The Annotated Numberless Word Problem

I recently modeled a numberless word problem in a 4th grade classroom. A few weeks later, I got an email about how the teachers were attempting to create and use some of their own, but they were encountering a problem…writing their own problems was harder than they thought!

They reached out to me for support, and I thought I’d share with you what I shared with them in case it’s helpful to anyone else creating their own numberless word problems.

1. Start with a problem

First things first, start with the problem you want to transform into a numberless word problem. Here’s the problem I started with for this example:

I type the problem on a slide, either in Powerpoint or Google Slides. You can create your problem on chart paper or on strips of paper if you’re working with a small group. I’m partial to digital slides because of some other features you’ll see later in the post.

2. Work backward

From here I create a copy of this slide and remove some of the information. Usually I start by removing the question.

Next I copy this new slide and again decide what information to remove. In this case I decided to remove the entire last sentence. That sentence dramatically changes our understanding of the situation. If you look at the slide below you’ll see that we know the total number of kids eating ice cream and the number of kids eating chocolate ice cream.

The situation is very open right now. The rest of the kids could be eating a variety of different flavors – vanilla, strawberry, chocolate chip. When I reveal the sentence that the rest of the students are eating vanilla ice cream, there’s a nice element of surprise because you aren’t necessarily expecting that the kids are only eating just two different flavors.

My next step is to remove one of the numbers. In this case I’ll take away the number of children eating chocolate ice cream.

Finally, I’ll remove the number in the first sentence to get me to the beginning of this problem. This is the first text students will read.

I structure my slides to minimize changes. I don’t want to overwhelm the students by revealing too much all at once. I will add new sentence, but I avoid changing language that’s already on the slide, if possible. More often than not I’m only changing a word like “some” into a specific quantity. There are rare instances where I’ll have to adjust a sentence as new information is added, but I try not to do that. I want the sentence structure to stay the same so that when the numbers are added that’s the only real change.

You might have noticed that I don’t include pictures on the slides with the text. This is intentional. I used to include pictures, but a colleague shared how distracting the pictures were for her students. Students were looking for meaning in them when they were only there essentially as decoration, with the intent that they would support visualizing. However, the pictures ended up confusing her students rather than helping because the students kept trying to make connections between the pictures and text. Since then I’ve avoided pictures on the text slides unless the picture is absolutely necessary.

3. Plan purposeful questions

The first step was to work backward to plan out each slide so that information is slowly revealed on each slide. Now it’s time to plan the questions I’m going to ask the students at each step along the way. I have two primary goals that I strive for in my questioning:

  1. I want students to visualize what the story is about as it unfolds. If they’re not “seeing” it, then they’re likely not making much sense of it.
  2. I want students to make guesses and estimates about quantities in the story using what they know about the situation and the relationships provided. I want them reasoning all along the way so that by the time they get to answering the question they are holding themselves accountable if their answer doesn’t make sense.

So now I go back through the slides in the order they will be presented and add the questions I plan to ask along the way.

Slide 1

Ask for a volunteer to read the story.

What are you picturing in your mind?
What do we know so far?
How many kids could be eating ice cream?
How many kids could be eating chocolate ice cream? Why do you say that?

Have students draw a quick sketch of the story so far.

Slide 2

Ask for a volunteer to read the slide.

What do we know now that we didn’t know before?
What does this tell us about the number of kids eating chocolate ice cream?

When a new slide is presented, I always ask a question to get students to state the new information. I’ve also worded this as, “What changed? What do we know now that we didn’t know before?”

Slide 3

Ask for a volunteer to read the slide.

What do we know now that we didn’t know before?
How does this number compare to our guesses? Does it make sense?
Are all of the kids eating chocolate ice cream?
What could the other kids be doing?

Slide 4

Ask for a volunteer to read the slide.

What do we know now that we didn’t know before?
What does this tell us about the number of kids eating vanilla ice cream? How do you know?

Have students draw another quick sketch of the story so far.

What question(s) could we ask about this math story?

Slide 5

What is the question asking?

Do you have all the information you need to answer that question?

Let students work on solving the problem. Confer with students as they work to look for strategies you want to bring up with the whole class.

4. The beginning and the end

Something I’ve been doing for the past year with numberless word problems is bookending them with visuals to add a little more texture to the experience.

The beginning

The first thing I do is find a high quality image or two to show the students and have them chat about before we dive into reading any text. My go-to website for images is Pixabay.

I type in a word or phrase related to the story problem, like ice cream, and more often than not I hit the jackpot:

I look for a photo that I think will capture kids’ attention and activate their prior knowledge of the context. It allows students who may be less familiar with a situation to hear the relevant language, such as ice cream, chocolate, vanilla, and cone, before we dive into reading the text.

Here’s the picture I ultimately chose to engage students at the start of this problem, along with some notes of how I’d facilitate the opening discussion with the students.

Image Source: https://pixabay.com/en/ice-ice-cream-milk-ice-cream-waffle-2367072/

What do you notice? What do you wonder? Give students 20-30 seconds of think/write time. Then let students share 1 noticing and 1 wondering with a partner. Finally let students share a few of their noticings and wonderings with the entire class. You may choose to record these in a t-chart, but it is not necessary for this problem.

Tell the students that today they are going to read a mathematical story about ice cream.

When I paste the picture on a slide, I always go into the Notes section of the slide and paste the source of the picture(s), usually the URL where I found it. On Pixabay, more often than not the photos have licenses allowing reuse.  You can find the license information to the right of each photo. I know in the privacy of your own classroom it feels easy to get away with grabbing whatever picture you can find on Google Images, but it’s good habit to pull legal photos to avoid unforseen issues down the road. And with amazing sites like Pixabay and Wikimedia Commons available, there’s no reason not to at least start by looking for freely available photos.

The ending

I’ve been making it a habit to close each numberless word problem with a short video. This serves two goals:

  1. It further builds students’ knowledge of the situation discussed. In the case of the problem I shared in this post, it was about kids eating ice cream so I found a short video of a kid making ice cream. Even if you can only find longer videos, you don’t have to show the whole thing. You could just watch the first minute (or whichever section is most relevant or interesting).
  2. It serves as a pay off for all of the hard work students just did to make sense of and solve the problem.
Here’s a link to the video I included in this problem

I’m sure you can guess where I go to find videos. YouTube has such an endless supply of videos, that I haven’t yet encountered a situation where I couldn’t find a video worth sharing. Sometimes it’s the first video and sometimes it’s the tenth, but it’s always there waiting to be discovered.

Final thoughts

Now that you’ve seen me put together this numberless word problem in pieces, here’s your chance to see the finished product. This link will take you to the slideshow for the finished product.

In the Notes section on some of the slides, you’ll see references to students sketching in boxes. I created a recording sheet to try out when I modeled a different problem recently. If you want to check out the recording sheet, here’s the link. I don’t have a lot of experience using it yet so I don’t want to say more about it right now, but I do want to share in case it’s helpful.

If you have any questions, don’t hesitate to reach out in the comments or tweet me @bstockus. And if you create your own problem, please share it with us on Twitter using the #numberlesswp hashtag.

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