 = 13.75.
C) The value of X is 2.5, because the scale factor from the larger shape to the smaller shape is divide by4. X's similar side on the larger shape is 10 centimeters in length, so 10 divided by 4 = 2.5.
D) Finding the measurement of angle B is easy. It is 68 degrees. This is because B and the other angle measuring 68 degrees are congruent. This means that they both have the same measurements.
Finding angles A and C is more challenging. They must be found using a number of clues.
First of all, since they are congruent, you know that they will have the same measurements.
Next, since an average parallelogram has angles adding up to 360 degrees, you must take the sum of the two angles whose measurements have already been determined (68 and 68, which adds up to 136) and subtract that sum from 360. 360-136 = 224.
Finally, since you now know that the sum of the two angles is 224, and that both angles must have the same measurements (because they are congruent) it only makes sense to divide 224 by 2.
224 divided by 2 = 112. Therefore, angles A and C are both 112 degrees.
In order to check that this is correct, all you have to do is add every angle together and see if the sum is 360. 68+68+112+112= 360 degrees. This proves that the angles have been measured correctly.
E) 1: The value of X can be found by using the scale factor. The scale factor from the larger shape to the smaller shape is divide by 1.75. So, since X's similar side on the larger shape is 1.75, the value of X is 1.
2: Again, the value can be found by using scale factors. Since Y's similar side on the smaller shape is 4, and 4 * 1.75.= 7, the value of Y is 7.
3: The area of the larger shape is 147. This is because the height adds up to 10.5, and the length is 14. 10.5 * 14 = 147.
4: The area of the smaller shape can be found by using the same strategy used during problem 3. The base of the smaller shape is 8, and the height is 6. 8 * 6 = 48, so the area of the smaller shape is 48.
**********************************************************************
Those are the answers to Labsheet 4.3, done in class today.
As previously mentioned, there will be a quiz on Thursday, so study hard!
Written by Grace T
Thursday, January 14, 2010
Daily Scribe ♥
A few days ago in class, we did problem 4.1 on page 61.
On number 1, we were trying to figure out the ratios of the rectangles. A is a 12 by 20. B is a 6 by 10. C is a 9 by 15 and D is a 6 by 20.
Ratio-Is a comparison between 2 quantities. ( For example, like lengths)
Problem 4.1
A1. Ratios
A- 3/5
B-3/5
C-3/5
D-3/10
2. A, B and c are all similar because they have the same angles and all have the same ratios. If the shapes are similar the ratios are equivalent.
3. From B to A the scale factor is 2. From the scale factor, I can tell that A's side lengths are 2x's longer than B's side lengths. (From b - c its 1.5) (from c- A its 1 1/3)
4. The scale factor shows us how much larger the shape will be and the ratio tells us if the shape is similar or not.
--------------------------------------------------
B1.
Paralelagram E- 10/8 = ratio-1.25 (68 degreese)
Paralelagram F=7.5/6= ratio-1.25 ( 52 degreese)
Paralelagram G=6/4.8=ratio- 1.25 (52 degreese)
2. Are all these paralelagrams similar???
The answer is no. You might think they are similar but they really are not. If you look at paralelagram E, It has a 68 degree angle. F and G have a 52 degree angle. They are not similar.
The scale factor from
E to F is .75
10/8 ---> 7.5/6 Scale factor is .75 =]
*************Even though the ratios of side lengths are equivalent and there is a scale factor between the parallelograms corresponding angles are different measures so only F and G are similar !!!!!**********************************************************************
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Moving on to a doggy named Duke ( number 26 on pg 70)
In the directions it says- Here is a picture of Duke, a real dog. The scale factor from Duke to the picture is 12.5%. Use an inch ruler to make any measures.
A. How long is Duke??
Duke is 44 Inches long.
How did we get that??- Well, if we take 12 1/2 % that is equivalent to 1/8 (which is the scale factor from Duke to his picture) So, we measure Duke and he is 5 1/2 inches wide. Now we take the scale factor from the picture to real Duke ( which is 8) and multiply that by 5 1/2 so 5 1/2 * 8= so the real Duke is 44 inches long. ☺
B. How tall is Duke?
Duke is 24 inches tall.
We got that by doing the same thing as A, but just switching numbers. So, we measured Duke with our rulers and he appears to be 3 inches tall so, we take 8 (the scale factor from the picture to the actual Duke) and multiply it by the 3 inches. 8*3=24 inches tall. The real Duke is 24 inches tall.
♥ Kelcey ♥ ☺☻☺☻☺☻☺☻☺☻☺☻☺☻☺☻☺☻☺☻☺☻☺☻☺☻
On number 1, we were trying to figure out the ratios of the rectangles. A is a 12 by 20. B is a 6 by 10. C is a 9 by 15 and D is a 6 by 20.
Ratio-Is a comparison between 2 quantities. ( For example, like lengths)
Problem 4.1
A1. Ratios
A- 3/5
B-3/5
C-3/5
D-3/10
2. A, B and c are all similar because they have the same angles and all have the same ratios. If the shapes are similar the ratios are equivalent.
3. From B to A the scale factor is 2. From the scale factor, I can tell that A's side lengths are 2x's longer than B's side lengths. (From b - c its 1.5) (from c- A its 1 1/3)
4. The scale factor shows us how much larger the shape will be and the ratio tells us if the shape is similar or not.
--------------------------------------------------
B1.
Paralelagram E- 10/8 = ratio-1.25 (68 degreese)
Paralelagram F=7.5/6= ratio-1.25 ( 52 degreese)
Paralelagram G=6/4.8=ratio- 1.25 (52 degreese)
2. Are all these paralelagrams similar???
The answer is no. You might think they are similar but they really are not. If you look at paralelagram E, It has a 68 degree angle. F and G have a 52 degree angle. They are not similar.
The scale factor from
E to F is .75
10/8 ---> 7.5/6 Scale factor is .75 =]
*************Even though the ratios of side lengths are equivalent and there is a scale factor between the parallelograms corresponding angles are different measures so only F and G are similar !!!!!**********************************************************************
-------------------------------------------------------------------------------------------------
Moving on to a doggy named Duke ( number 26 on pg 70)
In the directions it says- Here is a picture of Duke, a real dog. The scale factor from Duke to the picture is 12.5%. Use an inch ruler to make any measures.
A. How long is Duke??
Duke is 44 Inches long.
How did we get that??- Well, if we take 12 1/2 % that is equivalent to 1/8 (which is the scale factor from Duke to his picture) So, we measure Duke and he is 5 1/2 inches wide. Now we take the scale factor from the picture to real Duke ( which is 8) and multiply that by 5 1/2 so 5 1/2 * 8= so the real Duke is 44 inches long. ☺
B. How tall is Duke?
Duke is 24 inches tall.
We got that by doing the same thing as A, but just switching numbers. So, we measured Duke with our rulers and he appears to be 3 inches tall so, we take 8 (the scale factor from the picture to the actual Duke) and multiply it by the 3 inches. 8*3=24 inches tall. The real Duke is 24 inches tall.
♥ Kelcey ♥ ☺☻☺☻☺☻☺☻☺☻☺☻☺☻☺☻☺☻☺☻☺☻☺☻☺☻
Friday, January 8, 2010
Daily Scribe
Today in class we did a Math Reflection Gallery Walk which is like a rough draft for the Math Reflection final. It consisted of 3 questions:
1. How can you tell if two polygons are similar?
Answer: You can tell if two polygons are similar by whether or not they have a scale factor. For example, if one polygon had dimensions of 4 and 6 and another polygon had dimensions of 8 and 12, the 2nd polygon is 2x bigger (length wise) than the 1st polygon so the scale factor would be 2.
2. If two polygons are similar, how can you find the scale factor from one polygon to the other? Describe how you find the scale factor from the smaller figure to the enlarged figure. Then, describe how you find the scale factor from the larger figure to the smaller figure.
Answer: You find the scale factor from the smaller figure to the enlarged figure by figuring how many side lengths of the smaller figure go into the enlarged figure. Like, 4 and 6 with 8 and 12, 4 goes into 8 twice and 6 goes into 12 twice so the scale factor is 2. You can find the scale factor from the enlarged figure to the small figure because it's the reciprocal of the scale factor from small to large, so the scale factor is 1/2.
3. For parts (a)-(c), what does the scale factor between two similar figures tell you about the given measurements?
a. side lengths
b. perimeters
c. areas
Answer:
a. You can use the scale factor to find the similar side lengths because the side lengths would be the scale factor times the original side lengths to get the side lengths of the bigger polygon. For example, if one side is 4, you can do 4*2 to get 8 and 8 would be the length of the similar line.
b. You can use the scale factor to find the perimeter because the perimeter would be the scale factor times the perimeter to get the bigger polygon's perimeter. If the dimensions of the smaller polygon are 4 and 6, the perimeter is 20. And the dimensions of the larger polygon are 8 and 12, so the perimeter is 40. So that means the perimeter is 2x bigger than the smaller polygon, which is the same as the scale factor.
c. You can use the scale factor to find the area because the area of the larger polygon would be the scale factor squared times the area of the smaller area. So if the area of the smaller polygon (4 by 6) was 24, the area of the larger polygon (8 by 12) would be 96 because the scale factor is 2 and the 2 squared is 4 and 4*24 is 96. The way you can check is because 8*12 also equals 96 and its the same number so it is correct.
☺That is what we did in math today!☻
1. How can you tell if two polygons are similar?
Answer: You can tell if two polygons are similar by whether or not they have a scale factor. For example, if one polygon had dimensions of 4 and 6 and another polygon had dimensions of 8 and 12, the 2nd polygon is 2x bigger (length wise) than the 1st polygon so the scale factor would be 2.
2. If two polygons are similar, how can you find the scale factor from one polygon to the other? Describe how you find the scale factor from the smaller figure to the enlarged figure. Then, describe how you find the scale factor from the larger figure to the smaller figure.
Answer: You find the scale factor from the smaller figure to the enlarged figure by figuring how many side lengths of the smaller figure go into the enlarged figure. Like, 4 and 6 with 8 and 12, 4 goes into 8 twice and 6 goes into 12 twice so the scale factor is 2. You can find the scale factor from the enlarged figure to the small figure because it's the reciprocal of the scale factor from small to large, so the scale factor is 1/2.
3. For parts (a)-(c), what does the scale factor between two similar figures tell you about the given measurements?
a. side lengths
b. perimeters
c. areas
Answer:
a. You can use the scale factor to find the similar side lengths because the side lengths would be the scale factor times the original side lengths to get the side lengths of the bigger polygon. For example, if one side is 4, you can do 4*2 to get 8 and 8 would be the length of the similar line.
b. You can use the scale factor to find the perimeter because the perimeter would be the scale factor times the perimeter to get the bigger polygon's perimeter. If the dimensions of the smaller polygon are 4 and 6, the perimeter is 20. And the dimensions of the larger polygon are 8 and 12, so the perimeter is 40. So that means the perimeter is 2x bigger than the smaller polygon, which is the same as the scale factor.
c. You can use the scale factor to find the area because the area of the larger polygon would be the scale factor squared times the area of the smaller area. So if the area of the smaller polygon (4 by 6) was 24, the area of the larger polygon (8 by 12) would be 96 because the scale factor is 2 and the 2 squared is 4 and 4*24 is 96. The way you can check is because 8*12 also equals 96 and its the same number so it is correct.
☺That is what we did in math today!☻
Thursday, January 7, 2010
Wednesday, January 6, 2010
Daily Scribe
Today in class we worked on scale factors and similar shapes. We made similar rectangles and triangles for problems A and B for problem 3.3. We had to make new rectangles with the information that they gave us. One of the problems was to make rectangle A to a new rectangle with the scale factor of 2.5. To do this, you would have to first multiplied to base by 2.5, and then the height by 2.5. The new rectangle should be 2.5 times larger and still be similar to the original rectangle.
Another problem on 3.3 was to to make a new rectangle have and area 9 times larger than triangle B and still be similar. To figure this out, you would have to multiply the dimensions by 3. You would do this because to find out the area of a new shape, you would have to multiply the scale factor by itself, then multiply the area from the original shape by that.
This is what we worked on in class!!!
Another problem on 3.3 was to to make a new rectangle have and area 9 times larger than triangle B and still be similar. To figure this out, you would have to multiply the dimensions by 3. You would do this because to find out the area of a new shape, you would have to multiply the scale factor by itself, then multiply the area from the original shape by that.
This is what we worked on in class!!!
Tuesday, January 5, 2010
Today in class, we learned about the scale factor inside bigger shapes like triangles. We figured out a strategy to make perfect smaller shapes. First look at each side of the shape. Find the halfway point and make a dot. Do this for every side of the shape. Now connect the dots from each point so your lines make a smaller version of the original shape. The scale factor would be a fraction, because you are dividing the shape large to small. This would've been dividing the number too, but since the scale factor doesn't work in division, the factor would be less than 1. The area of the new perimeter is the scale factor times the original area squared. Example: the original area=2 and the scale factor from the original to the new is 3. 3 squared is 9, so the new area of the shape is 18. 2•(3^2)= 18.
Monday, January 4, 2010
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