Properties of Addition- WODB

Which of these doesn’t belong?

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Justify your reasoning.

Find a reason each box doesn’t belong.

What mathematical concept is this WODB having you think about?

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#MTBoS30- Day 9

Brute force…

 

This has really gotten me thinking lately.  I gave my students the following picture:

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Yesterday my students and I did a problem adding and subtracting a string of numbers.  We looked at patterns that could be found in it, and even looked at why PEMDAS is misleading for order of operations.  They were finding patterns, multiple strategies and stretching their thinking.  I had hoped that this would translate to this problem- I was wrong.

#1 answer (98% of my students): I counted all of the squares.

I had tick marks, numbers, dots, colors, everything on their sheets showing how they made a 1-1 relationship with each square to find the total.  I was very disheartened, but I didn’t show it.  I asked if anyone had an alternate method, and only received a small handful (of which I will also share).  So I asked those who counted all of the squares: “How many of you still find yourself using your fingers when computing?”  All of them raised their hand.  This is a strong indication of where my students are developmentally with math, and how I need to provide them with opportunities to explore and stretching their thoughts about problems.

So, I then reminded them about yesterday and asked what we looked at with that problem.  They talked about the patterns and multiple representations- they remembered the whole segment!  So I asked, “How can that apply here?”

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4+3+4+3+4+3+4 or 4×4 + 3×3

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1+3+5+7+5+3+1 or 2×1 + 2×3 + 2×5 +7

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15+4+4+1+1 or 15+ 2×1 + 2×4

And then the bomb dropped

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5X5 square

 

A student came up with this and blew the minds of every one of his peers.  I let it sit there for 3 minutes before I even uttered a word.  I could see each of them processing what just happened.  Finally one student’s face lit up and he said “Oh, I totally get it!”

It took a while for them to realize that yesterday was not a “one and done” day for math, that I will expect them to do this every day they are with me.  Tomorrow we will have another pattern and see how they do with that one.

Go out and drop the bomb on your class.

#MTBoS30- Day 8

Putting it all together- that’s what my mind is trying to do.  There are a TON (really, there are) of great resources out there on the web via the #MTBoS.  There are times where it is hard to decide what to use, when to use it and where.  I am lucky enough to be slightly flexible in my curriculum (and also a curse because of the nature of the placement).

So, because I teach in MN and there are no “perfectly pre-packaged curriculum” (really who even wants that truly?), my mind is trying to wrap itself around what would be a good meld of components in the classroom.  Like I mentioned in my previous post, Science Practices in Math, I really believe that restructuring the lesson layout will help not only my students, but others as well.

Currently I have been using parts of David Wees a2i, an online curriculum for Algebra 1 and 2 as well as Geometry.  It has opening activities and I like the exploration of topics.  I also need to work with resources I find great (and my students do too) such as Estimation 180, Open Middle, WOBD, Would you Rather, Visual Patterns.

This summer will be interesting for me…

Insurance Rates- Expected Value

Consider the following insurance quotes:

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If you plan on only having once accident in the next 5 years, with a damage total of $2,000, Which policy is better for you?

 

Is there an amount of damages where neither policy was the better deal? (That after 5 years, you would end up paying the same amount of money)

Create a statement that demonstrates which policy is the best deal over 5 years.

Science Practices in Math

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My thinking this year has been stretched a lot, and I’m still trying to decide what was good thinking and what wasn’t.  One thing that is really sticking in my head is the thought of learning methods I experienced in Science- and I’m wondering why I don’t implement them in Math.  I’m not sure how your science classes went but here is the breakdown of mine:

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This is very similar to Dan Meyer’s 3 Act Math, and perhaps one of the reasons I connect so well to that model.  Show me something cool like this that will set my brain on fire:

(and of course I wouldn’t show the whole video to begin with)

Now you have me, I’m totally hooked on what is going on in class.

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Once again, just like the 3 Act model, ask me about what is going on.

  1. What do you notice?
  2. What do you wonder?
  3. Why do you think that is?

Kids are endless fonts of ideas and knowledge, let their thinking dominate the classroom, not yours.  I typically record every student’s response (yes, every one that is appropriate for school- excluding those that imply language, race, sex, etc.).  Like I have blogged about before, students will hit upon what your learning concept of the day is, why give it to them when they can supply it themselves?

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After we had this great classroom discussion about what we saw, we learned about about it.  This is where formal instruction fits within the class.  This is where you implement your practice, whatever form it may take.  The secret here, everything was done in class.  This is NOT the time to assign homework, and if you run out of time that’s awesome!  It means you are sending students out the door with that itch in their brains about what just happened in class today.  Their brains will be kicking those ideas around all night, whether they know it or not.  Typically I see an increase in engagement and clarity of focus from students on the second day.


This is where normal instruction ends.  This is where the great story of Mathematics dies.  We practice, we know it, we show mastery of some degree.  We wait until the end of the week, month, semester or year to show that mastery on some formal assessment.  This is where we need to be like my science class and take it all one step farther…


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Take what you just learned about and test it out or apply it.  This could be a different spin on the activity that introduced the topic, or another application of the mathematics all together.

Take Fawn’s Barbie Bungee.  It seems to do this right?  It does, to some degree.  In Act 3, we learn if our calculations are correct, the same with Barbie Bungee.  The thing is, we are still building upon the knowledge we created in that lesson, even though we are justifying our thinking, we aren’t building new knowledge.  What do I mean?  I typically ended Barbie Bungee with this:

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This is my bungee jumper now, and there’s a lot more at stake with this one than our Barbies.  This one will create a huge mess in the room if I mess up.  I have never seen students so excited and purposeful about their calculations and mathematics than when I introduce the egg.

This whole process gives my students a whole new appreciation for this:

 

My newest mission: Don’t stop at practice, push students to reflect, rethink, reapply their mathematical knowledge for different experiences and scenarios.

Pi IS a Rational Number

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So this thing, this seemingly unrelated string of digits who has been given a very specific name Pi, why is it such an iconic topic for math?  We even go so far as to name a day after it, the fourteenth of March- Pi day.  I still get quite a few odd looks from my colleagues when I say that, to many Pi is something best forgotten or eaten.  This year is the first year I have struggled with the idea of celebrating Pi day with doing activities, and here’s why.

Pi is no longer Irrational.

What?  You are mad Bryan!  Why would you ever make such a false statement!  I know Pi is irrational, but in our current state of mathematics education Pi is most certainly NOT irrational to our students.  Test this in your own classroom, ask your students to write down Pi, and let me know what your results are.

Here’s mine:

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Pi is 3.14

To make things “easier” for students we have created a powerful misconception, that Pi terminates after two decimal places.  This is reinforced almost everywhere for students, how often have you seen (use 3.14 for ∏) in textbooks, on tests, or even given your students those instructions?  We are all guilty of it. Even though we throw out the catch phrase of “but remember this is an approximation”, this is heard and acknowledged just as much as “take out the trash before you go out with your friends”.  Some of my older students remember something about this (or perhaps it is because we talked about it 2 days ago…) and here is how they tried to explain that:

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They still think of Pi as 3.14, but add their own catch-all.  I will have to talk with them tomorrow about this thinking, how can you write 3.14 but then say it’s never ending?  I can see some of my students trying to restate however many decimal places of Pi they remember, but not one will use the word irrational.

I think about this as I also reflect on the Pi Day activity I had my students do on Monday (and now I am beginning to regret doing it).  It was an activity similar to Apple Pi from Illuminations, located here.  I had various circular objects in the classroom: soda cans, DVDs, rolls of tape, coasters, plates and cups.  Students measured the circumference and diameter of these and explored the ratio of the circumference to the diameter.

I had thought about posting pictures of the activity, other than the initial hurdle of students completing long division, students caught on quickly and completed their ratios without problems.  We looked at the class average and what that number looked like to them: Pi.  The problem- every student’s calculation was a rational number.  Some students only had to work through the hundredths place while others went beyond the hundred-trillionths.  But they all terminated or repeated.

I asked about Pi then, and I received answers similar to the first picture of this post.  No talk of terminating or repeating, just good ‘ol 3.14.  When I asked about number classifications, and we tried to classify their numbers I finally got some of my older students to talk about rational numbers.  They then had this itch back in their brains about irrational and Pi- but it was after a lengthy discussion before they realized that Pi might be something beyond 3.14.  When we got to irrational, they were lost.  Students still have problems envisioning this, how can a number go on forever without any sort of repetition or end?  All of their numbers did, and fairly quickly considering how “long” an irrational number is.  I asked them for ideas why they didn’t get Pi, and it was only my youngest student who cracked the problem- measurement.  Students are not accurate when doing measurements- in fact there isn’t anyone that I know of that can take a tape measure, make the measurements and not get a rational number.  We have to round somewhere, whether we realize it or not.  That rounding causes our ratio to become a rational number.  For one class I even atempted to be smart and use technology to create numbers for the circumference and diameter, only to find out that just like calculators, graphing technology is restricted by displaying place value and rounding- producing yet another rational number (but better approximation of Pi).

Is it good to take this journey, am I enforcing a misconception or am I creating an atmosphere where students can concretely learn, discuss outcomes, and translate into abstract thinking of a concept rationalized in the brain only?  Typically I thought it was, until here I am 2 days later and I get these responses for what Pi is.  In order to combat this deeply embedded concept I need to revisit this throughout the rest of the year, and hope that it filters down into their long-term memory.

When I get responses like the following:

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It further reminds me of how perhaps Pi day is a disservice to students, one that was shared by @delta_dc a few years ago here.  I had many students who were disappointed that they would not eat Pie that day, and disheartened that their expectation of a day that we created to help promote an important mathematical concept had degenerated into a homonym reminding them of a delicious dessert.

This year it seems, Pi day won…

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Always, Sometimes, Never Shapes

I know ASN has been around for a while, but I am still amazed at how such a small twist on asking students questions to determine knowledge will bring out such great thinking and understanding.  I remember going through class and having these two questions appear on a test:

True/False:

___ A rhombus is a square.

___ A square is a rhombus.

These types of questions were landmark at the time I’m sure, but when I think about how much knowledge I displayed by answering them- it’s very disappointing.  It’s a T/F question, which means even if I don’t have any clue what it is asking I have a 50% chance of getting it correct! (And who didn’t like having T/F Q’s on their tests?)  I write T or F, and there is no justification of how I know either statement is true.  Even if I was asked if it’s true my response would involve: “A square is always a rhombus”- a simple restatement of the original question.  My math teachers would either acknowledge my response or tell me I was wrong and why.  After being wrong enough, I finally figured out what I needed to say to get the answer right- not that I understood the distinction.  Now I ask students a much different type of question:

Always/Sometimes/Never (explain your response- how do you know?):

______ A rhombus could also be called a square.

The biggest shift in this question is that it is no longer presented as an absolute fact- something that students had to memorize.  Students have three choices, which means they need to consider all the possibilities of answers and the statement as well.  What does a rhombus look like, what it is properties, how many different possibilities of rhombuses are there?  They then go through the same process for squares.

Here are some responses by my students:

  1. Sometimes: because it is kind of shaped like a square.  People could refer to it as a square but other people may not.
  2. Sometimes because a square is classified as a shape with four equal sides.  If a rhombus has equal sides it is a square
  3. Sometimes because it’s a slanted diamond square
  4. Always because a rhombus is the same as a slanted square.

Now, I threw this out there to my students because I am going to start a geometry section and I wanted to get an idea of their understanding of shapes and properties.  Many students said that they didn’t remember anything about shapes and properties, and their justification of thinking supported that claim.  This question did give me a great idea on student’s background knowledge on it however, and I have a good idea of how I want my unit to begin.

These are other questions I plan on asking….

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I can’t wait to hear student thinking and the discussions that will occur from these.  Many of my students are afraid of Geometry- I have one who admitted he skipped class every day.  He hated proofs.  We talked about that as a class- and how more traditional teaching could lend a hand to their fears.  We also talked about how ASN went, and how we explained our reasoning and justifications.  I then told them that they just did a mini-proof.  Being comfortable with explaining why you do things in mathematics supports proof reasoning.  I also told them that I would show them how easy it is to transition from the group discussion we just had into proofs.  They relaxed and were good with that, I can’t wait to “see” their thinking over the next unit.

Let me know if you have any other favorite ASN geometry Q’s- my students enjoy hearing from you.