Category Archives: Math Teaching

The Slow Reveal

This year my colleague Regina Payne and I tried something new as we visited classrooms across our district – numberless graphs. Similar to a numberless word problem, you present a graph with no numbers and proceed to have a rich mathematical discussion as you slowly reveal more and more information on the graph. Early in the school year, I shared a Halloween-themed numberless graph, and I also wrote a blog post about it.

We briefly touched on this work in our session at the 2017 NCTM annual conference, and it’s been exciting to see my #MTBoS colleagues taking the idea and running with it in their schools! In case you don’t follow them – which will hopefully change after reading this post! – I want to share their work so you don’t miss out on all the great stuff they’re doing.

Kassia Wedekind

Kassia has written two wonderful blog posts about how she took our ideas and tinkered with them to create a data routine called Notice and Wonder Graphs. I like this name because it’s more inclusive than numberless graphs. When it comes to graphs, you might hide the numbers, but you could just as easily hide other parts of the graph first. It all depends on your goals and how you want the conversation to unfold. In Kassia’s first post, she shares this graph with students. Notice it has numbers, and little else.

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Curious what it’s about? Then check out Kassia’s post. I’m betting you’ll be quite surprised when you reach the final reveal.

I will share this snippet from her post:

I love this routine for many of the reasons that I love Brian’s numberless word problems–it slows the thinking down and focuses on sense-making rather than answer-getting.

But I also love it because it brings out the storytelling aspect of data. So often in school (especially elementary school!) we analyze fake data. Or, perhaps worse, we create the same “What is your favorite ice cream flavor?” graph year after year after year for no apparent purpose.

I’ve decided to make it a goal to think more about data as storytelling, data as a way to investigate the world, and data as a tool for action. In my next two posts (YES, people! I’m firing the ole blog back up again!) I’m going to delve into the idea that we can use data to discuss social justice ideas and critical literacy at the elementary level. I’m just dipping my toe into this waters, but I’m really excited about it!

And Kassia did just that! So far she’s followed up with one post where her students noticed and wondered about a graph showing the percent of drivers pulled over by police in 2011, by race. I love how the graph sparked a curiosity that got her students to dive more deeply into the data. How often does a graph about favorite desserts or our birthday months spark much of any curiosity?

Jenna Laib

Jenna shared a numberless graph that immediately got me curious! This is one she created to use with 6th grade students.

Jenna01

I can’t help but notice a bunch of dots grouped up at the beginning with a just few outliers streeetttcchhiiing across almost to the very end.

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Once she included some numbers, my first instinct was that this graph is about ages. Apparently I wasn’t alone in that assumption!

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And then there’s the final reveal.

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So why did Jenna create and share this graph? What was her mathematical goal?

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I especially loved this observation about how her students treated the dot at 55 before they had the full context about what the graph is really about.

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Chase Orton

Chase wrote a detailed post about how he worked with 2nd grade teachers to do a lesson study about interpreting graphs.

…there’s so many rich opportunities for meaningful student discourse about data.  That is, if it’s done right.  Most textbooks suck all the life out of the content.  Students need to understand that data tells a story; it has contextual meaning that is both cohesive and incomplete.  Students need to learn how to ask questions about data and to learn to identify information gaps.  In other words, students need to learn to be active mathematical agents rather than passive mathematical consumers.

Chase walks you through the lesson he and the teachers created and tried out in three different classrooms. I love how he details all of the steps and even shares the slides they used in case you want to use them in your own classroom.

He closes the post with a great list of noticings and wonderings about continuing this work going forward. Here are a couple of them about numberless graphs specifically:

  • We need to give students more choice and voice about how they make meaning of problems and which problems they choose to solve.  Numberless Data problems like these can be be a tool for that.
  • The missing information in the graph created more engagement.

A huge thank you to Kassia, Jenna, and Chase for trying out numberless graphs and sharing their experiences so we can all be inspired and learn from them. I can’t wait to see how this work continues to grow and develop next school year!

If you’re interested in reading more first-hand accounts of teachers using numberless word problems and graphs, be sure to check out the ever-growing blog post collection on my Numberless Word Problems page. I recently added a post by Kristen Acosta that I really like. I’m especially intrigued by a graphic organizer she created to help students record their thinking at various points during the numberless problem. Check it out!

Trick or Treat!

Now that I’ve completed sets of numberless word problems for all of the addition and subtraction CGI problem types, I wanted to do something fun.

This school year, my co-worker Regina Payne and I have been visiting the teachers in our Math Rocks cohort. One of the things they’ve been graciously letting us do is model how to facilitate a numberless word problems. In addition to making this a learning experience for the teachers, we’ve made it a learning experience for ourselves by putting a twist on the numberless word problem format.

Instead of your usual wordy word problem, we’ve been trying out problems that include visuals, specifically graphs. Instead of revealing numbers one at a time, we’ve been revealing parts of the graph. Let me walk you through an example I made tonight.

Here’s the graph I started with. I created it with some data I found on the Internet.

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If I threw this graph at a 4th or 5th grader along with a word problem, they would probably ignore what the graph is all about and just focus on getting the numbers they need for doing whatever computations they’ve decided to do. They would probably also ignore a vital piece of information – the scale that says “In Millions” – which means their answer is going to be about 1,000,000 times too small.

But what if we could change that by starting with something a little more accessible like this?

graph01

What do you notice? What do you wonder?

I’m guessing at least one student in the class would comment that it looks like a bar graph. Interesting. What do you think this bar graph could represent?

Oh, and you think a bar is missing in the middle. Interesting. What makes you say that?

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What new information was added to the graph? How does it change your thinking?

Oh, so there is a bar between Hershey’s and M&M’s. How tall do you think the bar for Snickers might be? Why do you say that?

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Now we know how tall the bar for Snickers is. How does that compare to our predictions?

Considering everything we know so far, what do you think this bar graph is about? What other information do we need in order to get the full story of this graph?

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What new information was added to the graph? How does it change your thinking about what this graph is about?

What are Sales? How do they relate to candy?

What does “In Millions” mean? How does that relate to Sales?

I know we don’t have any numbers yet, but what relationships do you see in the graph? What comparisons can you make?

graph05

What new information was added? How does it change your thinking?

Hmm, how many dollars in sales do you think each bar represents? How did you decide?

graph06

How do the actual numbers compare to your estimates?

What were the total sales for Reese’s in 2013? (I’d sneak in this question if I felt like the students needed a reminder about the scale being in millions.)

What are some other questions you could use answer using the data in this bar graph?

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What is this question asking?

How can you use the information in the graph to help you answer this question?

*****

I may or may not actually show that last slide. After reading this blog post by one of our instructional coaches Leilani Losli, I like the idea of letting the students generate and answer their own questions. In addition to being motivating for the students, it makes my time creating the graph well spent. I don’t want to spend a lot of time digging up data, making a graph, and then asking my students a whopping one question about it! That doesn’t motivate me to make more graphs. I  also want students to recognize that we can ask lots of different questions to make sense of data to better understand the story its telling.

Some thoughts before I close. This takes longer than your typical numberless word problem. There are a lot more reveals. Don’t be surprised if this takes you at least 15-20 minutes when you take into account all of the discussion. When I first do a graphing problem like this with a class, it’s worth the time. I like the extra scaffolding. Kids without a lot of sense making practice tend to be pretty terrible about paying attention to details in graphs, especially if their focus is on solving an accompanying word problem.

If I were to use this type of problem more frequently with a group of students, I could probably start to get away with fewer and fewer reveals. Remember, the numberless word problem routine is a scaffold not a crutch. My hope is that over time the students will develop good habits for attending to features and data in graphs on their own. If you’re looking for a transition to scaffold away from numberless and toward independence, you might start by showing the full graph and then have students notice and wonder about it before revealing the accompanying word problem.

If you’d like to try out this problem, here’s a link to a slideshow with all of the graph reveals. You’ll notice blank slides interspersed throughout. I’ve found that if you have a clicker or mouse that has a tendency to jump ahead a slide or two, the blank slide can prevent accidental reveals. It also helps with graphs because when I snip the pictures in they aren’t always exactly the same size. If the blank slides weren’t there, you’d probably notice the slight differences immediately, but clearing the screen between reveals mitigates that problem.

Happy Halloween!

Decisions, Decisions

This week our Math Rocks cohort met for the fourth time. We had two full days together in July, and we had our first after school session two weeks ago. One of our aims this year is to create a community of practice around an instructional routine, specifically the number talks routine. We spent a full day building a shared understanding of number talks back in July. You can read about that here. We also debriefed a bit about them during our session two weeks ago.

This week we put the spotlight on number talks again. We actually broke the group up by grade levels to focus our conversations. Regina led our K-2 teachers while I led our 3-5 teachers. The purpose of today’s session was to think about the decisions we have to make as teachers as we record students’ strategies. How do you accurately capture what a student is saying while at the same time creating a representation that everyone else in the class can analyze and potentially learn from?

We started the session with a little noticing and wondering about various representations of 65 – 32:

mr01

Very quickly someone brought up exactly what I was hoping for which is that some of the representations show similar strategies but in different ways. For example, the number line in the top left corner shows a strategy of counting back and so do the equations closer to the bottom right corner.

This discussion also led into another discussion about the constant difference strategy – what it is and how it works. It wasn’t exactly in my plans to go into detail about it this afternoon, but since my secondary goal for the day was to focus specifically on recording subtraction strategies, it seemed a worthwhile time investment.

After our discussions I shared the following two slides that I recreated from an amazing session I attended by Pam Harris back in May. (For the record, every session I attend with her is amazing.)

The first slide differentiates strategies from models. Basically, if you have students telling you their strategy is, “I did a number line,” and you’re cool with that, then you should read this slide closely:

mr02

The second slide differentiates tools for building relationships from tools for computation. This slide is crucial because it shows that while we want students to use tools like a hundred chart to learn about navigating numbers within 100, the goal is to eventually draw out worthwhile strategies, such as jumping forward and/or backward by 10s and then 1s.

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The strategy on the right that shows 32 + 30 followed by 62 + 3 is totally the type of strategy students should eventually do symbolically after building relationships with a tool like the hundred chart.

After blowing their minds with those two slides, I led them in a number talk of 52 – 37. During my recording of their strategies, I stopped a lot to talk about why I chose to do what I did, to solicit their feedback, and even to make some changes on the fly based on our discussion.

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For example, in the top right corner of the board I initially used equations to represent a compensation strategy. Someone asked if this could be modeled on a number line because she thought it might make more sense, so I did just that in the top left corner. By the time we were done they were like, “Oh, hey! That ends up looking like a strip diagram!”

It was amusing that the first strategies they shared involved constant difference. They were so excited about learning how the strategy worked that they wanted to give it a try. I didn’t want to quash their excitement by telling them that the strategy tends to work better, especially for students, when you adjust the second number to a multiple of ten. I wanted to stay focused on my goals for the day. We’ll discuss the strategy more in a future session.

(Unless you’re in Math Rocks and you’re reading this! In which case, see if you can figure out why that’s the case and share it at our next meeting.)

After some great discussion about recording a variety of strategies, we watched Kristin Gray in action leading a number talk of 61 – 27.

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We talked about how she recorded the students’ strategies. We also talked about some really lovely teacher moves that I made sure to draw attention to.

We wrapped up our time together talking about what new ideas they learned that they wanted to try out with their students. I had asked one of the teachers to lead us in another number talk, but we ran out of time so I think I’m going to have her do that at the start of our next session together. Hopefully everyone will have had some intentional experiences with recording strategies between now and then to draw on during that number talk.

Oh, another thing we talked about at various points during the session was how to lead students in the direction of certain strategies. This gets into problem strings, which may or may not happen in number talks depending on whom you talk to. Regardless, here are some we came up with. Can you figure out what strategies they might be leading students to notice and think about?

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Math Rocks Redux Part 2

At the end of July, @reginarocks and I kicked off our second Math Rocks cohort – a group of 30 or so elementary educators that meets for almost 30 hours across 7 months. Our first cohort, which ran last school year, was a success, but when it came time to plan for year 2, we definitely found ourselves wondering how we could provide an even better learning experience this year.

The other day I wrote about the tweaks we made to day 1 of Math Rocks. All in all, the tweaks were minor – you can read about that session here – but day 2 was completely overhauled! That’s what I’d like to write about today.

But first, let me bring you up to speed on some things that happened to influence my decisions about day 2. Last year, Regina and I delivered a lot of PD across a wide variety of topics and audiences  – diagnostic assessments with interventionists, fraction sense with grades 3-5 teachers, developing number concepts with grades K-2 teachers, weight and liquid volume measurement with grade 3 teachers, spiral review strategies for grades 3-5 – but the topic that seemed to resonate the most with our teachers was number talks. Across four half-day sessions, we ended up delivering an introduction to number talks to approximately 150 of our elementary teachers! I wrote about the experience here if you’d like to read about it.

Last year’s Math Rocks cohort also dove into number talks. As part of our work together we joined a book study of Making Number Talks Matter led by Kristin Gray and Crystal Morey. Our group loved it, but because the book study was mostly discussed online via Twitter and Teaching Channel forums, I realized later we didn’t do enough work in person to talk about and work through issues that came up to support our teachers as they took on this new practice.

Fast forward to Twitter Math Camp this summer, and I had the opportunity to take part in an incredible PD experience with David Wees, Jasper DeAntonio, and Katilin Ruggiero in their session titled “Rehearsing Instructional Routines Together.” You can access all of the slides and materials from the session on the Twitter Math Camp wiki here. Their session focused on teaching us the Contemplate then Calculate routine – which I now love! – but the structure of the PD itself is what captured my attention most. So much so that I borrowed liberally from their work when designing day 2 of Math Rocks!

Day 2 of Math Rocks followed this structure:

  • Regina, Jan, and I each model a number talk
  • Math Rocks participants unpack the components of a number talk
  • Math Rocks participants plan their own number talk in pairs or trios
  • Math Rocks participants rehearse their number talks for the group

All of this work drove us toward our two goals for the day:

  1. Dive deeply into the number talks routine
  2. Develop a community of practice that can more precisely talk about our teaching

The day started with Regina, Jan, and I each modeling a number talk. This was challenging to plan. One of the key pieces of David’s session at Twitter Math Camp was instructional routines. Contemplate then Calculate is a routine that is broken down into very discrete steps. In order to bring this to our teachers in my district, I had to think about what the steps of a number talk are supposed to be.

What ends up making this challenging is that what makes up a number talk is not universally agreed upon. A big point of contention has to do with how many problems you do in a number talk. Some people say number talks should focus on one problem and all the strategies used to solve that one problem, while others say a number talk can involve multiple problems to solve and discuss. Those that disagree say that having multiple problems is called a number string, not a number talk. Yay, semantics!

For the purposes of my work with my Math Rocks cohort, I opted to say a number talk can include more than one problem for the sheer fact that Sherry Parrish’s book Number Talks, which we have 6 copies of on all 34 of our elementary campuses, does present number talks as strings of problems. The sample number talks videos on the DVD all show teachers modeling strings, and all or nearly all of the sample pre-planned number talks that are shared in the book are strings as well. Knowing my teachers will be using Sherry Parrish’s book as a resource, I opted to define the routine as having multiple problems to solve, but I did not define how many problems.

When deciding what the components of our number talks instructional routine would be, I also consulted this document from Math Perspectives. Here’s how they delineate the routine:

NTs

Finally, I took all of that and simplified the number of steps to make the routine feel smooth and easy to follow. Here’s what I presented my teachers during day 2 of Math Rocks:

NTs-Components

Making the Math Rocks folks sit through three number talks might sound like overkill, but it served two purposes. First, we wanted to model number talks across grades to demonstrate that this routine is appropriate across the elementary grades. The three number talks we modeled came from mathematics in Kindergarten, 2nd grade, and 4th grade.

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Second, modeling so many number talks ensured we as a group had three shared experiences to draw upon when unpacking the routine later. We wanted the participants to really be able to unpack and analyze each of the components of the number talk, and in order to do that they needed to have seen each of the components enough times to have meaningful conversations about them.

After modeling the three number talks, we used the Ideas Carousel protocol to unpack the components of the number talks routine. (Just as a reminder, I borrowed liberally from David’s sessions. This protocol came from his session, too. )

Here’s how the protocol works. We made a poster for each of the components of the number talks routine, and participants chose a component to unpack. With their group, they recorded their understandings of the parts of that component, the rationale(s) for each of those parts, and any questions/wonderings they had.

Once the groups had a chance to dirty up their posters, they started rotating through the remaining posters. At each poster, they had to read the poster, check ideas that resonated with them, add new ideas, star ideas they wanted to discuss as a group, and circle the idea their group thought was most important on that poster.

After interacting with each poster, they took one last gallery walk through all of the posters before returning to their original poster. Once there, they read over their original comments and all of the extra things added by everyone else, and they marked anything that surprised them. Here are their completed posters:

Finally, as a group we talked through their wonderings, a-ha moments, and anything else that came up. It was such a rich conversation and demonstrated that we have a lot of interesting questions to explore this year. For example:

  • How do you do number talks in an intervention group that meets for only 30 minutes daily and is composed of students who are reluctant to participate or try out different strategies?
  • How do you modify number talks for emergent bilingual students? Sharing their strategies verbally may be too much of a challenge. What can we do to accommodate them?
  • How do you know what to record when students are talking about their strategies? How do you get better at that?

This really gets at one of our goals for this day of learning – creating a community of practice that can more precisely talk about our teaching. I don’t have all the answers for them, and how much more interesting is it that we as a group get to explore and discover our own answers through our experiences this year? We get to decide what works (and doesn’t) for our students, and we have a group of people to do that important work with.

Now that we had accomplished our other goal for the day – diving deeply into the number talks routine – we gave the participants time to plan their own number talks. We grouped them by grade levels to plan, though we did have one team composed of a special education teachers and two interventionists.

Finally, we had time for some of them to rehearse their number talks in front of the rest of the group. I reiterated a key thing David Wees said in our Twitter Math Camp session: the purpose of this rehearsal is not to coach individual teachers to be better at number talks. Rather it’s to give us as a group an experience where we can talk about the act of teaching. I like the meaning behind it, but I also think it helps take the pressure off the teachers. It’s not about any one person at the front of the room, it’s about how it gives everyone an experience and ability to talk about the very messy work of teaching.

All in all it was a very intense and focused day, but I loved it! I think this was just the right experience to kick off our time together over the next 7 months. I look forward to the conversations and support we’ll be able to provide one another going forward. What I’d like to do during the school year is have different participants plan and rehearse number talks so we can continue talking about the routine. I also want to spend some time focusing on how we record students’ strategies so that everyone can feel more confident in this area so they can be more intentional about how they are representing students’ strategies for the rest of the class to benefit from.

Thank you to David, Jasper, and Kaitlin for providing an awesome experience that I was able to take back and adapt for my teachers! Special thanks to Jasper for his elevator speech that encouraged me to attend his session instead of the one I was originally planning to attend.

 

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:

WhaleTailMath.PNG

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:

Pizza01Pizza02Pizza03Pizza04Pizza05Pizza06Pizza07Pizza08

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.

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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.

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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.

Pizza11.jpg

My First Three-Act Lesson

My co-worker Regina and I took a stab at our first three-act math lesson. Well, we took a stab at writing a lesson to provide some motivation for learning about measuring liquid volume, and it sort of morphed into a three-act lesson along the way. However we got there, it was fun to write, and the teachers we shared it with at a PD session in March really enjoyed it. Here’s hoping I get the chance to develop another three-act lesson sooner rather than later!

Writing this lesson came hot on the heels of spending a day with Dan Meyer at the recent Texas Association of Supervisors of Mathematics meeting. He offered some advice for designing engaging learning experiences that I couldn’t wait to try out:

  • Start a fight
  • Turn the math dial down
  • Create a headache

If you’re intrigued by his advice – and I hope you are – I recommend checking out his recent talk at NCTM. You’re only going to get about 45 minutes with his ideas about engagement instead of the 6 or so hours I got, but I guarantee it is still time well spent.

A Gallon of Ice

Standards

  • Texas: 3.7D and 3.7E
  • CCSS: 3.MD.2

Act 1

Watch the video.

  1. What do you notice? What do you wonder?
  2. How long do you think it will take for all of the ice to melt? Estimate – Write an estimate that is too low, an estimate that is too high, and your just right estimate.
  3. How much water will be in the jug after all the ice melts?

I recommend bringing in an empty milk jug so students can draw small mark and their initials on the side of the jug to show their estimate. Start a fight! The students will want to know if their answer is correct. I did this with teachers during a PD session, and they had quite a range of answers. At this point, the math dial is turned down low, so we did not talk about units of measurement, just an estimate of how high the water will fill the jug once the ice is melted.

JugLines

Act 2

Watch the video.

  1. How long did it take the ice to melt? (Sadly, it finished melting while I was sleeping, so the most precise answer we can give is longer than 11 hours but less than 20 hours, since I checked the jug again at 7:00am.)
  2. Whose estimate was closest to the actual height of the water in the jug? (Resolve the controversy!)
  3. How much water is in the jug? Estimate – Write an estimate that is too low, an estimate that is too high, and your just right estimate.

This is where you start to slowly turn up the math dial. Question 3 is a great question to find out what your students already know about units of volume. They might very well be stumped depending on their prior experiences. You might have them imagine other packages and containers that have liquids in them and think if there are any words they know that describe how much liquid is inside. It’s totally fine for the estimates to be sort of weak here.

The whole purpose of this question is to create a bit of a headache – get the class to a point where you (or your students!) can say, “I think we need to know a bit more about measuring liquids so we can come up with estimates we’ll feel confident about,” and then take a break from this three-act lesson to do some explorations of measuring liquid volume. After doing that, which might take a day or two, show the Act 2 video again and then give the students a chance to add on or revise their estimates.

Here are some estimates made by 3rd grade teachers at our PD session:

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I can tell the teachers were hooked when they reacted in shock when they found out I wasn’t going to reveal the answer right away. Just like with students, we took a detour away from this lesson. We wanted to spend a bit of time sharing ideas for how students can explore measurements of liquid volume. But they wanted to know the answer! One of them was really worried and wanted to make sure we would tell them before they left the PD session.

I couldn’t have been happier.

Act 3

All is revealed! Now that your students have some personal experiences with measuring liquids using various units and you’ve given them a chance to add on or revise their estimates, it’s time to find out the actual volume!

And of course I spilled some water! When I was first filling the jug, I had to cut a flap in the top to make the opening wider for ice cubes to fit. Unfortunately, I forgot about it when I was doing my first pour and water did not come out like I was expecting. Thankfully it was only a small amount.

There’s so much going on in this video! You’ve got quarts, and half gallons, and cups, and fractions of cups. All great stuff to talk about! But I purposefully tallied the number of cups throughout the video so that students could at least come up with 8 2/3 cups. However, this is a great opportunity to talk about how we can read measurements differently depending on our units. For example:

  • 8 2/3 cups
  • 1/2 gallon and 2/3 cup
  • 69 1/2 ounces
  • 2 quarts and 2/3 cup

This is different from making conversions; it’s more about the choices available when reading a measurement off a tool. You don’t have to go here, but I think it is important for students to know that they do have choices in how they read a measurement given the options provided by the tool. Learning that flexibility here is only going to help them when they start encountering questions related to measurement conversions down the road.

And that’s a wrap! If you try out this lesson in your own classroom, I’d love to hear about it in the comments.

 

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!