Homeschool Math Blog

These are answers to my earlier pos t where I asked if certain figures necessarily are parallelograms. The question was: Does the given information in each diagram guarantee that each is a parallelogram? Figure 1: This one you can't get around; it ends up being a parallelogram, actually a dandy rhombus. Let's prove it. You can notice it has lots of sides of the same length. If we draw a diagonal, we get two triangles with all kinds of same sides: The two triangles ABD and BCD end up having all three sides the same. So by the SSS triangle congruence theorem, they are congruent triangles. Hence, their corresponding angles are the same. I've marked the corresponding angles with the same colors. Actually the triangles are even isosceles so the blue and purple angles are even congruent... but we don't need that fact. To prove ABCD is a parallelogram, we need to prove its two sides are parallel. And for that, it's often handy to use the corresponding angle theorem: if c...

Continuing with the idea in my post about Squares that aren't squares? , let's look at the following "parallelograms". The question is the same: Does the given information in each diagram guarantee that each is a parallelogram? If you don't think so, your mission is to draw a quadrilateral with the given information but that clearly does NOT look like a parallelogram. Figure 1: Figure 2: Figure 3: Figure 4: Again, these problems let students practice logical reasoning, and also learn about parallelograms, of course. See answers here .

It's done! Finally! Took me some time to finish this review , perhaps because it involved three products: The book Geometry: A Guided Inquiry . As the name suggests, this book is based on letting students learn about theorems and their proofs in the setting of "guided inquiries" or interesting problems. It is quite unique in its approach. A Home Study Companion which includes solutions and about 300 interactive demonstrations Geometer's Sketchpad - dynamic geometry software. This review isn't just what you typically find on the web; someone called it an "exquisite in-depth review". It's fairly long... with sample pages and other pictures, examples, and more. I encourage you to read it even if you don't need a high school geometry book right now... because you'll get valuable insight just HOW GOOD geometry instruction can be, how the book handles proof, or what to think about an axiomatic vs. discovery based geometry text. Review of Geometry: ...

Continuing on a litlte bit more with my thoughts concerning Lochart's Lament . Lockhart starts out his lament with a comparison: WHAT IF music teaching only consisted of learning to write music , write notes on paper, and only after high school level would students be allowed to actually hear and make music? WHAT IT art instruction would consist of "paint by numbers" until high school, which is when they'd actually start applying paint... Lockhart remarks that if he wanted to destroy a child's natural curiosity and love of pattern-making, " ...I simply wouldn't have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education ." He calls school mathematics "pseudo-mathematics", where emphasis is on the accurate yet mindless manipulation of symbols. These are, of course, very strong words. I don't fully agree... I don't feel all that's done at school would be pseud...

By way of Natural Blogarithms, we can learn some "useful" proof techniques... For example: "Proof by vigorous handwaving: Works well in a classroom or seminar setting. Proof by forward reference: Reference is usually to a forthcoming paper of the author, which is often not as forthcoming as at first. Proof by funding: How could three different government agencies be wrong?" There's much more!

Resources These are some of the links I've added to my site recently. Maybe there's some that interest you. Mathopenref.com Free online textbook for high school geometry; not finished. GapMinder Visualizing human development trends (such as poverty, health, gaps, income on a global scale) via stunning, interactive statistical graphs. This is an interactive, dynamic tool and not just static graphs. Download the software or the reports for free. How to write proofs A 12-part tutorial on proof writing. Includes direct proof, proof by contradiction, proof by contrapositive, mathematical induction, if and only if, and proof strategies. Money Math Crystal clear tutorial on interest. Graph Mole A fun game about plotting points in coordinate plane. Plot points before the mole eats the vegetables. All sorts of sites to explore! But if those didn't fit your bill, if you're in need of a game or tutorial about specific math topic, check my link lists of online math resources; they...

Here recently I finished reading the two Dr. Math geometry books . (I will be writing a review of them, and I can say they're pretty good & inexpensive books!) The one meant for high school geometry had as its last chapter circles and theorems about circles. Well circles never were the strongest part of my mathematical knowledge, for whatever reason... (I think it stems from the fact that so big part of school geometry concentrates on calculating areas and volumes, and not on properties of figures.) So I wanted to brush up on circle theorems. I quickly read over the circle chapter on another geometry book I have, called " Geometry: A Guided Inquiry ". (I can recommend that book as well. It often asks the student to explore and try find theorems. See more info here. ) I don't know about you, but to me, reading & learning theorems and proofs can be enjoyable. First you maybe struggle to grasp it all, but afterwards there is a great satisfying feeling and admirat...

My child is dealing identifying Triangles as SAS,SSS,AAS,ASA and HL Theroms. She is also having trouble with the Flow proofs and Column proofs on explaing why 2 triangles are congruent. All of the theorems about proving that triangles (or other shapes) are congruent can be "translated" into a drawing problem: If I have my 'secret' triangle and I give you THESE pieces of information, can you reproduce my triangle? Can you do that every time, no matter what my triangle? We can describe a triangle using 6 pieces of information: the legths of the three sides, and the measures of the three angles. But you don't need all of those to be able to draw my secret triangle. Can you draw a copy of my triangle if I tell you that.... my triangle has a side 5 cm long, another side 6 cm long, and the angle between those sides is 29 degrees (I've given you S - A - S)? my triangle has a 30 degree angle, a 60 degree angle, and a 90 degree angle (I've given you A - A - A)? I...

How can you prove that irrational*non-zero rational number is equal to an irrational number? I suspect this is a university student asking me this. One important step in how to make proofs is to understand clearly what you're asked to prove. It often involves statements that are true for ALL numbers in a certain set (such as all real numbers, all rational numbers etc.). This one does not include the word "all" nor the word "any", but it still is of that type. I can reword it like this: Prove that any irrational number multiplied by any non-zero rational number is equal to some irrational number. Or, this way: Prove that for all irrational numbers x and for all rational numbers y excluding 0 the following is true: xy is an irrational number. This question basically ASKS for indirect proof. In indirect proof, we assume the OPPOSITE is true, and show that would lead to a contradiction. We are supposed to prove that if you take ANY irrational number times ANY non-z...

You might think, "Proof? You need it before high school geometry?" Sure! But, we are talking about a different form of 'proving' here. What IS proof, first of all? It is something that the person hearing or reading the 'proof' will become CONVINCED that whatever you're proving is, indeed, true. You will want to convince your youngsters or students that what YOU ARE telling them, is indeed true. That is 'proving', in a general sense. But it doesn't have to be on the same level as later. For example, when you are showing them how the multi-digit multiplication works and WHY it works (2 &times 371 is the same as 2 × 300 + 2 &times 70 + 2 &times 1), you are proving - or maybe we should say "justifying". When you take fraction manipulatives and demonstrate why 5 × 2/3 is 3 1/3, you are 'proving' - or demonstrating. (You take 5 times 2/3. You combine the thirds until you get 3 wholes and one third.) Oftentimes diagrams ...

I asked you in my previous post if what I wrote was a proof (Click here to read my 'proof') . Well, yes and no. It IS a fine proof if it was SPOKEN to someone while pointing to the various parts of the picture. But it isn't the best proof if it was written in a book. You probably had to spend some time figuring what I meant by "this line" and "that angle". Proof needs to COMMUNICATE clearly your thoughts. That's why we use "line segment AB" or AB in text. Then the other is you need to CONVINCE - to be logical in your reasoning. Also it's not enough to convince a fellow student but any sufficiently educated rational person - like your parents, your math teacher, and a mathematics professor. But, the form of the proof is not the most important thing. Numbering your arguments is not the most important thing. In my opinion, students don't need to write proofs in 2-column format if they want to write them as plain text (prose). I want ...

Another obstacle in high school math are the proofs in the geometry course. What is proof? Certainly, two-column proofs are not the only kind. In fact, they are mostly popular in high school geometry textbooks. Mathematicians, most often, just write their proofs out in sentences, and that's called "paragraph" proof (well their proofs usually take many many paragraphs worth of writing). Keith Devlin says in his book , "... being a proof means having the capacity to completely convince any sufficiently educated, intelligent, rational person..." Proof is about COMMUNICATING in a CONVINCING way. Remember those things: you need to COMMUNICATE (not just write a jumbled mess of symbols and numbers) in a CONVINCING way. Is this a proof? PROBLEM: If E is the midpoint of BD, and AE is as long as EC, prove that the two triangles are congruent. "Look at this picture that I drew. It's not drawn to scale or to be accurate. See, this line is as long as this line. An...

I want to discuss this topic because of a possible confusion. Recently in a blogpost I linked to the article Math anxiety , which mentions this math-related myth: MYTH #4: IN MATH, WHAT'S IMPORTANT IS GETTING THE RIGHT ANSWER. Then, in another recent post I linked to an article Things not to learn in school where the author strongly convinces us that getting 70% in a test is not enough since in real life you need to get it 100% right. So is it important to get the right answer when doing a mathematics problem? Well, yes but it depends. This is not a simple cut-and-dried question. It IS important if you're drilling multiplication facts. But many times it's not your focus. For example, when a student is learning to do long division or multiplying 4-digit numbers, the emphasis should be in learning the procedure and why it works. Calculation mistakes might yield a wrong answer, but if the way the division was done is right, then some credit should be given. And obviously all...

Last week I asked you two questions pertaining to the image below: 1) How does this connect with irrational numbers? Well, the two sides and the diagonal form a right triangle. You can use Pythagorean theorem to find the length of the diagonal. My picture doesn't have any lengths but I was thinking about having the side to be 1 (that's the simplest way). If both sides are 1 and diagonal is d , then Pythagorean theorem says: d 2 = 1 2 + 1 2 Solving that, you get d = √ 2 And, √ 2 is an irrational number. So if the sides of the square are 1, then the diagonal is an irrational number (square root of 2). 2) How does this connect with history of mathematics? Pythagoras was a philosopher in the 6th century B.C. who founded a philosophical school or 'cult' in southern Italy. They had some mystical beliefs, such as reality is mathematical in nature, certain symbols have mystical significance, that the whole cosmos is a scale and a number. Each number had its own personalit...

You know, some math history can help spark interest in whatever you're teaching, and enliven the math (make it 'live math' so to speak). So today I want to educate you just a little bit about Euclid and geometry. Did you know that a typical high school geometry course today with axioms, definitions, and theorems follows after the way Euclid presented geometry in his book Elements ... and that this happened around 300 B.C. in Alexandria! So the theorems your student is learning date back 2300 years! Euclid was great - not because he found many great theorems back in his time (he didn't), but because he organized all then known mathematics into a logical presentation in his book Elements . The geometry parts of Elements guided geometrical research for a long time after Euclid, and your school course of geometry is basically still based on this book. So Euclid gives his name to Euclidean geometry - also called plane geometry. In the beginning of his book, Euclid state...

Someone emailed me just recently with this question: "How do we know that pi is indeed non-repeating? Do we have proof? What is that proof?" (The person who emailed me this question is from Japan, based on the email address.) I think it's an excellent question! And I hope every eight-grader is thinking about that when they are first told about pi. You know, kids learn about pi when they are studying circles, and so they take several circles and measure the circumference and the diameter of them all, and they calculate the ratio and they get 3.3 or 3.1 or 3.45 or 3.274658 or whatever. I tell you a truth: In all your measuring you will never stumble upon the fact that pi is irrational. Proof: Because your measuring results are all rational numbers, and so you're diving a rational number by a rational number. It's not something that would be found with observation or exploration of that sort. Students are just plain announced the fact that this ratio is called Pi an...