Here are some more Mental Math questions. In a future post I will talk more about the concept of halfway numbers.
Mental Math Questions
Mental Math #6
1.Six minus two and three-fourths
2.Estimate the product of 57 and 81.
3.What is 50% of 73?
4.108 + 999
5.How many fourths in six and one-half?
6.1.7 / 100
7.What is the decimal equivalent of five-sixths?
8.What number is halfway between 138 and 190?
Mental Math #7
9.What is two-thirds of one fifth?
10.How many quarts in 5 gallons?
11.What is 25% of 46?
12.What is 2001 / 1000 ?
13.Four to the third power is what number?
14.11 feet is how many inches?
15.10 is what percent of 5?
16.Apples are 12 cents each and oranges are 13 cents each. What is the total cost of three of each?
Mental Math #8
17.Seven-eights minus three-fourths
18.What is 10% of 156?
19.Write two to the fourth power in factored form and in standard form.
20.How many thirds in eight and two-thirds?
21.What is one-fourth of three-fourths?
22.Find a number halfway between 88 and 106.
Mental Math #9
23.Estimate 82 times 66.
24.How many cups in 3 pints?
25.10.96 / 10
26.Five-sixths minus one-third. (answer in simplest form)
27.What is three-fifths of forty? .
28.Eight to the zero power, divided by eight.
29.If you left at 8:45 am and arrived at 11: 19 am, how long (time) was the trip?
30.Find a number halfway between 102 and 74.
Mental Math # 10
31.Bananas cost 10 cents each and pears cost 12 cents each. What is the total cost of three bananas and four pears?
32.What is 25% of 80?
33.Seven-eights times two.
34.Eleven-sixteenths minus one-half.
35.30 is two fifths of what number?
36.What number is halfway between 390 and 518?
Thursday, November 29, 2007
Tuesday, November 27, 2007
"USE YOUR BRAIN, NOT YOUR CALCULATOR."
MENTAL MATH
The ability to do math in your head is a skill that I don't believe we spend enough time on with students. If you think about it, when presented with a mathematical situation, most people would first try to do it in their head. For example, if a recipe serves 12 people and calls for 3/4 cups of flour, how much flour would you need if you wanted to serve 24 people? If an answer can't be found mentally, a person would likely try a calculator next. If a calculator weren't available, pencil and paper would be the next option. In this day and age, I believe kids (and adults) use calculators "reflexively". That means they use it before thinking whether or not the question could be answered more quickly just using their brain. I found over the space of a few months of working on Mental Math in class, I can get most kids to not use the calculator reflexively (at least in my room). This must be constantly reinforced in the regular classroom activity also. My constant refrain was "Use your brain, not your calculator."
When you begin doing Mental Math, many kids will want to use paper and pencil methods in their head. This must be discouraged or you won't get anywhere. For example, let's add 66+48. You DON'T want them adding 6+8, getting 14, putting down the 4 (in their mind), carrying the 1. Then they add 6+4, get 10 add the 1, get 11, put it next to the 4 (in their mind) and end up with 114. That is very inefficient for mental math. They could, however, add 70+50, getting 120 and then subtract 4 and subtract 2 (or just subtract 4+2), getting 114. Or they could add 60+40, getting 100 and then add 6+8, getting 14 and then adding 100+14. Or take 4 off of the 48, leaving 44 and add that to 66, getting 70, and then adding 40 and then 4 (or adding 44 all at once). Early in their mental math experiences let them come up with their own strategies, but also show them some efficient strategies that you know they can use in other situations.
The Mental Math questions I will eventually show (I used with 8th graders) are based on the work of Larry Leutzinger of Area Education Agency 7 in Cedar Falls, Iowa. I modified some of the questions and then added some of my own. He advocated using 5 questions a day. For me it is easier to do 8 questions every 2 to 3 days. Each teacher has to find what works best for him or her. I feel it is invaluable for middle school students to be forced to "use you brain, not your calculator." You then should see less "reflexive" use of the calculator.
My classroom procedure for this was the following: Students get out their Mental Math answer sheet, and stand up next to their desk. I give the question orally and then repeat it once. They think about while they are standing, then sit down to write their answer (all they can write is an answer), then stand back up. The standing and sitting is helpful for several reasons. First they get to move around a little. Second, you can observe who comes up with an answer quickly and who takes a while. Third, when everybody is back up, I can go on to the next question. Sometimes a question is difficult and I will give them a little longer. In some cases I will feel they have had enough time and I will move on to the next question even if some have not written an answer. I normally do not grade their performance on mental math. I do all of the questions this way. Then we go over each question in class. I have already prepared a transparency showing what I think is(are) the best way(s) to do the question and then see if they have any unique alternative methods. I usually ask for a show of hands as to who got each question right. Then at the end I might ask who got 8 out of 8,7 out of 8, and so on. The whole purpose is for students to develop the ability to think and reason mathematically and to be confident in using their brain to do math mentally. One other positive benefit to doing mental math this way is that it will increase the listening skills of the students since they do not see the questions, they hear them.
It is necessary to do Mental Math on a regular basis or you will lose whatever gains you have made. It takes awhile for students to gain confidence in their ability to do math in their head. Sometimes the reason that they are not successful is that the methods they are using are inefficient for mental work. You must show them a variety of ways to do math mentally, which will give them tools they can use, but it also builds their desire to think more creatively on their own. I have also seen the “spillover” of their increased confidence into the other areas of their math class. Of course if they aren't taught methods that will work, then the opposite consequences take place.
Here are some examples of Mental Math questions. In upcoming posts I will supply more questions.
MENTAL MATH QUESTIONS
Mental Math #1
1.58+16
2.How many inches are in 6 feet?
3.What is 50% of 54?
4.Mary has five times as much money as Bob. If Bob has 40 cents, how much does Mary have?
5.Five and one-third yards is how many feet?
6.A class has 24 students. Three-fourths of the class were participating in athletics. How many students in that class are participating in athletics?
Mental Math #2
7.Estimate the product of 196 and 52.
8.48+62
9.62 times .5
10.What is 3/8 of 40?
11.What is 2/3 of 21?
12.If a quart of milk costs 75 cents, what would be the cost of a gallon of milk?
13.20 feet is how many yards?
14.A trip takes two hours and forty minutes. If you need to be at your destination by 4:15 pm, you need to leave no later than what time?
Mental Math #3
15.27/2
16..939 (100)
17.66+48
18.Six and one-half minus two and one-fourth
19.If a dozen oranges cost $1.80, what would be the cost of six oranges?
20.165cm is equal to how many meters?
21.What is 75% of 32?
22.You make a purchase for $1. 72. If you pay with a $5 bill, how much change should you get back?
Mental Math #4
23.If a pound of candy costs $2.80, what would 4 oz. of that candy cost?
24.What is 10 squared divided by 5?
25.How many eights of an inch are in four inches?
26.You make a purchase for $5.73. You pay with a $10 bill. How much money should you get back?
27.Joe has three fourths as much money as Alice. If Alice has $30, how much does Joe have?
28.How many tenths in ten and one-half?
29.Two thirds minus one half.
30.The average classroom at SLMS has 22 students. If there are 20 classrooms in the school, how many students are in the school?
Mental Math #5
31.What is the reciprocal of 3?
32.A shirt costs $9.35 to make. How much would it cost to make 10 000 of those shirts?
33.How many inches are in 8 feet?
34.Write a quick estimate of the product of 55 and 47.
35. A special breakfast meal at McDonalds costs $2.50? What would five of those meals cost?
36. Three fourths plus one third.
37. A sweater normally costs $60 but is on sale for 20% off. What is the sale price?
38. What number is halfway between 16 and 30?
The ability to do math in your head is a skill that I don't believe we spend enough time on with students. If you think about it, when presented with a mathematical situation, most people would first try to do it in their head. For example, if a recipe serves 12 people and calls for 3/4 cups of flour, how much flour would you need if you wanted to serve 24 people? If an answer can't be found mentally, a person would likely try a calculator next. If a calculator weren't available, pencil and paper would be the next option. In this day and age, I believe kids (and adults) use calculators "reflexively". That means they use it before thinking whether or not the question could be answered more quickly just using their brain. I found over the space of a few months of working on Mental Math in class, I can get most kids to not use the calculator reflexively (at least in my room). This must be constantly reinforced in the regular classroom activity also. My constant refrain was "Use your brain, not your calculator."
When you begin doing Mental Math, many kids will want to use paper and pencil methods in their head. This must be discouraged or you won't get anywhere. For example, let's add 66+48. You DON'T want them adding 6+8, getting 14, putting down the 4 (in their mind), carrying the 1. Then they add 6+4, get 10 add the 1, get 11, put it next to the 4 (in their mind) and end up with 114. That is very inefficient for mental math. They could, however, add 70+50, getting 120 and then subtract 4 and subtract 2 (or just subtract 4+2), getting 114. Or they could add 60+40, getting 100 and then add 6+8, getting 14 and then adding 100+14. Or take 4 off of the 48, leaving 44 and add that to 66, getting 70, and then adding 40 and then 4 (or adding 44 all at once). Early in their mental math experiences let them come up with their own strategies, but also show them some efficient strategies that you know they can use in other situations.
The Mental Math questions I will eventually show (I used with 8th graders) are based on the work of Larry Leutzinger of Area Education Agency 7 in Cedar Falls, Iowa. I modified some of the questions and then added some of my own. He advocated using 5 questions a day. For me it is easier to do 8 questions every 2 to 3 days. Each teacher has to find what works best for him or her. I feel it is invaluable for middle school students to be forced to "use you brain, not your calculator." You then should see less "reflexive" use of the calculator.
My classroom procedure for this was the following: Students get out their Mental Math answer sheet, and stand up next to their desk. I give the question orally and then repeat it once. They think about while they are standing, then sit down to write their answer (all they can write is an answer), then stand back up. The standing and sitting is helpful for several reasons. First they get to move around a little. Second, you can observe who comes up with an answer quickly and who takes a while. Third, when everybody is back up, I can go on to the next question. Sometimes a question is difficult and I will give them a little longer. In some cases I will feel they have had enough time and I will move on to the next question even if some have not written an answer. I normally do not grade their performance on mental math. I do all of the questions this way. Then we go over each question in class. I have already prepared a transparency showing what I think is(are) the best way(s) to do the question and then see if they have any unique alternative methods. I usually ask for a show of hands as to who got each question right. Then at the end I might ask who got 8 out of 8,7 out of 8, and so on. The whole purpose is for students to develop the ability to think and reason mathematically and to be confident in using their brain to do math mentally. One other positive benefit to doing mental math this way is that it will increase the listening skills of the students since they do not see the questions, they hear them.
It is necessary to do Mental Math on a regular basis or you will lose whatever gains you have made. It takes awhile for students to gain confidence in their ability to do math in their head. Sometimes the reason that they are not successful is that the methods they are using are inefficient for mental work. You must show them a variety of ways to do math mentally, which will give them tools they can use, but it also builds their desire to think more creatively on their own. I have also seen the “spillover” of their increased confidence into the other areas of their math class. Of course if they aren't taught methods that will work, then the opposite consequences take place.
Here are some examples of Mental Math questions. In upcoming posts I will supply more questions.
MENTAL MATH QUESTIONS
Mental Math #1
1.58+16
2.How many inches are in 6 feet?
3.What is 50% of 54?
4.Mary has five times as much money as Bob. If Bob has 40 cents, how much does Mary have?
5.Five and one-third yards is how many feet?
6.A class has 24 students. Three-fourths of the class were participating in athletics. How many students in that class are participating in athletics?
Mental Math #2
7.Estimate the product of 196 and 52.
8.48+62
9.62 times .5
10.What is 3/8 of 40?
11.What is 2/3 of 21?
12.If a quart of milk costs 75 cents, what would be the cost of a gallon of milk?
13.20 feet is how many yards?
14.A trip takes two hours and forty minutes. If you need to be at your destination by 4:15 pm, you need to leave no later than what time?
Mental Math #3
15.27/2
16..939 (100)
17.66+48
18.Six and one-half minus two and one-fourth
19.If a dozen oranges cost $1.80, what would be the cost of six oranges?
20.165cm is equal to how many meters?
21.What is 75% of 32?
22.You make a purchase for $1. 72. If you pay with a $5 bill, how much change should you get back?
Mental Math #4
23.If a pound of candy costs $2.80, what would 4 oz. of that candy cost?
24.What is 10 squared divided by 5?
25.How many eights of an inch are in four inches?
26.You make a purchase for $5.73. You pay with a $10 bill. How much money should you get back?
27.Joe has three fourths as much money as Alice. If Alice has $30, how much does Joe have?
28.How many tenths in ten and one-half?
29.Two thirds minus one half.
30.The average classroom at SLMS has 22 students. If there are 20 classrooms in the school, how many students are in the school?
Mental Math #5
31.What is the reciprocal of 3?
32.A shirt costs $9.35 to make. How much would it cost to make 10 000 of those shirts?
33.How many inches are in 8 feet?
34.Write a quick estimate of the product of 55 and 47.
35. A special breakfast meal at McDonalds costs $2.50? What would five of those meals cost?
36. Three fourths plus one third.
37. A sweater normally costs $60 but is on sale for 20% off. What is the sale price?
38. What number is halfway between 16 and 30?
Saturday, November 24, 2007
More on the Kindle
For those of you interested in more information on the Kindle, I found a detailed article on the Newsweek website. It puts this device in greater perspective and describes some of the experiences the author has had in using it.
Here is the link.
Here is the link.
Wednesday, November 21, 2007
Reading Innovation
I recently heard and read about one of the most fascinating innovations in the history of mass communication. That's just my opinion of course. My opinion also is that the current price of this device will not allow it to become commonplace. When and if it does come down in price, I see a huge impact on education and society in general. It is exciting to be living during a time of technological innovation in which things like this can come into the marketplace.
Here is the link.
Here is the link.
Tuesday, November 20, 2007
Wave of the Future?
I came across this article in the New York Times. To me it suggests one of the possibilities for high school education that might seem to be impossible for most schools but with the proper motivation it may very well be possible. It is already being done in a few places with great success. When people know what can be done and the value of it, they may very well find a way to get it done.
Here is the link.
Here is the link.
Monday, November 19, 2007
Calculators
CALCULATOR USAGE ON TESTS AND QUIZZES
In some math classes, a teacher often faces a situation in which he or she is giving a quiz or test and which the teacher wants the student to use a calculator on part of the test but not on the other part of the test. I found a good way to handle this. You can do it in either order but I had the students place their calculators on the table in the front of my room and they took a marker out of a box on that front table. (I would put enough markers in the box so that each student could get one.) When they start the test they do all the questions on the test that I want them to do without a calculator. I would group those questions together or designate them in some way so that the student knew which ones they were. They had to do their work, if needed, with the marker, and write the answer(s) with the marker. When they had completed those questions, then they would put their test face down, take the marker back up to the front table, put it back in the box, and then pick up their calculator to finish the rest of the test.
When working with the calculator they show their work with pencil and write their answer(s) with pencil. Because their answers to the non-calculator questions are written with a marker they can't use their calculator to redo the question and change their answer (when they have their calculator, the marker is back in the box). Ink pens can be used instead of markers as long as students use a pencil with their calculator. This method works very well. Students should know ahead of time what kinds of questions they will be doing without the calculator (ideally when you are first teaching the skill). Also the day before the first time I use this method, I tell them about it and show them where they will put their calculator and where the markers will be and so on.
In some math classes, a teacher often faces a situation in which he or she is giving a quiz or test and which the teacher wants the student to use a calculator on part of the test but not on the other part of the test. I found a good way to handle this. You can do it in either order but I had the students place their calculators on the table in the front of my room and they took a marker out of a box on that front table. (I would put enough markers in the box so that each student could get one.) When they start the test they do all the questions on the test that I want them to do without a calculator. I would group those questions together or designate them in some way so that the student knew which ones they were. They had to do their work, if needed, with the marker, and write the answer(s) with the marker. When they had completed those questions, then they would put their test face down, take the marker back up to the front table, put it back in the box, and then pick up their calculator to finish the rest of the test.
When working with the calculator they show their work with pencil and write their answer(s) with pencil. Because their answers to the non-calculator questions are written with a marker they can't use their calculator to redo the question and change their answer (when they have their calculator, the marker is back in the box). Ink pens can be used instead of markers as long as students use a pencil with their calculator. This method works very well. Students should know ahead of time what kinds of questions they will be doing without the calculator (ideally when you are first teaching the skill). Also the day before the first time I use this method, I tell them about it and show them where they will put their calculator and where the markers will be and so on.
Saturday, November 17, 2007
E-commerce at Risk?
I came across an article that points out a potential flaw in the security of online transactions. It also illustrates one of the current uses of mathematics that is designed to protect those transactions. I do plan on continuing to buy things online until I hear otherwise. However I always told my students that, when someone says that a code or system is totally safe, that it isn't necessarily true. What man puts together, man can take apart and sometimes man can make an honest mistake.
Here is the link.
Here is the link.
Friday, November 16, 2007
Movie Review #1
ACROSS THE UNIVERSE
What in the world is a movie review, an amateur one at that, doing on “Practical and Creative Ideas for Math Teachers?” Well I just thought it would be something interesting and teachers are people too, right?
Being a long time fan of the Beatles I thought I should see this movie. I had read good and bad things about it. It turned out to be very enjoyable – but it did not “change my world.” The basic story is a very common one – lonely boy meets lonely girl, they fall in love, they fall out of love, they get back together. The End.
The various plot points are illustrated by about thirty Beatle songs, sung mostly by a group of young, talented actors. In many cases the arrangements are slightly different from the Beatle originals but still recognizable. Some lyrics are given more emphasis, some less, in service to the plot. A series of stunning visuals accompany the songs, again in moving the story forward, or at least sideways. The creativity of director Julie Taymor is commendable, as well as her audacity to try to use music that is sacred to many people in a way that some may not like. I did like and admire what she did. She shot for the moon, only made it into orbit, but certainly didn't stay on the ground. How many of us have made it into orbit?
The movie is actually a series of music videos, strung together by the love story of the plot. Bono has a good part singing, “I Am the Walrus”. Joe Cocker is involved in an enjoyable take on “Come Together.” Other segments built around, “Strawberry Fields Together”, “I Want You (She's So Heavy)”, Hey Jude”, and “Let it Be” were also highlights. I also liked the multiple Selma Hayeks, but not necessarily the setting of the song, “Happiness is a Warm Gun.” The actors playing a Janis Joplin type character (“Sadie”) and a Jimi Hendrix type character (“Jo Jo”) were very good. I do wish the writers would have done more with the title song, “Across the Universe”, one of my all time favorite John Lennon compositions.
Watching this movie reminded me once again of the genius of the Beatles and showed the timelessness of their music. In fact there were many other Beatle songs that could have been used, but there is only so much celluloid. One thing I did wonder about – did they write the story and then pick which songs would go with that story, or the other way around?
I read something the other day (“oh boy”) about the fact that a lot of tweens, after watching the movie, are going out to buy the soundtrack album. New Beatle fans are being created every day. This movie is like the popcorn that many people ate while watching it – very tasty, very enjoyable but, in the end, not nutritious. But not everything has to be nutritious. I did enjoy this movie and am glad I went to see it.
Numerous other professional reviews of “Across the Universe” can be found on the movie review sites on the web.
What in the world is a movie review, an amateur one at that, doing on “Practical and Creative Ideas for Math Teachers?” Well I just thought it would be something interesting and teachers are people too, right?
Being a long time fan of the Beatles I thought I should see this movie. I had read good and bad things about it. It turned out to be very enjoyable – but it did not “change my world.” The basic story is a very common one – lonely boy meets lonely girl, they fall in love, they fall out of love, they get back together. The End.
The various plot points are illustrated by about thirty Beatle songs, sung mostly by a group of young, talented actors. In many cases the arrangements are slightly different from the Beatle originals but still recognizable. Some lyrics are given more emphasis, some less, in service to the plot. A series of stunning visuals accompany the songs, again in moving the story forward, or at least sideways. The creativity of director Julie Taymor is commendable, as well as her audacity to try to use music that is sacred to many people in a way that some may not like. I did like and admire what she did. She shot for the moon, only made it into orbit, but certainly didn't stay on the ground. How many of us have made it into orbit?
The movie is actually a series of music videos, strung together by the love story of the plot. Bono has a good part singing, “I Am the Walrus”. Joe Cocker is involved in an enjoyable take on “Come Together.” Other segments built around, “Strawberry Fields Together”, “I Want You (She's So Heavy)”, Hey Jude”, and “Let it Be” were also highlights. I also liked the multiple Selma Hayeks, but not necessarily the setting of the song, “Happiness is a Warm Gun.” The actors playing a Janis Joplin type character (“Sadie”) and a Jimi Hendrix type character (“Jo Jo”) were very good. I do wish the writers would have done more with the title song, “Across the Universe”, one of my all time favorite John Lennon compositions.
Watching this movie reminded me once again of the genius of the Beatles and showed the timelessness of their music. In fact there were many other Beatle songs that could have been used, but there is only so much celluloid. One thing I did wonder about – did they write the story and then pick which songs would go with that story, or the other way around?
I read something the other day (“oh boy”) about the fact that a lot of tweens, after watching the movie, are going out to buy the soundtrack album. New Beatle fans are being created every day. This movie is like the popcorn that many people ate while watching it – very tasty, very enjoyable but, in the end, not nutritious. But not everything has to be nutritious. I did enjoy this movie and am glad I went to see it.
Numerous other professional reviews of “Across the Universe” can be found on the movie review sites on the web.
Thursday, November 15, 2007
Bush v. Gore part 2
A VERY CLOSE ELECTION
SOLUTION PLAN
Florida
Bush – 2 912 790 votes, Gore – 2 912 253 votes.
Total is 5 825 043 votes. Bush won by 537 votes. Using a proportion: 5 825 043 votes / 100yd. = 537 votes / X yd. Therefore:
5 825 043 X = 100 (537) Solving for X, we get
X = .009218816 yd.
.009218816 yd. multiplied by 36in./yd = .331877376 in.
4/16 = .25, 5/16 = .3125. 6/16 = 3/8 = .375
Therefore if all the votes in Florida were represented by a 100 yard distance and rounding to the nearest sixteenth of an inch, Mr. Bush won by about 5/16 of an inch.
Using the same information but with the 84 ft. basketball court and using a proportion, Mr. Bush won by .007743805 ft. or about 3/32 of an inch.
USA
Taking the same approach as above, Mr. Gore would have won on the football field by about 19 3/16 inches. On the basketball court Mr. Gore wins by about 5 12/32 inches.
An interesting way to use one of your answers to get another answer is as follows:
In Florida we calculated that Mr. Bush would have won by
about .331877376 in on a football field. We could use that answer to figure out the answer to a basketball court.
(100 yd. = 300 ft.) Again using a proportion:
300 ft./84 ft. = .331877376 in./ X in. Solving for X we get X = .092925665 in. which is virtually identical to the answer we got the other way (.092925666 in.)
If you want more information about this solution plan, write a comment at the bottom of this post.
SOLUTION PLAN
Florida
Bush – 2 912 790 votes, Gore – 2 912 253 votes.
Total is 5 825 043 votes. Bush won by 537 votes. Using a proportion: 5 825 043 votes / 100yd. = 537 votes / X yd. Therefore:
5 825 043 X = 100 (537) Solving for X, we get
X = .009218816 yd.
.009218816 yd. multiplied by 36in./yd = .331877376 in.
4/16 = .25, 5/16 = .3125. 6/16 = 3/8 = .375
Therefore if all the votes in Florida were represented by a 100 yard distance and rounding to the nearest sixteenth of an inch, Mr. Bush won by about 5/16 of an inch.
Using the same information but with the 84 ft. basketball court and using a proportion, Mr. Bush won by .007743805 ft. or about 3/32 of an inch.
USA
Taking the same approach as above, Mr. Gore would have won on the football field by about 19 3/16 inches. On the basketball court Mr. Gore wins by about 5 12/32 inches.
An interesting way to use one of your answers to get another answer is as follows:
In Florida we calculated that Mr. Bush would have won by
about .331877376 in on a football field. We could use that answer to figure out the answer to a basketball court.
(100 yd. = 300 ft.) Again using a proportion:
300 ft./84 ft. = .331877376 in./ X in. Solving for X we get X = .092925665 in. which is virtually identical to the answer we got the other way (.092925666 in.)
If you want more information about this solution plan, write a comment at the bottom of this post.
Tuesday, November 13, 2007
Brain Development
I saw an interesting article this morning on the New York Times website that I thought you might find interesting. It is self explanatory.
Here is the link:
It is exciting to think that as time goes on we will be able to find out more about how the brain actually works. That will enable us to figure out better ways of addressing the problems that teachers and parents observe in their children.
Here is the link:
It is exciting to think that as time goes on we will be able to find out more about how the brain actually works. That will enable us to figure out better ways of addressing the problems that teachers and parents observe in their children.
Sunday, November 11, 2007
Three Answers
RIGHT, WRONG, NOT WRONG
When students give answers to math questions they sometimes will give an answer that is not what the teacher really wants but it is equal to the correct answer or is written in a different way. For me, those answers fell into the "Not Wrong" category. That told the students two things. First and most important to them, their answer was not counted wrong. Second, there was something else I wanted for that answer. They got the benefit of not counting the answer wrong but they knew there was a different answer or a different way of writing the answer that I wanted them to understand.
There are answers that are "Right." There are answers that are "Wrong." There are answers that are "Not Wrong." Here is an example:
Let's say the student is asked to solve P=2L + 2W for L.
For me the "Right" answer is L=(P-2W)/2.
A "Wrong" answer could be L=P-W.
A "Not Wrong" answer could be L= -W + .5P.
Some teachers might very well consider that that last answer is just as "Right" as my first answer and that is fine. My point here is that there are situations in which the student may give an answer in a way that really isn't wrong but is not in the form that the teacher might want, in that situation, at that time. In the above example if the teacher had specifically told the students that there were to be no decimals or negative signs in a formula, then that "Not Wrong" answer would be "Wrong."
My experience with most students is that when they hear that an answer is "Not Wrong", they will attempt to understand the different type of answer that the teacher wants and try to do it that way next time.
There are other situations in which there are multiple right answers. A simple example would be the following:
What is the probability that a card randomly drawn from a standard deck is a diamond?
"Right" answers could be 13/52, 1/4, 25%, or .25. Of course if the directions had stated that the answer had to be a fraction in simplest form, then the only "Right" answer would be ¼.
The concept of "Not Wrong" might also apply to the method that a student uses to find an answer. It is good for students to know that there are multiple ways of solving a problem. However the teacher might want the student to use the most efficient method and/or a method that can be applied to many other problems. For example a student might use the Guess and Check method to successfully solve a certain problem. The teacher might call that method "Not Wrong" if they had wanted the student to use a more efficient strategy.
As you can imagine the concept of "Not Wrong" is in the eye of the beholder and the situation. What is "Not Wrong" to one teacher might be "Wrong" to another teacher and "Right" to another teacher. The concept of calling some answers or methods, “Not Wrong” can be an effective teaching tool that shows respect for student thinking but also teaches that there is something else they need to learn.
When students give answers to math questions they sometimes will give an answer that is not what the teacher really wants but it is equal to the correct answer or is written in a different way. For me, those answers fell into the "Not Wrong" category. That told the students two things. First and most important to them, their answer was not counted wrong. Second, there was something else I wanted for that answer. They got the benefit of not counting the answer wrong but they knew there was a different answer or a different way of writing the answer that I wanted them to understand.
There are answers that are "Right." There are answers that are "Wrong." There are answers that are "Not Wrong." Here is an example:
Let's say the student is asked to solve P=2L + 2W for L.
For me the "Right" answer is L=(P-2W)/2.
A "Wrong" answer could be L=P-W.
A "Not Wrong" answer could be L= -W + .5P.
Some teachers might very well consider that that last answer is just as "Right" as my first answer and that is fine. My point here is that there are situations in which the student may give an answer in a way that really isn't wrong but is not in the form that the teacher might want, in that situation, at that time. In the above example if the teacher had specifically told the students that there were to be no decimals or negative signs in a formula, then that "Not Wrong" answer would be "Wrong."
My experience with most students is that when they hear that an answer is "Not Wrong", they will attempt to understand the different type of answer that the teacher wants and try to do it that way next time.
There are other situations in which there are multiple right answers. A simple example would be the following:
What is the probability that a card randomly drawn from a standard deck is a diamond?
"Right" answers could be 13/52, 1/4, 25%, or .25. Of course if the directions had stated that the answer had to be a fraction in simplest form, then the only "Right" answer would be ¼.
The concept of "Not Wrong" might also apply to the method that a student uses to find an answer. It is good for students to know that there are multiple ways of solving a problem. However the teacher might want the student to use the most efficient method and/or a method that can be applied to many other problems. For example a student might use the Guess and Check method to successfully solve a certain problem. The teacher might call that method "Not Wrong" if they had wanted the student to use a more efficient strategy.
As you can imagine the concept of "Not Wrong" is in the eye of the beholder and the situation. What is "Not Wrong" to one teacher might be "Wrong" to another teacher and "Right" to another teacher. The concept of calling some answers or methods, “Not Wrong” can be an effective teaching tool that shows respect for student thinking but also teaches that there is something else they need to learn.
Thursday, November 1, 2007
Bush v. Gore
A VERY CLOSE ELECTION
The 2000 Presidential Election was one of the closest in history. To get some perspective on how close that election was, I came up with the following questions that I believed the students would find interesting, as well as using some good mathematics. This makes for a very good extra credit situation although it has lost some of its interest to today's students. It does suggest an approach that teachers could take with a similar situation in the future. The problem uses the visual images of a basketball court and a football field. This type of familiar image will help students understand the closeness of the vote. Older students and adults will also enjoy this.
In the state of Florida in 2000, the official “certified” voting results showed that George Bush got 2 912 790 votes and Al Gore received 2 912 253 votes. Imagine that the combined votes of these two candidates equaled the length of a football field, from goal line to goal line (100 yards). If the two candidates had received exactly the same number of votes, then they would each have 50 yards. However they didn't receive the same number of votes. Based on the vote totals above, Mr. Bush won Florida by what distance? Round your answer to the nearest sixteenth of an inch.
Using the same vote totals from above, assume the combined vote total is the length of a high school basketball court (84 feet). In this case, Mr. Bush won Florida by what distance? Round this answer to the nearest thirty-second of an inch.
In the total USA vote in 2000, Mr. Gore received
50 996 582 votes and Mr. Bush received 50 456 062 votes. Assuming the combined vote total of the two candidates is a football field, as above, Mr. Gore won by what distance? Round to the nearest sixteenth of an inch. Assuming the combined vote total is the length of the high school basketball court, as above, Mr. Gore won by what distance? Round to the nearest thirty-second of an inch.
The answers and my solution plan will appear in a future post.
Subscribe to:
Posts (Atom)