Category Archives: Math in the world/media

What’s a Shape?

Back in February, I came across a blog post from Telanna about shapes.  She saw a Twitter post from Sarah Caban asking a simplistic question.

 How would you define the word “shape”?

Not wanting to miss out on the bandwagon, I decided to jump in.  Considering that I have access to such a grade span, I patiently waited for the right time in each grade level’s curriculum to pop in on a few classrooms and have a conversation.    Each teacher that I chatted with was also intrigued with my master plan and wanted to see/hear the results.

Kindergarten

And so my journey of defining shapes began with Mrs. Z’s kindergarten in March.  She was right in the middle of her shapes unit (perfect timing) and so she asked the kiddos the question “What is a shape?”

Here’s a snap-shot of what was discussed….

  • Some shapes are big and small. 
  • sometimes round— circle or oval.
  • some are skinny/ thin
  • different sizes
  • star, heart, rectangle, square, triangle, diamond, hexagon
  • shapes have points and angles.  (T asked –do all shapes have points)
  • Not all shapes have points.
  • shapes you can trace or cut out.
  • Everything we color or write or draw is a shape.
  • Shapes are everywhere because they are. 

The kiddos keep having side conversations asking questions like “are they lines?” and “what about letters?”.  One child proclaimed “the sky is not a shape.”  Upon hearing this, another child replied, “but what’s in the sky?  Sun, Stars and Clouds”.

 

After the in-depth conversation, Mrs. Z asked them to get up and make shapes with their bodies.  First, they made a circle (or the attempt at a circle) and then a rectangle.  Some students ran up to me to show me the shapes with their fingers/hands.  

 

Third Grade

Fast Forward three weeks—->>>

My next door neighbor, Ms. N, teaches third grade and upon hearing about my shape quest,  invited me in to lead the discussion on shapes.  They were also in the middle of their geometry unit, so the kiddos wanted to impress me with their growing geometry vocabulary.  They also corrected me in that they are discussing POLYGONS, not shapes.  

  • something that has sides
  • has a vertex (corner of a shape)
  • has angles—> can be 90 degrees
  • has different sides
  • could be a quadrilateral
  • has to have more than 3 lines
  • has to be closed —-> all lines connecting
  • can be a polygon
  • shapes are all around us
  • convex —> shape doesn’t have a cave in it
  • can have a concave in it
  • rectangle can have opposite sides
  • can have parallel sides

 

Fifth Grade

OK OK OK.  I have a guilty conscience about this one.  I cheated.  Full admission of guilt.

 I didn’t have time to go to a fifth grade classroom.  Time became of the essence for the fifth grade teachers with reviewing for CAASPP testing.  

HOWEVER—- I have an a 11 year old son (5th grader) who was happy (** sarcasm**) to have a conversation with me about shapes.  Yes…this is what we do during our commute into work/school.

  • shapes are in everything
  • there’s no one thing that doesn’t have a shape
  • shapes are the building blocks of life. (how philosophical of my son)
  • have corners
  • they can have an infinite number of sides, but then that might turn into a circle
  • the sides are not always the same.
  • there are squares, rectangles, hexagons, circles, triangles, 
  • rectangle has uneven sides
  • square has all even sides

I asked what he meant by “uneven” and my son said that it was when one side was larger than the other.

Final Thoughts

Students in the primary grades start by being introduced to their shapes.  It becomes just identification which is the first level of learning.  By third grade, they are being exposed to more specific language and vocabulary.   This third grade wanted to impress me with their knowledge of geometry.  They had been testing out different shapes to see which would pass their definitions.  As for 5th grade, they have a broader view of what shapes are.  They have also explored 3 dimensional shapes as they discover volume.  

If I had a chance to follow up with each class, I could ask the question, “what does NOT make a shape?”  It would be a great contrast to their base knowledge.  It would challenge their thinking and we could probably have an in-depth conversation about their comprehension of shapes.  

A magnificent exploration.  Much thanks to Sarah Caban & Telannannalet.wordpress.com for the inspiration.

 

Until next time….

Kristen

 

Sorting in Kinder (a 3 Act)

My kindergarten collaborator, Stacy and I recently attended a 2 day workshop with Graham Fletcher and it re-ignited our passion for 3 act tasks/lessons.  She’s made it her goal to collaborate with me and create one task per topic.  I happily accept her challenge and told her, “GAME ON!”

The most recent topic in her curriculum was sorting.  This is a skill that we all take for granted.  We sort our trash into various recycling bins.  We sort through mail.  We sort our clothes while folding laundry.   How do we get little ones to understand how things are alike and yet different?

She already uses the “Which One Doesn’t Belong” routine and asks students “how would you sort these?” However, how can we bring this standard (K.MD.3 Classify objects into given categories; count the number of objects in each category and sort the categories by count.) to life in the form of a 3 act task?

Our answer was this….let’s give them a scenario they should be accustomed to.  

Act 1

 

As always we started with a notice and wonder routine.  

Notice –

  • I saw crayons
  • Math Wizard said “clean up”
  • He’s drawing a rainbow and boxes
  • Crayons are everywhere
  • I see markers
  • I heard the Math Wizard
  • She has a son ?!?!
  • He needs to pick up his stuff before school
  • It must be night time because it’s dark

Wonder –

  • Was he cleaning up to go to bed?
  • Was he cleaning up before dinner?
  • Was he cleaning up because he was done?
  • Does he have  a brother or a sister?

Act 2

I was really curious how Mrs Z was going to push their thinking beyond their notice and wonder.  She inquired further.  She showed the Act 2 picture. 

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“What would you do with that stuff?  What if Mrs Z said ‘clean up’? What would you do with it?  Where would you put the stuff?”

The students thought about her questions for a moment and slowly put their hands up.  One student piped up with “I’d put the pencils away”.   And Mrs Z next asked, “How?”

“The markers go together. The pencils go together and the crayons go together.”

“I WOULD ORGANIZE IT!”—>And there it was.  Just the answer we were looking for.  And that is a big word for this student.

And so we discussed how they would sort them.  Some students said by size.  Some students said by color.  One student said he’s organize them between caps and no caps (Markers have caps on them versus no caps.)

Usually at this part of the lesson, the students do some kind of calculations or reason out their answer.  How could they be expected to sort from a picture?   That’s where Mrs. Z comes in with her bag of tricks. Prior to the lesson, she made bags of pencils, colored pencils, and crayons.  Each group would be showing all the different ways to sort their bags.  Oh–let the games begin!

 

Mrs Z and I wandered around the room eager to see what the groups would do.  She informed me that they don’t work in groups too often, so she was curious of how this would go down.  

Here’s one group’s explanation.

Here’s another groups explanation.

 

At one point, we noticed that a group put all their pencils together.  We asked them how they could further sort this group.

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Just when I thought we were pretty much done, Mrs Z runs around a throws unifix cubes on to their tables.  The kiddos didn’t bat an eyelash and just incorporated them into their categories.  Here’s one groups way of organizing.  What do you notice about the picture?

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Act 3

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Final thoughts….

  • Having the students work in cooperative groups for this lesson gave us opportunity to see which roles the students would fall into.  You can see who lead the pack and who followed along. 
  • Student usually come up with more answers than you can anticipate, but we are never disappointed.
  • I love the hands-on exploration part.  We got to see how they were organizing their items.  

Until next time,

Kristen

Much Ado about Watermelons!

Day 2 of #SummerMathCamp

To feed our math brains at 8 am on a Tuesday morning in the summer, we showed this photo of some watermelons in the hopes of generating some conversation.

How many watermelons are there? How do you know?

watermelons
http://ntimages.weebly.com

This was just going to be 10 to 15 minutes of a notice and wonder conversation. Yea. We were wrong. Fifty minutes later we were still chatting about watermelons. Who knew a pile of cut up watermelons could keep 45 educators engrossed. Really, what is there to talk about—it’s just a bunch of watermelons, people.

carofmelon
Image from: weknowmemes.com

So, here’s what we chatted about.

We predicted that the most common answer would be to cut up two of the one-half sized pieces to make the missing one-fourth pieces, slide those around to make 4 whole watermelons. Then, add the other two halves to make another whole watermelon. So, the sum of 4 whole watermelons and the other whole make 5 watermelons. We thought that some version of this idea would be a good start to the day.

And that’s exactly what happened. One of the campers shared her version and just about every person in the room said, “Yup, I thought about it that way, too.”

Picture1

 

And then Shannon said, “I saw it another way. I saw 4 groups of 3/4 of a watermelon and then I added the 4 one-half sized pieces. So, 3 whole watermelons and two more means that there are 5 watermelons in the picture.” The conversation then moved to connecting Shannon’s use of the algorithm to the photo and then adding in the notation.

Picture2

Since many of the campers were happy to think about the quantity of watermelons using the algorithm in Shannon’s explanation, we thought we would pause here and push on the idea of equivalent representations. [We much preferred thinking of the watermelons as 3 groups of 1—written as 3 x 4/4.]

This provoked lots of conversation on how come we can do this. 

  • How does the image show (4 x 3/4) = (3 x 4/4)? 
  • How does the picture show Shannon’s equation: (4 x 3/4) + (4 x 1/2) = 5?

The third way that surfaced was to simply add all the pieces of the watermelons visible in the photo. We recorded it like this:  ¾ + ½ + ¾ + ½ +¾ + ½ + ¾ + ½

Picture3

Here are some of our takeaways:

  • If it’s important to the kids it MUST be important to us

We need to listen to what our kids are saying and not be on the lookout for the answer listed in the TE or for the student response that matches our preferred strategy.

  • Questions and their power

Your questions need to offer ALL kids a place in the conversation. So, in this situation we could have asked a couple of questions.

  1. How many watermelons are there?
  2. How many whole watermelons are there?

Which question lets the kids who say 8 be in the conversation?

  • Dots on ten frames and Photos of watermelons, almonds, and tangram puzzles

Math is an active subject—it’s interesting, irritating, perplexing, confusing and invigorating. It makes your head hurt when you are in the midst of the struggle and then you get to embrace the high fives when that last piece falls into place and the connection appears as a result of your hard work.

 

Until next time….

Kristen & Judy

 

Number Talk Images

Can I just say that I can’t get enough of visual math routines?  Or do you call it a number talk image?

Some call it a visual number talk.   It’s a picture that’s shown with a known quantity.  Students may start by making observations and ask questions that are lingering in their heads (AKA Notice/Wonder–thanks Annie Fetter!)  Once their questions are answered, they may try guessing a number that’s too low.  Next they’ll try to guess a number that’s too high.  The last number they’ll write down is an actual estimation.

Why should we do this?

  • Develops students’ understanding of quantity
  • Give numbers meaning
  • Help students see the relationships of numbers to one another
  • Support an understanding of how numbers operate

The conceptualization of quantity is foundational to number sense. As students’ abilities to visualize amounts improve, their number sense improves. Their strategies and mental math become efficient and quick.

Once I was introduced to this routine, I was ADDICTED!  It became a mission of mine to find my own pictures.  My goal was to develop a collection of everyday items (thanks Target!) that any other ordinary human would pass up.  Well, this April Fool has you covered.  Every now and then, I stumble across something and take pics like a crazy person.  And yes, I will sit there and count.

So here is something I came across during the after-Christmas sales. How many storage boxes are there?IMG_5834

Just in time for Valentine’s.  How many hearts are on the front side of this bag?

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If you are interested in more of these, I suggestion Andrew Stadel’s Estimation 180.  And if you are interested in more images ,Pierre Tranche’s Number Talk Images is really cool.

 

Oh…and not to leave you hanging.  There are 60 storage boxes and 388 hearts.

Kristen

 

It’s “Nacho” Business

I have a son who’s 10 years old.  He goes to school in the district in which I work.   When he was in third grade, I was asked to volunteer at his school’s Harvest Festival.   The third grade team was selling nachos.  “Sure, not a problem.  How bad can it be?”

What I walked into was in one word–EPIC.  A math teacher’s dream.  I was scheduled to only work a half hour.  I stayed the entire night. 

6 crock pots of nacho cheese were brewing.  Bags upon bags of nacho chips.  It was quite a production.  Selling nachos was “serious business.”

And so a math problem unfolded right in front of me.   The team of teachers was selling  cheese nachos for $2.00 and nachos with cheese and jalapeños for $3.00.  By the end of the night, the nacho leader proudly informed me that they had made $700 which was amazing. But the question remained…how many nachos did they really sell?

Let me clarify something from these pictures.  There are 8 desks/tables filled with nachos. Being there the entire night, these tables were fully filled twice.  (27 bowls of nachos can fit onto 1 table)

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Selling nachos with the third grade teachers
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Selling nachos with the third grade teachers
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27 nacho bowls can fit onto one table

 

I can’t eat, smell, or touch nachos without this experience being remembered.  My husband still has nacho burn marks on his hand.  However, we wouldn’t trade in that night for anything.

Kristen