# Inspiration – Summer Edition

As you may or may not know, I have a tendency to roam the seasonal aisle at Target, looking for mathematical inspiration. So far I’ve shared photos I’ve taken at Halloween, Valentine’s Day, and Easter. You can find them all here.

Today I was stopping by Target for some bug spray which just so happens to be next to the summer seasonal aisle. I couldn’t resist the urge to take a stroll and take some pictures. Here’s what I’ve got for you today.

How many large wooden dice are in the package?

It’s totally obvious, right? For younger students, maybe not so much. But even after everyone is in agreement that it’s 6, what do you think they’re going to say once you reveal the answer?

Not what you were expecting, is it? You probably thought I was wasting your time starting with such a simple image. So now you get to wonder, “Why/How are there only 5 dice in this package?” Perhaps this will help:

That burlap bag has to fit somewhere!

Let’s move on to another large wooden product. How many dominoes are in this pack?

It might be a little hard to tell from this perspective. Let’s look at it another way.

Barring any more burlap sacks, you might just have the answer. Before we find out, stop and think, what answers are reasonable? What answers are not reasonable?

Ok, time to check if you’re right.

No surprises here. Although after the first image, I probably had you second guessing yourself. There’s something to be said about the importance of how we sequence tasks.

Speaking of sequencing tasks, let’s move on to another one. How many light bulbs on this string of lights?

I really like this box because you get this tiny 2 by 3 window, and yet it’s such a perfect amount to be able to figure out the rest. This would be one I’d love to give students a copy of the picture and let them try to show their thinking by pointing or drawing circles on it.

Again, this is a great time to ask, what answers are reasonable? What answers are not reasonable? Assuming the light bulbs do create a rectangular array, there are definitely some answers that are more reasonable than others.

After some fun discussion about arrays, it’s time to check the actual amount.

So fun! Like I said, I love this image. Let’s look at another package that caught my eye.

How many pieces of sidewalk chalk in this box?

I was pleasantly surprised to find that Crayola put arrays on top of all their summer art supplies. It’s like they were designed to inspire mathematical conversation! Granted, the box doesn’t give it away that the dots represent the pieces of chalk, I wouldn’t point it out to students. I’d let them wonder and make assumptions about it. It’ll turn out that their assumptions are completely right, and how satisfying that will be for them!

Since we’re talking about arrays, which means we’re talking about multiplication, let’s shift gears a bit to look at some equal groups.

How many plastic chairs in this stack?

And to throw a wrench into what looks to be a simple counting exercise, how much would it cost to buy the whole stack?

Now students have got some interesting choices about how they calculate the cost. The fact that half the stack is blue and half the stack is red is just icing on the mathematical discussion cake.

My final image from the summer seasonal aisle has been a real head scratcher for me.

How many water balloons do you estimate are in this package?

What is an estimate that is too low?

What is an estimate that is too high?

What is your estimate? How did you come up with that?

Take a look at the box from another angle, and see if you want to revise your estimate at all.

We clearly have groups – eight of them to be precise – but the question I’m not entirely sure about is whether there are eight equal groups. Maybe? And if there are equal groups, then there are certain answers that are more reasonable than others.

I’ll give you a moment to think about why this is confusing me a bit.

Assuming there is an equal amount of each color, this doesn’t make any sense! But then I noticed the small white tag on the set of purple balloons.

Oh! That explains it. There’re only 260 balloons in here so…no, that still doesn’t work if these are eight equal groups.

Oh, then maybe it’s 5 more than 265 so it’s actually 270 so…no, that doesn’t work either. So I’m left to conclude that either this is not a pack with eight equal groups or there is some funny math going on! Sadly, \$25 is a bit steep to satisfy my curiosity. If any of you purchase this pack and want to count balloons, I’d love to get the full story.

And with that, my tour of the summer seasonal aisle comes to an end. If you’re just finishing the school year, bookmark this post to revisit when school gets back in session. What a fun way to start the year! If you’re still going strong, then I hope you’re able to use these to spark some fun, mathematical discussions in your classrooms.

# Order All The Pizzas!

In Dan Meyer’s recent talk at NCTM, he shared some contrived examples of “real world” math, including this one about congruent triangles found on the tail of an orca:

Pretty ridiculous, right?

But then some days you really do find some math out in the real world, and you can’t help but snap a picture:

I mean, holy cow! So many boxes – and one would presume – so many pizzas! I couldn’t help but take a picture and share on Twitter. The photo grabbed the attention of a few folks:

What makes this image so much more compelling than the whale tail? Both are photographs and therefore “real world.” Both have connections to math concepts. And yet one is ridiculous (not in a good way) while the other prompts thoughtful notice and wondering.

To me the difference has to do with two things – novelty and narrative. While there is a tourism industry around whale watching in person, there is nothing particularly novel about seeing a photo of a whale’s tail sticking out of the water. In addition, the textbook photo doesn’t even hint at a story. It’s a tail. It’s sticking out of the water. It’s likely going to go back in the water. Even worse, that flimsy narrative has nothing at all to do with congruent triangles.

The pizza picture, on the other hand, is extremely novel, assuming you don’t work at a pizza parlor. So much so that I felt compelled to not only stop and take a picture but also post it on Twitter for others to see. The picture taunts you with a narrative. What’s going on here? Why are there so many pizza boxes stacked on this table?

I couldn’t help but get to the bottom of it.

As I ate lunch, I watched as the guy put together even more pizza boxes. He eventually spread over two tables, and he kept consulting these long receipts.

I couldn’t help myself. I finally went over and asked who the order was for. It turns out a hospital had ordered 78 pizzas. 78!! Not only that, they had an order for 88 pizzas that afternoon followed by another order of 78 pizzas. And(!) they had an order for 88 pizzas the night before.

I asked how long it would take to make all 78 pizzas. I couldn’t believe my ears when she told me an hour to make them all and 40 minutes to bake them. Holy cow! 78 pizzas in less than two hours?! It just boggles the mind.

And why is a hospital ordering so many pizzas? Here’s a wonderful idea shared on Twitter. I hope it’s true.

Novelty and narrative, two factors that make the real world real and interesting to talk about in math class.

If you happen to want to share this with your students to see what they notice and wonder, here’s the final photo I took of all the boxes stacked up:

And here’s a photo with some additional information about the sizes of pizza and the number of slices. By the way, all of the pizzas in this order were large.

# Inspiration

Tonight I hosted #ElemMathChat and our topic was inspiration. Specifically, what inspires you as you’re planning for and teaching math?

One place I’ve found a great deal of inspiration is the seasonal aisle at Target. Honestly, inspiration can be found at just about any store, but the seasonal aisle is a particularly rich source of inspiration because it taps into the novelty and appeal of holidays.

“What can I do with this?” That’s the question I carried with me as I wandered the Easter aisle this week, wondering what mathematics I could draw out of the colorful assortment of products around me. I shared a few examples during #ElemMathChat tonight. I’ll share those here along with several more examples I couldn’t squeeze into the hour-long chat.

If you’d like even more examples, check out these posts I wrote around Halloween and Valentine’s Day:

As you’re reading this post, I challenge you to continually ask yourself “What can I do with this?” because you might notice something I didn’t and be inspired to ask a different question or draw out different mathematical ideas. If that’s the case, I’d love to hear about it in the comments!

Let’s get started!

## Jelly Beans

How many jelly beans are in this bag? What is an estimate that is too HIGH? Too LOW? Just right?

When estimating, our goal is to come up with a reasonable guess. The reasonableness comes from our guess lying within a particular range of numbers that makes sense. You could easily say that your “too low” guess is 1 because you know there is more than 1 jelly bean in the bag. You could also say your “too high” guess is 10,000 because it is unlikely there are 10,000 jelly beans in this one bag. But those are just cop out answers, not reasonable estimates. They don’t demonstrate any understanding of what makes sense given the picture of the bag and the window showing some of the jelly beans.

If you share this picture with your students, see if you can get them to take risks as they estimate. For example, I can count about 12 jelly beans in the bag’s window. I’m going to guess there are at least 10 groups of 12 jelly beans in the entire bag for a low-ball estimate of 120 jelly beans. However, I don’t think there’s enough room for 25 groups of 12 jelly beans in the bag, so my high-ball estimate is 300 jelly beans. I think the actual number is somewhere in the middle around 200 jelly beans.

See how much more narrow my range is? I think the number of jelly beans is somewhere between 120 and 300 jelly beans. In some ways that’s still a fairly broad range, but it’s so much more reasonable (and riskier!) than saying there are between 1 and 10,000 jelly beans in the bag.

And now for the reveal:

Notice I didn’t give the actual answer. I’d want my students to use the information provided to find out about how many jelly beans are actually in the bag. Depending on the grade, this could be a great impromptu number talk to find the product of 23 × 9.

We’ve talked about one bag of jelly beans, but let’s compare that to some others. Which of these bags do you think has the least jelly beans? The most? How do you know? (Click the pictures to enlarge them.)

After some discussion and estimating, reveal this image for the SweetTarts bag. How does this bag compare to the Nerds jelly beans? Can you compare without calculating?

Some students will likely calculate the products regardless, but I would want to make sure it also came out that both packages have 9 servings. The serving size in the SweetTarts bag is larger so the total amount of jelly beans in that bag is greater than in the Nerds bag. In other words, 31 × 9 > 23 × 9 because you are multiplying 9 by a greater number in the first expression, so the resulting product will be greater.

After that discussion, it’s time to reveal the answer for the third bag. A challenge to students: Can you compare the quantity in this bag to the other two without calculating the actual product?

## Which One Doesn’t Belong?

If you’ve never checked out the site Which One Doesn’t Belong?, I highly recommend it. The basic gist is that students are presented four images and they have to choose one and justify why it doesn’t belong with the other three. The twist is that there isn’t one right answer. You can make a case for why any of the four pictures doesn’t belong with the other three.

Look at the four pictures below. Find a reason why each one doesn’t belong.

And here’s another example, this time involving candy:

You’ll notice I’m not providing answers, because there isn’t one right answer! To quote Christopher Danielson, “It’s not about being right. It’s about being true.”

## Chocolate Bunnies

Here are some questions that came to my mind:

• How many chocolate bunnies are left? Can you find the number in another way?
• How many chocolate bunnies have been sold? Can you find the number in another way?
• If each bunny costs 75¢, how much will it cost to buy the remaining bunnies?
• What fraction of each package has (not) been sold?

## Peeps

How many Peeps are in this package? What is an estimate that is too HIGH? Too LOW? Just right?

The quantity is smaller and you can see so many of them that I would want students to be very narrow in their range of estimates and very clear in their justifications.

We know it’s a number divisible by 3 because there are three rows. We also know there are at least 3 Peeps in each row – we can see those! I would estimate 12 (four per row) is too low and 18 (six per row) is too high. My just right estimate therefore is 15 because I think there’s room for more than 4 in each row but not enough room for 6.

This might be a tad controversial because some folks associate estimating with numbers that end in 0 or 5, such as 25, 75, 100, 900. However, given the facts – three rows – I know the total number has to be divisible by 3. That means estimates like 12, 15, and 18 make much more sense to me than 10 or 20. That’s not to say that 10 and 20 are unreasonable estimates – they’re decent in this example – but I’m not going to limit myself to just those numbers given what I know about the configuration of Peeps.

And here’s the reveal:

But it doesn’t end there! Now that you know the quantity in one package, what can you tell me about the number of Peeps in this case?

And to take it another step further, here’s the price of one package. How much would it cost to buy half the case? How many Peeps would I be getting?

I love the layering in this example because it starts out so simple – estimating how many Peeps in one pack – but it really takes off from there with a few added details.

## Easter Eggs

How many eggs in my hand? What is an estimate that is too HIGH? Too LOW? Just right?

This one is trickier because the eggs are not arranged neatly like the Peeps. In this case I’m probably going to use numbers like 5, 10, or 20 to make my estimates.

However, this question is also a bit tricky because of how I worded the question. Did you notice?

Let’s take a look at the front of the package.

Students might be drawn quickly to 18 as the answer, but that’s not quite it. If you read carefully, it says “18 colored eggs and one golden egg” which brings the total to 19. But that’s not quite right either. I asked how many eggs in my hand, and if you’re noticing the shape of the container, there are actually 20 eggs in my hand. Sneaky!

So, if there are 20 eggs in my hand, how many colored eggs inside these 5 containers? (I would say “on this shelf” but students might get caught up in the fact that you can see there are more containers in the back. I want to focus just on the five up front.)

This is another chance for an impromptu number talk. I especially like how it can build off the discussion about the number of eggs from the previous image. You can start with 20 × 5 and back up to remove the 5 large egg containers (I asked about the colored eggs inside) and the 5 golden eggs (I asked about the colored eggs, and the packaging does not include gold as a colored egg. This is semantics though, so I might accept these in the total since gold is a color.)

Now that we’ve talked a bit about this package, let’s do some comparing. Which would you rather buy – one package of the eggs we just talked about or two packs that each have 12 eggs in them.

In case you missed it, the price for the package on the left is \$5.00. It’s printed on the label. The price for the packages on the right is 89¢ each. (I would probably ignore the Buy One, Get One 50% Off unless you wanted to take into account that wrinkle.)

Notice I didn’t ask, “Which is cheaper?” I asked, “Which would you rather buy?” On cost alone the two dozen eggs is significantly cheaper, but there are some definite perks to the \$5.00 package. Again, it’s about being true, not correct. So as long as students are able to defend their choice, that’s what matters.

For this next one I would probably change up the question and ask, “Which is the better deal – 1 pack of 48 eggs or 4 packs of 12 eggs?”

The price you see in the left picture – \$2.50 – is the cost of 1 pack of 48 eggs. Ignoring the buy one, get one 50% off, the left picture is a clear bargain. However, this might be a good time to tell students that for every one pack of 12 eggs, you get a second for half off. Then I would challenge them to determine the price of 4 packs given that discount. It’s definitely a closer answer when you take that into account!

## Coconut Macaroons

I don’t know that I associate coconut with Easter, but I had to share these packages of coconut that caught my eye in the Easter aisle.

How many cups of shredded coconut in this package? What is an estimate that is too HIGH? Too LOW? Just right?

Here’s the reveal, which is why these packages caught my eye:

Such an oddly specific amount! So if I bought all of these bags of coconut, how many cups of coconut would I be getting? How much would the three bags cost?

There’s a recipe for coconut macaroons on the back of the package. If I bought three bags of coconut, how many cookies could I make?

I like this because students have to wade through a lot of information to find what they need. Oftentimes in math problems we make needed information stand out or we don’t provide any distractions at all. It’s good to make students work for it a bit like they would have to do in the real world if they wanted to bake these cookies.

Another question I thought of is, “How long does it take to make 3 dozen macaroons?” This provides another opportunity for reading the recipe to search for relevant information. Students might just add 15 minutes and 20 minutes, but that’s only if you can fit all 36 cookies in the oven at the same time. If you only have one baking sheet that can hold 12 cookies at a time – which is about all I can do at home – then how long will it take? What if you could squeeze 18 cookies on a cookie sheet? How much time would you save?

## Miscellaneous

I’m going to close out this post with a final set of pictures that might inspire you to share them with your students and prompt some mathematical discussions. (Click the pictures to enlarge them.)

I get a kick out of this last one because it’s pretty easy to tell how many candies are in the package.

I can foresee some really interesting discussion when you reveal what the packaging says about the number of candies contained within.

## Final Thoughts

Please feel free to use these pictures with your students. I’d love to hear about the conversations they spark. If you get inspired to use them in ways I didn’t think about, please share in the comments. That way we can all learn and get ideas from one another!

# My Favorite: Holidays at Target

Here we are in Week 2 of the ExploreMTBoS 2016 Blogging Initiative! This week’s challenge is to blog about one of my favorite things. During this school year, one of my favorite things has been visiting Target during the holidays. The holiday-themed merchandise is rich with mathematical possibilities! I already wrote three posts about a treasure trove of images from Halloween:

Valentine’s Day is around the corner, and I snapped some photos this evening to share with you. I’m going to cover a range of mathematical skills – mostly centered around estimation –  from Kinder through about Grade 6 to show you just how versatile this stuff is!

These first two images are good for estimating quantity. You can estimate the quantities individually. Don’t forget to ask students to estimate an answer that is TOO HIGH and one that is TOO LOW in addition to their actual estimate. Coming up with a reasonable range takes a lot of practice! You could also show students both images at the same time and ask, “Which package has more?”

I forgot to snap a picture of the answers, but I can tell you there are 15 bouncy balls and 24 eraser rings.

Here’s another one. How many Kisses are in the box?

I was kind of surprised that the answer wasn’t an even number like 10 or 12. This just seems oddly specific.

Students tend to estimate better when the quantities are smaller. Here’s a larger quantity package to up the challenge a bit. How many gumballs are in the bag?

I was kind of surprised to find out the answer myself.

This next one is tricky! How many truffles are in the box? Go ahead and make an estimate.

Now that you’ve made your estimate, I’d like to show you how deceptive product packaging can be. Would you like to revise your estimate?

And now for the reveal. How does your estimate compare to the actual amount?

The first few images dealt with disorganized quantities. Once we move into organization, the thinking can extend into multiplicative reasoning. The great thing is that it doesn’t have to! Students can find the total by counting by 1s, skip counting, and/or using multiplication.

There are several questions you can ask about these pictures. They’re of the same box. I just gave different perspective. I’d probably show the almost-front view first to see what kids think before showing the top-down view.

• How many boxes of chocolate were in the case when it was full?
• How many boxes of chocolate are left?
• How many boxes of chocolate are gone?

Here’s another package that could prove a bit tricky for some students. How many heart stickers are in this package?

Students might notice that the package says 2 sheets. If they don’t, you might show them the package from a different perspective.

And finally, you can reveal the total.

This next package can be shown one of two ways depending on how much challenge you want to provide the students. Even with some of the hearts covered, students can still reason about the total quantity.

This next one could simply be used to ask how many squares of chocolate are in the box, but what I’d really like to know is how many ounces/grams of chocolate are in the box.

After some estimating, you could show your students this and let them flex their decimal computation skills to find the total.

However, the reveal is likely to raise some eyebrows.

And finally, you can do some more decimal calculations with this final product. How much would it cost to buy all of the boxes shown?

And if you bought all 6 boxes, how many ounces of chocolate would you be getting?

Ten minutes in the holiday aisle and my iPhone are all it took to gather this wealth of math questions can now be shared with students. Even better, I didn’t have to purchase any of these products! Even better than that, I can go back for every major holiday to capture new images that will feel timely and relevant!

By the way, feel free to use any and all of these images with your own students. They’re fairly low quality so I don’t recommend printing them, but they should look just fine projected or shown on a screen.

Happy Valentine’s Day!

# 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?

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?

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?

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.)

# 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.