Category Archives: Math in the world/media

Mills, Bills, and Trill

The first week of school lesson plans usually consist of getting-to-know-you type activities.  In planning, I figured it would be a great time for exploring in math.  I hemmed and hawed about it.  At first I thought I’d do a 3 act lesson.  They are always exciting and engaging, but I couldn’t think or find one that these kids hadn’t seen in 5th grade.  

But then I found this……

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This was a picture I found a long while ago and kept it in my files, never knowing what to do with it.  I saw the math in it, but didn’t have a solid idea.  However, over the summer, inspiration struck and I got to work.  

Have you ever asked your students about millions, billions, or trillions?  Do they truly understand the magnitude of these ginormous numbers?  It was worth a conversations.

Part one of our investigation was showing the above picture and simply asking “what do you notice and what you wonder?”

My students recognized notice/wonder which thrilled me.  Some perked right up when they saw those questions (which made me smile).  I noticed that my students focussed on the picture itself.  Some of my higher thinkers started pondering the saying.  Finally, we got to the question I was looking for….which was how many pictures are in a video.

Here’s the information we were looking for…Screenshot 2017-08-19 21.24.24.png

We decided we needed to figure out how many pictures were in a second, then a minute, and so on.  They determined that one second of video would be worth 2400 words.  And the video would have to be less than 1 minute.  One minute of video would be 1,440,000 words.   Not bad for my 6th graders.  They were getting the hang out this.  

 

Then I asked them how long it would take them to draw a billion circles.  Saw this at a workshop given by Graham Fletcher back in January and I finally found a use for it. The students were thrilled to investigate this for sure.   

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With much enthusiasm, the students were jumping with energy on this one.   One girl raised her hand and asked, “are were really doing a billion?”  I looked at the clock and responded, “we got a few hours left so I don’t see why not.”  Their eyes got bigger.  

First we took guesses.  I felt like I was on the Price is Right.  “I’ll say 7 hours…I’ll say 7 hours and 15 minutes.”  One group kept whispering about the question and concluded that it should take about 1 second per circle.  I loved this observation because I could tell they were not only thinking, but trying to make sense of the problem.   The class concluded that we should just try it for a minute.    Before we got started though, there was more discussion on how big of a circle they should draw.  Wouldn’t that be a factor in how many they could draw in a minute?

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I started the timer and the students worked vigorously on their circles.  The looks on their faces was priceless.  They focussed, they concentrated, they were super-serious about getting this done.

When the timer went off, they counted up their circles.  We took a poll to see how many circles were drawn and the range was between 80 to 120.  For our purposes, we kept using 100.  Next, we figured out how many circles in an hour, a day, and then a year.  Finding out how many circles in a year was a bit of a doozy for them, but they persevered.  I was also happy to report that they knew how to read their place value very well.  Whew!

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We even discussed what would 2 years looked like, and they told me it would only be one hundred  million circles.  “That’s not enough.”

I revealed the answer and they were floored.  I’m not going to reveal the answer in this post because I want to leave a little mystery to the question.  Go ahead…do the math yourselves.  (Answer is in the lesson PDF below).

We weren’t done yet.  For the last piece to this investigation,  I remembered this Twitter pic from Mark Chubb.   And I also saw a ripe opportunity to break out the clothesline.  I asked my students “Where would 1 million go?  Where would 1 billion go?”

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After much debate and moving cards back and forth, this is what they agreed upon.  I was puzzled because they didn’t really know the relationship between billion and trillion.  I let them watch a video I found on Youtube and some referenced that when they were validating their answers.

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However, I wanted to push them further.  I inquired why each of the cards was equally spaced.   Were there an equal amount of numbers between each number?  Some answered yes.  Some were completely puzzled.  Rather than beat the horse to death, I wrote a few new numbers on cards to see if they could show me the relationship.  I wrote 0, 10, 100, 1000.     The conversation got a little more exciting because my students were dealing with numbers they could relate to a bit more.  Unfortunately I forgot to take a picture of the final clothesline and specifically where they moved the cards.  

Final thoughts…

  • We hear how the US and other countries have trillion dollar debts, but how many people truly know the size of that number.  It’s quite “ginormous” but how does it relate to millions and billions?
  • The students loved that they got to “investigate” something tangible.  They could all draw circles.  It was an easy task that they all handled.  It was hilarious to see them take it so seriously.  
  • The idea of the trillion clothesline was spot on with the investigation, but I saw many blank stares because the students couldn’t really fathom a trillion.  Instead of beating the idea to a pulp, we went with something they were familiar with based on their discussion and reasoning.  
  • This showed their lack of knowledge in how 10 related to 100 and so on. 
  • Considering this was the 2nd day of school, this showed that we will be making math fun, accessible, and worthy of their attention.   Many of my students said this was their favorite thing of the day.  

 

Here’s the lesson if you’d like to try it out for yourself.

 

Until next time,

Kristen

 

 

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?

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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.

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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.”

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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.

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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

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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