Geometry has too many words: Practice with triangles

I know I mentioned this in class, but I'll say it again: Geometry has too many dang words. How am I, much less my students, supposed to remember all the terms? 

isoceles. scalene. rhombus. humanoid*. polygon.

* just making sure you're paying attention

I'll look up lots of games and tips and tricks as my career progresses, I'm sure. But meanwhile, I need to make sure I have a rock-solid understanding of the terms themselves--beginning with that basic geometric shape: the triangle.

You will NOT break me.

Let's start with the angles themselves.

A right angle is an angle at 90 degrees. We mark it with a little square, like this:

That's intuitive to me. I don't even need to think about it. So that's one down.Next:

An acute angle is less than 90 degrees . . .

. . . and an obtuse angle is greater than 90 degrees (but less than 180 degrees--duh, since 180 degrees would be a a straight line):

 Again, these terms make intuitive sense to me:

• Acute sounds like a sharp word: "Acute!"And skinny angles look sharper to me. Plus, we use the word acute to mean "sharp," as in "acute pain."

•  Obtuse sounds like a big, slow word: "Obtuuuuuse." And obtuse angles are big. Plus, we use the word obtuse to mean "blunted," or the opposite of sharp, as in, "because he hadn't read the piece, his analysis was somewhat obtuse." 

So what's the problem, then? Looks like I have these terms down, amiright? Well, hold the phone:

A

B

C

D

E

 
 Are you confused yet? I am.

I know an acute triangle has all acute angles and that an obtuse triangle has one obtuse angle. Just by looking at these five triangles, the only one that's obvious to me is E: It's an obtuse triangle because the bottom left angle clearly measures greater than 90 degrees:

 

E

I would guess that triangle A is also obtuse, because I can see clearly that the bottom right angle is wider than 90 degrees:

A

 

Triangle A is a good segue into what confuses me the most about angles/triangles, though: Which way do we measure them? 

What on earth are you talking about, Miss B?

I'll show you what I mean. Here's triangle A again--but now I've rotated it:

A

Let's look at this lower left angle. Clearly it's acute, right? Because we see what 90 degrees looks like, and this angle is less than 90 degrees.

But what about this lower right angle?

A

Again, we see what 90 degrees looks like. But the y axis of the lower right angle seems like it would create an angle greater than 90 degrees to me, because it's to the left of the 90-degree y axis. And yet, the angle doesn't look obtuse. Clearly I'm missing something.

A little obtuse, are we?

 I only see one difference between the lower left angle and the lower right angle: the direction of the x axis. The lower left angle has the x axis going to the right . . . 

A

. . . and the lower right angle has the x axis going to the left: 

A

So what if the correct way to measure an angle is to rotate it until the x axis is going to the right? Let me practice a little with my good friends at--say it with me--Khan Academy, and see if my theory holds up.

Here's a practice set for identifying triangles by angles. Here we go:

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

 

Wahoo!

 I'm not gonna lie--I'm glad to get that achievement. But only two of those were even remotely challenging: questions 5 and 7 (the others had the angle measurements in the drawings). Still, I am delighted to report that by keeping my new rule in mind, I was able to identify those angles correctly!

And bonus: I have a trick to help my students who struggle with how to measure angles. Now, I don't know why this rule works . . . but it's good to have a little mystery in our lives:

There is more to know. . . .

 Thanks for stopping by!

 

Where have all the factors gone? Finding the LCD through factor trees

I've read your blogs. I've asked in class (a couple times, even). But I still can't figure out why.

Why do I not have to repeat all the factors to find the least common denominator (LCD)?

I know some of us have also been stumped by this question.

But for those of us who haven't, let me explain:

                               

 Let's pretend we're adding

We need to make a common denominator. We could just multiply 12 x 15 and use a denominator of 180, but that's embarrassingly huge. We could list out multiples and see what the first common one is (12, 24, 48, 60  and 15, 30, 45, 60 shows us 60 is the first one in common)--boooring. FACTOR TREE!

                                        

So here's my problem: How do I know which of these factors--2 x 2 x 3  and 3 x 5--to include in the least common denominator? Well, I could use them all, of course:

2 x 2 x 3 x 3 x 5

Which gives me what? 180. That's a common denominator, sure, because all we're doing is multiplying all the factors together, which is the same as multiplying the numbers themselves together, which we already tried (see "embarrassingly huge," above).

We're looking for the least common denominator. What if we only use the common factors, maybe? Well, the only factor (besides 1) that 12 and 15 have in common is 3, so clearly that doesn't work.

Maybe the factors that repeat cancel each other out? 

2 x 2 x 3 x 3 x 5 = 2 x 2 x 5 = 20

20 is not a common factor of 12 and 15. So there went that idea.

I'm stumped. Which means, here at Miss B's Math Trip, that it's time to head over to Khan Academy!!

Let's take a look at this video on finding common denominators:

And just like that, there it is, starting at 1:50. Seriously. Take a look. I'll wait.

The video really does explain this concept perfectly (for me, anyway). But I'll have to learn to paraphrase it for my students anyway, so here goes:

 

Let's go back to our example of finding the LCD for 5/12 and 4/15. We recall we used factor trees to get the factors of each number:

                                       

And we know we have to find a common multiple of both 12 and 15 for our denominator:

• In order to be divisible by 12, the denominator has to have factors of 2, 2, and 3.
  

• In order to be divisible by 15, the denominator has to have the factors of 3 and 5.
  

• So as long as the denominator has a 2, a 2, a 3 (both the 12 and the 15 have a 3), and a 5 in its factors, it will be divisible by both 12 and 15, because each of the factors will be there.

Remember, we want to find the lowest common denominator. And the way to do that? Just multiply the factors we know have to be there:

2 x 2 x 3x 5 = 60.

Ssssssnap!

Just to be sure I can do this in practice, though, let me do a practice set (and yes, get that sweet Level Up!)

I did have to switch up the questions a little, since the set practices other ways of finding the LCD besides using factor trees. But I got some good practice in. For example:

Which I solved like this:

And now I have officially Leveled Up!

NICE.

Thanks for stopping by!

So you say division is repeated subtraction? I'd like to see you prove it!

The truth is, subtraction is challenging. I was glad to hear Dr. Moldavan confirm as much in class. I have the procedural knowledge down pat (if I didn't at my age, I'd have bigger problems), but now that I'm thinking in terms of understanding instead of performing, I'm seeing that I don't have the comfort level with subtraction that I'd like if I'm going to  teach this fundamental concept.

I don't think I'd find it useful to practice a bunch of subtraction problems. But I would like to explore in more depth what it means to say that "division is repeated subtraction."

Maybe I should see what other resources are out there besides Khan Academy. But on the other hand . . . why fix what isn't broken, am I right? 

What can I say? I'm a loyalist.

So let's start at the beginning--with their video The idea of division:

Well, that WAS interesting. This video explains that you can think of 24 ÷ 3 as either 3 equal groups within 24, or as groups of 3 within 24.(This concept reminds me of Bonnie Jeanne's post on arrays and how you can flip them on their sides).

So on to repeated subtraction.

Wait . . . 

The Khan Academy search bar doesn't bring up anything for repeated subtraction?!

You betrayed my loyalty, Khan Academy!

After taking a moment to pick up the shattered pieces of my math life, I'm ready to move on. . . .

One of the things I've learned in this course is that unlike most other subjects, I learn math best when someone explains it to me. Videos are much more useful to me than textbooks are in this context. So I'll go straight to the source for All Things Video: YouTube.

Whoa--here's a link to a video combining two of the core concepts I'm exploring on my PLP: Division as Repeated Subtraction Using a Number Line!

Oooh. I really like her description of division using the example of 6 ÷ 2:

"This question is asking you, 'How many times does this number [2] fit into this number [6]?' "

I might reword her question as:

 

"How many groups of 2 can you divide 6 into?" Because that wording includes the word divide--and we're teaching division.

 

Okay, so I see in the video that the presenter is subtracting 2s to get to 0 . But I still don't get why division is considered repeated subtraction; after all, couldn't I just as easily start at 0 and see how many 2s I have to add to get to 6?

 

Let's see what else we can find. . . . Check out this video--Interpret division using repeated subtraction:

 

This video is only three minutes long. But it has two key phrases that break this concept open for me.

Phrase 1 is at 2:15: "We had ten objects. We wanted to divide by two. Meaning we wanted to make groups of two, or give two to each group." 

•  When I give two to each group, I am subtracting 2 each time. I am giving those twos away. Giving something away and subtracting it are the same to me conceptually. And now I understand why we would say "division is repeated subtraction."
  

Phrase 2 is at 2:49: "The quotient is how many times we subtracted the divisor."

• This is one of those simple phrases that one person might gloss right over, but that could shift someone else's understanding (namely mine). Basically, the answer to any division problem (quotient) is how many times the second number (divisor) fits into the first number (dividend).

 
But perhaps most importantly, now I understand something else: "multiplication is repeated addition" is very different from "division is repeated subtraction":

 

• Multiplication actually is repeated addition: 3 x 3 is simply a shorter way of saying 3 + 3 + 3. But

• Division is not necessarily repeated subtraction: For example, you can also start from 0 and add your divisor until you hit your dividend to get to the quotient--which does not use subtraction at all. 

 

In other words: "multiplication is repeated addition" is a mathematical fact. "Division is repeated subtraction" is a mathematical process.

 

How cool is that?!

 

Adding and subtracting negative numbers

It's kind of subtle, but if you've been following along with this blog at all, you may have noticed . . . negative numbers are not in my comfort zone. So the more I learn about them, the better. Today I'm focusing on adding and subtracting negative numbers--which is only fitting, since I started this exploration after getting that problem in our problem set.

Again, this is gonna be a shocker, but . . . we're going to the Khan Academy!

Not surprisingly, they have content that looks like it could be useful. Here's a video on adding and subtracting negative numbers. Let's take a look:

Okay, that wasn't particularly helpful. It's basically the exact same thing I did on my first blog entry about negative numbers. It does make me feel a little better, though, that I couldn't understand in that first entry how to use the number line to show -3 - (-5): The answer is, You can't--not without converting it to -3 + 5, anyway. Which is what he does at 2:40.

Maybe it will help if I can explain why subtracting a negative number is the same as adding a positive number? I was able to get there in my first post on this subject by using examples and logical argument, but truth be told? I still don't understand why subtracting a negative is the same as adding a positive. Maybe this video will help? After all, I'm not a betting man, but it is called  subtracting a negative = adding a positive:

That does help a little. The concept is,

• if Steve has a net worth of negative $3 (so -3, basically),

• and his uncle wants to help him out by taking away that negative net worth (so -3 - [-3], basically),

• then the easiest way to do that is by giving Steve $3 (so -3 + 3, basically).

I'd probably avoid using the term "net worth" with young kids, though. Speaking of which: When are students supposed to learn this? Khan Academy usually has the Common Core standard listed at the top of a lesson, but that's not the case here. Let me do a little digging. . . .

Oh.

That's much later in the curriculum than I thought. Seventh graders should be a lot easier to teach negative numbers to than second- or third graders. I feel better already! I may even feel good enough to move on to another topic. But let me do some practice first--which, Ill admit, I mostly want to do because I want to get to that sweet, sweet Level 2 of Mastery! 

So close!

What can I say?

 I'm feeling a little cocky now, so let me try a problem set with negative numbers but in a way I haven't studied them yet--negative symbol as opposite:

Ouch.

 I'll keep trying!