I recently saw an interesting article in the Washington Post on some elementary math teachers incorporating more algebraic concepts in their teaching. The article also talked about some developments in the areas of educating and training math teachers to enable them to serve their students more effectively.
Here is the link.
Wednesday, December 26, 2007
Thursday, December 20, 2007
Mom's Recipe
My mom is a great cook as well as a great person. I thought I would let you know about her recipe for Cavatelli. It is extremely tasty and I have never known anyone who doesn't like it. Combine it with a salad, garlic bread, and beverage of your choice and you have a great meal! This recipe serves about six - unless me and my brothers are there in which it serves about three. Anyway, if you try it I know you'll like it.
Cavatelli
1 lb. Ground Beef
1 lb. Italian Sausage
1onion chopped
1 Green Pepper chopped
1 garlic bud
Saute all together
Add:
2 cans chopped tomatoes
16 oz. can tomato sauce
6 oz. can tomato paste
1 tsp sugar
1/2 tsp salt
1/2 tsp pepper
1 tsp basil
1 tsp oregano
1 tbls. parmesan cheese
Simmer for 2 hours
Prepare 3 kinds of pasta (shells,penne,etc.) 1 lb. Total
Mix with sauce, put in baking dish, bake at 350 until hot and bubbly. Top with shredded mozarella cheese. Heat until cheese melts.
ENJOY!!
Cavatelli
1 lb. Ground Beef
1 lb. Italian Sausage
1onion chopped
1 Green Pepper chopped
1 garlic bud
Saute all together
Add:
2 cans chopped tomatoes
16 oz. can tomato sauce
6 oz. can tomato paste
1 tsp sugar
1/2 tsp salt
1/2 tsp pepper
1 tsp basil
1 tsp oregano
1 tbls. parmesan cheese
Simmer for 2 hours
Prepare 3 kinds of pasta (shells,penne,etc.) 1 lb. Total
Mix with sauce, put in baking dish, bake at 350 until hot and bubbly. Top with shredded mozarella cheese. Heat until cheese melts.
ENJOY!!
Monday, December 17, 2007
I couldn't resist.
Because of my recent posts on mental math, the article linked below had a very interesting title. The story really isn't about the type of mental math that I have been talking about, but it does report on a pretty surprising test result. The students at Duke University that participated in this study probably hope that no one ever finds out that they were involved!
Here is the link.
Here is the link.
Thursday, December 13, 2007
Using Math to Save Money and Make Decisions
The following article contains a lot of useful information that people can use to help make some of the most common money decisions that they have in life. It would also be something that high school teachers can use with senior students, or college teachers with their students, because they will be making these types of decisions in their near future. The article also provides links to other websites that have more detailed information.
Here is the link.
Here is the link.
Wednesday, December 12, 2007
"EXTREME" Mental Math
Some of my recent posts have been on the subject of mental math. Well have you ever spent any time trying to mentally figure out the 13th root of a randomly selected 200 digit number? (For those that don't know the 13th root of a number would be a number that when taken to the 13th power would equal the given number - for example 2 is the 13th root of 8192 because two to the thirteenth power is 8192.) I recently read a story of a young man who did just that in a little over one minute!
Here is the link.
Here is the link.
Saturday, December 8, 2007
Basketball Math
BASKETBALL COMBINED
FIELD GOAL PERCENTAGE
During my thirty-three year math teaching career, I spent thirty of those years coaching basketball. One of the things that I found interesting was that when the three point shot became part of the game, it changed the concept of what was a good field goal percentage (meaning what percent of the field goals attempted were successful). Most high school coaches were happy if their team made at least 45% of their field goal attempts when all shots made were worth two points. The advent of the three point shot changed that because the team scored three points for a made shot from behind the arc as opposed to two points for a made shot from inside the arc. So making a three point shot was naturally more valuable to a team than making a two point shot.
The made three point shot is worth 1.5 times a much as a made two point shot so the three point percentage should be multiplied by 1.5 to fairly compare it to the percentage from two point range. So if a team or player shot 40% from three point range that would be equal to shooting 60% from two point range. As an example: making four out of ten (40%) from three point range would score 12 points (in ten shots) and making six out of ten (60%) from two point range would also score 12 points (in ten shots). Therefore if a coach would be happy with 45% shooting from two point range they would have to be happy with 30% shooting from three point range.
In the reality of basketball there might be some slight differences in those two situations. Three point shooters are less likely to be fouled than two point shooters. Also missed three point shots are not as often rebounded for baskets as are the missed two point shots. (One thing that some people consider to be negative about the three point rule is that the “in between” shot [between the lane and the three point arc] has almost completely disappeared from the game at the high school and collegiate level.) However for the balance of this discussion I am not going to consider these realities.
One day while I was watching a game I started to wonder if there was a way to take all of the field goal attempts and field goals made and come up with one percentage which reflected the different value of the shots. In other words it would not be realistic to simply add the two point shots made and the three point shots made and divide by the total number of field goals attempted. Here is why: let's say a team made ten of twenty from two point range (50% and scoring 20 points) and made five of twenty from three point range (25% and scoring 15 points). Just adding them normally that would be 15 made shots in 40 attempts which is 37.5%. However the team actually scored 35 points with those 40 shots and if they had made those same 15 out of 40 shots under the old rules (no shot worth three points) they would have only scored 30 points.
After some experimentation, thinking, and calculating, I finally came up with a way to actually figure out a combined field goal percentage method that accurately reflected the value of a three point shot as well as the two point shot. What it actually does is to give you a field goal percentage that would accurately reflect how many points you scored with your field goal attempts if all the attempts were from two point range. The original formula I had wasn't “user friendly” so I made an adjustment that was mathematically equal but much easier to use. Here is the “friendly” formula:
[Total points scored from all field goals] divided by [Twice the total number of all field goal attempts] Convert the resulting decimal to a percent.
In the above example the team scored 35 points with 40 shots so it would be 35 divided by (2x40) or 35 / 80 which is .4375 or 43.75%. That is higher than the 37.5 % which would be the normal way and that reflects those five made three pointers scored 15 points not ten points.
The interested reader might want to comment on why this formula works. I think I know but I would be anxious to hear what you have to say about it.
Here is one additional challenge. Can someone come up with a formula that also takes into account the free throws made (worth one point of course) and the free throws attempted along with all the field goals made and attempted? In other words, one percentage that would reflect one point shots, two point shots and three point shots. It is possible that no one in the world has ever done that, maybe because they had many other better things to do! As I am typing this, I do have an idea on what the formula would be but I would like to hear what other people think. If you figure it out or find that someone else has figured it out, please let everyone know by posting a comment on this blog.
One other way that basketball coaches and/or fans could look at this information is to simply compute what I would call the “PPS” for their team. “PPS” stands for “Points Per Shot.” Take the total number of points scored by a team and divide by the total number of shots (free throws, two point shots, and three point shots) that the team took during the game. Here is an example: A team makes 12 of 18 free throws scoring 12 points, 20 of 45 two point shots scoring 40 points, and 5 of 17 three point shots scoring 15 points. Therefore the team scored a total of 67 points and took a total of 80 shots. The PPS would be 67 / 80 which is .8375. So the team averaged .8375 points for each shot. That decimal should not be converted to a percent.
Let me know what you think, if you are still awake and make it this far!
FIELD GOAL PERCENTAGE
During my thirty-three year math teaching career, I spent thirty of those years coaching basketball. One of the things that I found interesting was that when the three point shot became part of the game, it changed the concept of what was a good field goal percentage (meaning what percent of the field goals attempted were successful). Most high school coaches were happy if their team made at least 45% of their field goal attempts when all shots made were worth two points. The advent of the three point shot changed that because the team scored three points for a made shot from behind the arc as opposed to two points for a made shot from inside the arc. So making a three point shot was naturally more valuable to a team than making a two point shot.
The made three point shot is worth 1.5 times a much as a made two point shot so the three point percentage should be multiplied by 1.5 to fairly compare it to the percentage from two point range. So if a team or player shot 40% from three point range that would be equal to shooting 60% from two point range. As an example: making four out of ten (40%) from three point range would score 12 points (in ten shots) and making six out of ten (60%) from two point range would also score 12 points (in ten shots). Therefore if a coach would be happy with 45% shooting from two point range they would have to be happy with 30% shooting from three point range.
In the reality of basketball there might be some slight differences in those two situations. Three point shooters are less likely to be fouled than two point shooters. Also missed three point shots are not as often rebounded for baskets as are the missed two point shots. (One thing that some people consider to be negative about the three point rule is that the “in between” shot [between the lane and the three point arc] has almost completely disappeared from the game at the high school and collegiate level.) However for the balance of this discussion I am not going to consider these realities.
One day while I was watching a game I started to wonder if there was a way to take all of the field goal attempts and field goals made and come up with one percentage which reflected the different value of the shots. In other words it would not be realistic to simply add the two point shots made and the three point shots made and divide by the total number of field goals attempted. Here is why: let's say a team made ten of twenty from two point range (50% and scoring 20 points) and made five of twenty from three point range (25% and scoring 15 points). Just adding them normally that would be 15 made shots in 40 attempts which is 37.5%. However the team actually scored 35 points with those 40 shots and if they had made those same 15 out of 40 shots under the old rules (no shot worth three points) they would have only scored 30 points.
After some experimentation, thinking, and calculating, I finally came up with a way to actually figure out a combined field goal percentage method that accurately reflected the value of a three point shot as well as the two point shot. What it actually does is to give you a field goal percentage that would accurately reflect how many points you scored with your field goal attempts if all the attempts were from two point range. The original formula I had wasn't “user friendly” so I made an adjustment that was mathematically equal but much easier to use. Here is the “friendly” formula:
[Total points scored from all field goals] divided by [Twice the total number of all field goal attempts] Convert the resulting decimal to a percent.
In the above example the team scored 35 points with 40 shots so it would be 35 divided by (2x40) or 35 / 80 which is .4375 or 43.75%. That is higher than the 37.5 % which would be the normal way and that reflects those five made three pointers scored 15 points not ten points.
The interested reader might want to comment on why this formula works. I think I know but I would be anxious to hear what you have to say about it.
Here is one additional challenge. Can someone come up with a formula that also takes into account the free throws made (worth one point of course) and the free throws attempted along with all the field goals made and attempted? In other words, one percentage that would reflect one point shots, two point shots and three point shots. It is possible that no one in the world has ever done that, maybe because they had many other better things to do! As I am typing this, I do have an idea on what the formula would be but I would like to hear what other people think. If you figure it out or find that someone else has figured it out, please let everyone know by posting a comment on this blog.
One other way that basketball coaches and/or fans could look at this information is to simply compute what I would call the “PPS” for their team. “PPS” stands for “Points Per Shot.” Take the total number of points scored by a team and divide by the total number of shots (free throws, two point shots, and three point shots) that the team took during the game. Here is an example: A team makes 12 of 18 free throws scoring 12 points, 20 of 45 two point shots scoring 40 points, and 5 of 17 three point shots scoring 15 points. Therefore the team scored a total of 67 points and took a total of 80 shots. The PPS would be 67 / 80 which is .8375. So the team averaged .8375 points for each shot. That decimal should not be converted to a percent.
Let me know what you think, if you are still awake and make it this far!
Wednesday, December 5, 2007
Encouraging news
This is an article from Thomas Friedman, the author of "The World is Flat" and the winner of three Pulitzer prizes. I am encouraged by this information.
Here is the link.
Here is the link.
Monday, December 3, 2007
"USE YOUR BRAIN, NOT YOUR CALCULATOR." part 3
Here is a final group of mental math questions. In the near future, I will talk more in depth about the concept and use of "halfway numbers."
Mental Math
Mental Math # 11
8.What is two-thirds of four-fifths?
9.40 is what percent of 1O? Ten is what percent of forty?
10.How many thirds in 60?
11.8.4 /1000
12.163 cm is how many meters?
13.What is 15% of $12?
Mental Math #12
16.What is three-fourths of three-fourths?
17.One and one-half minus one-fourth?
18.How many quarts in 12 gallons?
19.If marbles are $1.20 per dozen, what is the cost of 3 marbles?
20.9 is what percent of 6? 6 is what percent of 9?
21.What is halfway between 490 and 522?
22.A trip lasted from 7:50 am to 2:05 pm. How long was the trip?
23.Which is better in percentage - making 8 out of 14 free throws or making 12 out of 25 free throws?
Mental Math #13
24.4.3+9.8
25.202-39
26.Round 68.561 to the nearest tenth.
27.Five minus one and seven-eights
28.9.6 / 1000
29.How many pints in one gallon?
30.What is four to the third power in factored form? Standard form?
31.What would be a 15% tip on a restaurant bill of $21.50?
Mental Math # 14
32.218 to the zero power equals what number?
33.What is one-half of seven eights?
34.What is 40% of 40?
35.Round 4.088 to the nearest tenth.
36.Eleven and one-fourth minus two and one-half.
37. 981-208.
38.A trip lasted from 9:45 am to 2:20 pm. How long is the trip?
39.Approximate a 15% tip on a restaurant bill of $26.49.
Mental Math #15
40.4 is what percent of 20?
41.20 is what percent of 4?
42.Two-fifths minus one third.
43.How many quarts are in two and one-half gallons?
44.$4.85 (10 000).
45.Two fifths minus one third.
Mental Math
Mental Math # 11
8.What is two-thirds of four-fifths?
9.40 is what percent of 1O? Ten is what percent of forty?
10.How many thirds in 60?
11.8.4 /1000
12.163 cm is how many meters?
13.What is 15% of $12?
Mental Math #12
16.What is three-fourths of three-fourths?
17.One and one-half minus one-fourth?
18.How many quarts in 12 gallons?
19.If marbles are $1.20 per dozen, what is the cost of 3 marbles?
20.9 is what percent of 6? 6 is what percent of 9?
21.What is halfway between 490 and 522?
22.A trip lasted from 7:50 am to 2:05 pm. How long was the trip?
23.Which is better in percentage - making 8 out of 14 free throws or making 12 out of 25 free throws?
Mental Math #13
24.4.3+9.8
25.202-39
26.Round 68.561 to the nearest tenth.
27.Five minus one and seven-eights
28.9.6 / 1000
29.How many pints in one gallon?
30.What is four to the third power in factored form? Standard form?
31.What would be a 15% tip on a restaurant bill of $21.50?
Mental Math # 14
32.218 to the zero power equals what number?
33.What is one-half of seven eights?
34.What is 40% of 40?
35.Round 4.088 to the nearest tenth.
36.Eleven and one-fourth minus two and one-half.
37. 981-208.
38.A trip lasted from 9:45 am to 2:20 pm. How long is the trip?
39.Approximate a 15% tip on a restaurant bill of $26.49.
Mental Math #15
40.4 is what percent of 20?
41.20 is what percent of 4?
42.Two-fifths minus one third.
43.How many quarts are in two and one-half gallons?
44.$4.85 (10 000).
45.Two fifths minus one third.
Sunday, December 2, 2007
Math and Science Education Collaborative Initiative
In the December 2nd edition of the Des Moines Register there is an article on the Iowa Board of Regents' Mathematics and Science Education Collaborative Initiative, a joint effort of the three state universities, to strengthen math and science education in the state. I cannot give you a link to the article but it is on the desmoinesregister.com website right now. Go to that site and click on "opinion" and look for the article by Benjamin Allen (President of UNI).
Another way to find out the information of this very important program would be at this link
When you are on that page you can click on "news release" and "funding proposal".
If possible read the introductory article in the Register first.
I think this is an exciting program that has the potential to make a big difference in math and science education and the recruitment of many new and talented teachers.
Another way to find out the information of this very important program would be at this link
When you are on that page you can click on "news release" and "funding proposal".
If possible read the introductory article in the Register first.
I think this is an exciting program that has the potential to make a big difference in math and science education and the recruitment of many new and talented teachers.
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