Two Cats and Two Tortoise

Yesterday my wonderful co-workers threw us an adoption shower, and thanks to them our daughter is an even richer girl if you measure wealth in books.

One of the gifts was from Mary Beth. She said it’s a favorite thing to do with a favorite counting book, Rooster’s Off to See the World by Eric Carle.

Included with the book was a baggie that contained a small rectangular board and a bunch of small cards with animal pictures. My daughter pulled out the baggie and asked, “What’s this?” I said it was something we could look at while we read the book. My daughter didn’t want to wait to read the book, so this morning while I made breakfast, she plopped down on the kitchen floor to explore the baggie of cards on her own.

Within a few minutes I heard her say, “There’s two cats and two tortoise!” I looked over to see that she had filled her card with animal pictures. And sure enough, the card had two pictures of cats and two pictures of tortoises (or turtles, I’m not sure which yet). I like that all of the counting we do throughout our day has led her to notice and count things on her own without any prompting from me.

After she was sure I had seen the pictures, she cleared off the card and said, “I want to do more.” She put all the cards back in the baggie and started filling the rectangular board again. One thing that was really interesting to me was how she naturally made two rows of three pictures on her card.

When she finished filling the board up a second time I asked, “What do you notice?” I was curious if she would count again or if she would tell me something different. Her response was, “One chicken, one cat, and two fish.” A few seconds later she exclaimed, “And two frogs!”

Then she cleared the board again and started filling it with new animal cards. This time she chose only cards with fish on them. Her observation at this point was, “There’s a lot of fish on here!”

She started digging through the baggie for a few seconds before saying, “I need one more fish.” We haven’t really talked about “one more” very much so it was so interesting to hear her say that. Since she kept laying out the cards in the same arrangement, I’m assuming she could tell there was room for just one more card.

Picking randomly from the baggie wasn’t working so she pulled all the cards out and spread them out on the floor. She couldn’t find a fish so she changed the activity. She asked me, “Where’s the tortoise? Touch it.”

I dutifully touched the tortoise, and that was the end of the activity. She put all the cards back in the baggie and moved on to looking at another one of the books she received yesterday.

The one thing that came to mind as I watched all this unfold was Christopher Danielson’s message: Let the children play:

We adults have a responsibility to let the children play. We can be there to listen to their ideas as they do. We can play in parallel by getting our own egg cartons out and filling these cartons with our own ideas.

But when we tell kids to “make a pattern” or “use the colors”, we are asking the children to fill that carton with our ideas, rather than allowing them to explore their own.

I could easily have overtaken the activity by asking questions such as:

• How many tortoises are on the board?
• Are there more cats or fish in the baggie?
• How many animal cards are not on the board?

All of these are great math questions, but they’re MY great questions, not my daughter’s. I want her to develop her own questions and curiosities to explore. In the end, it was much more fascinating and rewarding for me to see the ways she came up with to explore the cards and to share the things she noticed about them.

Thank you, Mary Beth, for the wonderful gift!

Looking For (and Finding!) Math All Around: Part 3

Welcome to the third and final post in this series of examples of math from the real world – specifically the Halloween aisle at Target. In the first post we looked at packages of 4 party favors, and in the second post we looked at packages of 6 or 8. Today we’re going to move into estimating using packages with larger quantities, and I’ll wrap up by sharing a few bonus images that didn’t fit anywhere else.

If you aren’t familiar with Andrew Stadel’s website Estimation 180, that’s what gave me the inspiration for taking and sharing the pictures in this post. He has a great Ed Talk from this summer’s California Teachers Summit that you should check out.

Here’s a quick rundown of Andrew’s main points:

• Students often lack the access to opportunities to strengthen their number sense.
• Estimation is a gateway to better number sense.
• Estimation is important because it’s an opportunity to take numbers and make sense of things around us.
• Have students estimate by giving them a visual, asking them a simple question, avoiding guesses, and justifying estimates with reasoning and context clues.
• Make estimation accessible by having students create their own estimation activities.

When you visit Estimation 180, you’ll see that every estimation activity asks a simple question that fall into three categories:

• How long…?
• How many…?
• How much…?

Then he guides students to estimate using reasoning, not guesses. He always asks students to make an estimate that is too high and an estimate that is too low before asking them for their actual estimate. Finally, students have to provide a reason they chose their estimate.

I say all of this because if you want to get the most out of the pictures I share in this post, then you’ll want to follow this same structure or something similar to ensure students are truly processing the activity and not randomly guessing. And with all that said, let’s get to the pictures! (Click a picture to see and/or save a larger version of it.)

Halloween Estimation 1

How many fingers are in the bag?

Before you look at the reveal, you may want to answer the following questions:

• What’s too LOW?
• What’s too HIGH?
• What is your estimate?
• What is your reasoning?

Okay, here’s the reveal:

24 fingers

Halloween Estimation 2

How many skull erasers are in the bag?

I suggest asking the same questions that you did for the fingers. It may seem redundant, but what we’re going for is repeated reasoning through repeated questioning. What regularities will students begin to notice the more they estimate using those guiding questions?

I’d also like you to think about these two questions:

• Do you think the number of skull erasers in the bag is greater or less than the number of fingers that were in the previous bag?
• What is your reasoning?

And here’s the reveal:

60 erasers (Were you expecting it to be more than double the number of fingers?)

Halloween Estimation 3

How many erasers in the pack?

What strategies would you use to estimate here? Technically, your students could slowly count every eraser, so you may want to mesh this estimation with a quick images routine – show the picture long enough that students can get a mental image, but not so long that they can count one by one.

Here’s the reveal:

18 erasers

Halloween Estimation 4

How many party favors in the pack?

This is another picture that could benefit from the quick images routine of showing the picture just long enough for students to get a mental image. You may even want to show it a second time to give students a chance to revise their thinking, but still keep it short enough that they can’t count one by one. I especially like how students can use color to help estimate with this picture.

Here’s the reveal:

50 party favors

Halloween Estimation 5

How many stickers in this pack?

In order to make a better estimate, you might like some additional information:

How does the side view help you make a more reasonable estimate?

I can give you a bit more information:

How could the measurements help you estimate the total number of stickers?

And, finally, the reveal:

120 stickers (Were you close?)

If you want to make your own estimation activities for your students, it’s really that easy. Find something that comes in a pack, cover or hide the total quantity if it’s written on the pack, and provide something for students to use as a benchmark. In the case of the previous pictures you could see all or some of the items in the pack to help get a sense of the size of each object.

As promised, I have some bonus pictures to share before signing off. These pictures didn’t fit with the other sections I wrote about, but I still wanted to share them.

Bonus Pictures

I like this picture because there are so many different ways students could find the total number of pumpkins. I also like that some students may notice the tall white pumpkin while others may only see the 3 by 3 array of pumpkins. It reminds me of a similar visual prompt Joe Schwartz shared in a post he wrote about the Notice and Wonder strategy. Scroll down to the section in his post that says “Grade 1.” What I liked was all the different number sentences the teacher recorded to show all the different ways students saw the quantities in his picture.

We saw several examples of arrays in the previous two posts. This is a much larger total than those examples. I like how the rows are spread apart from each other to draw attention to them. However, the columns also stand out because the color of gem is the same within each column. So much to talk about structure here, along with multiplication and fractions.

I considered putting this final image in with the estimation pictures. I didn’t hide how many bubble sticks are in each package, but that doesn’t necessarily tell you how many bubble sticks are in the whole box! I like that there are 10 packs in the two left columns and 1 pack all by itself. It’s such a natural way to show 24 × 11 broken apart into 24 × 10 and 24 × 1. (And I didn’t even plan it. This is how the box was arranged.)

That wraps up my blog series on finding math in the world around us. In addition to getting a slew of pictures you can use in your classroom, I hope it sparks your curiosity the next time you’re out and about, or even hanging around close to home. Maybe something will catch your mathematical eye. if so, snap a picture to share and discuss with your class. Then come back here and leave a comment to let me know about it. I’d love to hear your stories.

Looking For (and Finding!) Math All Around: Part 2

In my previous post, I shared images of various Halloween party favors packaged in groups of 4. Today I’m going to share packages of 6 and 8. (Click a picture to see and/or save a larger version of it.) Without further ado:

What sorts of things do you think students would notice as they looked at this picture? Here are some things I’m noticing:

• There are 6 bouncy balls.
• There are two groups of 3 bouncy balls if you look at the columns.
• There are three groups of 2 bouncy balls if you look at the rows.
• The top 4 bouncy balls have the same layout as the packages of 4 items from the previous post. It just looks like 2 more have been added at the bottom.
• None of the colors repeat, so if I think about fractions I could say that 1/6 of the bouncy balls are yellow.
• On the other hand, 0/6 of the bouncy balls are blue.
• Half the bouncy balls are in each column.
• One third of the bouncy balls are in each row.

This is a great time to mention something to be mindful of when using noticing and wondering. You can go in with a plan that your students will notice some particular mathematical idea you have in mind when you share an image, but that is not a guarantee that they will notice it. Especially if students are new to the practice of noticing and wondering, don’t be surprised or discouraged if their observations are not as rich as you were hoping. Perhaps they haven’t had a lot of practice noticing math before. Give them ample opportunities, and honor everyone’s noticings and wonderings even if they don’t match your desired noticings and wonderings.

I like these next two pictures because they show an arrangement of 6 in one column.

When we talk about multiplication, we often think of multiple groups/rows/columns/piles/etc. (emphasis on all those nouns being plural). However, we can’t neglect showing students models of one group of the quantity. If students can only model multiplication with 6 if there are 2 or more groups of 6, then there is a hole in their understanding of multiplication. If on a previous day you talked with your students about the bouncy ball picture being represented by 3 × 2 and 2 × 3, then these pictures are a great opportunity to talk about how to represent one column of mustaches or one column of bats using the multiplication expressions 1 × 6 and 6 × 1. I like that the package of bats also brings in other multiplicative relationships if you think about the number of eyes or the number of wings on all the bats.

It’s fascinating how differently a quantity of 6 can be packaged. This next one was the most interesting to me.

I wonder why they didn’t package the lizards separately like they did with the bouncy balls. Instead, you have two compartments with 3 lizards in each compartment. Whereas in the bouncy ball picture students might talk about rows and columns, this image likely steers conversation to the idea of groups instead.

And again, you can get into even more relationships if you think about the number of legs on all the lizards or the number of eyes. Would students realize the number of eyes is the same for both the package of lizards and the package of bats? How might they prove it to you? Would they understand the layout in the package isn’t affecting the total number of eyes since there are 6 creatures with 2 eyes each in each package? Often things that are obvious to adults are not at all obvious to young children.

Let’s move on to packages of 8. Here’s another example that’s great for talking about one group of a quantity.

Would students think of this as a row or column? In the images of the mustaches and bats, they were stacked one on top of each other in a column, but now we have one row of 8 pencils that can still be represented by a 1 × multiplication expression. Digging a little deeper, they might notice there are sub-groups of 2 within the package which can lead to more discussions about multiplication, or perhaps even fractions. Within this one package, each design makes up 1/4 of the package. If I bought 2 packages of pencils, would 1/4 of all my pencils have skulls?

Here are two final images of packages of 8.

I’ll leave you to notice and wonder about them individually and in comparison to the previous packages we’ve observed. What are you noticing and wondering? What might your students notice and wonder about them? What math topics could these pictures spark discussion of in your classroom?

After looking at all of these pictures, my final wondering for today is this: Why do party favors seem to always be packaged in even numbers like 4, 6, and 8? What products can you think of that are packaged using an odd number of items? Is there a practical reason to package using even or odd numbers?

In my next post we’ll look at packages with larger numbers of items and we’ll even get to do a bit of estimating. Stay tuned!

Looking For (and Finding!) Math All Around: Part 1

We often tell ourselves and our students that math is all around us, but that can ring hollow if you’re someone who looks around and, to be quite honest, you don’t really see it. I’ve been guilty of this myself. In the past I didn’t know what to look for – I didn’t know what “mattered” – so I didn’t really see it.

So recently I started challenging myself to find and share examples of math in the world around me in the hopes of showing others where to start finding it for themselves. I’ve been sharing pictures on my work Twitter account (@RRElemMath) so you can go there if you want to see the random pics I’ve shared so far this school year.

What got me kick started on this mission was taking part in the #mathphoto15 challenge that spanned this past summer. You can scroll through the hashtag to see a huge collection of photos people from all over the world shared on a variety of math topics throughout June, July, and August. You can learn more about the challenge on the official website. There’s even a section called In the Classroom where you can share how you’ve used some of the photos yourself with teachers or students.

Most of the math photos I’ve taken since school started have been at Target. Stores are such rich environments for math noticings, and walking through the store today, the Halloween section was a veritable cornucopia of math imagery. I took so many photos today that I decided to share them across a few posts rather than tweet them out randomly on my Twitter account where they might get lost in the noise. I also wanted to take the opportunity to share mathematical ideas I saw as well as ideas for conversations these photos might spark with elementary school students.

My hope is that browsing through these posts might inspire you to share some of these pictures with your students. Even better, I’d love for you to be inspired to start taking your own photos to share and discuss with your students.

One of the easiest things you can do with just about any of these pictures is to have your students spend time noticing and wondering about them. I recently wrote a post about this for my school district. You can read that here. I learned about the routine from Max Ray-Riek’s book Powerful Problem Solving. In case you don’t have his book handy, you’re in luck because you can read more about the strategy in this short PDF.

This first series of photos all have to do with the number 4. (Click a picture to see and/or save a larger version of it.)

As you look at the images, what do you think a primary grade student would say about them? Hopefully they would all be able to tell you there are 4 things in each package, but what do you think they would say if you asked them, “How do you know?”

Would they say, “I counted 1, 2, 3, 4”? Do you think any of them would notice the rows of 2 and say, “I saw 2 and 2, and I know that’s 4”? Would you be surprised to hear, “It looks like you could make a square out of the 4 bracelets or 4 yo-yos”?

The quantity may be small, but that doesn’t mean there isn’t room to notice, wonder, and discuss.

If you showed them one picture a day, they might start to notice how 4 is always arranged in roughly the same way. What are they going to say when you show them this?

And what if you show them this next image, but instead of worrying about quantity you ask them, “Which one doesn’t belong?” (h/t wodb.ca)

Would they notice the cat notepad is the only one with an orange background? Would they notice the Trick Or Treat notepad is the only one with words? How else might they justify the other notepads not belonging?

Could you do this same activity with any of the other pictures of 4 objects? Don’t worry if you don’t necessarily have answers right away for why each one doesn’t belong. The point is to give your students a chance to think of and, more importantly, justify their own reason.

I have one final picture of 4 that I love because it offers up so much to talk about.

At first it looks just like the image of bracelets or yo-yos, but if students take some time noticing and wondering, one of them is bound to mention the spots on the back of the spiders. This might lead into a wondering about how many spots there are altogether. Someone else might notice the eight legs on each spider, which again might lead to a wondering about how many legs there are altogether.

The addition strategies for finding the total quantities of dots are excellent work for first and second grade students, and the multiplicative work determining the total number of legs is a great fit for second or third grade students. And all of this work is still perfectly appropriate for Kindergarten students because of the concrete image in front of them. Kindergarten students may not multiply 8 by 4 to find the total number of legs, but that doesn’t prevent them from finding the total all the same.

That’s all I have for this post, but come back next time to see some pictures of 6 and 8 in a variety of interesting arrangements.