Angles of Elevation and Depression
Read 7-5 – this is just further application of SOH-CAH-TOA. Final Test in 2 days!
The alchemist sees Sally lurking outside his shop, and after a brief chat, hires her on as a servant. Today, she must fetch sulfur from the inland cliffs. After a half a day's trek, she reaches a cliff edge that looks down on broken rock dotted with yellow sulfur. If the cliff is 200 feet from the sulfur rocks, and the angle of depression is around 10 degrees, about how high is the cliff edge?
In 7-5, work problems 4-20 even, 28-44 even. Finally, go outside and find the height of the apartments next door! You'll need to find the angle from your eye to the building - talk to me to get help with the Angle of Elevation Device.
Extra Credit: # 19, 21-25 all. (21 is hard – you may need to draw a picture AND write an equation)
Monday, January 18, 2010
Friday, January 15, 2010
Geometry: 7-4 (Friday)
Trigonometry!
Read 7-4 – Now we can find every part of any right triangle, but the ratios are extremely difficult to memorize! Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. Look at the Key Concept and Example 1 until you get the basic idea, and then write SOH-CAH-TOA or some other reminder in your notes. Say it to yourself a few times to start memorizing it. You'll use this one a lot.
As the sun creeps into the cellar entrance, Sally wakes up and goes wandering through town. She comes across an alchemist, and likes the look of their sign. The “A” is a right triangle 16 inches high and 4 inches wide. What is the measure of the angle at the top? What is the length of the hypotenuse?
In 7-4, work problems 6-16 even, 17, 32-46 even, 49-52 ALL, 70-82 even.
Extra Credit: # 59-61 all, 66, and the following:
Refer to #59. A second is a very small angle measurement equal to 1/60th of a minute, which is equal to 1/60th of a degree. A parsec is a parallax second, in other words the distance from Earth an object would have to be to have a stellar parallax of 1 second, or 1/(60*60) = 1/3600 degree. How many astronomical units are there in a parsec? If NASA scientists find alien life 197 parsecs away on Betelgeuse, how many astronomical units is that?
Read 7-4 – Now we can find every part of any right triangle, but the ratios are extremely difficult to memorize! Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. Look at the Key Concept and Example 1 until you get the basic idea, and then write SOH-CAH-TOA or some other reminder in your notes. Say it to yourself a few times to start memorizing it. You'll use this one a lot.
As the sun creeps into the cellar entrance, Sally wakes up and goes wandering through town. She comes across an alchemist, and likes the look of their sign. The “A” is a right triangle 16 inches high and 4 inches wide. What is the measure of the angle at the top? What is the length of the hypotenuse?
In 7-4, work problems 6-16 even, 17, 32-46 even, 49-52 ALL, 70-82 even.
Extra Credit: # 59-61 all, 66, and the following:
Refer to #59. A second is a very small angle measurement equal to 1/60th of a minute, which is equal to 1/60th of a degree. A parsec is a parallax second, in other words the distance from Earth an object would have to be to have a stellar parallax of 1 second, or 1/(60*60) = 1/3600 degree. How many astronomical units are there in a parsec? If NASA scientists find alien life 197 parsecs away on Betelgeuse, how many astronomical units is that?
Thursday, January 14, 2010
Geometry: 7-2 and 7-3 (Thursday)
Special Properties of Right Triangles
Today is the last 2-section day!
1. Read 7-2: We can now use the converse of the Pythagorean Theorem! Feel your powers increase tenfold!
2. Sally can't sleep, so she wanders aimlessly through the dark town. One area seems to have been burnt by fire, and a wall, surrounded by charred wood and threatening to collapse, has been propped up by a beam. The beam is 9 feet long, and touches the wall 5 feet up and the ground 7 feet out. Is the wall standing upright?
3. In 7-2, work problems 6-12 even, 20, 28, 40, 52, 60, and 62.
4. Read 7-3: Two special right triangles with specific leg-leg-hypotenuse ratios.
Sally wanders around the collapsed house, and finds the stairs down to the cellar. A support has collapsed diagonally across the 3 foot wide cellar entrance. She finds cheeses and salted meat inside, eats, and falls asleep. If the collapsed entrance is a 30-60-90 triangle, how tall is it?
5. In 7-3, work problems 8-12 even, 18-22 even (draw 20 & 22!), 32, 40, 52, 54.
Extra Credit: In 7-2 # 36, 37, 42. In 7-3, # 24-28 even.
Today is the last 2-section day!
1. Read 7-2: We can now use the converse of the Pythagorean Theorem! Feel your powers increase tenfold!
2. Sally can't sleep, so she wanders aimlessly through the dark town. One area seems to have been burnt by fire, and a wall, surrounded by charred wood and threatening to collapse, has been propped up by a beam. The beam is 9 feet long, and touches the wall 5 feet up and the ground 7 feet out. Is the wall standing upright?
3. In 7-2, work problems 6-12 even, 20, 28, 40, 52, 60, and 62.
4. Read 7-3: Two special right triangles with specific leg-leg-hypotenuse ratios.
Sally wanders around the collapsed house, and finds the stairs down to the cellar. A support has collapsed diagonally across the 3 foot wide cellar entrance. She finds cheeses and salted meat inside, eats, and falls asleep. If the collapsed entrance is a 30-60-90 triangle, how tall is it?
5. In 7-3, work problems 8-12 even, 18-22 even (draw 20 & 22!), 32, 40, 52, 54.
Extra Credit: In 7-2 # 36, 37, 42. In 7-3, # 24-28 even.
Wednesday, January 13, 2010
Geometry: 7-1 (Wednesday)
First, take the Chapter 6 Quiz! If you need a few minutes to study, we can take some time to do that.
Read 7-1: Geometric means can be written as proportions, or as square roots. Make sure you understand both! This section can be confusing - ask me if it's not clear!
Unable to find food, Sally tries to sleep in the stable again, and ponders the roof (a right triangle with the right angle at the top), with its single vertical support (an altitude). If the roof is 13 feet wide, with 9 feet on one side of the support and 4 feet on the other side, how tall is the support?
In 7-1, work problems 6-12 even (for 12, draw it – this is ONE triangle, you just need to move the altitude), 36, 38, 42, 43, 50, 58-66 even. Finally, do the activity on p. 349 and answer questions 1-3 (yes, that means you'll do the activity twice – you can draw squares on your assignment paper, or get a separate sheet – ask me for a ruler if you need one)
Extra Credit # 28, 33, 34, 44, 45 (draw this out with a specific triangle, and remember the rules for similarity)
Read 7-1: Geometric means can be written as proportions, or as square roots. Make sure you understand both! This section can be confusing - ask me if it's not clear!
Unable to find food, Sally tries to sleep in the stable again, and ponders the roof (a right triangle with the right angle at the top), with its single vertical support (an altitude). If the roof is 13 feet wide, with 9 feet on one side of the support and 4 feet on the other side, how tall is the support?
In 7-1, work problems 6-12 even (for 12, draw it – this is ONE triangle, you just need to move the altitude), 36, 38, 42, 43, 50, 58-66 even. Finally, do the activity on p. 349 and answer questions 1-3 (yes, that means you'll do the activity twice – you can draw squares on your assignment paper, or get a separate sheet – ask me for a ruler if you need one)
Extra Credit # 28, 33, 34, 44, 45 (draw this out with a specific triangle, and remember the rules for similarity)
Tuesday, January 12, 2010
Geometry: 6-5 (Tuesday)
Today we're looking at more proportions between similar triangles. We'll have a quiz tomorrow, so keep those notes up to date and study tonight!
Read 6-5
After waking, Sally remembers she's in a strange town, in a stable. The roof of the stable is shaped the same as the roof of the inn next door. It has a short, steep side, and a long, shallow side. A vertical beam runs from the bottom to the top of each roof. If the stable's roof is 10 feet wide with a 6 foot beam, and the inn's roof is 25 feet wide, how tall is the inn's beam?
In 6-5, work problems 2-6 even, 12-18 even, 24-28 even, 34, and 40-46 even.
Extra Credit p. 324 # 1-5 – Sirpenski Triangles, an introduction to Fractals (you don't need to draw the triangles, if you can calculate in your head)
If you have time, check out these 3D fractals.
http://skytopia.com/project/fractal/mandelbulb.html
Additional Extra Credit: I'm planning on skipping Fractals. However, anyone interested may do 6-6 #4, 7-13 all, and 42 for a half an assignment's worth of extra credit.
FINALLY, if you do 6-6 first, you can also turn in a full-page fractal of your design (something iterative, at least mostly self-similar, and school appropriate) for another 5 points. It should include at least 5 stages.
Read 6-5
After waking, Sally remembers she's in a strange town, in a stable. The roof of the stable is shaped the same as the roof of the inn next door. It has a short, steep side, and a long, shallow side. A vertical beam runs from the bottom to the top of each roof. If the stable's roof is 10 feet wide with a 6 foot beam, and the inn's roof is 25 feet wide, how tall is the inn's beam?
In 6-5, work problems 2-6 even, 12-18 even, 24-28 even, 34, and 40-46 even.
Extra Credit p. 324 # 1-5 – Sirpenski Triangles, an introduction to Fractals (you don't need to draw the triangles, if you can calculate in your head)
If you have time, check out these 3D fractals.
http://skytopia.com/project/fractal/mandelbulb.html
Additional Extra Credit: I'm planning on skipping Fractals. However, anyone interested may do 6-6 #4, 7-13 all, and 42 for a half an assignment's worth of extra credit.
FINALLY, if you do 6-6 first, you can also turn in a full-page fractal of your design (something iterative, at least mostly self-similar, and school appropriate) for another 5 points. It should include at least 5 stages.
Monday, January 11, 2010
Geometry: 6-4 (Monday)
Today is Parallels and Proportions!
Read 6-4 – Now we can cut triangles into proportional sections! Tomorrow we'll look at more parts of triangles, and then Wednesday we'll have a quiz over all of Ch. 6.
As Blackbeard's flotilla resupplies at Santo Domingo, on the island of Hispaniola, Gunnel Sally slips out into town. By the time she returns, the ships are gone! She sleeps in a stable, and has a terrible dream. A giant is running after her! Its right leg is longer than the left and swings out sideways, but both of its knees brush the rooftops as it runs after her. When she wakes up, she finds comfort in triangles and proportions. What can be said about the monster's lower and upper legs?
In 6-4, work problems 4-16 even, 20-24 even, 28, 46-60 even.
Extra Credit # 18, 26, 35-37
Read 6-4 – Now we can cut triangles into proportional sections! Tomorrow we'll look at more parts of triangles, and then Wednesday we'll have a quiz over all of Ch. 6.
As Blackbeard's flotilla resupplies at Santo Domingo, on the island of Hispaniola, Gunnel Sally slips out into town. By the time she returns, the ships are gone! She sleeps in a stable, and has a terrible dream. A giant is running after her! Its right leg is longer than the left and swings out sideways, but both of its knees brush the rooftops as it runs after her. When she wakes up, she finds comfort in triangles and proportions. What can be said about the monster's lower and upper legs?
In 6-4, work problems 4-16 even, 20-24 even, 28, 46-60 even.
Extra Credit # 18, 26, 35-37
Friday, January 08, 2010
Geometry: 6-3 (Friday)
Similar Triangles
Read 6-3 – Keep your binder up to date with these theorems! VERY similar to the congruence theorems, but instead of ASA and AAS we simply have AA – any two congruent angles makes it similar!
The mainmast on Gunnel Sally's ship is topped by a triangular sail. A reefline cuts across it, and that line is exactly parallel to the base of the triangle. Is the triangular top section of the sail similar to the whole triangular sail? Why or why not?
In 6-3 work problems 2-10 even, 20, 26-32 even (on 28, start by looking for more right angles and parallel lines), 36, 40, and 50-60 even.
Extra Credit # 41, 42 (think about the city context), 45, and 46.
Read 6-3 – Keep your binder up to date with these theorems! VERY similar to the congruence theorems, but instead of ASA and AAS we simply have AA – any two congruent angles makes it similar!
The mainmast on Gunnel Sally's ship is topped by a triangular sail. A reefline cuts across it, and that line is exactly parallel to the base of the triangle. Is the triangular top section of the sail similar to the whole triangular sail? Why or why not?
In 6-3 work problems 2-10 even, 20, 26-32 even (on 28, start by looking for more right angles and parallel lines), 36, 40, and 50-60 even.
Extra Credit # 41, 42 (think about the city context), 45, and 46.
Thursday, January 07, 2010
Geometry: 6-1 and 6-2 (Thursday)
Take the Chapter 5 Quiz!
6-1 and 6-2: Proportions and Similarity
1. Read 6-1 – ratios are simply another way of writing fractions.
2. Blackbeard comes across a merchant fleet. This fleet is heavily armed, with 500 cannons among its ships. Blackbeard's ships only boast 300 cannons. What is the ratio of Blackbeard's cannons to the merchants' cannons?
3. In 6-1 work problems 2, 4, 16, 26 (look back at proportions), 40, 44, 46.
4. Gunnel Sally enjoys poking around the captain's cabin while he's not there. Her favorite object is a scale model of the ship. If it's a perfect reproduction, and a 50 foot tall mast is 10 inches on the model, how wide would the captain's cabin be, if it's 10 feet wide in real life?
5. Read 6-2 – we'll use ratios to see if two polygons are similar.
6. In 6-2 work problems 4-8 even, 28, 32, 36, 52, 62, and 73.
Extra Credit: 6-1 # 23-25. 6-2 # 16, 26, and 44.
6-1 and 6-2: Proportions and Similarity
1. Read 6-1 – ratios are simply another way of writing fractions.
2. Blackbeard comes across a merchant fleet. This fleet is heavily armed, with 500 cannons among its ships. Blackbeard's ships only boast 300 cannons. What is the ratio of Blackbeard's cannons to the merchants' cannons?
3. In 6-1 work problems 2, 4, 16, 26 (look back at proportions), 40, 44, 46.
4. Gunnel Sally enjoys poking around the captain's cabin while he's not there. Her favorite object is a scale model of the ship. If it's a perfect reproduction, and a 50 foot tall mast is 10 inches on the model, how wide would the captain's cabin be, if it's 10 feet wide in real life?
5. Read 6-2 – we'll use ratios to see if two polygons are similar.
6. In 6-2 work problems 4-8 even, 28, 32, 36, 52, 62, and 73.
Extra Credit: 6-1 # 23-25. 6-2 # 16, 26, and 44.
Tuesday, January 05, 2010
Geometry: 5-4 and 5-5 (Tuesday)
Triangle Inequalities
Spock: "But Mister Kemp, we aren't finished with yesterday's assignment. Now we're supposed to do a double-assignment today, too?"
Mister Kemp: "I know. Trust me, it'll be okay. Tomorrow we'll have some time to finish it all up."
Spock: "Yay! I'm going to get on Farmville and-"
Mister Kemp: "THAT BEING SAID, the Chapter 5 quiz will be first thing on Thursday, and you will need to be ready for it! And these last two assignments are slightly longer than usual. Use your time wisely!"
1. Blackbeard takes his fleet from Maracaibo up to Kingston, and then heads for Nassau in The Bahamas. If the trip to Kingston is 500 miles, and from there to Nassau is another 600 miles, what's the furthest it could possibly be from Maracaibo directly to Nassau?
2. Read 5-4 – Keep track of these theorems in your binder! Make sure you understand the concepts before moving on to the problems! Get my help if you need it.
3. In 5-4, work problems 2-10 even, 44, 52.
4. Read 5-5
5. The cannons on the new ship can be raised to aim higher in the air. Gunnel Sally notices two cannons next to each other, the first raised to a high angle and the second cannon at a very low angle, almost parallel to the deck. Which cannon's mouth will be farther from the deck, and which theorem proves this?
6. In 5-5, work problems 4, 6, 12-16 even.
Extra Credit: 5-4 # 34, 36, and 46. 5-5 # 8, 9, 28, and 30.
Spock: "But Mister Kemp, we aren't finished with yesterday's assignment. Now we're supposed to do a double-assignment today, too?"
Mister Kemp: "I know. Trust me, it'll be okay. Tomorrow we'll have some time to finish it all up."
Spock: "Yay! I'm going to get on Farmville and-"
Mister Kemp: "THAT BEING SAID, the Chapter 5 quiz will be first thing on Thursday, and you will need to be ready for it! And these last two assignments are slightly longer than usual. Use your time wisely!"
1. Blackbeard takes his fleet from Maracaibo up to Kingston, and then heads for Nassau in The Bahamas. If the trip to Kingston is 500 miles, and from there to Nassau is another 600 miles, what's the furthest it could possibly be from Maracaibo directly to Nassau?
2. Read 5-4 – Keep track of these theorems in your binder! Make sure you understand the concepts before moving on to the problems! Get my help if you need it.
3. In 5-4, work problems 2-10 even, 44, 52.
4. Read 5-5
5. The cannons on the new ship can be raised to aim higher in the air. Gunnel Sally notices two cannons next to each other, the first raised to a high angle and the second cannon at a very low angle, almost parallel to the deck. Which cannon's mouth will be farther from the deck, and which theorem proves this?
6. In 5-5, work problems 4, 6, 12-16 even.
Extra Credit: 5-4 # 34, 36, and 46. 5-5 # 8, 9, 28, and 30.
Monday, January 04, 2010
Geometry: 5-3 (Monday)
Review and Indirect Proofs
In 5-1, work problems 14, 23, and 26. Make sure to show your work!
In 5-2, work problems 20, 30, 42, and 48. Show all your work here, too!
Read 5-3
In several instances, the entire universe has almost been destroyed, but in each case Spock stepped in to save everything. Using indirect reasoning and this fact, write a short informal proof that the universe needs Spock to exist (i.e. for the universe to exist, Spock must exist).
The 5-3 problems are mostly small - just remember, I tend to pick the big ones to grade: In 5-3, work problems 4-8 even, 14-18 even, 22, 24, 38, 42-48 even.
Extra Credit: # 12, 31, 36, 40 (on #40 you'll need to prove the two halves are congruent).
In 5-1, work problems 14, 23, and 26. Make sure to show your work!
In 5-2, work problems 20, 30, 42, and 48. Show all your work here, too!
Read 5-3
In several instances, the entire universe has almost been destroyed, but in each case Spock stepped in to save everything. Using indirect reasoning and this fact, write a short informal proof that the universe needs Spock to exist (i.e. for the universe to exist, Spock must exist).
The 5-3 problems are mostly small - just remember, I tend to pick the big ones to grade: In 5-3, work problems 4-8 even, 14-18 even, 22, 24, 38, 42-48 even.
Extra Credit: # 12, 31, 36, 40 (on #40 you'll need to prove the two halves are congruent).
Subscribe to:
Posts (Atom)