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.
7 comments:
Love the idea about introducing proofs, every little bit helps. I teach 5th grade but I am always trying to show the students where they will see certain concepts in the future. Any familiarity with a subject makes it a little less scary down the road.
I will warn you, I liked your lesson. I also loved proofs and math. But, your students are going to be absolutely freaked out. I have zero doubt about that. New math is uncomfortable math for many students. Stick with it through the end and some of them will get the big picture. You are not looking for mastery here, right? Introduce and accept the results, don't worry that some of them won't get it. (My advice anyways, I may be an idiot)
Are there any easier proofs you could use? Most of them should know the properties of triangles by now, could you prove congruency? My thinking with triangles is the students should know it AND you could show it visually. Just seems like triangles are more visual and comfortable than exponents, but you should know what the students are comfortable with. Good luck and I'll come back looking for a recap of the lesson.
There are certainly easier proofs. I chose exponents because this is a pre-algebra class and they are currently working with simple linear and quadratic equations, although they are not yet familiar with these terms. They ought to know about triangles and congruency by now, this is true. I put some angle problems on the worksheet, but I didn't know if I could count on them remembering stuff from past years. I have watched them expand x^2 and x^3, which is why I chose to tackle exponents. I also wanted to prove something that was not obvious. I'll post an update after I teach the lesson.
Why not introduce proofs in an easier manner? Let the students know that proofs are nothing to be scared of? Then, prove your exponents?
Just thinking out loud here.
What did you want them to do here? "For each question, circle the correct answer." Did you want them to determine if it is a statement?
How long are the class periods? This seems like a lot of material to cover in one session.
Looking forward to hearing how it went.
Jackie,
This true/false "quiz" was an afterthought. I was trying to come up with a quick way to get the idea of a provable statement across. I was hoping that they would see that only #4 can be provably true or false, because it is a statement. I realize this is a lot of info to cover in an hour. Ideally it would be two lessons. We'll see how it goes.
Ah. I'd also include whatever oral directions you plan to give on the worksheet itself (sometimes the cherubs don't actually listen to the directions).
Good luck with your lesson. Will you be working with this mentor teacher for the long term?
I would love to hear how it went! Were they able to grapple with the abstract?
Post a Comment