Consider the following phrases: "4 by 6" and "3 into 12." To which of the four arithmetic operations are these referring? If you said, multiplication and division, respectively, I can conclude one thing. You are not from India.
I recently began tutoring a young woman who immigrated here from India last year. Through a series of unfortunate events, she has experienced a 6 year gap in her education. At 18 years old, she is only eligible to attend public school for one more year. After that, she must pursue a GED. She is actually a very capable young mathematician, although she needs to build confidence.
During our first few lessons, she appeared to be confusing multiplication with division. After speaking with her father, I discovered that I was the one who was confused. Or rather, we were suffering from a miscommunication. Evidently, in India, the word "by" denotes division and "into" refers to multiplication.
Isn't that delightfully fascinating? I think so.
Tuesday, June 16, 2009
Monday, June 15, 2009
Certification or Bust
This past Wednesday I met with my graduate student cohort for the first time. There are roughly 20 of us and it seems we come from all walks of life. There are older professionals changing careers and eager young graduates alike. All in all, I think it's going to be good mix of personalities and perspectives.
Our first meeting was predominantly informational. We had a chance to introduce ourselves, or rather to be introduced by one of our peers. Beyond that, it was mostly paperwork and scheduling for the upcoming semester. We were given a few assignments, which I have begun exploring, and I am hoping to begin posting again at Pencils Down, as a secondary sounding board. I know I've promised that before, but this time I really mean it.
Our first meeting was predominantly informational. We had a chance to introduce ourselves, or rather to be introduced by one of our peers. Beyond that, it was mostly paperwork and scheduling for the upcoming semester. We were given a few assignments, which I have begun exploring, and I am hoping to begin posting again at Pencils Down, as a secondary sounding board. I know I've promised that before, but this time I really mean it.
Tuesday, January 6, 2009
Pythagorean Lesson Update
I know the last few loyal followers have been holding their breath in anticipation of the results of my first efforts as a teacher. Sorry for the wait.
I was fortunate enough to teach this lesson three different times. As would be expected, each attempt resulted in a unique outcome. For my first effort, I had Sequence 2. This was the group of kids with whom I had spent the least amount of time. Although the three pre-algebra sections are not deliberately grouped by ability, a definite caste system seems to have materialized with Sequence 2 performing near the bottom. Still, I decided to teach my lesson as-is, despite advice to the contrary from my placement teacher. It turned out including the proof was a bad idea, as she had warned. It took way more time than I had estimated and consequently, I was not able to stress key aspects of the lesson. In looking at student work, I found that I had not clearly communicated the fact that c2 refers specifically to the hypotenuse and that it always goes in the same place in the theorem. For example, a2 does not equal b2 plus c2. I also discovered that my instructions on the handout could have been clearer, and that the students would have benefited greatly from working more examples together in class.
Next up was Sequence 5. I had worked with a few of these students before, so I was definitely more comfortable and my sentences flowed with greater clarity. I decided to cut the proof from the lesson and spend more time working through examples. Student work greatly improved as a result, but there was still some confusion over the order of the variables in the theorem, with several students jumbling the equation. Exit Slip feedback suggested that I had not stressed that the relationship only works for right triangles.
For the final group of the day, I had Sequence 4, which was the group that I had observed for EDU320. The lesson went much more smoothly as a result of the relationship I had established with those students. The simple fact that I was able to call them by name made things much easier. Despite my disappointment over skipping the proof earlier, I left it out again so that I could work even more sample problems and stress the importance of keeping terms in the proper order. One interesting thing did happen that I had not expected, though. I had made many efforts to make this a culturally sensitive lesson, and I made mention of great geometers from China, Egypt, Greece, and the Mayan culture. When I was mentioning the Egyptians method of using ropes to measure distance, I inadvertently suggested that this was an antiquated technology. One student in the class who is of South American heritage raised his hand to inform me that it many parts of the world, this is still the preferred method. He was pleased at my mention of the Mayas, and even took an opportunity to teach me some Mayan words, but it was clear that I had erred and should be careful of that in the future. He is typically a problem student, prone to gang related discipline problems, but his interest in his Mayan heritage helped him to focus on this activity. He did not finish the entire worksheet, but what he did do was correct and flawlessly organized.
In general, I thought it was a success. The students said that they enjoyed the hands-on activity with the ropes, though in the future, I would eliminate the 5-12-13 rope, as it was clearly too complicated for them. If I had another day, I would definitely revisit the proof, but only after the students had the main idea down cold. Access to a Powerpoint projector would have been helpful, but I know that I can not depend on technology, so it is probably better that I learn to work without it.
I was fortunate enough to teach this lesson three different times. As would be expected, each attempt resulted in a unique outcome. For my first effort, I had Sequence 2. This was the group of kids with whom I had spent the least amount of time. Although the three pre-algebra sections are not deliberately grouped by ability, a definite caste system seems to have materialized with Sequence 2 performing near the bottom. Still, I decided to teach my lesson as-is, despite advice to the contrary from my placement teacher. It turned out including the proof was a bad idea, as she had warned. It took way more time than I had estimated and consequently, I was not able to stress key aspects of the lesson. In looking at student work, I found that I had not clearly communicated the fact that c2 refers specifically to the hypotenuse and that it always goes in the same place in the theorem. For example, a2 does not equal b2 plus c2. I also discovered that my instructions on the handout could have been clearer, and that the students would have benefited greatly from working more examples together in class.
Next up was Sequence 5. I had worked with a few of these students before, so I was definitely more comfortable and my sentences flowed with greater clarity. I decided to cut the proof from the lesson and spend more time working through examples. Student work greatly improved as a result, but there was still some confusion over the order of the variables in the theorem, with several students jumbling the equation. Exit Slip feedback suggested that I had not stressed that the relationship only works for right triangles.
For the final group of the day, I had Sequence 4, which was the group that I had observed for EDU320. The lesson went much more smoothly as a result of the relationship I had established with those students. The simple fact that I was able to call them by name made things much easier. Despite my disappointment over skipping the proof earlier, I left it out again so that I could work even more sample problems and stress the importance of keeping terms in the proper order. One interesting thing did happen that I had not expected, though. I had made many efforts to make this a culturally sensitive lesson, and I made mention of great geometers from China, Egypt, Greece, and the Mayan culture. When I was mentioning the Egyptians method of using ropes to measure distance, I inadvertently suggested that this was an antiquated technology. One student in the class who is of South American heritage raised his hand to inform me that it many parts of the world, this is still the preferred method. He was pleased at my mention of the Mayas, and even took an opportunity to teach me some Mayan words, but it was clear that I had erred and should be careful of that in the future. He is typically a problem student, prone to gang related discipline problems, but his interest in his Mayan heritage helped him to focus on this activity. He did not finish the entire worksheet, but what he did do was correct and flawlessly organized.
In general, I thought it was a success. The students said that they enjoyed the hands-on activity with the ropes, though in the future, I would eliminate the 5-12-13 rope, as it was clearly too complicated for them. If I had another day, I would definitely revisit the proof, but only after the students had the main idea down cold. Access to a Powerpoint projector would have been helpful, but I know that I can not depend on technology, so it is probably better that I learn to work without it.
Sunday, December 7, 2008
Change of Plans
So there's been a change of plans since the last post. My mentor teacher decided that my lesson on proofs was going to be way over the students heads. I am inclined to disagree, but it's her class, so I'm happy to comply with her wishes. I'm going to teach a lesson on the Pythagorean Theorem instead. As an activity, I made some of those knotted ropes the ancient Egytians used to survey land. We're going to scout out the foundation of a pyramid and hopefully find some Pythagorean triples in the process. After that, I'm going to introduce the actual equation and do the following informal proof.
Each of the four right triangles are of equal size with area equal to
½AB
The A-side angle and B-side angle of each of these triangles are complementary angles, so each of the angles of the blue area in the middle is a right angle, making this area a square with side length C. The area of this square is C2. Thus the area of everything together is given by:
However, as the large square has sides of length A + B, we can calculate its area as (A + B)2. We can expand this to A2 + 2AB + B2. So
4(½AB) + C2 = A2 + 2AB + B2
2AB + C2 = A2 + 2AB + B2
C2 = A2 + B2
Q.E.D.
So that's the plan for now. I teach the lesson this coming Friday. I'll let you know how it goes. Right now, I'm off to take the Praxis I. Wish me luck.
Monday, November 24, 2008
My First Lesson Plan
So I am about to teach my very first lesson ever. My mentor teacher is giving me the reigns of her 8th grade math class for one day and she's letting me talk about anything I want. Since I am only teaching one lesson which in no way required to link to their current unit, I have made a crazy decision. When I was back in school, the biggest road block for me in math was the sudden and unforeseen appearance of proofs in geometry. It has always bothered me that we wait so long to introduce this subject. I understand that students may not be cognitively ready to handle this concept until then, but I still feel that if we wait until they are 100% ready, then we've probably waited too long. I think it makes more sense to start broaching the subject as early as possible. To that end, I have decided to do just that. The following is a handout that I have prepared for my lesson. I do not intend to read this word for word, but these are the concepts I want to cover and in this order. If anyone is still reading this, let me know what you think regarding scaffolding, differentiation, and all those other buzz words. (Note: The true/false "quiz" at the end will be handed out immediately after the introductory remarks and before the discussion of statements.)
Mathematical Proof
The word proof means something a little bit different to mathematicians than it does to normal people. This is not surprising, since we know that anyone who willingly chooses to study math is odd, to say the least. For the average person walking down the street, if a statement seems reasonable and comes from a reliable source, then the statement is assumed to be true. Let’s say you turn on the evening news and hear that President-Elect Obama has officially selected his Secretary of State. There’s no obvious reason why you shouldn’t believe the story. We know that the newsroom carefully checks all the facts, and we were expecting Mr. Obama to begin choosing his cabinet soon anyway. So we are willing to accept the statement as true.
Other people would be a little more suspicious. Scientists, for example, test a hypothesis by performing the same experiment many times. If the result is always the same, no matter how many times they repeat the experiment, then they say that the hypothesis must be true
Mathematicians are the most stubborn of all. For them, it doesn’t matter how many times the experiment is run or how many supporting examples there are. There is always the possibility that if you run one more test or check one more example, that the hypothesis will prove false. So instead, what they do is carefully build an argument that proves the statement has to be true in all possible cases. This is similar to the way a lawyer might build a case or a film critic might write a review.
In this lesson, we will take a closer look at how mathematicians build proofs and why it is important for us to learn them.
How do we decide what to prove?
Before we can set about building our argument, we have to decide what it is we are trying to prove. That usually begins with a statement. A mathematician defines a statement pretty much the same way as everyone else. It has to be a declarative sentence. For example, which of the following are declarative sentences?
What are we having for dinner?
Birds have feathers.
I love math.
Bao, please close the door.
The square root of 9 is 3.
The first sentence is a question. Questions sometimes lead to interesting discoveries, but there is no way for us to prove a question by itself. The fourth sentence is a command, also known as an imperative. It tells someone what to do. Again, there is nothing for us to prove. The rest of the sentences are all declarative.
We also require that the sentence have an objective truth value, meaning it is either true or false. If the sentence states an opinion, there is no way for us to prove it true. Which of the following declarative sentences are statements?
I love math.
The square root of 9 is 3.
September 12, 2027 will be a Friday.
Mr. L’s lesson is boring.
The first and last sentences are matters of opinion. There is no way to prove them true or false, no matter how strongly you may agree or disagree. Both of the middle two sentences are statements. The second sentence is obviously a statement, and we know immediately that it is true. What about the third sentence? This is also a statement, because we know that it must be either true or false, even if we don’t immediately know which one.
Now that we know how to recognize a provable statement, let’s look at how to begin building our argument.
How do we build an argument?
There are a lot of different kinds of proof strategies, but the one you will probably use the most is the direct proof. In a direct proof, you start with something you know to be true and build the argument one fact at a time until to reach the desired conclusion. How many facts you use depends on how complicated the statement is and who your audience is. It is a good idea to design your proof for someone who knows less about math than you do.
The facts you will use to construct your proof will basically come in three forms: definitions, axioms, and theorems. A definition is just what it sounds like. Let’s say you want to prove a statement related to a triangle, you must first know what a triangle is. An axiom, sometimes called a postulate, is a statement so basic that it doesn’t need to be proven. For example, the statement “the product of any number multiplied by one is equal to the original number (x · 1 = x) is an axiom. That’s just the way multiplication works. There’s no way for us to prove this; we just have to assume that it’s true. A theorem is a statement that has already been proven, and math textbooks are full of them. Once a statement has been proven true, we are then free to use that statement to prove other things.
The tricky part is deciding what facts to use and in what order to put them. Let’s look at the example of the next page.
What if you were asked to prove that dogs are mammals? Which of the following sentences might you use to build your case and why?
Dogs have a backbone.
Dogs have four legs.
Dogs have either fur or hair.
A mammal is a warm-blooded animal that has a backbone.
Mammals are the only animals that have hair or fur.
All dogs go to heaven.
Dogs can learn over 200 human words.
Choosing the right facts to use in your argument is trickier than it might seem. Sometimes you will come across a true statement that is related to the topic, which doesn’t help you prove your point. For example, the last sentence is true. Scientists have demonstrated that some dogs can respond to more than 200 different commands, but that doesn’t help us decide if they are classified as mammals. The same is true of the fact that dogs have four legs, since we know that lizards have four legs but are reptiles. Each line of your proof should move you closer to your goal, but don’t be afraid if you hit a dead end. When that happens, just go back a step or two and take a different path.
Once you have some idea of the kinds of facts you will need, you have to figure out where your proof should begin. Sometimes that will be obvious by looking at the statement you are trying to prove. For example, look at the statement “If the student gets an A on the final exam, then the student will get an A in the class.” In order to prove this is true, we can begin by assuming that the student gets an A on the final exam. But sometimes, the starting place is not immediately clear. In that case, it becomes a judgment call. Where you choose to start may depend on your audience. If you are proving something to your teacher, you might choose to start differently than if you were proving something to a classmate. Generally, it is a good idea to assume that your audience knows almost nothing about math.
Let’s look at an example. What if you were asked to prove that ice cubes float in water? Where might you begin? It would probably be a good idea to make sure that you explain what we mean by the word float. So a good starting point would be the following statement: “When we say an object floats in a particular fluid, we mean that if we were to drag that object to the bottom and release it, it would be pushed to the surface.” Once we have that, we can start adding more details. The following sentences, when placed in the correct order, will complete the proof. In what order do you think they should go?
1. When we say an object floats in a particular fluid, we mean that if we were to drag that object to the bottom and release it, it would be pushed to the surface.
2. Since 917 is less than 1000, we know that ice cubes float.
3. The density of liquid water is 1000 kg per cubic meter.
4. For objects to float, the object must be less dense than the fluid.
5. The density of water in solid form (ice) is about 917 kg per cubic meter.
We have already said that the first sentence will be our starting point, so we can set that one aside. The second sentence restates what we are trying to prove, so it makes a good conclusion. Now we just have to worry about the other three. According to the fourth statement, in order to prove that something floats, we must first prove something about its density. We don’t even need to know what density is, just that the object’s density is less than water. Statements three and five give us key information about the density of both water and ice. So we may conclude that the statements belong in the following order: 1,4,5,3,2, You could make the argument that statements 5 and 3 could be flipped, which is certainly true. However, it is a good idea to match the natural order given in the previous statements. Since both 2 and 5 mention the density of ice first, we should put statement 5 before statement 3.
What does any of this have to do with math?
So far we have been avoiding dealing with actual math. Now that you know a little more about proofs, we can look at an actual mathematical example. The following proof deals with a topic that is not normally presented until Algebra II. DO NOT BE ALARMED!!!
None of what we have been talking about is usually covered until Geometry, so you’re already way ahead anyway.
Before we can deal with the proof, we must first look at the following theorem.
Theorem A: For any number x , xa · xb = xa+b.
Remember, a theorem is a statement that we already know to be true. Somebody has proven it for us and put it here for us to use. But we still need to make sure that we understand what it is saying. Let’s look at an example.
Example: x3 · x4 = x3+4 or x7
We know that x3 = x · x · x and x4 = x · x · x · x
So we are saying that x3 · x4 = (x · x · x) · (x · x · x · x) = x · x · x · x · x · x · x = x7
Once you understand what the theorem is telling you, it isn’t that scary. You have already been working a lot with x2, and even though these powers are larger, the idea is exactly the same. Now that we understand how to apply the theorem, we can use it to prove other statements. For example, let’s look at how to prove the following:
For any number x, x0 = 1.
You are probably starting to get a headache. You know what x3 means. That’s just x multiplied by itself three times. We can do the same with x4 and x5, but what in the world do we mean by x0? DO NOT BE ALARMED!!! This is exactly the same question that mathematicians once asked themselves. Then they answered it with the following proof:
To Prove: For any number x, x0 = 1.
We are told that x is a number. This tells us that we can apply Theorem A to the problem since that theorem is true for all numbers. Begin by raising x to some arbitrary power, say x7. Then using the theorem, we can rewrite x7 as x7 · x0 since
x7 · x0 = x7+0 = x7.
We know by equality that x7 · x0 = x7. In other words, if we multiply x7 by x0, we get back x7. There is only one number that when multiplied by another number returns that second number. That number is the number 1, therefore, we have proven that x0 = 1. ▪
The little ▪ symbol at the end of the proof shows that you are finished. You do not have to use that symbol. You could simply write “The End” if you like, just as long as your reader knows that you are finished proving whatever you are trying to prove.
So far, all of our proofs have been written in paragraph form. Although there is nothing wrong with this, in school we commonly use an organizational device known as the two-column proof. As you’ve probably guessed, they have two columns. On one side you have a list of statements and on the other you have the reasoning behind them. The following is an example of what the previous proof would look like as a two-column proof:
Argument Reason Why
1. The unknown x is a number. 1. We are given this in the problem.
2. x7 · x0 = x7+0 2. By Theorem A.
3. x7 · x0 = x7 3. By addition.
4. x0 = 1 4. By the multiplicative identity. ▪
Are we done yet?
The short answer is “yes.” You now know some of the basic concepts of mathematical proofs, and even though you won’t see any of this again until high school, you can start using some of the ideas now. Whenever you are asked to solve a word problem, you can practice using some what you have just learned to defend your answer. You can also use the idea of proof outside of math class. When you are writing essays in language arts or social studies, you can use the same steps to build your argument.
The remaining pages of this handout are full of examples for practice. You will be asked to pick out provable statements, put statements in proper order, and even to build a few simple proofs. The more you practice now, the easier this will be in the future.
Name__________________ Sequence___
1) Give an example of a statement. (Remember that for it to be a statement, it must be provably true or false.)
2) Put the following statements in order to prove that water is a liquid at room temperature.
Room temperature is defined as 70° F.
The temperature at which a liquid becomes a solid is called the freezing point.
There are only three states of matter: solid, liquid, or gas.
Since 70 is greater than 32 and less than 212, water must be a liquid at room temperature.
The temperature at which a liquid becomes a gas is called the boiling point.
The freezing point of water is 32° F.
The boiling point of water is 212° F.
3) Fill in the blanks with the appropriate letters.
To Prove: If 4x + 10 = 38, then x = 7.
We begin by assuming that 4x + 10 = 38. By the subtraction property of equality, we can subtract 10 from both sides of the equation. (_____) Then using the (______), we can divide both sides of the equation by 4. (______) Therefore, we have proven that if 4x + 10 = 38, then x = 7. ▪
a. multiplicative identity
b. This gives us 4x = 28.
c. This leaves us with x = 7.
d. division property of equality
e. This gives us 4x = 38.
4) Convert the finished proof from Exercise 3 into a two-column proof.
Argument Reason Why
1. 1.
2. 2.
3. 3.
4. 4. ▪
Mathematical Proof
The word proof means something a little bit different to mathematicians than it does to normal people. This is not surprising, since we know that anyone who willingly chooses to study math is odd, to say the least. For the average person walking down the street, if a statement seems reasonable and comes from a reliable source, then the statement is assumed to be true. Let’s say you turn on the evening news and hear that President-Elect Obama has officially selected his Secretary of State. There’s no obvious reason why you shouldn’t believe the story. We know that the newsroom carefully checks all the facts, and we were expecting Mr. Obama to begin choosing his cabinet soon anyway. So we are willing to accept the statement as true.
Other people would be a little more suspicious. Scientists, for example, test a hypothesis by performing the same experiment many times. If the result is always the same, no matter how many times they repeat the experiment, then they say that the hypothesis must be true
Mathematicians are the most stubborn of all. For them, it doesn’t matter how many times the experiment is run or how many supporting examples there are. There is always the possibility that if you run one more test or check one more example, that the hypothesis will prove false. So instead, what they do is carefully build an argument that proves the statement has to be true in all possible cases. This is similar to the way a lawyer might build a case or a film critic might write a review.
In this lesson, we will take a closer look at how mathematicians build proofs and why it is important for us to learn them.
How do we decide what to prove?
Before we can set about building our argument, we have to decide what it is we are trying to prove. That usually begins with a statement. A mathematician defines a statement pretty much the same way as everyone else. It has to be a declarative sentence. For example, which of the following are declarative sentences?
What are we having for dinner?
Birds have feathers.
I love math.
Bao, please close the door.
The square root of 9 is 3.
The first sentence is a question. Questions sometimes lead to interesting discoveries, but there is no way for us to prove a question by itself. The fourth sentence is a command, also known as an imperative. It tells someone what to do. Again, there is nothing for us to prove. The rest of the sentences are all declarative.
We also require that the sentence have an objective truth value, meaning it is either true or false. If the sentence states an opinion, there is no way for us to prove it true. Which of the following declarative sentences are statements?
I love math.
The square root of 9 is 3.
September 12, 2027 will be a Friday.
Mr. L’s lesson is boring.
The first and last sentences are matters of opinion. There is no way to prove them true or false, no matter how strongly you may agree or disagree. Both of the middle two sentences are statements. The second sentence is obviously a statement, and we know immediately that it is true. What about the third sentence? This is also a statement, because we know that it must be either true or false, even if we don’t immediately know which one.
Now that we know how to recognize a provable statement, let’s look at how to begin building our argument.
How do we build an argument?
There are a lot of different kinds of proof strategies, but the one you will probably use the most is the direct proof. In a direct proof, you start with something you know to be true and build the argument one fact at a time until to reach the desired conclusion. How many facts you use depends on how complicated the statement is and who your audience is. It is a good idea to design your proof for someone who knows less about math than you do.
The facts you will use to construct your proof will basically come in three forms: definitions, axioms, and theorems. A definition is just what it sounds like. Let’s say you want to prove a statement related to a triangle, you must first know what a triangle is. An axiom, sometimes called a postulate, is a statement so basic that it doesn’t need to be proven. For example, the statement “the product of any number multiplied by one is equal to the original number (x · 1 = x) is an axiom. That’s just the way multiplication works. There’s no way for us to prove this; we just have to assume that it’s true. A theorem is a statement that has already been proven, and math textbooks are full of them. Once a statement has been proven true, we are then free to use that statement to prove other things.
The tricky part is deciding what facts to use and in what order to put them. Let’s look at the example of the next page.
What if you were asked to prove that dogs are mammals? Which of the following sentences might you use to build your case and why?
Dogs have a backbone.
Dogs have four legs.
Dogs have either fur or hair.
A mammal is a warm-blooded animal that has a backbone.
Mammals are the only animals that have hair or fur.
All dogs go to heaven.
Dogs can learn over 200 human words.
Choosing the right facts to use in your argument is trickier than it might seem. Sometimes you will come across a true statement that is related to the topic, which doesn’t help you prove your point. For example, the last sentence is true. Scientists have demonstrated that some dogs can respond to more than 200 different commands, but that doesn’t help us decide if they are classified as mammals. The same is true of the fact that dogs have four legs, since we know that lizards have four legs but are reptiles. Each line of your proof should move you closer to your goal, but don’t be afraid if you hit a dead end. When that happens, just go back a step or two and take a different path.
Once you have some idea of the kinds of facts you will need, you have to figure out where your proof should begin. Sometimes that will be obvious by looking at the statement you are trying to prove. For example, look at the statement “If the student gets an A on the final exam, then the student will get an A in the class.” In order to prove this is true, we can begin by assuming that the student gets an A on the final exam. But sometimes, the starting place is not immediately clear. In that case, it becomes a judgment call. Where you choose to start may depend on your audience. If you are proving something to your teacher, you might choose to start differently than if you were proving something to a classmate. Generally, it is a good idea to assume that your audience knows almost nothing about math.
Let’s look at an example. What if you were asked to prove that ice cubes float in water? Where might you begin? It would probably be a good idea to make sure that you explain what we mean by the word float. So a good starting point would be the following statement: “When we say an object floats in a particular fluid, we mean that if we were to drag that object to the bottom and release it, it would be pushed to the surface.” Once we have that, we can start adding more details. The following sentences, when placed in the correct order, will complete the proof. In what order do you think they should go?
1. When we say an object floats in a particular fluid, we mean that if we were to drag that object to the bottom and release it, it would be pushed to the surface.
2. Since 917 is less than 1000, we know that ice cubes float.
3. The density of liquid water is 1000 kg per cubic meter.
4. For objects to float, the object must be less dense than the fluid.
5. The density of water in solid form (ice) is about 917 kg per cubic meter.
We have already said that the first sentence will be our starting point, so we can set that one aside. The second sentence restates what we are trying to prove, so it makes a good conclusion. Now we just have to worry about the other three. According to the fourth statement, in order to prove that something floats, we must first prove something about its density. We don’t even need to know what density is, just that the object’s density is less than water. Statements three and five give us key information about the density of both water and ice. So we may conclude that the statements belong in the following order: 1,4,5,3,2, You could make the argument that statements 5 and 3 could be flipped, which is certainly true. However, it is a good idea to match the natural order given in the previous statements. Since both 2 and 5 mention the density of ice first, we should put statement 5 before statement 3.
What does any of this have to do with math?
So far we have been avoiding dealing with actual math. Now that you know a little more about proofs, we can look at an actual mathematical example. The following proof deals with a topic that is not normally presented until Algebra II. DO NOT BE ALARMED!!!
None of what we have been talking about is usually covered until Geometry, so you’re already way ahead anyway.
Before we can deal with the proof, we must first look at the following theorem.
Theorem A: For any number x , xa · xb = xa+b.
Remember, a theorem is a statement that we already know to be true. Somebody has proven it for us and put it here for us to use. But we still need to make sure that we understand what it is saying. Let’s look at an example.
Example: x3 · x4 = x3+4 or x7
We know that x3 = x · x · x and x4 = x · x · x · x
So we are saying that x3 · x4 = (x · x · x) · (x · x · x · x) = x · x · x · x · x · x · x = x7
Once you understand what the theorem is telling you, it isn’t that scary. You have already been working a lot with x2, and even though these powers are larger, the idea is exactly the same. Now that we understand how to apply the theorem, we can use it to prove other statements. For example, let’s look at how to prove the following:
For any number x, x0 = 1.
You are probably starting to get a headache. You know what x3 means. That’s just x multiplied by itself three times. We can do the same with x4 and x5, but what in the world do we mean by x0? DO NOT BE ALARMED!!! This is exactly the same question that mathematicians once asked themselves. Then they answered it with the following proof:
To Prove: For any number x, x0 = 1.
We are told that x is a number. This tells us that we can apply Theorem A to the problem since that theorem is true for all numbers. Begin by raising x to some arbitrary power, say x7. Then using the theorem, we can rewrite x7 as x7 · x0 since
x7 · x0 = x7+0 = x7.
We know by equality that x7 · x0 = x7. In other words, if we multiply x7 by x0, we get back x7. There is only one number that when multiplied by another number returns that second number. That number is the number 1, therefore, we have proven that x0 = 1. ▪
The little ▪ symbol at the end of the proof shows that you are finished. You do not have to use that symbol. You could simply write “The End” if you like, just as long as your reader knows that you are finished proving whatever you are trying to prove.
So far, all of our proofs have been written in paragraph form. Although there is nothing wrong with this, in school we commonly use an organizational device known as the two-column proof. As you’ve probably guessed, they have two columns. On one side you have a list of statements and on the other you have the reasoning behind them. The following is an example of what the previous proof would look like as a two-column proof:
Argument Reason Why
1. The unknown x is a number. 1. We are given this in the problem.
2. x7 · x0 = x7+0 2. By Theorem A.
3. x7 · x0 = x7 3. By addition.
4. x0 = 1 4. By the multiplicative identity. ▪
Are we done yet?
The short answer is “yes.” You now know some of the basic concepts of mathematical proofs, and even though you won’t see any of this again until high school, you can start using some of the ideas now. Whenever you are asked to solve a word problem, you can practice using some what you have just learned to defend your answer. You can also use the idea of proof outside of math class. When you are writing essays in language arts or social studies, you can use the same steps to build your argument.
The remaining pages of this handout are full of examples for practice. You will be asked to pick out provable statements, put statements in proper order, and even to build a few simple proofs. The more you practice now, the easier this will be in the future.
Name__________________ Sequence___
1) Give an example of a statement. (Remember that for it to be a statement, it must be provably true or false.)
2) Put the following statements in order to prove that water is a liquid at room temperature.
Room temperature is defined as 70° F.
The temperature at which a liquid becomes a solid is called the freezing point.
There are only three states of matter: solid, liquid, or gas.
Since 70 is greater than 32 and less than 212, water must be a liquid at room temperature.
The temperature at which a liquid becomes a gas is called the boiling point.
The freezing point of water is 32° F.
The boiling point of water is 212° F.
3) Fill in the blanks with the appropriate letters.
To Prove: If 4x + 10 = 38, then x = 7.
We begin by assuming that 4x + 10 = 38. By the subtraction property of equality, we can subtract 10 from both sides of the equation. (_____) Then using the (______), we can divide both sides of the equation by 4. (______) Therefore, we have proven that if 4x + 10 = 38, then x = 7. ▪
a. multiplicative identity
b. This gives us 4x = 28.
c. This leaves us with x = 7.
d. division property of equality
e. This gives us 4x = 38.
4) Convert the finished proof from Exercise 3 into a two-column proof.
Argument Reason Why
1. 1.
2. 2.
3. 3.
4. 4. ▪
5) Prove that if x + 3 = 15, then x = 12. (Hint: Look at Exercises 3 and 4.)
6) Prove that the area of the inner square is exactly half the area of the outer square.
(Hint: Don’t be afraid to draw more lines if it will help.)
7) Congruent angles are angles that have the same measure. Prove that if the two horizontal lines are parallel, then angle 1 is congruent to angle 8. ( Hint: Use alternate interior angles.
8) Prove the three angles of a triangle sum to 180°. (Hint: Draw a line through C that is parallel to line AB, then think about supplementary angles.)
Name_______________________ Sequence_____
For each question, circle the correct answer.
1) Give an example of an integer. True or False.
2) How do we simplify 5x + 3y – (-2x) ? True or False
3) Portland, Maine is the best city in the US. True or False.
4) The first three questions are impossible to answer. True or False.
Tuesday, November 18, 2008
The Other Maine: Faces of Homelessness
The following is a reflection on a series of articles recounting tales of homelessness in the state of Maine.
This is the sixth time I’ve started this reflection. I’m too angry to know exactly how to begin, but the due date is fast approaching, so I’d better get something on paper. The articles on Maine’s homeless children have induced a state of frustration in me so powerful as to dampen my regular flow of wit. The situation seems hopeless to me. One of the best and worst things about being human is that our ability to act deliberately in total disregard for natural instinct makes us think that we can “fix” laws of nature. Try as we might, we can not legislate away the fact that all systems have selection pressures and that not every member of that system is going to survive.
When I moved here to Maine, I had to get a new driver’s license. I also had to get a dog license, which I had never had before, despite have pets my whole life. In the same office, I saw applications for hunting licenses, fishing licenses, business licenses, and marriage licenses. You need a license to start a fire in city limits, to broadcast on a radio station, or to practice law. But anyone with a working set of genitalia can have children. We don’t get to license that. It’s a natural right endowed upon us by the universe. But it’s a right that carries with it a tremendous amount of responsibility and therein lays the problem.
The children are blameless in this struggle. As a social species, we feel compassion for them, knowing full well that they are the victims of their parents’ bad choices. We want to help, so we pass laws or enact assistance programs. But nothing works. The system is simply too big with too many cracks. The price of progress is that our family group is just too extended for us to help one another anymore. We are largely on our own and some of us are bound to fail.
It is a question of inheritance and education, really. For example, by monetary standards, I am neither rich nor poor. I live from paycheck to paycheck, and though I am comfortable now, I am one disaster away from having to renegotiate. But I have a huge safety net underneath, because no matter what happens, I always have my inheritance. Don’t misunderstand; there is no money to be had. I am not to be the beneficiary of some familial fortune. My inheritance is the power than comes from a superior education. It is my firm belief that if I were to be stripped naked, blindfolded, and dumped anywhere on the globe, that I would have the requisite skills to quickly rebuild a life. Nothing short of massive head trauma can steal that from me.
It occurs to me that my exception has proved the rule, that even an eternal “have” like me could be turned into a “have-not” with a quick crushing blow to the occipital lobe. So how do I respond to that realization? Can we chalk these scenarios up to bad luck and count our own blessings or are we obligated to help in any way we can? I suspect our humanity obligates us to the latter course. As admirable as that instinct may be, it dooms us to a certain amount of frustration and failure. I have chosen to combat this social ill, along with all other systemic malfunctions, in the only way I know how- as an educator. A good education can provide a measure of relief that no government assistance program can. It is the only solution I see as being effective, so that is how I choose to do my part.
This is the sixth time I’ve started this reflection. I’m too angry to know exactly how to begin, but the due date is fast approaching, so I’d better get something on paper. The articles on Maine’s homeless children have induced a state of frustration in me so powerful as to dampen my regular flow of wit. The situation seems hopeless to me. One of the best and worst things about being human is that our ability to act deliberately in total disregard for natural instinct makes us think that we can “fix” laws of nature. Try as we might, we can not legislate away the fact that all systems have selection pressures and that not every member of that system is going to survive.
When I moved here to Maine, I had to get a new driver’s license. I also had to get a dog license, which I had never had before, despite have pets my whole life. In the same office, I saw applications for hunting licenses, fishing licenses, business licenses, and marriage licenses. You need a license to start a fire in city limits, to broadcast on a radio station, or to practice law. But anyone with a working set of genitalia can have children. We don’t get to license that. It’s a natural right endowed upon us by the universe. But it’s a right that carries with it a tremendous amount of responsibility and therein lays the problem.
The children are blameless in this struggle. As a social species, we feel compassion for them, knowing full well that they are the victims of their parents’ bad choices. We want to help, so we pass laws or enact assistance programs. But nothing works. The system is simply too big with too many cracks. The price of progress is that our family group is just too extended for us to help one another anymore. We are largely on our own and some of us are bound to fail.
It is a question of inheritance and education, really. For example, by monetary standards, I am neither rich nor poor. I live from paycheck to paycheck, and though I am comfortable now, I am one disaster away from having to renegotiate. But I have a huge safety net underneath, because no matter what happens, I always have my inheritance. Don’t misunderstand; there is no money to be had. I am not to be the beneficiary of some familial fortune. My inheritance is the power than comes from a superior education. It is my firm belief that if I were to be stripped naked, blindfolded, and dumped anywhere on the globe, that I would have the requisite skills to quickly rebuild a life. Nothing short of massive head trauma can steal that from me.
It occurs to me that my exception has proved the rule, that even an eternal “have” like me could be turned into a “have-not” with a quick crushing blow to the occipital lobe. So how do I respond to that realization? Can we chalk these scenarios up to bad luck and count our own blessings or are we obligated to help in any way we can? I suspect our humanity obligates us to the latter course. As admirable as that instinct may be, it dooms us to a certain amount of frustration and failure. I have chosen to combat this social ill, along with all other systemic malfunctions, in the only way I know how- as an educator. A good education can provide a measure of relief that no government assistance program can. It is the only solution I see as being effective, so that is how I choose to do my part.
Monday, October 20, 2008
Whose History?
Those who can not learn from history are doomed to repeat it.
-George
Santayana
Same shit, different day.
-Steven King, Dreamcatcher
Why do we study history? Scholars assure us that history informs both our present and our future, that while the universe is vast with possibility, mankind tends to tread familiar and well-worn paths. Great generals painstakingly recreate battles waged beyond living memory to better understand the nature of warfare and to prepare for future engagements. Scientists use data gleaned from the past to predict tomorrow’s reality. To be sure, the past constantly nips at the heels of the present. But is that really why we so dutifully record our stories for the historians of tomorrow? Perhaps, it is. Maybe humanity, the only species on the planet known to understand its own mortality, compiles these complex annals for practical reasons. But I doubt it. It strikes me that a far more primal imperative is at work. In short, we love a good story.
Since before history was history, men and women have been telling stories. We tell stories about the gods and about the heavens and about the creatures of the earth. But mostly, we tell stories about other people. In evolutionary terms, we have been singing our own praises since we strayed from the safety of the trees and started roaming the African savannahs. History has mostly been an oral tradition, passed down from generation to generation as campfire tales and bedtime stories. Apprentice bards learned their craft from tribal elders as tales were honed and polished to suit the tongues of the tellers. It was very much more an art than a science, and a certain creative license was expected and encouraged. Then along came the written word, and suddenly that which was ethereal and fleeting achieved a degree of permanence that changed the discipline forever.
Once a story gets written down, it is much harder to edit. Books can be burned and edicts decreed, but some vestige will always remain. With this realization, the study of history took an egocentric turn. To the victor go the spoils, and no spoil is of greater importance than the ability to calcify one’s own version of the tale. This incontrovertible truth lies at the heart of every multicultural historical debate. Truth, like beauty, is in the eye of the beholder. The history textbooks in our classrooms typically have a lot of ground to cover. The authors are limited to one version of each story, and too often, that version is the only one most students will ever hear.
To truly understand history in a way that will move mankind toward enlightenment, we must be willing to listen to all sides, to construct our truth from all the facts at hand. As educators, it is our duty to teach our students to think critically and take nothing for granted. But that does not mean throwing out the historical baby with the bath water. Historians like Howard Zinn would have us simply substitute one half-truth for another. It is the responsibility of a scientific observer to remain as impartial as possible, to acquire objective evidence before drawing any conclusion or backing any agenda. I have read Zinn’s A Peoples History, and objective it is not. There is a clear and unapologetic agenda of tearing down heroes and championing the downtrodden. If history is a popularity contest, Zinn is backing the kid with coke-bottle lenses and acne.
According to Alejandro Segura-Mora, teachers are “cultural workers” who “can, and should, challenge white supremacist values.” I agree with this in as much as I believe we should teach our students to question everything, including ourselves. It is unfortunate that students expend so much effort parsing out the “right” answers. I am lucky enough to be training in fields like mathematics and physics, where subjectivity rarely comes into play, and “right” answers are even possible. For the rest of the world, such a concept is meaningless. For historians and poets, if you think you know the answer, you probably don’t. Here in America, where we argue black and white, it is especially easy to overlook the myriad shades of gray. As the current bearers of the mantle of imperialism the stretches back for millennia, we must force ourselves to take a hard look in the mirror. To those who mock opponents of the PATRIOT Act as overly dramatic, I offer the Alien and Sedition Laws and the Japanese Internment. To those who celebrate the low prices of Wal-Mart, I reply with sweat shops and abusive child labor. These comparative histories are no less powerful for my having heard them before. They are essential to our understanding of civilization and the stories must be told.
What we should not do, what I refuse to do, is to promote any one set of values of another. My job is to teach mathematics, and it is not an easy one. I have no problem with demonstrating how mathematical problem solving can be used to inform cultural debates, but it is not my place to inject my own politics, even if they be the politics of multi-cultural awareness and equality. Our goal should be to strip away bias, not to replace it with our own. As much as I despise the dogma of “white supremacy,” I would prefer that they at least be white supremacists that can model linear equations, follow statistical arguments, and demonstrate abstract reasoning ability. To that end, I will resort to whatever strategies I deem effective, including allegedly “sexist, racist, culturally insensitive, and contemptuous” games like Oregon Trail.
There is little doubt in my mind that this popular simulation exhibits each of these qualities, but then so does history. I freely admit to spending many hours attempting to cross the Great Plains in my digital wagon train, although in the interest of full disclosure, I never made it to the Oregon frontier. On the occasional attempts where I avoided falling victim to dysentery, my adolescent male fascination with violence sidetracked me toward extended squirrel hunting expeditions. Even so, I enjoyed learning through the game for no other reason than that it was fun. That’s why we play games after all-for fun. Part of that fun comes in the challenge of winning. If the game can not be won, the wind slips out of our sails. No one would want to play a game called Trail of Tears. Slave Trade 2 won’t be flying off the shelves. Imagine spending countless hours watching your computer avatar lying side by side with countless others in the underbelly of a slave ship with the ultimate goal of arriving triumphantly in the cotton fields of the agrarian American south. How is that supposed to facilitate the love of learning? It can’t. So I guess we’ll just have to settle for the “contemptuous” Oregon Trail and remember that it is only a supplement to a balanced curriculum, not a primary source.
The issue of cultural bias is unavoidable in history class, but it exists in one form or another in all subjects. Though we can never eradicate those prejudices completely, by discussing them openly and honestly, we can significantly curtail their influence. As teachers, we should content ourselves with sparking that debate, knowing that it is often more important to ask questions than to find answers.
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