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.

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?

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?

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?

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?

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?

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?

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?

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!

Sink your teeth into data. Don’t just nibble.

Looking for math all around started as a challenge I made for myself and I’m realizing it’s becoming a full-fledged theme for my year. When I had to think of a topic to moderate this week’s #ElemMathChat, I started by asking myself, “What’s a topic we haven’t talked about since the chat started in August 2014?” After some brainstorming, I eventually came up with analyzing data. What a great topic for my theme! I don’t think I could throw a rock without hitting some data in the world around me.

In fact, as I was fleshing out the topic for the chat, I was regularly checking some real-world data online. After a long dry spell, we finally got some rain in Austin. And by “some rain” I mean a deluge. On a couple days last month it just kept pouring and pouring. Throughout each day it rained, I found myself checking our neighborhood weather station on Weather Underground to see how much rain had fallen. By the time October was over, we had received 10.3 inches of rain in my neighborhood! That simple piece of data became the catalyst for tonight’s #ElemMathChat.

I started digging into rainfall data for October, then rainfall data for other months, and finally I expanded my data dive into other cities in and out of Texas. When I was done, I had a spreadsheet full of various tables of data that I wanted to share in my chat. To make this chat work, I realized I needed to be intentional about how I shared the data in order to tell a coherent story. I also wanted to create a variety of data displays that would match the various data displays students encounter across grades K-5. As an aside, I think #ElemMathChat sometimes leans a bit heavy on content for grades 3-5, so I was trying to be mindful to show some graphs that could be analyzed in a Kinder or 1st grade classroom.

It took several nights to research, create graphs, and pull it all together to make a story, and in the end I’m proud enough of the final result that I wanted to capture it on my blog.

Before starting my data story, I shared the following guiding questions that tied into my primary goals for the chat.

My Data Story

Our story begins with the piece of data that started it all. I asked the participants to tell me what they noticed and wondered about this statement.

What do you notice and wonder?

Many people wondered how this amount of rain compared to other cities. Funny you should ask.

What do you notice and wonder as you look at this pictograph?

One thing I noticed is that I accidentally left the key off the graph. Oops! Each picture is meant to represent 1 inch of rain. Despite my mistake, several people liked that the missing key invited students to wonder about what the pictures represent. That sounds like such a wonderful conversation to me that I opted to leave the key off when sharing the picture in this blog post.

I had a little fun with this graph because I had to decide which cities to include. I decided to focus on other state capitals, but the question became, which ones? When I noticed how many start with A, I decided that was more interesting than picking random capitals. It just so happens that all the other capitals on this pictograph are all on the East coast, so I wonder if it would have been better to choose capitals with greater geographic diversity. In the end this is just a fun way to get our story started so I’m okay with what I chose.

Next we moved from cities outside of Texas to cities inside of Texas, specifically along the I-35 corridor from San Antonio to Waco.

What do you notice and wonder as you look at the October rainfall totals for these cities?

Now that I shared two different graphs, what questions could you ask students about these graphs? What math skills can students bring to bear to interpret and further understand the data in these two graphs?

One thing that we often do with graphs found in textbooks and tests is ask one question about them and then move on. How unfortunate! There’s so much rich information to dig into here. One of my key points for tonight’s chat was reiterating something I read by Steve Leinwand about mining data. Ask a variety of questions about data displays. Sink your teeth into them; don’t just take a small nibble.

The one thing that stood out to me and many others in the chat was how little rain San Antonio received. The difference between San Antonio and New Braunfels is quite striking considering how close they are to each other.

Other people felt that Austin’s rain wasn’t fitting with a general trend in the data. I didn’t want to get into it in the chat, but I’ve noticed the rainfall in my neighborhood tends to be less than other parts of the city. Our weather station recorded 10.3 inches for October but others in Austin clocked in at closer to 13 inches of rain. I thought about using the larger number, but because the catalyst for this whole story was my weather station’s data, I opted to stick with that. By the way, I don’t think it’s an issue with our weather station’s rain gauge. Over the years there have been many instances of rainfall in other parts of the city while my neighborhood in north Austin remains bone dry.

Now that we’ve looked at rainfall in and out of Texas, it’s time to drop a bit of a bombshell. With this new information, what story is the data telling so far?

Here’s what I see as the story so far: Austin received 10.3 inches of rain in October, which was a lot compared to areas outside of Texas, but fairly common for our area in Texas. Not only was this a lot of rain, but it also fell in a very short amount of time, 6 days.

Next, I asked for help. Now that you know it rained only 6 days in October, which data display would you choose to represent October rainfall?

Option 1

Option 2

Most people preferred option 2 because it shows the full picture of October. That was surprising to hear. In my mind, because we just saw the picture graph showing that it only rained 6 days in October, I didn’t feel option 2 was needed. I already know it didn’t rain on very many days, so why waste the space with all those days showing 0 inches of rain? Option 1 puts the focus squarely on analyzing the rainfall on the days where it actually rained. In the end there’s no “right” answer, it all comes down to how you justify showing what you choose to show.

We’re nearing the end of our story. There are two more graphs remaining. What does this next graph add to our story? What is one question your students could answer based on this data?

I love looking for relationships so here are the questions I came up with:

• Where do you see the relationship “three times as much” represented in this graph?
• Where do you see the relationship “half as much” represented in this graph?

I especially like wondering what students will come up with because both questions have more than one correct answer.

And now for the last graph. How does this close out our data story?

Here’s a follow up question for you. What could be the sequel to the story I just told? How could you and your students explore and tell the sequel? What other data stories could your students explore and tell?

I closed the chat, and I’ll close this post, with two key points I want everyone to take away from this conversation.