Here’s a lesson that begins with common objects that students are familiar with (milk cartons), incorporates giving students time for some hands-on exploration (folding Origami squares), introduces a digital tool to model something I couldn’t do in any other way (thanks to Desmos), and more. I’ve taught the lesson twice and, in this blog, I tell the story of what happened the second time with photos to help describe what actually happened. Enjoy. Part 1. Comparing Quart and Half-Gallon Milk ContainersI began the lesson by showing the class two empty milk cartons with their tops cut off and as ..read more

I realize that using “half as big” in my question isn’t mathematically precise, but it’s how I posed the question recently to students, first to fourth graders and then to fifth graders. I wanted the students to try and visualize what two squares would look like, one centered within the other, with the area of the smaller square one-half the area of the larger square. Before I presented the problem to students, I reached out for digital help to math friends. I had a thought that was sort of a-teacher-can-dream request. What if I projected two squares as shown below and ask a question that I t ..read more

In my previous blog, Number Lines―A Lesson in the Planning Stage, I wrote about how I made a detailed five-step lesson plan based on a Tweet I had seen 7 years ago. (Clearly the lesson plan had ample time to marinate.) Here I write about what actually happened when I taught the lesson to fourth graders. Which parts of my five-step plan did I stick to and what changes did I make as I was teaching? What were my reflections afterward? On the written assignment, what did the students show about their understanding? This is the board near the end of the lesson. Read on to learn how we got here ..read more

I typically write blogs after teaching a lesson to share what I did and what I learned. This blog is different. I’m deep into planning a lesson for fourth graders about place value and number lines. It’s a lesson that I’ve never tried before with students but that I’ve been thinking about for a while. The inspiration for the lesson is a Tweet from when Twitter was still Twitter. Mark Chubb posted it in 2016, evidence that I’ve truly been thinking about this lesson for a long while. When I first saw the Tweet, I struggled to think about where 1 billion would go. It wasn’t obvious to me. Using ..read more

Silent Math is a pedagogical routine I’ve used for many years. I can’t remember exactly when I first learned about Silent Math lessons. I think it was when I observed a classroom lesson in the beginning years of my teaching. I don’t remember the teacher or the specific lesson, but I remember being impressed by the students’ engagement and enthusiasm. I’ve used Silent Math from time to time ever since. How Silent Math WorksThe routine is simple: After the teacher draws a star on the board, no one talks. Not even the teacher. And then, when the teacher erases the star, the class discusses what ..read more

I’m in the weeds mulling about teaching division. When introducing division, what’s the role of contextual problems? Should we start with interpreting division as sharing or grouping? Is it better to introduce problems without remainders first or with and without remainders together? How do I make the connection to multiplication clear and useful? I’ve been thinking about these questions for a while. And by “for a while,” I’m talking years. I find mulling with others useful. That’s what I did for this lesson. Jenna Laib is a K–8 math specialist in Brookline, Massachusetts, and she and I ..read more

Does every experience in life trigger a math experience? Sometimes it seems so. It did recently when a package arrived with a gift I had ordered for a friend’s birthday. It’s a cordless lamp. I have one that sits on our kitchen table and I love it. You tap the top and the light goes on. Tap again and it gets brighter. And tap once more for its brightest setting. When the light starts dimming, I just plug in the charge cord and connect to a charger. It’s terrific. About My Cordless LampI bought our cordless lamp after seeing a similar one at our next-door neighbor’s house. I did a search onlin ..read more

The photo below shows the board at the end of the fraction lesson I taught. Well, not exactly at the end of the lesson, but just before I gave the students an assignment so that I could get feedback about their thinking as a result of the lesson. More about that later. As with many lessons, there was much to think about. Before the LessonA few days before, the students completed an assignment that served as a pre-assessment for the lesson. I asked them to draw and label representations of two fractions―4/5 and 5/4. Here are a few samples of their work. This student used a rectangular repres ..read more

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