Day 52: Arithmetic Sequences

In Algebra I use Dan’s Super Stairs to introduce Arithmetic Sequences. I used this same activity in Geometry which can be found on Day 15 and wrote about how I work Three-Act problems here.

The highlight of the day was when two students from two different periods recognized the rule n(n+1) in a way that I was not expecting…

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Most students used a form of arithmetic sequence to add 1+2+3+4+…+21 then doubled it. When I asked for the total students for 1,000 stairs this method came up short and directed them towards different solution options.

Day 46: In-N-Out

Today students started Day 1 of In-N-Out by Robert Kaplinsky.  My whole department does this activity to kick off functions in Algebra 1. I did this activity last year and have made a few modifications…

We started by talking about our favorite fast food places. Since Montana doesn’t have any In-N-Out it is nice for a student to mention the place so it doesn’t come across as forced. Last year we started with the 100 x 100 pictures. I mixed things up this year and showed students a video instead.

At this point we dive into 3-Act world.

Write down the first question that comes to your mind.

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The +6 shows that 6 other students found that question interesting.

In this period the main question How much would a 100 x 100 cost? didn’t come up. That’s ok. I helped them along.

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We took estimates. Sometimes I get crazy low estimates and that doesn’t bother me. However, this year I have been feeling that students are starting to take advantage of the amount of openness they have in math… So I took a page from Andrew’s comment here and asked students “Take a minute to look at the range of estimates we have, convince why any of them might not make sense” 

I didn’t just blow off the fact that Alex’s name was attached to one of the estimates, after someone called him out, I gave him a chance to respond.

After I showed students what a cheeseburger and double double look like.

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And asked them what they noticed about each.

They came up with pretty much everything; same number of buns on each, same condiments, different number of patties and cheese.

Students created a wish list of information they wanted to answer the question:

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I added a new piece here that I REALLY liked… by asking students: what would you do with this information?

I then gave them this picture and set them loose to answer the question.

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About half the class was done in two minutes.

There were two main misconceptions I saw….

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Instead of showing these pictures right off the bat I asked students “I am seeing a lot of different calculations. If we use a double double to find the price, shouldn’t it be the same as using a single cheese burger to find the price?”

They agreed. Then we worked through $2.65(100) and $1.75(100). They weren’t the same. We talked about what they were calculating.

******A big difference this year… This incorrect answer seemed to be discouraging. Students weren’t even sure if the 100 x 100 existed. So at this point in time I showed them the picture

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I loved the reactions and students had a good idea of how they needed to calculate the price.

In some classes we reached final calculations and looked at the receipt. I am saving that for the continuation of this post… coming tomorrow!

Day 33: Compound Candy Corn

Today in Geometry things were insane. We spent most of the day working through a couple proofs, things moved very slow. After class I didn’t feel too great about how things went. The proofs were sloppy (on my end) and it felt like students were taking a 360-nose-dive.

I know exactly why the day took a turn for the worse. I didn’t even buy what I was teaching. The way I worked through one proof followed the textbook’s logic. I would NEVER have worked through it that way. But, with the influence of the textbook (which isn’t a bad way, just different) I convinced myself to teach against my gut. I feel better about things now that I recognize this issue.

Anyways, Algebra was awesome today! Courtesy of Estimation180

I started by throwing this up on the board:

day33.2Students wrote down their two guesses and we recorded ’em.

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I asked students to word it as a range of numbers BETWEEN two values; which I would change next year. You can always add more structure later on. I showed them the answer then we moved on to this one:

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After their guesses were recorded, we wrote a. and b. as inequalities and recorded guesses.

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This was exactly how students worded things. Pretty cool to see the concept emerging without any promoting on my end.

At this point I used Jared’s as an example and introduced a little more structure: 320 < x < 450

We looked at the answer then watched the video on how many scoops it would take to fill the jar. I asked students to write this as “x placed between two numbers”

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In every single class there was an inequality that was either < 15 or 15 <, it was fun to see students call each other out for being wrong by not including the 15 and fixing their estimates.

After this point I introduced some more structure and vocabulary, we first talked about what a graph of Aubrey’s estimate would look like then graphed it.

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After, students practiced graphing and solving a few and we talked about the difference behind what we mean when we use AND or OR in a normal conversation;

Eric invited me over for dinner tonight, he said that would would be having dinner and dessert, what should I expect to have?”

He also let me in on what we are having; mac and cheese or lasagna, what should I expect to eat”

Kinda cheesy….

Oh gosh….

But it seemed to help in understanding the difference between AND and OR.

Day 28: half-your-age-plus-seven

I have been at a stand still in finding activities/resources for introducing inequalities that both perplex students and transition naturally into the concept. Last year I started by putting x>3 up on the board and scaffoled my way up to multi-step inequalities. I wasn’t happy with the results; students had a very small base of conceptual understanding, which did not transition smoothly into linear inequalities.

I took a risk today and developed my own introduction. I really wanted to focus on letting students ask the question and approach the concept from a different direction.

I threw this up on the board;

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Then asked students if they had ever heard of the half-your-age-plus-seven rule. Not a single student had. They calculated half Brad Pitt’s age plus seven; 32. I asked them if this had any meaning? A lot of students didn’t believe he was 50, so we Googled it.

After I told them he is married to Angelina Jolee;

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“Brad Pitt is all good according to our rule” I told them.

At this point things started falling into place, we talked about what the rule might mean and decided that it would be creepy if Brad Pitt dated anyone younger then 32.

Up next…

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Professor Xavier they shouted out. Well our rule says 43.5 is the cut off. We talked about possible ways to work around the decimal; round up, round down, keep it. I let the students decide.

We went through who he is married to and finished up with the one:

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I had students write a sentence about what our rule meant, it took a while for it to evolve but they eventually came up with:

It is creepy for Mila Kunis to date anyone under the age of 22.5.

We have been working to translating between words and math so I asked students to translate this into an expression.

If z is Mila,

then z > 22.5

After we talked a little more about if we should include 22.5 in the range of ages then moved into a few notes about inequalities and how to graph them.

Each period I taught the lesson it improved and I tweaked things around. I know there is still plenty to change and the tons of potential to further develop the lesson. What is just as important for me though is I was excited about the lesson. I was not excited about lecturing on how to interpret x > 3. Students feed off that excitement and it becomes a better experience for everyone involved.

Day 27: 9/10 of a cent

Geometry took a concept test today and worked through a few logic puzzles. After Mrs. Murphy’s Laundry they were loving these.

In Algebra I am just starting the wrap up the first chapter. For some reason today’s lesson is another of my favorites. It doesn’t require much but students always go crazy over it.

This year I approached the lesson in a 3-Act sorta way.

I started by throwing this up on the projector:

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Students have a short conversation over anything they notice, eventually the 9 comes up. Some students though it was a G for gallons, others had never actually paid attention to it.

“How much did gas cost today?” 

Every. Single. Student. replied with $3.43.

Those students who picked up on the 9/10 of a cent thing quickly self corrected and said $3.44. However, most students didn’t think that 9/10 of a cent even mattered; it added up to less than 20 cents per trip. After a while, we started talking about currency and how there are no coins/bills for tenths of cents.

Students were perplexed at this point; why would companies even do this?

They started talking about how it 9/10 of a cent does actually matter, but only really when you are talking about thousands of gallons of gas.

I chime in with “how much extra each year do you think the gas companies make by charging 9/10 of a cent?” – Gotta strike when you can.

We wrote down guesses, attached names to them, and put them up on the board. After, students thought about what info they would need to solve this problem: How many gallons are sold per year is really the only thing.

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They converted a few numbers and did the calculations; it turned out to be in the billions.

Here is my favorite part;

How big is a billion? 1,000,000,000

How many millions are in a billion? Hundred millions? Thousands?

Well I decided to look a little deeper into what a billion actually is. In this word document I have 1 billion represented as groups of 10,000″

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Students initial reaction was “No big deal”.

Then I hold the down arrow for about 30 seconds.

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After a while students look at the bottom left…

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Day 26: Emperor’s Banquet

It is always a good week when I have back to back lessons from Fawn’s site. Geometry presented and wrapped up Mrs. Murphy’s Laundry with group presentations. As the period went on, the presentations continued to develop, which was great to see.

In Algebra, students worked through The Emperor’s Banquet. I put the problem up on the projector and had students read it then work by themselves for five minutes to make sense of the problem. After, they discussed their progress with their neighbors.

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I saw a lot of students taking this direction. A few went up and shared how they decided where to sit if there were 5 guests at the banquet.

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In my eyes this image shows a great example of first step in developing algebraic thinking on the student’s end. After this point students slowly added mathematical framework to the problem. It was extremely difficult for me to step back and just let things happened. But after a while numbers started appearing.

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What happened in green was awesome; a huge shift in structure, which makes communication much easier on the student’s end.

IMG_0362More patience on my end paid off and tables started emerging; another big step, this time into abstraction.

Conjectures started coming out and we created a table on the board. After about 10 minutes the table reached 17 guests and most of the conjectures had counterexamples. So we started searching for patterns different ways (in green).

day26At this point I felt very stretched in my ability to provide further scaffolding. In a few periods, students noticed linear patterns between 3, 6, and 12 guests. Which was extended to answer almost every case (excluding primes) ie. where should you sit if there are 18 guests? For 9 guests the last seat is 2, so double both of those, 18 guests, the last seat is 4. One co-worker suggested going in the direction of piecewise, and helping students see that certain rules work for certain number of guests; interesting!

I went into today’s lesson with the hope that students would look deeper into patterns, but what came out of today was much richer on both my end and the student’s.

**This problem is also great for Algebra 2 and even pre-calc… I believe it can be modeled by a logarithmic composed with a floor function **

Day 22: Rates + follow up

Concept tests in both classes today. Student’s scores are always going up which is great to see. In Geometry we finished up some practice on conditional statements and inductive reasoning. I have not been following the textbook’s sequencing of these things and feel like it is going well. With logic, students need to practice bits and pieces of all kinds of reasoning at the same time rather than simply going through them as different concepts.

In Algebra I started with a number talk:

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I swear I didn’t encourage any of this! Not even absolute value…

Last year I gave the same question but on May 18th or something and got this:

18 +0

17 + 1

16 + 2….

When we do number talks/visual patterns I tell students that I am only interested in how they are thinking about things, not the end result. It was really funky to begin with and students were uncomfortable with a focus other than the answer.

What I saw in the number talk today validated everything I have been doing this year.

After, students ranked the following from most expensive to least expensive.

day21.2A few students were hesitant and wanted to know how much of each, I kinda sorta ignored them. Then showed this:

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This slide got them fired up; a bottle of water for $17.99?? What? Then they started asking what each of these were measured in.

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Then they went to the power of Google to convert each into mL and re-rank the items. (Shout out to Dan for this idea)

A lot of times the blog post ends here and there isn’t much talk about the follow up. Well here is what I did.

In their notes, students wrote down the definition and examples of Ratio, Rate, and Unit Rate. We talked a little about how each of these tied into the activity. The vocab helps students communicate what they are finding; rather than saying I divided ______ by _______ they can now say I found the rate of $/mL and so on. The vocabulary is actually useful rather than just meaningless writing.

After the test I put up a few practice problems and on Monday we will continue to practice these concepts with the idea of proportions coming to surface.