Construction Video
The video linked above is good. It's quick but goes through all the basics. Make sure you know how to do each of them as they provide the steps for the tougher ones (e.g. incenter, inscribed equilateral triangle, ...)
We'll go through those at the end of classes until Tuesday.
Wednesday, May 18, 2011
Saturday, March 12, 2011
Refreshers and Tips for 1-9
OK, y’all, I trust that you know the formulae for area and circumference of a circle. As I said in class, one way to keep them straight is to remember that area is always given in square units which means that a unit had to be squared. (cm x cm = cm2) So, area is the formula with the squared radius.
You have to complete section 1-9, largely on your own. Treat it like a proof packet which will have to be turned in. Much of it is review from middle school math but I’ll give you tips in case you’ve forgotten the details.
1-7) Basics (Draw a quick sketch for 6 and 7)
8-11.) Read directions. Don’t use a calculator.
12-15) I changed the directions to read “Find the circumference of the circle to 4 significant figures.” Refer to the sheet I gave you in class for help with significant figures. Use a calculator.
16-19.) Draw these out. For some sides, you’ll need the distance formula.
20-26.) Be careful with units. They must be the same. Give two answers, one in each unit given.
27-32.) Again, leave pi in your answer and give the units.
33-36.) Use the pi key on your calculator unless you happen to know pi to 9 decimal places. This time, give the area to 5 significant figures.
37-38.) What’s it going to be – addition or subtraction? You choose. One way is clearly easier for #38 but you can do it either way.
39.) Part c is good general knowledge to have. A square foot is a square with 1-foot sides.
40-46.) SKIP
47-49.) I know these are simple to do mentally but I’d like to see you create solvable equations from formulae. For each of these, follow this algorithm:
a. Top line – formula (No values substituted into the variables yet.)
b. Next line – fill in the parts that you know, making an equation
c. Solve, keeping your = signs lined up.
d. You will need to repeat this twice for 50c.
50.) Use good ole’ common sense for part a and b. For part c, this is an important skill to have. Pay attention to this… it says to “write a formula for the area of a square – “ (A) – “in terms of its perimeter. “- (P) This means that your formula will start with A= and can only have a P for its variable. The s variable will be the bridge from one to the other. Solve for s in the perimeter equation and then substitute what you get into the area equation.
51.) It helps to label each face of a solid with a number in order to draw its net.
52.) Use inches throughout the problem and show the units in each step. As mentioned above, units act just like variables. When you multiply inches by inches, you have inches2. If you divide inches2 by inches, you’ll have inches.
53-54.) SKIP (too easy)
55-56.) Just do it.
57.) Read the rounding part carefully. ( If I round a number to the nearest ten, that number must end with at least one zero. If I round a number to the nearest hundred, it will end with two zeros.)
58.) Good algebra practice. Show the expression that you used to find the answer.
59-62) SKIP
63.) Don’t forget that you’re first finding a diameter. Read carefully to decide whether or not to calculate pi.
64.) Refer to class notes on using your calculator to make tables of values. Specifically:
a.) Are you in FCN mode? You should be.
b.) In this mode, the variable will always be x. Your work will have b’s (for the base) but change them on your paper to x’s.
c.) Write an equation for the height of this corral in terms of the base (x). Hint: There’s 100 feet of fencing available so you will write an equation that looks like this: ______ + h = 100. Look at the picture. Solve this equation for h. (In other words, get h alone.)
d.) We enter equations into the calculator with the y= key. Enter your equation for h. (Clear whatever is on your y= screen.)
e.) Use TBLSET to start at 1 and to make steps of 1. Quit TBLSET. Show your table. (2ND GRAPH) Do you know what the columns represent?
f.) Now you want another column for the area of the corral. You will enter an area equation as y2. You will need to multiply every value of x to its corresponding y1 value. There is not a key for y variables but they’re in there. Find the VARS key. (VARS – YVARS – Y1)
g.) So your y2 equation will be y2 = xy1.
h.) When you press TBL now, you will have three columns. The first is the list of possible integral bases, the second is the corresponding heights if you’re using all 100 feet of fencing and the third is the corresponding areas. Isn’t that cool?!
i.) Find the greatest area and give its dimensions.
LOOKING AT THE TABLE, WRITE ALL THE VALUES ON YOUR CALCULATOR SCREEN WHEN THE HIGHEST AREA IS RIGHT IN THE MIDDLE.
j.) To see the graph, you have to adjust your window. We know that the x-values (the possible bases) range from 0 to 100. Well, not really but that’s a decent range. The area values range from 0 to something around 1400. (I’m not telling.) To set my window (the part of the graph that we can view), I enter the smallest and largest x’s for Xmin and Xmax. Then divide the difference by 10 to get XStep. Do the same for y’s. Reset your window to those numbers.
k.) We’re told to graph the base on the horizontal axis and the area on the vertical. We need to turn off the height equation so we don’t see it when we graph. Just go to the y= screen, highlight the = sign for y1, and press ENTER. This should un-highlight the = sign and turn off this graph.
DRAW A PICTURE OF YOUR GRAPH AND GIVE THE DIMENSIONS OF YOUR VIEWING WINDOW.
PHEW……….. Aren’t you glad that most of this assignment was so easy? This will seem easy to you once you’re halfway through advanced algebra next year.
65.) Awesome question. Easy to answer but harder to comprehend.
66-68.) I like this one too. Don’t be afraid to use variables as if they were numbers. They ARE numbers for us. Of course your answers will have variables in them. For #67, you could look back at your work for 50c and see if it worked.
69.) This is a cool problem. Now that you know the Pythagorean Theorem so well, you can find two squares with radicals in their sides whose areas add up to an 11 X 11 square. Find three squares with integral sides that add up to the 11 x 11 square. I just used guess and check. I didn’t find four squares but I did find more than four that added up to the right answer.
70.) Let’s use four significant digits for the answer. Think carefully about the radius of the two circular ends of the field. This problem isn’t bad, huh?!
Thursday, March 3, 2011
I was intending to post tips but I don't want to give too much because this is for extra credit - in other words, not required. Let's just cover the basics:
1. An equilateral triangle is made by starting a working line of random length and keeping the compass set to that radius the whole time. One X is all it takes.
2. An inscribed circle comes from finding the incenter, that is, finding the intersection of the angle bisectors. It is easier to do the circumcenter first though because you can use a perpendicular segment to find the perfect length for the inside circle. But, hey, wait a minute, isn't the triangle supposed to be equilateral for the first construction?! There's some connection between incenter and circumcenter for an equilateral triangle.
3. To construct a square, it might be easiest to construct two congruent diagonals that are perpendicular to each other. Faster than working with sides.
4. Rhombi (isn't that the plural of rhombus??) are easiest to draw with diagonals too. Remember the question on the quarter exam?
5. Remember the short cut for the circumcenter of a right triangle.
6. Warning: If you google "How to construct an equilateral triangle inside a circle", you might come to a schnazzy site that looks cool but goes the long way. You'd be better off starting at the center and lightly drawing a radius. Construct a 120 degree angle on it and.... You can divide the circle into thirds and connect the dots.
Hope this helps a little (and not any more!)
Mrs. Dant
1. An equilateral triangle is made by starting a working line of random length and keeping the compass set to that radius the whole time. One X is all it takes.
2. An inscribed circle comes from finding the incenter, that is, finding the intersection of the angle bisectors. It is easier to do the circumcenter first though because you can use a perpendicular segment to find the perfect length for the inside circle. But, hey, wait a minute, isn't the triangle supposed to be equilateral for the first construction?! There's some connection between incenter and circumcenter for an equilateral triangle.
3. To construct a square, it might be easiest to construct two congruent diagonals that are perpendicular to each other. Faster than working with sides.
4. Rhombi (isn't that the plural of rhombus??) are easiest to draw with diagonals too. Remember the question on the quarter exam?
5. Remember the short cut for the circumcenter of a right triangle.
6. Warning: If you google "How to construct an equilateral triangle inside a circle", you might come to a schnazzy site that looks cool but goes the long way. You'd be better off starting at the center and lightly drawing a radius. Construct a 120 degree angle on it and.... You can divide the circle into thirds and connect the dots.
Hope this helps a little (and not any more!)
Mrs. Dant
Thursday, January 27, 2011
Tips for Packet due Monday
Ok, here goes. I explained several of these in class so I will be brief on those.
#3. Do NOT set the expressions equal to each other. That would only happen in a rhombus.
#4. If you work with slopes first, you'll have to do two calculations. That will tell you parallelogram (?) and possibly rectangle. Regardless, you'll have to do a couple of distance calculations to see if you're dealing with a rhombus. Remember, rhombus + rectangle = square.
#5. You should have A = ....... and P = ......... In part c, the word evaluate means to find a value for A and P.
#6b. Kites?! What are they, I'd like to know. Do we take Wikipedia as a reliable source???? According to them, a kite may be a square. OK, so our textbook loses by Wiki rules.
#7. Focus on the angles that I numbered in class. Converse of AIA is useful.
#8. You already have a p-gram. To make it a rhombus, you only have to get two consecutive sides congruent. If you prove the overlapping triangles congruent, you've got CPCTC and you're almost done.
#9. Five steps, if you include the parallels in the Given. You could put Draw steps into the beginning of the proof which would add on two steps. It would be a professional geometer's way to go.
#10. Still in the A exercises... don't overthink it. I like an equidistance theorem to make this a slick proof.
#11. I have 9 steps because I just put in the fact that AE is NOT parallel to DC in the ending triple play. For a reason, just put Diagram. About that triple play, it includes one parallel statement, one non-parallel statement and one congruence statement. Wrap those up and you're done.
#12. Ah, one of my favorites. Here's a helpful hint... if I have to subtract two numbers from a third number but I only know the sum of those two numbers, can't I subtract the sum? In other words, I can subtract x and y from z even if I don't know what x and y each are. If I know their sum, it's all the same.
#13. No hints needed. Be observant.
#14. I worked on four slopes and two distances. You have other options. Keep in mind that slopes give a little more bang for the buck. They prove parallelograms and, if you're lucky, show perpendicular segments too. Show all work.
#15. Long proof. First focus on Triangle PQA. That will give you sides that you need in a transitive step. Then switch over to Triangle RAS. That will give you angles in a transitive step. Once you have the two angles at vertex S congruent, you're done. You'll need AIA and "If sides, then angles."
#16. Discussed in class. For the base, don't think horizontal.
#17. Draw the special quads and use their diagonals as mirrors.
#18. Took 10 steps for me. After I used the Given stating "Segments are drawn...", I followed up with simple parallel statements using the right letters. That makes the first Prove a breeze.
For the second part, do you remember which letter patterns we can use to describe alternate interior, corresponding, same-side interiors, ...? For example, SSI's take on a U-shape. Looking at letter patterns might help you. It is fine to say BC is parallel to MP, even though BC is an extension of the parallelogram. I figured that as a Restate (or Given would be fine).
#19. Discussed in class. Don't forget to finish up with the Perimeter.
#20. How can you figure out (x, y). One is already on the diagram, the other comes from the fact that the point is on a line.
#21. Area = pi r squared
#22. Answers will vary. But we all know what shape will result.
#23. After our circumcenter/centroid work, this is child's play, right? Midpoints are averages of x's and y's.
#24. Reflection of a point over an axis only changes the sign of one coordinate. So label D appropriately. Then set up an algebra equation by setting the slopes equal. Once you set up that equation, the answer might jump off your paper (or you can cross multiply).
#25. Probability of two things happening = (probability of first) X (probability of second) When you figure out the probability of choosing one true statement, assume SUCCESS. This means that there is one less true statement left in the pot. The first probability will have a denominator of 4. The second will have a denominator of 3.
#26. See #24 for reflection tips. I did two slopes and counted two distances.
#27. I proved two overlapping triangles congruent to get angles of one isosceles triangle. Then a quick subtraction property closes it up.
#28. Same old stuff.
#29. Imagine the whole figure shaded. What's its area? Subtract out the white part. And speaking of that white part, you know that x must be less than some quantity. So x squared will be less than that quantity squared. (And of course you might throw in there that x must be greater than 0.)
FEEL FREE TO LEAVE A COMMENT SO THAT I KNOW WHO WAS HERE.
#3. Do NOT set the expressions equal to each other. That would only happen in a rhombus.
#4. If you work with slopes first, you'll have to do two calculations. That will tell you parallelogram (?) and possibly rectangle. Regardless, you'll have to do a couple of distance calculations to see if you're dealing with a rhombus. Remember, rhombus + rectangle = square.
#5. You should have A = ....... and P = ......... In part c, the word evaluate means to find a value for A and P.
#6b. Kites?! What are they, I'd like to know. Do we take Wikipedia as a reliable source???? According to them, a kite may be a square. OK, so our textbook loses by Wiki rules.
#7. Focus on the angles that I numbered in class. Converse of AIA is useful.
#8. You already have a p-gram. To make it a rhombus, you only have to get two consecutive sides congruent. If you prove the overlapping triangles congruent, you've got CPCTC and you're almost done.
#9. Five steps, if you include the parallels in the Given. You could put Draw steps into the beginning of the proof which would add on two steps. It would be a professional geometer's way to go.
#10. Still in the A exercises... don't overthink it. I like an equidistance theorem to make this a slick proof.
#11. I have 9 steps because I just put in the fact that AE is NOT parallel to DC in the ending triple play. For a reason, just put Diagram. About that triple play, it includes one parallel statement, one non-parallel statement and one congruence statement. Wrap those up and you're done.
#12. Ah, one of my favorites. Here's a helpful hint... if I have to subtract two numbers from a third number but I only know the sum of those two numbers, can't I subtract the sum? In other words, I can subtract x and y from z even if I don't know what x and y each are. If I know their sum, it's all the same.
#13. No hints needed. Be observant.
#14. I worked on four slopes and two distances. You have other options. Keep in mind that slopes give a little more bang for the buck. They prove parallelograms and, if you're lucky, show perpendicular segments too. Show all work.
#15. Long proof. First focus on Triangle PQA. That will give you sides that you need in a transitive step. Then switch over to Triangle RAS. That will give you angles in a transitive step. Once you have the two angles at vertex S congruent, you're done. You'll need AIA and "If sides, then angles."
#16. Discussed in class. For the base, don't think horizontal.
#17. Draw the special quads and use their diagonals as mirrors.
#18. Took 10 steps for me. After I used the Given stating "Segments are drawn...", I followed up with simple parallel statements using the right letters. That makes the first Prove a breeze.
For the second part, do you remember which letter patterns we can use to describe alternate interior, corresponding, same-side interiors, ...? For example, SSI's take on a U-shape. Looking at letter patterns might help you. It is fine to say BC is parallel to MP, even though BC is an extension of the parallelogram. I figured that as a Restate (or Given would be fine).
#19. Discussed in class. Don't forget to finish up with the Perimeter.
#20. How can you figure out (x, y). One is already on the diagram, the other comes from the fact that the point is on a line.
#21. Area = pi r squared
#22. Answers will vary. But we all know what shape will result.
#23. After our circumcenter/centroid work, this is child's play, right? Midpoints are averages of x's and y's.
#24. Reflection of a point over an axis only changes the sign of one coordinate. So label D appropriately. Then set up an algebra equation by setting the slopes equal. Once you set up that equation, the answer might jump off your paper (or you can cross multiply).
#25. Probability of two things happening = (probability of first) X (probability of second) When you figure out the probability of choosing one true statement, assume SUCCESS. This means that there is one less true statement left in the pot. The first probability will have a denominator of 4. The second will have a denominator of 3.
#26. See #24 for reflection tips. I did two slopes and counted two distances.
#27. I proved two overlapping triangles congruent to get angles of one isosceles triangle. Then a quick subtraction property closes it up.
#28. Same old stuff.
#29. Imagine the whole figure shaded. What's its area? Subtract out the white part. And speaking of that white part, you know that x must be less than some quantity. So x squared will be less than that quantity squared. (And of course you might throw in there that x must be greater than 0.)
FEEL FREE TO LEAVE A COMMENT SO THAT I KNOW WHO WAS HERE.
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