Saturday, August 2, 2014
Monday, April 7, 2014
Quiz Understanding Check on Circles
Not for a grade, just to see what is known and unknown before we embark on circle area:
Try this 5 question quiz, see how you do. I want to know what you could figure out without guessing, what you already knew, and what you completely had to guess on. In the comment section post how you did in this fashion:
1. Guessed/knew/figured
2. Guessed/knew/figured
3. Guessed/knew/figured
4. Guessed/knew/figured
5. Guessed/knew/figured
10 minute task max.
Try this 5 question quiz, see how you do. I want to know what you could figure out without guessing, what you already knew, and what you completely had to guess on. In the comment section post how you did in this fashion:
1. Guessed/knew/figured
2. Guessed/knew/figured
3. Guessed/knew/figured
4. Guessed/knew/figured
5. Guessed/knew/figured
10 minute task max.
Wednesday, April 2, 2014
What is your Square Footprint?
Each of you has a primary residence, maybe two! We're going to find out what proportion of that residence is occupied by you. Prior to a house being constructed, a blueprint of the design of the interior is determined. Along with that blueprint comes an amount of square footage (area) the living space takes up. This IS NOT just the walls around the outside of the house; that's the land area the home occupies. The square footage is livable space on the interior of the home. Bedrooms, bathrooms, living/family rooms, kitchen, studies, and for our project, closets. It does not include hallways, garages, unfinished basements, sheds, backyards, exposed porches, or kitchen pantries.
Your task is this: determine how much of the house is space designated just for you in square footage. First, you need to make a fairly accurate diagram of the interior of the home's living spaces. 3 floors? Then you'll have 3 diagrams to make (you'll live). It should look something like this:
Obviously your diagram won't be as defined as this image of a blueprint, but you get the idea. You'll then need to measure, using a tape measure, every living space in the house/apartment. Put the actual measurements inside the diagram. In other words, if your living room mesaures 19'X12', then put the 228 sq.ft. number inside that room in the diagram. After getting the square footage of each and every living space within and written in the appropriate spaces, then add them all up! that's the total square footage of your home (again, don't forget the closets). Lastly, you'll need to take the square footage of your room/closet, and then divide that total by the total square footage of the house. So if your room/closet is 175 sq.ft., and the whole house is 2400 sq.ft., then 175/2400= 7.3% of the house. Got it?
If perchance you share a room with a sibling/relative/stranger, then you'll just simply cut the percentage in half (or thirds or whatever). Ask me if you're not sure how to approach that. If you have multiple residences, then you have choices! Pick which home works better for you for this particular project. No tape measure? Talk to me about that and we'll try to resolve it.
To complete:
1) Diagram your entire house, floor by floor, separately.
2) Measure the square footage of each and every living area as defined above.
3) Put those measurements in the diagram appropriately.
4) Determine your Square Footprint. How much of the house is "yours"?
Your task is this: determine how much of the house is space designated just for you in square footage. First, you need to make a fairly accurate diagram of the interior of the home's living spaces. 3 floors? Then you'll have 3 diagrams to make (you'll live). It should look something like this:
Obviously your diagram won't be as defined as this image of a blueprint, but you get the idea. You'll then need to measure, using a tape measure, every living space in the house/apartment. Put the actual measurements inside the diagram. In other words, if your living room mesaures 19'X12', then put the 228 sq.ft. number inside that room in the diagram. After getting the square footage of each and every living space within and written in the appropriate spaces, then add them all up! that's the total square footage of your home (again, don't forget the closets). Lastly, you'll need to take the square footage of your room/closet, and then divide that total by the total square footage of the house. So if your room/closet is 175 sq.ft., and the whole house is 2400 sq.ft., then 175/2400= 7.3% of the house. Got it?
If perchance you share a room with a sibling/relative/stranger, then you'll just simply cut the percentage in half (or thirds or whatever). Ask me if you're not sure how to approach that. If you have multiple residences, then you have choices! Pick which home works better for you for this particular project. No tape measure? Talk to me about that and we'll try to resolve it.
To complete:
1) Diagram your entire house, floor by floor, separately.
2) Measure the square footage of each and every living area as defined above.
3) Put those measurements in the diagram appropriately.
4) Determine your Square Footprint. How much of the house is "yours"?
Wednesday, February 26, 2014
Circles all Lined Up
All right, we've looked at the rules for circles, and there are lots of them. It doesn't ever seem to be more than cutting a number in half mathematically, but there sure are a lot of things to remember. Let's see if you can create one yourself. Using some circle as a base (like a can or something), put in to it every rule we've learned in circles such that all the angles and arcs are appropriately marked and measured. Inscribed angles, tangents intersecting with diameters, congruent chords cutting off equivalent arcs, central angles, etc...
Your task is then to turn in to solvable problems, where enough information is given for another student to then solve for missing arcs and angles. There should be a minimum of 10 angles/arcs to solve for. An example is posted below this text. If you're not sure on how to proceed after looking at my example, I'll be glad to show you another physical copy in the classroom, just ask. This is going to be a high value HW assignment, so choosing to skip this one will be harmful to your overall grade. I appreciate your efforts!
Completion of this HW assignment is three-fold:
1) Draw the circle, put in the appropriate segments/lines and figure all the angle/arc measures for your self.
2) Take out a lot of the measures, leaving just enough for another student to be able to solve it themselves as best they can.
3) Actually give that drawing for the other student to solve (trade, if you will) and see if indeed if it is solvable. Remember, there is a minimum of 10 'answers' for the partner to try to get.
-
Your task is then to turn in to solvable problems, where enough information is given for another student to then solve for missing arcs and angles. There should be a minimum of 10 angles/arcs to solve for. An example is posted below this text. If you're not sure on how to proceed after looking at my example, I'll be glad to show you another physical copy in the classroom, just ask. This is going to be a high value HW assignment, so choosing to skip this one will be harmful to your overall grade. I appreciate your efforts!
Completion of this HW assignment is three-fold:
1) Draw the circle, put in the appropriate segments/lines and figure all the angle/arc measures for your self.
2) Take out a lot of the measures, leaving just enough for another student to be able to solve it themselves as best they can.
3) Actually give that drawing for the other student to solve (trade, if you will) and see if indeed if it is solvable. Remember, there is a minimum of 10 'answers' for the partner to try to get.
-
Saturday, February 1, 2014
Sin/Cos/Tan AKA: Trigonometric Ratios
Right triangles really are everywhere, the importance of them in building design cannot be overstated, and you also can find them on your trip into the mountains (and out of) on some road signs! Ever seen the one that states a warning to truckers that the 'grade' is 6%? This one:
What is this 'grade'? Sometimes these signs are accompanied with an actual percent, as this one is:
Okay, it says 7%. Percent of what? Well, when you travel through the mountains, you often see signs that say things like "Trucks check brakes -- 10% grade" or "6% grade -- Trucks use right lane only." These numbers obviously have something to do with the steepness of the road, but their exact meaning is a mystery to most drivers.
The grade of something is simply a measure of its rise over its run. To understand rise and run, it helps to think of the hill as a big right triangle (a triangle with a 90-degree angle), like this:
The rise is the length of side B, or the height of the hill. The run is the length of side A, the horizontal measure of the hill at ground level. So, if you rose 100 feet over a horizontal distance of 1,000 feet, rise over run would equal 100 divided by 1,000, or 0.1. To get the percent grade, you simply multiply by 100, which gives you 10%. It doesn't matter whether you use feet, meters, miles or kilometers -- if you know how far the road rises in a given horizontal distance, you can calculate the percent grade.
In common practice, people often refer to percent grade as the rise divided by the distance you would travel going up the hill (side C), rather than the horizontal distance (side A). If you have an odometer and an altimeter, this is pretty easy to calculate. You check the altitude at the starting point and reset the odometer trip meter. You climb a certain distance and divide the change in altitude by the miles you've traveled. This is not technically the grade, but for normal roads that aren't very steep, it ends up being pretty close because the horizontal distance and the length of the actual road are nearly the same.
To calculate percent grade exactly, you need to figure out the horizontal distance traveled (A). Since you know B and C, you can calculate A using the Pythagorean theorem -- "the square of the hypotenuse is equal to the sum of the squares of the other two sides," or:
C2 = A2+ B2
This means that:
A = SquareRoot (C2 - B2)
If you had driven 1,000 miles down the road and risen 100 miles, the horizontal distance would be the square root of 990,000, which is approximately 994.99 (rounded up).
So what good does all of this do you? First of all, the percent grade gives you a relative sense of how steep the hill is. If you've climbed a hill designated as having a 5% grade, for example, you'll know approximately what to expect from any other 5% grade hill.
. Percent grade also gives you enough information to figure out the grade angle -- the angle of the hill's ascent (angle d in the diagram below). If you know trigonometry, you've probably recognized that rise divided by run is equal to the tangent of angle d. To calculate the angle of ascent when you know the percent grade, you simply take the arc tangent of the grade (the inverse of the tangent). So, if you have a 10% grade, you look up the arc tangent of 0.1 and find that the angle is 5.71 degrees.
Roads are measured in percent grade rather than degrees for two reasons:
· You don't have to have a special calculator or trig tables to calculate percent grade.
· If you know the percent grade, it's easy to calculate the approximate distance you have risen or fallen simply by looking at your odometer. If you have coasted down a mile-long hill on a 10% grade, you know you have fallen about a tenth of a mile in vertical height.
What is this 'grade'? Sometimes these signs are accompanied with an actual percent, as this one is:
The grade of something is simply a measure of its rise over its run. To understand rise and run, it helps to think of the hill as a big right triangle (a triangle with a 90-degree angle), like this:
The rise is the length of side B, or the height of the hill. The run is the length of side A, the horizontal measure of the hill at ground level. So, if you rose 100 feet over a horizontal distance of 1,000 feet, rise over run would equal 100 divided by 1,000, or 0.1. To get the percent grade, you simply multiply by 100, which gives you 10%. It doesn't matter whether you use feet, meters, miles or kilometers -- if you know how far the road rises in a given horizontal distance, you can calculate the percent grade.
In common practice, people often refer to percent grade as the rise divided by the distance you would travel going up the hill (side C), rather than the horizontal distance (side A). If you have an odometer and an altimeter, this is pretty easy to calculate. You check the altitude at the starting point and reset the odometer trip meter. You climb a certain distance and divide the change in altitude by the miles you've traveled. This is not technically the grade, but for normal roads that aren't very steep, it ends up being pretty close because the horizontal distance and the length of the actual road are nearly the same.
To calculate percent grade exactly, you need to figure out the horizontal distance traveled (A). Since you know B and C, you can calculate A using the Pythagorean theorem -- "the square of the hypotenuse is equal to the sum of the squares of the other two sides," or:
C2 = A2+ B2
This means that:
A = SquareRoot (C2 - B2)
If you had driven 1,000 miles down the road and risen 100 miles, the horizontal distance would be the square root of 990,000, which is approximately 994.99 (rounded up).
So what good does all of this do you? First of all, the percent grade gives you a relative sense of how steep the hill is. If you've climbed a hill designated as having a 5% grade, for example, you'll know approximately what to expect from any other 5% grade hill.
. Percent grade also gives you enough information to figure out the grade angle -- the angle of the hill's ascent (angle d in the diagram below). If you know trigonometry, you've probably recognized that rise divided by run is equal to the tangent of angle d. To calculate the angle of ascent when you know the percent grade, you simply take the arc tangent of the grade (the inverse of the tangent). So, if you have a 10% grade, you look up the arc tangent of 0.1 and find that the angle is 5.71 degrees.
Roads are measured in percent grade rather than degrees for two reasons:
· You don't have to have a special calculator or trig tables to calculate percent grade.
· If you know the percent grade, it's easy to calculate the approximate distance you have risen or fallen simply by looking at your odometer. If you have coasted down a mile-long hill on a 10% grade, you know you have fallen about a tenth of a mile in vertical height.
So, what would I like you to try? I would like you to construct a 'hill' that has approximately a 10% grade. Lots of choices here, you could use our new fallen snow(!), maybe Play-doh, cardboard and tape, dozens of books, or whatever you can think of. Ultimately, when you've completed this, I'd like you to either literally bring it intact into the classroom (which seems to me to be somewhat unlikely, but if you can...) or take an image of it and bring that in. Obviously, I'll discuss this in class a bit with you on Monday to ensure we all are on the same page, and I will gladly give advice/hints as to its completion, so don't panic. And I understand the explanation I just gave is slightly confusing, so I'll also make that as clear as possible. But hopefully you'll find this moderately entertaining, and have fun constructing!! [link]
Wednesday, January 15, 2014
a2 + b2 = c2
The Pythagorean Theorem!
Sweet. I know this! I can do it! Now... show that you can! Take the three squares we cut out in class, and reconstruct the smaller 2 to match exactly the larger one. You can show this by placing them next to each other, but it does need to be evident in the post that you indeed have successfully reconstructed the 2 to match exactly the third larger square. Good luck!
.JPG)
Sweet. I know this! I can do it! Now... show that you can! Take the three squares we cut out in class, and reconstruct the smaller 2 to match exactly the larger one. You can show this by placing them next to each other, but it does need to be evident in the post that you indeed have successfully reconstructed the 2 to match exactly the third larger square. Good luck!
.JPG)
Notice how I aligned the orange (6X6 square) and the purple (8X8 square) as a 10X10 square to match the green one Your turn! It appears the Blogger won't allow photos to be posted in the comments, so you'll need to bring the evidence to class on Friday. 3 ways:
1) Take a photo and bring that
2) Bring the actual squares and display them on your desk in the appropriate fashion.
3) Post a link here in the comment I could open that would show me your finished result.
Friday, December 13, 2013
NOT MANDATORY! DO ONLY IF YOU CHOOSE TO!
The Triangle Inequality Theorem states:
"The sum of the lengths of any two sides of a triangle is greater than the length of the third side."A corollary to the theorem (an add-on if you will) is that the perpendicular segment from a point to a line is the shortest segment from the point to the line.Your task:
Create a short (30-60 second) video detailing this theorem and its corollary. The video is not you just reciting the theorem, or you drawing it out. It is showing the actuality of the rule. (hint-- think streets and sidewalks and the such, you know, lines that are real.)
Have a narrative of some sort to your video, again, don't just define it. Live it.
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