Time Zones

Griffin is 13 years old and seems to be coming to the end of that early adolescent phase of rejecting everything those around him hold dear. Engaging him in math talk has taken more finesse in this phase of life.

Mostly it has involved giving him responsibility for things that involve making calculations. When he was little, we could talk collaboratively about how many tangerines are in a 3 pound bag and discuss whether this would be enough to last the family a week. Now I tend to put him in charge of getting enough tangerines to last us a week. He still has to do the same thinking, but he’s in charge.

This is not enough tangerines for a week at our house. (By the way, which is more?)

From time to time, though, we still put a mathematical idea up for discussion, and as he ages through adolescence, these conversations happen a bit more often. Yet he is still wary. Nevertheless, I persist.

We have been watching the Olympics, and we have wondered about which events are happening as we watch them, and which ones happened earlier (yet somehow happened “tomorrow”!)

Griffin was thinking about time zones, and about their implications for traveling as we wrapped up an evening this week, and made preparations for the next day.

Griffin (13 years old): So they’re 14 hours ahead of us?

Me: Yes.

G: You’d get a lot of jet lag, huh?

Me: Yeah. Maybe not as much as it looks like, though. Maybe it’s just 10 hours’ worth, going the other way.

There is a bit of a puzzled silence.

G: Wait. Really?

Me: Yeah. Well, plus a day.

G: Wait. Is this one of your mathy talks?

Me: Absolutely not.

If you’re reading this, Griff, I’m sorry (sort of). I am totally busted.

Me: Yeah. 14 hours ahead is the same as 10 hours behind, right? Just going the other way.

G: But the day would be wrong.

Me: Yeah. You have to add a day, but you don’t get jet lag because the day changes, you get jet lag because the time of day does.

G: Maybe.

He returns to packing his lunch. I go back to whatever I was doing. Putting turtles in boxes, probably.

A couple minutes later…

G: So the east coast is 23 hours behind us?

So What Do We Learn?

Keep trying. Opportunities to talk about numbers, shapes, and patterns present themselves. Seize them and do not stop. Ask questions, think out loud. Don’t worry about whether any particular conversation goes anywhere. Just keep at it.

Birthday Chocolate

Today is my birthday. Griffin (12 years old) gave me three chocolate bars as a gift. He gave me candy because he is deeply aware of its value in life. He gave me dark chocolate because he knows it’s my preference.

He is frequently disturbed by how slowly I eat these gifts of candy he gives me.

Here’s how my after-work greeting went this evening.

Griffin (12 years old): Happy birthday, Dad.

Me: Thanks.

G: One thing I’ve noticed about you is that you eat the candy I give you incredibly slowly.

Me: I know. But actually I ate half of one today.

G: Half of a bar, or half of all the bars?

Me: Half of one bar. And then maybe I’ll have another half tomorrow.

G: Oh brother.

Me: And since I know 3 divided by 1/2 is 6…

G: You ate one-sixth of it.

Me: And it’ll last me 6 days.

Having arrived home a bit chilly and damp from the bike ride in the 45° rain, I went downstairs for a shower and he returned to his iPod.

So What Do We Learn?

I haven’t written a lot about this boy recently because he is in a phase of rejecting everything the adults around him care about. All adolescents go through some form of this. He is doing it with gusto.

In any case, the groundwork we’ve laid in the early years has paid off. When math is useful for his purposes, he will use it. Here, he wanted to prove his point that I am a painfully slow candy consumer. That made it important to clarify that I had not eaten half of my candy, but only half of one bar of candy.

We play around with units like this frequently. It has contributed to both children’s place value understanding, as well as their fraction work.

Starting the Conversation

Ask frequently about the units that are attached to the numbers in your lives. When you’re cooking, ask, Should we use 3 eggs or 3 dozen eggs? Ask about how many pieces of candy a pack of Whoppers is at Halloween.

Look at these pictures—one at a time—and ask How many? Challenge yourselves to find different numbers, and different units. (For example, there are 15 avocado halves, 7.5 avocados, 8 pits, 7 holes, and 1 cutting board).

 

Tessalation: A great new book

Tessalation is a terrific new picture book by Emily Grosvenor. The story involves a little girl whose mother needs a bit of peace and quiet, so sends her outside to play. While outside, Tessa (get it?) notices shapes fitting together without gaps everywhere she looks.

I helped sponsor Tessalation on Kickstarter this spring, and our hard copies arrived last week. Naturally Tabitha (9 years old) and I read it together right away.

Here are some of the things Tabitha, Griffin (11 years old) and I noticed and discussed while reading it, and afterwards:

  • The turtles are delightful.
  • While they are somewhat different turtles from the ones we’ve played with around the house for the last year, they have an important characteristic in common—two noses and two tails come together in both tessellations.
  • There are tessellating leaves that look an awful lot like some shapes I’ve made and we’ve played with a number of times. We saw kites and hexagons and triangles in the leaves just as we have in the pink quadrilaterals below.
  • We wondered whether this object counts as a tessellation. (It’s not from the book, but Tessa set a great example for us to notice and ask about tessellations in our world.)

2016-07-11 17.32.02

All in all, Tessalation is perfectly aligned with the Talking Math with Your Kids spirit. It creates a richly structured and playful space for parents and children to notice things and to converse. The language is fun. The images are beautiful. Tabitha and I highly recommend it.


Quick notes: Tessalation will be a component of August’s Summer of Math box. It’s not too late to sign up! Also, we’ll soon have a Tessalation/Tiling Turtles combo pack available. You can order the book right now from Waldorf Books, and e-books from Amazon.

 

A tale of two conversations

Here are two conversations about hot chocolate.

The first one didn’t happen. The second one did. Read them both, then I’ll tell you about their meaning.

Both conversations begin on a cold November night in Minnesota. Unseasonably cold. Fourteen degrees, to be precise (–10 Celsius).

A cup of hot chocolate

Zero marshmallows for me on this cold night.

Tabitha (7 years old), Griffin (10 years old) and I get in the car to head for Tabitha’s basketball practice.

What might have been

Me: Wow! It is cold!

Tabitha (7 years old): You know what you do when it’s cold? You make hot chocolate.

Me: Ooooo! Good idea! We can do that when we get back home after practice.

T: Does it count as dessert?

Me: If you have marshmallows in it, it does.

T: I won’t have any marshmallows, then. So I can have some Jell-O.

Griffin agrees that this is the way to go, and the conversation moves on to other things.

What actually happened

Me: Wow! It is cold!

Tabitha (7 years old): You know what you do when it’s cold? You make hot chocolate.

Me: Ooooo! Good idea! We can do that when we get back home after practice.

T: Does it count as dessert?

Me: If you have two marshmallows in it, it does.

T: I’ll have zero marshmallows in mine, then, so I can have some Jell-O.

Griffin (10 years old): I’ll have one marshmallow, and a small serving of Jell-O. Wait, no! I know! I’ll cut a marshmallow in half!

I presume that this is in order to maximize his allowable Jell-O serving, while still retaining some marshmallow in his hot chocolate. It’s a scheme nearly as complicated as credit default swaps.

So what do we learn?

One small difference changed the course of the conversation—my use of a number word. I could have said, “It counts as dessert if you have marshmallows in it.” But I did say, “It counts as dessert if you have two marshmallows in it.”

Using numbers—two marshmallows instead of just marshmallows—invited the children to talk about numbers. It invited them to use numbers to maximize their benefit. It invited them to think about numbers.

This invitation is important.

A few years back, researchers paid careful attention to the ways preschool teachers talked with their students. Those teachers who used more number words and concepts as they talked with children stimulated greater growth in math than those who used less math talk.

This was not a study about math instruction; it was a study about the math language that these teachers used when they weren’t teaching math. “Yes, you three may help me.” versus “Yes, you may help me.” is the sort of difference that matters.

Using number words and math concepts in everyday speech invites children to notice and to think about number. That’s what Talking Math with Your Kids is all about.

Link to full study ($)

Cookies under constraints

A propos of nothing one day, I ask Griffin (9 years old at the time, finishing up fourth grade) a question.

Me: Griff, imagine you are baking cookies and you need \frac{3}{4} cup of sugar, but you only have a \frac{1}{2} cup measure. How would you get \frac{3}{4} cup?

He thinks about this for a moment.

Griffin (9 years old): You put \frac{1}{2} cup of whatever you’re measuring.

Me: Sugar.

G: Does it matter?

Me: No. I suppose not.

The conversation could end here and I would be delighted. But it does not end here.

G: You put that into the bowl, then you fill the cup halfway and put that in.

Me: And that’s \frac{3}{4} cup?

G: Yes.

Me: How do you know?

G: Because \frac{3}{4} is a half, and then half of a half.

Me: Yeah. That is what you just described. How do you know that that’s right?

G: Like a square. If you shade in half of it, and then half of what’s left, that’s the same as shading \frac{3}{4} of it.

squares

So What Do We Learn?

One question division helps answer is how many of this are in that? My question of Griffin asked how many halves are in three-fourths? This is a division question.

Griffin may not know that it is a division question. That is fine. He is thinking about a specific example of how many of this are in that? This will lead to good things further down the line.

That he sees “sugar” as a non-essential detail of the story is lovely. This will serve him well.

Griffin’s mental image for this task is a common one. He can see three fourths of a square in his mind, and he can see that this is the same as one-and-a-half halves of a square.

Finally, we learn (because I am about to tell you) that this scenario could never really happen when baking in our home. I have an awesome set of measuring cups (pictured below): \frac{1}{4}, \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, 1 and 1 \frac{1}{2}. (A friend—and friend of the project—has pledged to donate her \frac{1}{5} cup measure to the Talking Math with Your Kids cause.)

Stack of measuring cups

Starting the Conversation

There are so many ways to raise the question how many of this are in that? Measure each other in inches, wonder how many feet tall that is. Count your quarters, wonder how many dollars that is. Repeat with nickels, or dimes. Bake a batch of cookies using only the \frac{1}{2} cup measure.

And you can read through previous division posts for more ideas.

Math in the alphabet

The children attended a well-run chess day camp this summer. Good people running things; a warm and welcoming atmosphere. Lots of varied activities to keep kids’ bodies engaged as well as their minds.

Sadly, this takes place on the complete opposite end of the Metro area from where we live. We had to drive all the way across St Paul, Minneapolis and deep into St Louis Park during rush hour. Ugh.

This led, one day, to my trying to find a topic of conversation to keep at least one of the children occupied while we drove home. I recount for you this conversation below.

Me: Tabitha. Can I ask you a question?

Tabitha (7 years old): Sure.

Me: What letter comes before I in the alphabet?

T: H. That was kind of an easy question.

I love that she has turned into a critic. If I am not challenging her, she calls me on it.

What she has not seemed to notice yet is that these questions she deems easy are just my openers for the good stuff.

Me: Yeah. Here’s a harder one. What letter comes two letters before S?

There is a fairly long pause here. This is a harder question because of how most of us know the alphabet—forwards. If we want to know what is 2 less than 71, it is not so hard to count backwards. We have lots of experience counting backwards. But we don’t have so much experience saying the alphabet backwards, so we need to make up a strategy.

T: Q and R.

Letter squares for q, r and s

Me: Q is two letters before S, yes. Now you ask me one.

T: What letter comes after Z?

Brilliant. What a great question. I wish I had thought of it myself.

Letter squares for w, x, y, z and three blank squares

Me: Oooooo. Good one. I say A. I say it starts over.

T: Nope.

Griffin has been listening in but not participating. He sees his chance to get in on the action.

Griffin (9 years old): Negative A.

Me: Wouldn’t that be what comes before A?

G: No. It comes after Z. It’s negative A.

T: Nope. Not that either.

Me: OK, then. I am stumped.

T: Nothing.

Me: Huh?

T: Nothing. No letter comes after Z.

So what do we learn?

This is a more sophisticated version of another mathy letters conversation I had with Tabitha a while back. Back then, we were trying to figure out which of two letters comes first in the alphabet. Here, we are more paying careful attention to precise placement (two letters before, not just before).

The other interesting thing going on is our three different ideas about what comes after the end.

My idea: After the end, we go back to the beginning, like the days of the week.

Tabitha’s idea: There is nothing after the end. It just ends.

Griffin’s idea: The end is like zero. When you get to the end, you repeat what you already had, only using negatives.

It is OK that we didn’t resolve who is right.

Starting the conversation

About a year ago, I started making a habit of having the kids ask me the next question. I highly recommend it.

You know how your children are always testing the limits of rules in everyday life? Like you say, “Do not touch” and they see how close they can get their finger to the forbidden object without actually touching it? That is normal and necessary behavior on the part of children.

They will do it in the world of ideas, too. Tabitha did not choose “What letter comes after Z” at random. She chose it because she knew it would be interesting to talk about. It probably would not have occurred to me to ask it. Our conversation was richer because she did.

 

Nights of camping

The following conversation took place in the run-up to our annual summer camping trip recently.

Rachel has no interest in camping, so this ritual is all mine. I started the little ones young with a one-night trip within an hour from home so that we could come home if it’s a total disaster. As they have aged and we have developed our routines, we have gone further afield, exploring wide-ranging Minnesota state parks for two-night stays. We added a weekend fall trip, too.

Last summer, the kids began to ask why “we only go for two nights”.

Ladies and gentlemen, when the kids ask that question, you know you’re doing it right.

So this summer we are expanding to three nights. Tabitha was thinking about that change the other day.

I am straightening some things on the front porch, sweeping and tidying. Not thinking about anything in particular.

Tabitha (7 years old): If we’re going for three nights, is that 2 days and 2 half-days?

Me: Yes.

A few seconds pass.

I realize that I have an opportunity here.

Me: How did you think about that?

T: Every night is a day, except the last one, when we go home.

Me: What if we went for a whole week’s worth of nights? What if we went camping for 7 nights?

T: Easy. Six days.

Me: And?

T: Two half-days.

Me: OK. Ready for a hard one?

T: Yeah!

Me: There are 365 days in a year. So what if we went camping for 365 nights?

T: [slowly] Three…hundred…sixty…four!

Me: Nice!

T: I can even do 400.

Me: You mean 400 nights of camping? You know how many days that would be?

T: Yeah.

Me: All right. Tell me.

She does.

Later, she is in the shower. I am not-so-closely supervising nearby. I get an idea.

Me: Tabitha, what if we wanted seven days of camping?

T: How many nights?

Me: Right.

T: Eight. Am I right?

Me: I can’t trick you at all, can I?

T: Ask me another!

Again, a sign that things are going well. Contrast with her claim a couple years back, “Sometimes I don’t want to tell you about numbers because it’s just going to turn into a big Daddy math talk!”

I have to think hard to dig up something that will be more challenging for her.

Me: You want a hard one? A really hard one?

T: Yes!

Me: Last year, we went camping twice. Altogether, we camped 4 nights. How many days did we have?

T: Three…five…

It turns out that Griffin is lingering in hallway outside the bathroom. He chimes in.

Griffin (9 years old): Four.

Me: Two days, and four half-days.

G: Right. That’s four.

Me: But she’s thinking about it as four half-days, since they aren’t attached to each other. I can see an argument either way.

This summer’s trip was to Lake of the Woods in the far northern reaches of Minnesota.

Griffin posing with an oversized walleye statue in Baudette, MN

So what do we learn?

It may surprise some readers that I have filed this conversation under Algebra.

Like many of the other algebra posts, we are not using x or y, or making graphs or solving for variables. Instead we are thinking about a relationship, and about what that relationship looks like for a wide variety of numbers.

The relationship we are working with here is a simple one: one less. Tabitha had noticed that the number of full days we camp is one less than the number of nights we camp. She had even generalized the idea—notice that she didn’t count the days individually. She said, “Every night is a day, except the last one.” This answer doesn’t depend on any particular number of days; it works for all numbers of days.

What I did in this conversation was help her to apply this idea. By asking her about a wide range of numbers of days, she got to feel the power of her generalization. That is algebra.

The other important part here was continuing the conversation while she showered. Thinking in reverse is an important mathematical skill. We had started with how many days do we get with a certain number of nights? I moved us to how many nights do we need for a certain number of days? The fancy math word for the relationship between these two questions is inverse.

Starting the conversation

Camping trips, vacations, trips to grandma’s house…these are all opportunities to have the conversation we had. If your child doesn’t ask about it, you can ask your child. We are going to grandma’s house for three nights—how many days will you have to play with your cousins while we’re there?

More generally, there are two Talking Math with Your Kids moves I want to emphasize.

  1. It took me a moment to notice that Tabitha had offered me an opening for conversation. I was thinking about something else at the time. When I noticed it, I put those other thoughts aside to talk, ask and listen. That part of the conversation took probably 2 minutes. We can all spare 2 minutes to get our kids’ minds working. We just need to notice the opportunities.
  2. I followed up later on. Following up is good for two reasons: It lets you and your child examine an idea more deeply, and it helps cement memory of the conversation. We remember something we revisit multiple times better than something we only think about once.

The equal sign

It has been a long, busy semester for me in my community college work. Many interesting and productive projects, lots of interesting and challenging teaching problems.

But I am tired. Wiped out and exhausted.

So I devised a plan the other evening when Tabitha needed to finish her first-grade math homework. I would lie on the daybed on the porch with my eyes closed while she worked at the adjacent table. I could answer any questions she might have without opening my eyes. (Seriously, parents—you may mock me, but can you honestly say you haven’t tried something similar?)

This plan worked beautifully for about five minutes.

She was working through some addition facts when it occurred to me that I had never asked her one of my favorite math questions. So I wrote the following in my notebook.

Eight plus four equals box plus five

Me: What goes in the box?

Tabitha (7 years old): (reading aloud in a mumble to herself) Eight plus four is…

Hey! This doesn’t make any sense!

Me: Why not?

T: 8 plus 4 is something, then plus 5?

Me: What does the equal sign mean?

T: Is. Like 2 plus 2 is 4.

Me: What about this? Would it make sense to write 2 plus 2 equals 3 plus 1?

T: No!

I let it go and we move on with our evening.

Later on, though, after putting on jammies but before toothbrushing, I follow up.

Me: Tabitha, I want to ask you a follow up math question.

T: OK.

Me: Does it make sense to say 2 plus 2 is the same as 3 plus 1?

T: Yes! Of course! Easy!

Me: Can I let you in on a little secret?

T: A secret secret? Or not really a secret?

Me: Not really a secret. But something you might not know.

T: [rolls eyes] OK.

Me: The equal sign means “is the same as”.

T: Of course! I know that!

Me: But that means it would be OK to say that 2 plus 2 equals 3 plus 1.

T: Oh.

So what do we learn?

This is kind of a big deal.

We train children to think that the equal sign means and now write the answer. Arithmetic worksheets reinforce this idea. Calculators do too. (What button do you press to perform a computation on a typical calculator? The equal sign!)

But doing algebra requires that we understand the equal sign to mean is the same as or has the same value as.

Tabitha is in first grade, though, so she has lots of time to learn the correct meaning, right?

Sadly, older students in U.S. schools do worse on the task I gave Tabitha than younger ones do.

The good news is this: If we are aware that children may develop the wrong idea about the equal sign, it is easy to help them to get it right.

You can follow Tabitha’s and my adventures in equality in the coming weeks.

Starting the conversation

If you have a school-aged child of any age, pose that task above. No judgment. No hints. Report your results below. It’ll be fun!

Postscript

Coincidentally, a fourth-grade teacher wrote up his class’s explorations in equality today. If you’re interested in what this can look like in school (easily adaptable for homeschool), head on over.

Legos

The dirty little secret about Legos is how very many pieces there are to be cleaned up after building. And how very ugly the clean-up battle can become.

I try to keep calm. I try to turn the clean-up battle into math talk from time to time.

Here is how it played out a while back, when Griffin was 7 years old (nearly 8).

I send Griffin upstairs to clean up a big Lego mess. A while later, he returns.

Me: Did you pick those Legos up?

Griffin (7, nearly 8 years old): Yes. Well…not all of them.

Me: What fraction did you pick up? More or less than half?

G: Three fourths. No. One fourth. No….I’d say one third.

One third is more than one fourth.

Me: How do you know that?

Griffin draws a picture like the one below.

legos.1

G: You see that it’s bigger.

Me: OK. Do want to hear how I know it’s bigger?

G: Yeah. How?

Me: If we had one big cookie and I had to share it with three people or share it with four people, I’d get more if I only shared with three.

G: Yeah.

Me: Which is bigger: three fourths, or two thirds?

He draws a picture like the one below (the original is lost to history).

legos.2

G: Three fourths is.

Me: How do you know?

G: [referring to his picture]: It would be here [indicating the heavy line pointing to the lower left] if it were two thirds, and there’s more shaded in, so three fourths is bigger.

So what do we learn?

It seems to be easier to engage children in comparisons than in precise computations. I have a lot more success with Which is more? questions than I do with Exactly how much/how many? questions. This is especially true with fractions, where precise computations are often quite tedious.

In any case, the simple follow up question, How do you know that? is a powerful one. That is where the conversation happens nearly every time.

At this point Griffin is accustomed to being asked this question, which means he is pretty good at doing so. It seems natural to talk about how he knows something. If you are just starting these conversations with your own child, expect to have to ask on 10 or 15 separate occasions before it starts to become natural for them. This is just like introducing a new food. Repeated exposure is key.

Starting the conversation

This is classic lemonade-from-lemons parenting. I knew Griffin would not have completed his assigned task. We could fight about it, or we could talk about it. I chose to talk about it, and save the power struggle for a bit later.

You can do the same. What fraction dressed are you? What fraction of your teeth did you brush? How much of your body was touched by soap? Et cetera. Have a little fun with the math first. Then make them finish the job.

Bedtime

In the thick of the holiday season last month, Griffin (9) turned his attention to the mythical midnight on New Year’s Eve.

Griffin (9 years old): Can we stay up until midnight on New Year’s Eve?

midnight

It is a nice touch that he includes his sister in this request. They may argue but they are a team.

Me: No.

G: We can when we’re 12.

At one point a year or two ago, he forced me to commit to the age at which he will be allowed to stay up to welcome the new year. The future me may be angry about the commitment I made, but for now it is paying dividends.

Me: Yes.

G: When I’m 12, I’ll get to stay up a lot later every night, since my bedtime is a half hour later every year. I’ll be able to stay up until….10.

Me: We’ll see about that.

Griffin restates his rule. I am pretty sure that I never endorsed this rule, but he is committed to it. I figure that I may as well exploit it.

Me: By your rule, what will your bedtime be when you’re 18?

G: Twelve to eighteen….so…three more hours…One o’clock!

Me: I should have asked about an odd number. How about when you’re 21?

G: Two-thirty in the morning.

Me: By your logic, what would have been your bedtime when you were 1?

G: Well, at 8 it was 8, so…4 hours…Four o’clock!

Me: I disagree. I don’t think it should be 4 hours from 8 to 1.

Griffin defends his answer.

G: It’s eight years, which is 4 hours.

Me: I don’t think 1 is 8 years younger than 8.

G: Oh. Yeah. Four thirty.

Then when I was 3, it would have been six o’clock.

He pauses thoughtfully.

G: Did I go to bed at six when I was three?

Me: I honestly don’t know. We make sure your bedtime matches your need for sleep.

So what do we learn?

This conversation exemplifies an important Talking Math with Your Kids principle—use whatever interests your children as an opportunity to talk about math. Another related principle is to use the conversation you don’t want to be having as a launching point for math talk.

I assure you that I am a relatively normal parent; I dread discussions of bedtimes. There is so much opportunity for whining, wheedling and comparisons to classmates. I understand that these classmates are mythical, but my children assure me that they all stay up later.

The key moment in this conversation was when I grabbed the bull (as it were) by the horns and asked him to apply his rule. What will your bedtime be when you are 18? That question got us talking about some good math, and it turned a potential power struggle into a fun conversation.

Using a rule to make a prediction is an important aspect of algebra. Griffin’s half hour later for every year older rule is a wonderful example of rates. I asked him to predict both forwards (what will be your bedtime when you are 18?) and backwards (what was your bedtime when you were 1?)

I missed two opportunities here.

It is a good idea to ask questions that make kids think in the opposite direction. So I should have asked something like How old will you be when your bedtime is midnight? This would force Griffin to think bedtime-to-age when he has been thinking age-to-bedtime.

The other missed opportunity is to play with the silliness of extending this rule too far. I could have asked, How old will you be when your bedtime is 8:00 p.m. again? If his bedtime keeps getting later, it’s going to come back around. Could he work out that this would require 48 years? Or would he reject the question as silly and put a limit on his rule? Either way, it’s a productive math talk to have.

Starting the conversation

Anytime your child wants to enter a negotiation, there is an opportunity to turn it into a math talk. How many M&Ms can I have for dessert? How many pumpkins can we buy? When can I stay up later? How many friends can I have at my sleepover? All of these and more are opportunities to ask what if questions involving rates, predictions, past and future quantities.

So don’t dread the negotiations. Take advantage of them!