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.

Pizza09.PNG

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.

Pizza10.PNG

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:

JugEstimates.PNG

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.

 

Weighty Matters

This year I won a grant from our district’s Partners In Education Foundation. (Yay!) With the money, I was able to purchase quite a few platform scales for every third grade team in our district. Today I got to visit a class using the scales, and I got to see the amazing Julie Hooper teach a lesson I developed with my partner Regina. It was so much fun!

The class started with a computation warm-up which made my math heart happy. It was so amazing to listen to Julie’s students solve the problem in so many different ways. They were so comfortable doing it, too. You can tell they have internalized the idea that they are able to solve problems in ways that make sense to them.

After the warm-up, the class dove into the day’s lesson. Julie started by asking the students to name things that are heavy and things that are light.

She asked some thought provoking questions after they had compiled their list.

  • Is 100 pounds heavy to you?
  • Do you think it’s heavy to a weight lifter?
  • Are big things always heavy?

I love how the conversation got the students thinking about their current conceptions of weight.

Next, the students had the opportunity to explore two different scales. Julie asked them to notice and wonder as they tried out the scales. I noticed that 3rd grade students *love* to put as many items as they can on the scale all at once. They couldn’t believe how much it took on the larger scale to make the dial move.

After having some time to explore, Julie asked the class to think about which scale they would use to measure different objects in the room. The reason for this is because one scale can measure weight up to 11 pounds while the other can only measure up to 2 pounds. She was curious to see if students had already started noticing that the bigger scale would measure heavier things while the smaller scale would max out unless the objects were lighter.

After all of this exploring, Julie brought the class together to focus on the scale and to make connections between the scale and the number line. The class talked about whole number connections first, but then she drilled down to fractions and mixed numbers.

Finally, Julie asked the students what unit of weight they thought the fractional parts might represent. Someone volunteered ounces. Then she asked a wonderful question: “How many ounces do you think are in a pound?” Many students thought there must be 8 ounces in a pound, which makes sense given the number of parts between 4 and 5, but then she transitioned to the other scale to see what students would notice.

She wants the students to figure out that there are 16 ounces in a pound, but unfortunately she ran out of time for the day. I did like that the final comment from a student was, “That scale goes up to 4 pounds.” Just wait until they continue their work tomorrow!

Thank you to Julie for letting me spend an hour learning with her students!

 

18 Amazing Things

This school year, my partner Regina and I facilitated an 8-month long cohort of math educators called Math Rocks. You can read about our first two days together in July here and here. I also wrote this post if you want a peek into our learning during the school year.

As our time together drew to a close, I asked each participant to write a blog post reflecting on Math Rocks. Today I went back through their reflections to pull out snippets to share to encourage other teachers in our district to apply for next year’s cohort. You can read more snippets from their reflections here.

Deb

Can I just tell you how amazing it is to read this? In my first few years of teaching, the path of my career and my understanding of math were steered in entirely new directions when I took part in an extended PD experience. I can’t express to you how much it means to me to be in a position now where I’m able to learn with and from teachers on their own journeys through teaching and math. I couldn’t be happier.

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!

Purposeful Numberless Word Problems

This year I read Sherry Parrish’s Number Talks from cover to cover as I prepared to deliver introductory PD sessions to K-2 and 3-5 teachers in November. She outlines five key components of number talks; you can read about them here. One of the components in particular came to the forefront of my thinking the past few days: purposeful computation problems. I’ll get back to that in a moment.

It all started when I got an email the other day asking whether I have a bank of numberless word problems I could share with a teacher. Sadly, I don’t have a bank to share, but it immediately got me thinking of putting one together. That led to me wondering what such a bank would look like: How would it be organized? By grade level? By problem type? By operation?

That brought to mind a resource I used last year when developing an extended PD program for our district interventionists: the Institute of Education Sciences practice guide Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools. The guide lays out 8 recommendations. I was reminded of this one:

Recommendation 4. Interventions should include instruction on solving word problems that is based on common underlying structures.

Students who have difficulties in mathematics typically experience severe difficulties in solving word problems related to the mathematics concepts and operations they are learning. This is a major impediment for future success in any math-related discipline.

Based on the importance of building proficiency and the convergent findings from a body of high-quality research, the panel recommends that interventions include systematic explicit instruction on solving word problems, using the problems’ underlying structure. Simple problems give meaning to mathematical operations such as subtraction or multiplication. When students are taught the underlying structure of a word problem, they not only have greater success in problem solving but can also gain insight into the deeper mathematical ideas in word problems.

(You can read the full recommendation here.)

And it was this recommendation that ultimately reminded me of the part of Sherry Parrish’s book where she talked about purposeful computation problems:

“Crafting problems that guide students to focus on mathematical relationships is an essential part of number talks that is used to build mathematical understanding and knowledge…a mixture of random problems…do not lend themselves to a common strategy. [They] may be used as practice for mental computation, but [they] do not initiate a common focus for a number talk discussion.”

All of this shaped my thoughts on how I should proceed if I were to create a bank of numberless word problems to share. Don’t get me wrong, the numberless word problem routine can be used at any time with any problem as needed. However, the purpose is to provide scaffolding, and we should provide scaffolding with a clear instructional end goal in mind. We’re not building ladders to nowhere!

The end goal, as I see it, is that we’re trying to support students so they can identify for themselves the structure of the problems they’re solving so they can successfully choose the operation or operations they need to use to determine the correct answer.

In order to reach that goal, we need to be very intentional in our work, in our selection of problems to pose to students. We need to differentiate practice for solving problems from purposefully selecting problems that initiate a common focus for problem solving.

What Sherry Parrish does to achieve this goal with regards to number talks is she creates problem strings and groups them by anticipated computation strategy. I didn’t create problem strings, per se, but what I did do was create small banks of word problems that all fit into the same problem type category. I’m utilizing the problem types shared in Children’s Mathematics: Cognitively Guided Instruction.

CGI

Here’s the document the image came from. It’s a quick read if you’re new to Cognitively Guided Instruction or if you want a quick brush up.

So far I’ve put together sets of 10 problems for all of the problem types related to joining situations. I plugged in numbers for the problems, but you can just as easily change them for your students. I did try to always select numbers that were as realistic as possible for the situation.

My goal is to make problem sets for all of the CGI problem types to help get teachers started if they want to do some focused work on helping students build understanding of the underlying structure of word problems.

I created these problems using the sample contexts provided by Howard County Public Schools. They’re simple, but what I like is that they help illustrate the operations in a wide variety of contexts. Addition can be found in situations about mice, insects, the dentist, the ocean, penguins, and space, to name a few.

As you read through the problems from a given problem type, it might seem blatantly obvious how all of the problems are related, but young students don’t always attend to the same features that adults do. Without sufficient experience, they may not realize what aspects of a problem make addition the operation of choice. We need to give them repeated, intentional opportunities to look for and make use of structure (SMP7).

Even though I’m creating sets of 10 problems for each problem type, I’m not recommending that a teacher should pick a problem type and run through all 10 problems in one go. I might only do 3-4 of the problems over a few days and then switch to a new problem type and do 3-4 of that problem type for a few days.

After students have worked on at least 2 problem types, then I would stop and do an activity that checks to see if students are beginning to be able to identify and differentiate the structure of the problems. Maybe give them three problems, 2 from one problem type and 1 from another. Ask, “Which two problems are of the same type?” or “Which one doesn’t belong?” The idea being that teachers should alternate between focused work on a particular problem type and opportunities for students to consolidate their understanding among multiple problem types.

On each slide in the problem banks, I suggest questions that the teacher could ask to help students make sense of the situation and the underlying structure. The rich discussion the class is able to have with the reveal of each new slide is just as essential as the slow reveal of information.

You may not need to ask all the questions on each slide. Also, you might come up with some of your own questions based on the discussion going on in your class. Do what makes sense to you and your goals for your students. I just wanted to provide some examples in case a teacher wasn’t quite sure how to facilitate a discussion of each slide for a given problem.

Creating these problem sets has prompted me to make a page on my blog dedicated to numberless word problems. You can find that here. I’ll post new problem sets there as they’re created. My current goal is to focus on creating problem sets for all of the CGI problem types. When that is complete, then I’d like to come back and tackle multi-step problems which are really just combinations of one or more problem types. After that I might tackle problems that incorporate irrelevant information provided in the problem itself or provided in a graph or table.

I’ve got quite a lot of work cut out for me!

 

 

Writing Numberless Word Problems

So you came across my post on numberless word problems, you got excited by the idea, but you’re left wondering, “Where does he get the problems from?” Good question! I thought it was high time I answer it.

For starters, I try to avoid writing problems from scratch whenever possible. I can do it, and I have done it on numerous occasions, but I’ll be honest, it’s mentally exhausting if you have to write more than one or two problems in one sitting! It takes a lot of work to think of context after context for a variety of math topics, especially if you don’t want to feel like you’re reusing the same context over and over again.

I’ll let you in on a secret. More often than not, I base my questions on existing questions out in the world. I don’t reuse them wholesale, partly because I don’t want to infringe on copyright and partly because I don’t want to deprive teachers in my district of an existing problem they could be using with their students.

I always change names and numbers, and as needed I tweak the contexts and questions. This is so much easier than writing problems from scratch! Basing my problems on existing problems makes me feel like I’m starting 10-20 steps ahead of where I would have otherwise!

I’ll share a few problems I’ve created to give you an idea of what I’m talking about. I based all three of them off grade 3 2015 STAAR sample questions released by the Texas Education Agency.

Problem 1

Here’s the original problem:

 

Question3

First, I thought about how I could adjust the problem to make it my own:

  • I decided to change the character to Jenise.
  • I changed “flowers” to “carrot plants.”
  • I changed 21 to 24. I did this intentionally because 24 has so many factors. You’ll see how this plays out when you get to the sample questions I created later.
  • I removed the number 3 altogether. Again, this plays out later when I created questions about the situation.

Note: This step is only necessary if you want to create a unique problem. The released tests are free to be used, so you could just as easily convert this exact problem into a numberless word problem. Again, I don’t want to steal resources from my teachers so I’m opting to change this into a new problem.

Next, I think about how I want to scaffold presenting the information in the problem. I create slides, one for each phase of revealing information. Remember, the purpose of a numberless word problem is to give students an opportunity to collaboratively identify and make sense of mathematical relationships in a situation before being presented with a question. There are several factors that dictate how much or how little new information to present on each slide:

  • Students’ attention span
  • Students’ familiarity with the type of situation being presented
  • Students’ familiarity with the math concepts involved in the situation

Here’s how I broke down this question into 4 slides:

Slide 13-1

Slide 23-2

Slide 33-3

Slide 43-4

Thinking this would be used in a 3rd grade classroom, I opted to break it down quite a bit to draw emphasis on the language of “rows” and “same number in each row.” If I already knew my students were comfortable connecting this language to multiplication and division, then I probably would have combined slides 2 and 3 into one slide.

At this point, I stop and think about what question I want to ask about the full situation on slide 4. If I were a teacher, I might select a question and keep it in my pocket. After discussing slide 4, I’d ask my students what questions they think could be asked about this situation. Students need opportunities to generate problems for themselves, not just be told the problems we expect them to solve. I could allow them to answer their own question before answering the one I had planned (or instead of!).

Here are a few questions I generated that I might ask about this situation:

3-q

This is where changing 21 to 24 in the problem adds some richness to the potential questions I could ask about this situation. This is also the reason I removed the number 3 from the original problem. Not specifying the number of rows allowed me more flexibility to ask about either the number of rows or the number of plants in each row.

Problem 2

Here’s the original problem:

Question2

I like this problem, so I didn’t want to change it too much. Here are the changes I decided to make. Remember, I always change names and numbers; context and question are tweaked as necessary.

  • I changed the character to Mrs. Prentice.
  • I changed the food from “yoghurt cups” to “pints of ice cream.”
  • I changed the flavors to chocolate, strawberry, and vanilla.
  • I changed all three numbers. However, I noted that there was a way to make ten (6 + 4) in the ones, tens, and hundreds places across the 3 numbers, so I tried to create a similar structure in my numbers with 3 + 7.

With those changes, here’s how I scaffolded the problem across 5 slides:

Slide 12-1

Slide 22-2

Slide 32-3

Slide 42-4

Slide 52-5

Depending on my students, I might have combined slides 4 and 5. Keeping them separate means I can play it safe. I can reveal each number one at a time, but I can also breeze through slides 3 and 4 if the situation warrants it and spend more time talking about all three numbers on slide 5.

And finally, it’s time to think of some potential questions that can be asked about this situation:

2-q

By the way, this is a great time to point out that I don’t have to pick just one! I spent valuable time crafting the situation and my students will spend valuable time making sense of the situation. Milk it for all it’s worth!

I could pose one question today for students to solve and discuss. Tomorrow we could revisit the same situation, maybe just talking about slide 5 together to jog our memories, and then I could give them another question to solve about this situation. I could even pose 2-3 questions and let the students choose which one they want to solve. Be creative!

Problem 3

Here’s the original problem:

Question1

I like this one because I’m able to take a 3rd grade problem and make it fit concepts for grades 3-5. In this case, I didn’t change as much of the original problem because the context is so simple. Here are the 3 slides I created to scaffold presenting the information:

Slide 11-1

Slide 21-2

Slide 31-3

It’s important to remember that the power of a numberless word problem lies in the conversation students have as you reveal each new piece of information. That conversation is driven by the questions you ask as more and more information is revealed. Here are sample questions you could use as you discuss each slide of a numberless word problem:

  • What do you know?
  • What information have you been given?
  • What do you understand about the information given?
  • What kind of problem could this be?
  • What information do you know now?
  • Does this new information help you?
  • What does the new information tell you?
  • How does the new information change or support your thinking?
  • What operation(s) does this situation make you think about?
  • What kinds of questions could be asked about this situation? (This can be asked on several slides, not just the final one.)

The fun part for this particular situation was thinking of all the different questions I could ask:

1-q

So there you have it – three very different examples of numberless word problems. As cool as I think numberless word problems are, please note that not every problem needs to be a numberless word problem. We have to be intentional about when and how much we provide scaffolding to our students. However, knowing about this type of problem is a great tool to have in your belt when you’re looking for ways to help your students develop a deeper understanding of the mathematical relationships in real life situations.

If you have any questions, please don’t hesitate to ask in the comments!