## Wednesday, January 13, 2016

### Integers [temperatures]

Yesterday, I co-taught an integers activity with a colleague. It was a blast! Before I share the lesson, I'll back up and give the backstory on the context. During winter break, I ventured over to Brian Head, Utah to do some snowboarding. I knew it was going to be cold so I went into the trip with the intention to frequently check my phone's weather app and take screenshots of temperatures. I figured I might be able to make an activity out of it and/or use it with 6th graders at some point when discussing integers. (official lesson page with resources)

My fellow displayed this slide and asked, "What do you notice? What do you wonder?"
(the 3 in the lower right is the slide number)

Students noticed and wondered great things. Here are just a few:
• What's the temperature at 9am?
• Why is it warmer on the days it is supposed to snow?
• Thursday is the only day with a negative temperature.
• It's 4:13 am.
• It's zero degrees at 5am.
• It's cold!
• How cold does it get?
We established that -4 degrees Fahrenheit is cold, below zero, and the temperature at 4:13 am. Let's plot this on a vertical number line today, just like a thermometer. Does -4 degrees go above or below -5 on the vertical number line?

I told students that we're going to show them five more times and their temperatures throughout the day. Most importantly, I asked students to first predict the temperatures at those given times (tap into student intuition). Here are the times:
• 6:00 am
• 7:00 am
• 9:30 am
• 2:30 pm
• 8:00 pm
Essentially, we're tapping into student intuition, a free resource in our classrooms. I want them to predict the story of temperatures and degree change for the remainder of the day. If anyone has experienced winter weather, they know it gets cold at night and warmer during the day, possibly peaking midday. It's a small part of the activity to keep it moving along and gain student investment.

Here come the temperatures. For each time and temperature revealed, here's what were going to do:
• Plot the temperature on your vertical number line.
• Find the degree change between the last temperature given.
• At the end, we'll find the largest difference in temperature during the day.

Here's a few of our whiteboard representations:

This was a simple and fun context to work with integers and the vertical number line. I also took screenshots of the temperatures in Celsius and might be able to make a Math 8 activity out of it. Here's the desmos rough draft.

The best part for me (as a teacher) was listening to students make sense of the temperature changes and explaining their thinking. There were so many opportunities to help students with their vocabulary. For example, when asked, "what's the difference between 12 degrees and -8 degrees?" it was interesting to hear how students wanted to change -8 to a positive in order to add it to 12. There was our intro to absolute value and a number's distance from zero. Love it!

One student came up to me on his way out and showed me his paper,
"Hey Mr. Stadel, I predicted the temperature correctly for each time!"
High-five!
I asked, "Do you want to pick my Powerball lottery numbers for this week?"
He declined. Drat.

Again, official lesson page with resources here.

Brrrrrrr! it's cold,
225

1. Great work. I will forward this on to my 6th and 7th grade colleagues.

I was having a conversation with my 8th grade teaching colleague who was born in France and thought that European students / people have less trouble with negatives because they are in celcius and since zero is freezing they are used to seeing negative numbers more and also they use a vertical number line much more often.

And don't get me started about how if we, as Americans, used the metric system as our standard and how students would adapt and be more fluent with the base 10 number system.

1. I hear ya, brother.
Personally, I think we (in America and even in warmer states) can totally introduce children to negative temperature before 6th grade. If my 5-year old can tell me that -6 degrees is below zero, then I think it's most beneficial we tap into this intuition more in 6th grade math.

2. This is great Andrew and thanks for curating all these pieces into a lessons page. What a great opportunity for students to building automaticity with integers in context. Cheers!

3. Great Job Andrew!!!!!!

4. I've been talking a lot with Michael Pershan, Andrew Gael, and Aran Glancy about integers this year. We've been discussing the benefits and drawbacks of various models or contexts for integers, among other things.

Temperature is a good context in some ways because a lot of kids (at least in the North) have some experience with negative temperatures, but there are drawbacks as well.

For example, positive and negative temperatures are not distinct objects, and they don't "cancel" to zero. If you think of a problem like 5+(-3), it's hard to model that specific problem with temperature. It's much easier to model -3 +5 by saying the temperature was at -3 and then rose by 5 degrees. But if you have two distinct objects like anchors and floats, balloons and sandbags, or assets and debts, then -3+5 and 5+(-3) are equally easy to model. The anchors and floats cancel with each other, so you can evaluate the remaining objects. Also, 0 degrees Fahrenheit is an arbitrary point, which is kind of annoying. Celsius would be much better, as Martin mentions above.

I wrote a post about integer contexts a while back, but I've learned a TON since then. I might write it up again.

But to your more general point about student intuition, I totally agree. The more we can get kids to keep the rest of their brains on when they enter "Math World" the better.

1. I forgot one thing: Vertical Number Lines FTW!

I think that the vertical number line is the best model for integers. It's much more intuitive for students than the horizontal number line, where left is up and right is down. Or is it the other way around? If only there were some model where up was up and down was down!

2. Hi Kent,

Thanks for sharing your thoughts. I completely agree with temperature being a poor context to model integer relationships. Shame on me if I didn't make the learning objectives of this activity clear. The learning objectives were to place integers on a vertical number line and discuss the change in temperature between two temperatures.

I would not use temperature (as a false context) to model -3+5 with a student. If any connection with temperatures could be made, it might be the average temperature over a span of time, but even then I'm not sure I'd go there with students.

Yes, vertical number lines for the win regarding integers.
Thanks for the input.

5. Just something to point out: the screenshots shared in the folder are over 3 days. I had the kids plot points from 4 different screen shots, thinking that they were over one day. I wanted them to see the trend of the whole day. This didn't work. So next time, I'll just give them screenshots from the SAME day. If that makes sense.