Tag Archives: WODB

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.

My closing thought for the chat was this:

Q0-FinalThought

“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?

Q1

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:

Q1-Answer

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?

Q2-Answer2

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.

Q3

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

Q6

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

Take a look at this picture of two boxes of chocolate bunnies and ask yourself, “What questions could I ask about this?”

Q4

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?

Q5a

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:

Q5a-Answer

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?

Q5b

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?

Q5c

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?

Q7

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.

Q7-Answer

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.)

Q7b

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.

Q7c

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?”

Q9

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?

Q8a

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

Q8a-Answer

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?

Q8b-2

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?

Q8b-1

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.)

 

Bunnies

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

Q9d

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

Ferrero

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!

Go Big or Go Home: Math Rocks Day 1

My brain is full! I just finished two amazing days of PD with about 30 educators in my district. I promised I’d blog about it, and I need to because I just have so much going on in my head right now. Like I said, my brain is full!

This school year, I’m leading an advanced course with elementary teachers in my district. I didn’t really have any guidance beyond that, so it was left to me and my co-worker Regina to set some goals and make a plan. All we started with was a name: Math Rocks. And that’s only because our district already offers an advanced course called Reading Rocks.

Back in May, Regina and I put together an application and asked teachers to apply for this course that has never existed before. Amazingly enough, about 36 people took the time to apply. We read through their applications and selected 24 educators to be in our inaugural class. What I like about it is that we have a wide variety of folks – general education teachers K-5, a few instructional coaches, a TAG teacher, and a few interventionists. And within that group we have dual language teachers and inclusion teachers. They are so diverse; I’m excited about the varied perspectives they’ll bring to our work.

We kicked off the course yesterday and today. We’ll continue our work online for the next month before school starts. Once the school year begins, we’ll meet every other Thursday after school throughout the fall semester. We’ll continue into the spring semester with a final meeting in early February. It’s going to be awesome!

But let’s get back to the first two days. This is the most we’ll ever be together in one place: 12 intense hours across two days.

We opened the first day with a little estimation from Andrew Stadel’s Estimation 180. We of course did the task that started it all: How tall is Mr. Stadel?

After everyone made their estimates, we had them take a walk. Every time we asked a new question they had to find a new partner and introduce themselves. We went through the usual Estimation 180 questions:

  • What is an estimate that is too LOW?
  • What is an estimate that is too HIGH?
  • What is your estimate?

We also added some questions of our own:

  • Where’s the math?
  • Which grade levels could do this activity?
  • Which process standards did you use?

Take A Walk

This was a great way to get everyone up and moving at 8:30 in the morning, but it also started something they weren’t going to be aware of immediately. One thing I did very intentionally throughout the two days was embed FREE resources from my online PLC, the Math Twitter Blogosphere (MTBoS). Unbeknownst to everyone, one of my primary goals for the course is to connect them with this inspiring community. And what better way to entice them than by taking these two days to show off some of the rich resources this community creates and shares freely?

Community Circle

After our getting-to-know-you activity, we moved into a community circle. Regina set the tone by talking about why our district is excited about and invested in this course. Then everyone went around to introduce themselves to the group and talk a bit about why they chose to apply for the course. Their reasons varied, but there were some overriding themes. For many of us in the group, math is not a subject we loved as a kid. In fact, several folks went so far as to say they hated it growing up. On the bright side, these same folks want their students to have better experiences with math than they did. Everyone agreed that math is a rich subject, and they want their students to experience and appreciate that richness.

Their stories during the community circle provided a nice segue into our next activity. We asked the participants to reflect on their own experiences learning math. They had to choose three images that came to mind that symbolize what math was like to them as a student and sketch them on a blank sheet of paper. When everyone was finished, we did a gallery walk.

IMG_9587 IMG_9586 IMG_9585 IMG_9584 IMG_9583 IMG_9582 IMG_9581 IMG_9580 IMG_9579

There were a few recurring themes here as well. Many pictures showed formulas with variables. People said that they remembered being told to use these formulas because they would “work” but they never understood what they meant or why they were using them. Many pictures also showed numerous worksheets, indicating that math was more about quantity of problems than quality of reasoning or understanding. For those that said they disliked math as a child, we talked about when that started happening, and the group was split over it being Algebra or Geometry.

By the way, I’m sharing a lot of the negative experiences, mostly because I felt like I was hearing those most, but I do have to say that there were some voices of folks who did like math as a kid or they grew to like it as they got into higher grades. So negative stories were definitely not universal, which was encouraging.

After debriefing these experiences, we watched Tracy Zager’s talk from Shadow Con 2015. This was basically a small teacher-led mini-conference in the “shadow” of NCTM Boston (hence the name). All of the talks given at Shadow Con are available on the website, along with a facilitator’s guide if you’re interested in utilizing any of the videos in your own PD. Two of the videos really struck a chord with me and ended up becoming the inspiration for our two course goals.

Tracy’s video is called Breaking the Cycle. Here’s a short synopsis. I could write a whole blog post about this video and my thoughts on it, but really you should take 15 minutes and watch it for yourself. It’s powerful stuff.

The majority of elementary school teachers had negative experiences as math students, and many continue to dislike or avoid mathematics as adults. We’ll look at how we can better understand and support our colleagues, so they can reframe their personal relationships with math and teach better than they were taught.

We watched the video, debriefed, and then I shared our first goal for Math Rocks: Relationships.

MathRocksGoal1

We want our participants to focus on building relationships this year with:

  • their teammates,
  • their administrators,
  • me and Regina,
  • with their students, and
  • with other educators.

We also want them to build their relationship and their students’ relationships with mathematics.

To help them start working on this goal, we took Tracy’s call to action from the end of the video. Each participant chose a word from a word cloud that shows how mathematician’s describe math. Over the course of the next month, as they attend PD and prepare for the start of the school year, their mission is to plan for math instruction with that word as an inspiration and guide. We’ll revisit how this went when we meet back in September.

WordCloud

And then it was time for lunch. Whew! We crammed a lot in that morning.

After lunch we did a little math courtesy of Mary Bourrassa’s Which One Doesn’t Belong? If you’re unfamiliar with this site, students are presented an image of four things. They have to answer one question, “Which one doesn’t belong?” The fun part is that you can justify a reason why each one doesn’t belong. Here’s the one we did as a group:

Everyone had to pick one picture that doesn’t belong and go stand in a corner with other people who chose the same picture. Once they were grouped, they discussed with one another to see if their justifications were the same, and then we shared out as a group. Here are some of their reasonings:

  • The quarters don’t belong because they equal a whole dollar. The value of each of the other three pictures equals part of a dollar (4 cents, 5 cents, 40 cents).
  • The quarters don’t belong because the word you say for their value (one dollar, one hundred cents) doesn’t start with “f” like in the other three pictures (four, five, and forty cents).
  • The pennies don’t belong because they are not the same color as the other coins.
  • The pennies don’t belong because they are the only coin where the heads face right instead of left.
  • The nickel doesn’t belong because there is only one.
  • The dimes don’t belong because they are the only one where the tails side is showing.
  • The dimes don’t belong because the value of a dime has a 0 in the ones place. All the other coins have some number of ones in the ones place (5 ones in 25, 1 one in 1, 5 ones in 5).

Like Estimation 180, this activity was included intentionally because this is yet another FREE resource created by the MTBoS (pronounced “mit-boss”). It’s actually inspired by another FREE resource created by someone in the MTBoS, the Building Better Shapes Book by Christopher Danielson.

After talking about money, we prepared to watch Kristin Gray’s talk from Shadow Con. Hers is called Be Genuinely Curious, and you should take a few minutes to watch it for yourself:

When students enter our classroom, we ask them to be genuinely curious about the material they are learning each day: curious about numbers and their properties, about mathematical relationships, about why various patterns emerge, but do we, as teachers, bring that same curiosity to our classes? Through our own curiosities, we can gain a deeper understanding of our content and learn to follow the lead of our students in building productive, engaging and safe mathematical learning experiences. As teachers, if we are as genuinely curious about our work each day as we hope the students are about theirs, awesome things happen!

Again, we watched the video, debriefed, and then I shared our second goal for Math Rocks: Curiosity.

MathRocks2

We want participants to use their time in this course to get curious about mathematics, about teaching, and about their students. We also want them to find ways to spark their students’ curiosity about mathematics.

When you’re curious about something, you need resources to help you resolve your curiosities. I didn’t want the folks in this course to feel like we were going to leave them hanging. That’s when I formally introduced the MTBoS.

MTBoS

I told them the story of how I joined the MTBoS back in August 2012. (On a side note, it’s hard to believe I’m approaching my third anniversary of being part of this amazing community of educators!) This is a community that prides itself on freely sharing and supporting one another. If the educators in Math Rocks really want to take their math teaching to the next level, getting connected to a network like the MTBoS is the way to go.

One of the amazing things the MTBoS has done to help new members join and get started is to create Explore MTBoS. Periodically, the group kicks off an initiative to help new members start blogs and Twitter accounts. Unfortunately, there isn’t an initiative starting up right when Math Rocks is starting, so I started one up myself. I created a blog where I tailored the existing missions from Explore MTBoS to guide our group as they become members of this online PLC. We did the first two missions to wrap up the first day of Math Rocks. Each person had to make a blog and create a Twitter account.

I’ll admit, I was super stoked about this, but I’ll be honest that I threw more than a few people way out of their comfort zone that afternoon. Despite that, they still made their accounts, wrote their first blog posts, and sent out their first tweets. I am so proud of them for taking these steps, and I am eager to see where it leads from here.

That wraps up Day 1, our first 6 hours together. I’ll share Day 2 in another post.