Average value

I worked hard on today’s calculus lesson and I’m so pleased with how it went!

This weekend I gathered (what I found to be) interesting statistics from Statista.com and then worked with regressions to make equations that matched the stats. This was no easy task, but I wanted the numbers we found in class to be accurate, or at least close to reality.

I made nine different scenarios and put the kids in groups of four: two kids to work with the physical model and two kids to work on the calculations.

The idea of average value is (1) find the integral (area under the curve for positive functions). The kids did this with peas:

Then (2) create a rectangle with the same area (so use the same peas) and same base. Your new height is the average value of that function:

The kids made approximations in that first step and then the calculus/calculations came in the next two steps.

I had all the groups put their findings on the board so we could discuss.

One of my favorite moments was talking about that last stat. “Who has the reported violent crimes stat? Hold up your graph.”

“What do you all notice!”

“It’s decreasing!”

One class started applauding.

I almost started crying.

“That’s right. Don’t you believe for one second that this world is getting worse. We have mathematical proof that, on the whole, we are progressing and moving forward.”

The world is being restored. And if I didn’t believe that, why on earth would I invest this much in the future?

I loved this lesson because it was your classic low entry, high ceiling. We got to see some scenarios that are unsolvable without calculus. And my kinesthetic learners got to control the lesson today.

I love when I see growth in myself.

It’s a marathon–not a sprint–isn’t it, friends?

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Lifting

One of my precalc kids told me today that I remind her of Miss Honey from Matilda.

“Stop! She’s my hero!” It elicited a more emotional response than I think she bargained for, but she went on nonetheless…

“When I walk into this room, I feel immediately calm, uplifted, and safe.”

You. All.

As is normal when people pay me a compliment, I just kinda of sat there with my mouth agape, looking like a fool and not knowing quite how to respond to such kindness.

What can I say: I’ve always needed time to process my emotions. I’m certain my son is the same way. His poor father.

But the more I thought about it, the more I realized her words made me feel calm, uplifted, and safe.

My quote of the week last week was “We rise by lifting others.”

How these Loves of mine lift me. How they come underneath me and say, “You are enough and you’re not too much.”

I told them at the beginning of the year that it wasn’t an accident that they were in my class.

That’s true.

It’s no accident I’m with them.

But it’s also no accident they’re with me.

We belong to one another.

Estimation Skills

I blew my kids’ minds today. I am working on getting them to think about numbers in relation to each other in order to build estimation skills. I want them to notice when their computation is way off. Right now they aren’t noticing it at all.

So we sorted some numbers onto number lines. The same numbers onto two different lines. One line went from 260-280. The other from 200-300. We had a lot of good discussion as kids moved numbers into place. Lots of talking about what numbers it might be close too and which would it be closer to – conversations we have when they are right and when they aren’t.

number lines

At the end I pointed out that the numbers on the bottom, between 261 and 279, were the same numbers as on the top line (which they knew because they had each dragged one number up to each line but they clearly hadn’t really thought about it). They sat there staring at the number lines for a bit, trying to process the idea that the small section of the bottom one was the same as the entire top one.

Hopefully it will help them notice more about relative distances on number lines in the future!

(I was super impressed with the first kid who place a number. She chose 275 and was able to place it halfway between 270 and 280 without any trouble. I was worried about the bottom number line though because she doesn’t always feel comfortable and confident in math. She knew exactly where to put it and basically described how she knew as knowing it had to go halfway through that side of the number line. We talked a bit to be sure everyone understood and connected it to quarters – knowing 250 would be in the middle so 275 would be in the middle of the right half of the line.)

Windows

Last semester I had 37 kids in fifth hour. I begged them–any of them–to switch into my second hour at semester if their schedules allowed and if it wasn’t too much of a headache on their counselors’ parts.

My fifth hour is now down to 35.

It’s something.

One of my former fifth hour students took me seriously and moved heaven and earth to be in my second hour. “I feel like I would learn so much more in a smaller class.”

Can you tell our legislators that?

“Absolutely! Class size matters. If you can switch, do it. It’ll be better.”

She had to do some finagling but her counselor was able to switch her into my second hour and then she is in service learning (for me) fifth hour, so she gets double the math.

Today I asked this young lady how the switch has been so far. “Was it worth it?”

She started laughing. “I actually finish all my homework. And now…people are coming to me for help. And I’m like, ‘Yeah! I can help!'”

Oh be still my heart.

This is why I feel like a failure every day in fifth hour. Because how can you possibly make sure every kid is ok? How can you teach a valuable, rich calculus lesson in as little time as possible so that you can get away from the front of the room and back to the troops to see how things are going (both mathematically and on a personal level). When I ask my fifth hour kids what I can help with, I know some of them lie to me and say “nothing” because they don’t want to take time away from others.

It kills me.

They deserve better.

But…

My second hour is much smaller and those kids don’t hesitate to ask questions because they know there’s enough time–enough of me–to go around for everyone.

That’s the kind of education every kid deserves.

Every kid.

Not just the ones who can afford schools that can limit class sizes to fifteen.

I’m thankful one of my kids was able to move to second hour so she can get the attention she deserves.

And I’m thankful she now knows the feeling of being the friend who can help. Because that feeling is pretty priceless.

*****

For extra credit on each precalc test, I give my kids a mathematician to research with four tasks:

  1. Find a quote by the mathematician.
  2. Restate the quote in your own words.
  3. State why you liked/disliked this quote.
  4. Find three pieces of interesting biographical information.
  • Today their extra credit on Andrew Wiles was due. This was one of the submissions, and I thought it was too good not to share:
  • Written by a sophomore in high school.

    She gets it. ❤️

    *****

    One of my students from last year came in today during my plan, just to decompress.

    “I just like being in this room.”

    “It’s the windows,” I assured him.

    But all joking aside, I love it when my kids come back, when I know they feel safe and welcome…when they know Room 2704 will always hold them–all of them. That they are always enough in that room and never too much.

    This room has held and carried so much.

    And I am so thankful for its four walls.

    And also its two windows.

    Hand lettering and dance

    I have a sophomore in precalc who I am convinced is my son in thirteen years. I tell him that often, too. I don’t know that he appreciates being compared to an almost-three-year-old, but I can’t help myself. I find them both so mischievous and so lovable. So smart and yet…there are more important things for them to do than what’s on my agenda for the day.

    I digress.

    On Canvas, our learning management system, you can upload an icon for each class that then shows up on your and the kids’ dashboards. I couldn’t find an image I liked after a quick Google image search so I made my own:

    Today, this said student must have noticed it for the first time (in his defense, it’s new since January, not new since August) and he brought it to me in amazement.

    “WHERE did you get this? Did you make it? How?”

    I hate when I catch myself stereotyping but I definitely did in that moment. And obviously his lovableness grew even more.

    The cuteness did not stop there, however.

    Towards the end of class, I was explaining the difference between Calc AB and Calc BC and then also going into what options they have for math classes their senior year after finishing one of the AP Calcs their junior year.

    I looked over at my easel and–wouldn’t you know it–a certain someone had found something better to do: practice his own hand lettering:

    The one on the right is his imitation of mine. Not bad at all, am I right?

    After the bell, I was met with a “Can you explain that whole Calc 2/3 thing again? I wasn’t listening.” Naturally.

    But how can you be annoyed when his hand lettering is so good?

    You can’t.

    It’s my first year teaching classes with only sophomores and I have to say–their goofiness, their authenticity, and their kindness have captured my heart. I love being with them and watching them navigate high school. They’re amazing.

    *****

    After school got out, two of my calc girls ran breathlessly into my room.

    “We have a very important question for you, Mrs. Peterson!”

    “Oh?”

    They explained that they will be in a dance routine at an international pageant tomorrow night and wanted to know if I could make it.

    I know we have something special when they’re ready to share their passions with me.

    Moreover, they know me well because they know I love dance and other cultures.

    These are my people. They belong to me and I to them and that right there is heaven on earth. I know it is.

    Math Test

    I rarely give my kids math tests as I have many other ways I am assessing them on a daily basis. But there are state, district, school, and team-created tests that I do give. Today was one of those days. My team had worked hard to create an assessment for our last unit (on multiplication and division) and my kiddos took it today. Normally this wouldn’t be one good thing for me.

    But today fascinated me. It is a challenge for my students to work completely independently and without talking to one another because we so rarely do so. Today they jumped in to the work really well. The most beautiful part was seeing them get up to grab scratch paper or math tools (unifix cubes, ones/tens/hundreds blocks, calculators, etc.) as needed. These kiddos know themselves as mathematicians. They asked me to clarify questions as needed. They advocated for themselves in order to do their best. They are fabulous.

    My people

    I was reminded today how much I love this community, my people…

    Overheard in first hour: “I LOVE first hour! I LIVE for first hour!”

    Mind you, this quote came from someone who loves math *maybe* as much as I do so it has more to do with the fact that he got to do math in first hour than anything else. But still. How can you not love it?

    *****

    We introduced the Fundamental Theorem of Calculus yesterday. I feel I would lose most (maybe all) of my students if I went over a proof, so instead I try to appeal to their intuition.

    Today one of my students said that wasn’t good enough for him.

    First person in six years of teaching calculus who has said that.

    I adore him for it and sent a proof to him individually.

    *****

    We played Graphles to Graphles today in Precalc. Afterwards a student said, “This really helped me understand domain and range so much better.”

    *****

    I’m on the committee to help decide our district teacher of the year (don’t be impressed: the chair of the committee is the mom of a current student). As we were going around introducing ourselves, I recognized one of the board member’s names.

    “Are you R’s dad?!” (a student I had in precalc and calc who is now a senior in college)

    “I am! You taught my son so well. He made a five on the AP Calculus Exam because of you,” he announced to the room.

    (1) What dad remembers his son’s AP scores from four years ago? I’ll tell you: a great one.

    (2) What dad remembers his son’s teachers from four years ago? Again, a great one.

    (3) I usually don’t take credit for my kids’ accomplishments, but when it comes to the AP Exam, yes, I absolutely will. And you’re welcome. It’s a team effort.

    *****

    Today a student from last year was on my heart. It felt a little random but I just thought I’d pray for him as I wondered what he was up to his senior year.

    Less than two hours later, he stopped by to fill me in on his year.

    Sometimes I wonder where God is and does he care about…any of it.

    And then sometimes I don’t have to wonder at all.