Tag Archives: math

Math Warm-Ups January 14-18, 2013

This week’s warm-ups are geometry related, as we are at the beginning of a 2D geometry unit now.  My hope was that they were review, since they’ve had similar units for many years.  And for the most part, they were.  Oh, and you’ll notice there are only 3 this week; the morning schedules were a little rushed on Tuesday and Friday, so we skipped them those days.

Monday

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And speaking of having done it before, my kindergartener is actually talking about these very things right now, too!  When I shared this question with him, he was able to tell me which were polygons and which were not!  The definition was simpler in kindergarten, but the idea was the same.  Cool!

Wednesday

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This one was pretty easy, but was a conversation starter for that day.  It helped us take the next step to putting quadrilaterals into categories.

Thursday

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When we first talked about this one, we were unclear about the definition for a parallelogram.  We spent our group time on Thursday clarifying this.

What do you know about geometry?  What questions would you ask for a geometry math warm-up?  Share some with us in the comments!

 

ActivActivity–January 16-17, 2013

Remember when we were in the middle of our fraction unit and we were dividing?  Think about the problem about bows: Avery had 6 yards of ribbon.  He’s making bows for packages, and each bow uses 1/3 of a yard of ribbon.  How many bows can he make?  Then think about how we all said 1/18 for the answer?  I’ll remind you of how we solved it:

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Today your job is to create a fraction problem with division.  I’m not going to tell you what the answer has to be–I want you to figure out that part, too!  You can divide a whole number by a fraction (like this one), or you can divide a fraction by a whole number if you remember how to do that.  Remember to make sure your story makes sense, that you have an answer that is LOGICAL, and that you sign your names so we know who’s amazing thinking it is! Good luck, friends!

ActivActivity–January 14-15, 2013

Today’s ActivActivity is related to the Math Warm-up we had the other day:

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Your job today is to come up with a story that uses FRACTIONS and has an answer of 3/4.  Work together with  your group to write one, and if you have time after that one, write another!  Leave your story in the comments for this post, and be sure to leave your names!

Feast Week Part 2: How I Learned Fractional Parts Without Thinking About Pizza

In case you missed Feast Week Part 1, check it out here.

Feast Week was born, and we had decided what (and how) we were going to teach that big, deep list of concepts about fractions.  We utilized the UbD template for planning the unit, focusing on what we wanted the outcomes to be and then how we’d get them there.

And then we told our kids about it.  And they were BEYOND excited!  We were giddy about the plan, and my students were as eager as me to start our fraction work so we could head down the road toward the beginning of the actual Feast Week.  And just as we had hoped, this was just the motivation that 81 5th graders needed to get through a really hard unit on fractions.

But first we had to learn about fractions.  The unit was broken down into eight big ideas:

1. What are fractions anyway?

2. How are fractions related and equivalent to percents and how can they be used to solve problems?

3. How do you find fractional parts of a group (i.e. what is 2/5 of 30 students)?

4. How do you add and subtract fractions?

5. How can you multiply a whole number by a fraction? What does it mean and why would I need to do it in real life? (As a side note: this one was cool, because it is the same as finding the fractional part of a group–they just didn’t know that back at the beginning of the unit)

6. How can you multiply a fraction by a fraction?  What does it mean and why would I need to do it in real life?

7. How can you divide a whole number by a fraction? What does it mean and why would I need to do it in real life?

8. How can you divide a fraction by a whole number? What does it mean and why would I need to do it in real life?

The really fabulous (yes, I know I say that word a lot, and yes, I do it on purpose 🙂 ) thing about this unit was how many times I heard the words “Wow, this is easy!”  And how surprised so many kids were that it was easy.  For some reason, fractions is a four-letter-word to most people and honestly, I think that’s why so many of us (including me!) had so much trouble figuring them out.

We use Investigations as a math resource in our district, and I have always loved the way it works through math concepts–always starting with the why before showing the how.  And it was no different with fractions.  We did not start with straight number problems where we colored in pies that were the same amount, or with “this is the algorithm for adding fractions.”  We started with the why–or the “what” really.  What is a fraction, and how does it relate to percents, which are something that everyone already knows about.

Our fraction unit introduced many graphic organizers for kiddos to use to represent their math thinking, and the first one we used was a 10 by 10 grid.  We used it to find fourths (again, going back to something they already know), and figured out what fractions and percents we knew from those: 1/4 is 25%, 2/4 is equivalent to 1/2 and 50%, 3/4 is 75%.  Then I blew their minds when I showed them how they could find eighths on the same grid.  Yep, even though 8 is not a factor of 100.  Again, we had them think about what they knew and how they could use that knowledge to figure out something they didn’t know.  (I’ll let you stop right now and see if you can figure out how to do it.  Go ahead, I’ll even give you a 10 by 1o grid to use.)

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Yeah, so I’m sure you’ve figured it out, but I’ll show you anyhow:  If you find fourths, then think about what an 1/8 is.  It’s half of a 1/4, right?  Yep, 1/4=2/8.  Since you know that 1/4 is 25%, then you can easily figure out that 1/8 is the same as 12 1/2%.  Now you can use that to figure out the percent that is equivalent to any fourth or eighth, just by adding more of them.  3/8 is 37 1/2% because you know 2/8 (1/4) is 25% and then 1/8 is 12 1/2%.  Crazy, right?  I LOVE THIS PART!!  It’s so freeing to kids who have thought all along that fractions are impossible, too hard for them, some secret that they haven’t been told.  But now it’s just another puzzle–and they have the pieces to help them solve it!  This fraction/percent equivalence plays a HUGE part in the whole rest of the unit, so we spend lots of time at the beginning working with those numbers in different ways to help get it sold in their minds.  They had a chart they used as a resource, as well, throughout the unit.  Math doesn’t have to be a mystery.  It isn’t something you have to memorize.  You have tools and you just have to know when and how to use them!

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Another organizer we used were 4 x 6 and 5 x 12 rectangles.  They’re arrays, just like the 10 x 10 grids, but work better for other numbers that have factors like 3, 4, 5 and 6 (thirds, fourths, fifths, sixths, etc.). We could use them to find the fractional parts of almost any number that way.

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These were used when we moved on to finding thirds and sixths (which you can do with percents, as well, too).  These were cool, too, when they figured out that 1/3 was 33 1/3% and that 1/6 is half of that.  That’s a crazy question: what is 1/2 of 33 1/3?  Wish I would have recorded them figuring out that it’s 16 2/3%. Really.  It is.  Try it.  Here, let me show you:

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The first time I did that it heard my head.  The second and third times it did, too.  Yeah, I’ll admit it.  Some of the things I ask my kids to do seemed crazy in the beginning.  Mainly because it’s not how I learned it, but this time around it totally makes sense.   I wonder what it would have been like to do math like this when I was a kid…

(Ready for Part 3?  Find it here.)

Feast Week Part 1: The Birth of Feast Week

First of all, Happy New Year!  I don’t know when you’re reading this, but I’m writing it during Winter Break, on New Year’s Eve Eve.  This is a time of year I both love and hate: the fresh start that comes both personally and professionally in January is one of my favorite things–there is an air of anticipation of new and wonderful things to come; the fact that Spring Break isn’t for another two months is a little disheartening.  Winter can be long in Missouri.

That being said, I am excited to tell you a story about what was happening in my classroom–well all the 5th grade classrooms at my school really–during the months of November and December.  It was a fun and exciting time in our school, full of learning and anticipation; an eagerness that had nothing to do with holidays or vacations.  We were doing hard work, focused on something that at that time seemed like it was forever in the future: Feast Week 2012.

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This year I have a whole team full of new friends, and with that comes new ways of doing things–mainly just because we have all done it differently in our previous teaching “lives,” and because we want to plan new things together.  So…when it came time to talk about fractions and how we were going to teach that dreaded fabulous unit, we knew it was something we wanted to do together.

First we looked at what we had.  I have taught a fraction unit of some sort for the last 7 or 8 years, in 4th and 5th grade.  Previously, we really just had to get our friends to a solid understanding of what “fraction” means (part of a whole), and be able to use fraction/percent equivalents to solve problems related to parts of a group.  There was also a small part that included adding and subtracting fractions, using the equivalents as the basis (rather than finding common denominators, which is a common practice).

This year, however, our school district is really trying to dig into the new Common Core Standards–hoping to get a feel for what they ask of our kids and how they’ll change things for us as teachers.  This is happening most deeply in math; all of our curriculum and rubrics were rewoven to match the CCSS this past summer.

Now, instead of just the basic foundation like I mentioned previously, our kids have to be able to do this:

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Can I be honest here for a minute and tell you that we were a little FREAKED OUT by all of that!  Unfortunately, until you get to know the CCSS really well, and dig into what they mean and are actually asking your kids to do, I find that they are written in a really complicated way.  Needless to say, the first time we even read those expectations we were scared: how were we supposed to get 10- and 11-year-olds to be able to do those things (and do them well, with a deep understanding) if we couldn’t even understand what the standards said?

So after we picked our jaws up off the floor, dried our tears, and got our heart rates back to a somewhat normal rate, we sat down to figure out just how we were going to tackle these things with our students. We began with the belief that they could do it, we could do it and we were going to do it well.  We wanted to do it in a meaningful, authentic and real-life way that would help build a “forever and always” understanding, rather than just an “I-get-this-now-but-will-forget-it-after-I-take-the-test-next-week” understanding.   That meant rewriting assessments, possibly reworking assignment and activities and rethinking our own working knowledge of fractions.

And so Feast Week was born.  It began as an assessment idea, really, but quickly melded into more of a celebration–a culminating activity that would incorporate all that we expected our kids to know and be able to do.  It was to take place the last full week before Winter Break, and would include all that goes into creating a Winter Feast–planning, shopping, cooking, and then of course, eating!  We based it on an activity I had done in previous years around Thanksgiving where I had students use the circulars from the grocery stores to plan dinner for their family.  In that scenario, however, the whole situation was hypothetical.  In this reincarnation, it was for real.  We set the 5th Grade Fraction Feast to take place as our Winter Party, and the kiddos were entirely responsible for making it happen.  Talk about real-life.  Authentic.  Engaging.  Motivating.

And yes, it was.  None of it was easy.   And yes, I can admit there may have been some tears shed along the way.  But we made it, and yes, it was FABULOUS!

Hopefully you’ll hang on for the rest of the story of Feast Week! I promise it’ll be worth your time.  🙂

(**Be sure to read Part 2 here!**)

Math Warm-Ups Nov. 26-30, 2012

This week we had five whole days of school!  And even better, we had five math warm-ups!  Check ’em out!

Monday

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Tuesday

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This warm-up question illustrates just how authentic and real-life I try to make these.  This one really came from a conversation I had with a group as they were working on the problem I gave during work time on Monday.  It was just where I wanted us to go, and so presenting it as a warm-up made sense.  And when it can be suggested as a kid’s idea instead of mine–which it really was, anyway–that’s even better.

 

Wednesday

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As they did these warm-ups, the focus was on finding common denominators to help them add.  Rather than finding the LCM, however, I want them to connect these to work we’ve already done with fraction-percent equivalents; they know they can double or halve certain fractions to make other ones.  Mostly we’ve worked with thirds, fourths, fifths, sixths, eighths, and tenths, but I threw that 25 and 50 one in there to see if they could transfer the thinking to a similar problem.  They could. 🙂

 

Thursday

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This one reminded (or introduced to some) us of important vocabulary of improper fractions and mixed numbers.  As we added these, they focused on changing the mixed number to an improper fraction, adding them, and then reducing it to simplest terms.  Notice how the last one has the fraction circled?  It was on this problem that someone figured out that we could add the whole numbers and then just add the fractions and put them back together.  Smart, huh?  Again–this was so much more meaningful that they discovered it on their own, than if I just told them that they could do that as a shortcut.

 

Friday

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These problems encouraged my friends to focus on the strategy we had discovered at the end of the warm-up the day before, and also threw in some vocab we already knew (sums).  By the end of this conversation, there some kids who were smiling, which was nice considering all the frowns I’d seen early on in the week.  🙂

On a side note…sorry for the mess of the charts this week.  I ran out of chart paper and had to use the backs, too!  It’s resourceful, right?  Or just messy…not sure which. 🙂

 

 

Math Warm-ups Nov. 5-9, 2012

This was a slim week for math warm-ups.  We didn’t have school on Tuesday because of Election Day, and then I guess since we were in between units, there were really any good questions that came to mind.  That being said, I feel like I should apologize for these; they may not be very helpful to those of you who were here to get ideas.  Next will be better.  I promise. 🙂

 

I think this one was from Monday.  It was related to an assessment that my friends had finished the week before.  I was out of the classroom for a meeting when they did it, so partly this question was to get a better feel for how they perceived their performance on it.  I’m glad I asked, because their words told me more than the note from the sub about how it had gone.  They felt better than it had at first seemed they did.

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I don’t think we had another warm-up until Friday, and this was it.  At the beginning of a unit, I usually ask this same type of tell-me-what-you-already-know question. 🙂

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Thanks for reading!

Math Warm-Ups Oct. 29-Nov. 2, 2012

This week’s warm-ups–already!  Aren’t you proud?!  Another week of review of things we’ve already done, but needed to tweak a little bit.

Monday

This warm-up was in response to a reflection I had them write the Friday before.  I asked them to tell me what they’d learned during Box Factory, how their thinking had changed, and what questions they still had.  These were ones that came up on several kiddos’ papers.  This was an opportunity for me to see what other kids’ ideas were before I just answered the questions for them.  And what I had hoped would happen did–the other kids in the class gave responses that cleared it up for us all. 🙂

Tuesday

This equation gave us a good reminder of the order of operations, and how you have to DIVIDE or MULTIPLY before you ADD or SUBTRACT.  We ended up with answers of 4 and 16, and had to discuss which one was correct.

Wednesday

This one was from Halloween, can you tell?

Thursday

The discussion of “which comes first” came up again, only this time the answer was that it didn’t matter since both of the signs were division.  You would just do the in order from left to right.

Fridays are a little crazy in our classroom in the morning because many of us come in later due to Instrumental Music, so no Warm-Up today.  Enjoy working through these and let us know what you think.  🙂

What’s All This “Box Factory” Business?–Part 1

You may have heard me or my students mention the Box Factory lately, and wondered what in the world we were talking about.  Let me tell you about this fabulous math work we’ve been doing lately.
In 5th grade, we have a unit on 3D geometry, focused around finding volume of different kinds of rectangular prisms and figuring out a formula for how to do this (l x w x h or b x h).  This year we incorporated a unit by Cathy Fosnot, which created a context for this learning.  Enter the “Box Factory.”

The basic premise of the investigation is that kids work in a box factory and have to figure out certain things related to volume and surface area (although these things are not specifically named until later in the unit).  There were three parts, and kids worked small groups to investigate the answers to these questions:
1.  If the box factory wanted to create boxes that held 24 items, how many different boxes could they create?  What would the dimensions be of those boxes? Which box would be the cheapest one to produce? There were 16 possible answers to this question, and the students used cubes, graph paper, equations, drawings, or whatever necessary to figure it out.  They had to then create a poster to show their strategies and explain their thinking to show to the other groups.

2. How much cardboard would you need to cover each of these boxes? This one extended the conversation into surface area, and invited students to now look at the outside of the box, instead of just the inside.  Most groups figured out that if they used the formula (2 x L) + (2 x W) + (2 x H) to determine how much cardboard they’d need.  The cheapest boxes to make would be ones that are closest to the shape of a cube, as opposed to a long, skinny box.

3.  If the factory created three sizes of cube-shaped boxes–2 x 2 x 2, 3 x 3 x 3, and 4 x 4 x4, how many units could each hold?  If it costs 12 cents per unit, how much would each box cost?  This one looked at the inside again, and added another layer of multiplication (with money) to figure out the final answers.

 

All throughout this investigation (which goes for about 10 days), the focus is on kids discovering strategies for volume, rather than just giving it to them.  Through the posters they create and the Math Congress conversations we share, they are also working on sharing and representing their thinking.  They are learning how to make their representations clear and concise so that other people can understand exactly what they did.

This poster-sharing part is not new to me in Math Workshop.  But Fosnot’s unit added a layer I’ve never thought of before in math–revision.  Much like when mathematicians publish proofs (and like we’d just spent time on in Writer’s Workshop!), students were able to get feedback from others on what worked, what was confusing, what they should add or take away.  They they had the opportunity to revise and edit their posters before they shared.  With each new poster they created, they added new ways of showing their thinking clearly.   They did this by discussing with their group, and then leaving suggestions on post-its.  We used the “Plus-delta” model to share something we liked and something we’d change:

 

So by the third time around, we were pretty great at showing thinking on our posters.  Even though you didn’t see all the steps, you can still appreciate the clarity and organization of these:

 

 

 

 

 

 

Have you ever done a Cathy Fosnot unit before?  How have you used revision and feedback in math to clarify thinking? What strategies do you use for teaching volume? We’d love to hear about it!

Math Warm-Ups Sept. 17-21, 2012

I think I said it in my last post, but golly–being sick is hard!  Someone around my house has had a cough for at least a month now, and I’ve had one for at least 2 weeks!  Hence my absence from blog-world for that long.  All I could stand to do was fall into my bed and sleep when I got home.  Luckily, though, I’ve been to the doctor and gotten some medicine and am on the mend.  So here I am. 🙂

The horrible thing about not writing for two weeks is that I have a million and one things to post about that have happened, but I’m not really sure how and when I will get them all done–because new things keep getting added everyday!  So I guess I’ll start here and see where it takes me.

Last week we were working on division in math, but then added in some practice (an introduction really) with order of operations.  Here’s what our warm-ups looked like:

Monday

Tuesday

I was gone and so don’t have a pic of this one, but they solved the problem that they wrote the story for on Monday, using an efficient strategy.

Wednesday

When we started, the red parentheses were not there, and so many were not sure which part to do first.  This was the introduction to using order of operations.

Thursday

The original problem is what’s written in pink, and the brown is our work after we talked about how to tackle it.

Friday

I love how complicated this one is!  This one tricked many friends, but we’re getting a hang of it now that we’ve done it several days in a row.  This was a brand new concept to most mathematicians in my room.

How did you answer our math warm ups?  What can you tell us about using Order of Operations to solve problems?  Any advice you can give us?  Thanks for leaving your comments to help continue our learning. 🙂