Category Archives: Math

Math Problem

We did this a few weeks ago. I’ve just been a bad blogger recently. Bad, bad blogger. My apologies.

The video shows kids being genuine mathematicians. Sharing ideas, debating, attempting to prove their solutions. Good fun…and probably the last video evidence of how well this group of students works together. At least for this school year!

Speed Dating

The first day back from a week off is the perfect opportunity to subject the kids to a “What did you do on your holiday?” bore fest. How can you spice it up? Speed dating! The kids rotated, seeing half of their classmates, spending 30 seconds detailing their highlights, then listening to their partner’s, then rotating to the next person. This was funny, first and foremost, but also a good listening test. The speed of the first 13 seconds of the video may be nauseating for some of you, but then we got into a good math problem about the time required to complete the rotation. Watch as Lexi, Karl, Edward, and Lev share there thoughts (in the 2nd video). After that thought-provoking debate, we went around the entire class, recalling what we could about each person’s holiday.

Math debate…

Our Infinite Number System

It was apparent last week that some of our students still don’t fully grasp that concept, that our number system has no end, and that we can express any value, no matter how big or small, just by arranging digits and decimals accordingly. Recent investigations, therefore, have been aimed at fixing that. We had fun this morning with the following virtual manipulative, and more importantly, I think some students started to understand the implications of place value for the first time. Click on the ‘Zoomable Number Line” link below and ask your child to explain the rules to a couple of games/challenges we came up with today. Good conversation starter, parents, as well as an opportunity to see what your child understands.

Zoomable Number Line

Ronnie’s Back…

…and on FIRE! We had a blast today learning about Pascal’s Triangle with our favorite gift-giving mathematician. I had NO idea about all the patterns in this famous triangle. Thanks, once again, for enlightening and inspiring us, Ronnie…and for continuing to show us how much FUN math is!

Follow this link if you want to know more. LOTS of information about Pascal’s Triangle, all the cool patterns, etc….and believe me, Ronnie covered almost all of it! The man is ‘energy’ on two legs…

Numbers…Big Numbers

We’re overdue for a glimpse into the latest math news!

I am the first to admit that not all math classes go as well as today’s did. While I always strive to inspire the type of discussions, conjectures, debates, reasoning, revising of hypotheses, etc. that you’ll see in the videos, it doesn’t always happen.

The first video shows students playing Target Number and/or simplifying an expression. It’s a good opportunity for me to assess the students’ problem solving, computation, and communication skills, but more importantly, it’s a good opportunity for the kids to simply play with numbers. It’s also interesting to see how different students react to the “video phone.” For some, the camera means the end of conversation – “Oh…camera…too embarrassed to keep talking.” For others, it’s an invitation to talk more – “Hey! Cameras are rolling…time for me to shine!” Funny.

Some students got close on the Target Number, but we’ll leave it open since nobody found an exact set of steps that would lead to 784. (Challenge to the audience! Can you solve it? Can you write an expression that would use each of the randomly chosen numbers 2, 1, 7, 5, 6, 10, 3, once and only once, that could be simplified to 784? If so, please share!) Only a few students attempted to simplify the LONG expression, and the solution you see below (on the right) is from Lydia. The solution on the left was an attempt from a group of boys who only realized their error (they did 2 to the power of 8 rather than 7) while presenting it to the class. Perfect!

target number expression

We then moved into an activity designed to inspire analysis of our number system. We built a 10,000 chart and started asking simple questions about it. “If one is here, where is 10,000? Where is 100? Where is 5,000?” Responses were fascinating, and as students explained and defended their choices, our overall understanding increased. We will continue adding “landmark numbers” to the chart and asking students to locate random numbers – 2,347, for example – and explain how they did so.

Parents,

Investigation in Volume

Parents, this post is meant for all of our valued fans, but it is especially meant for you so that related conversations/investigations can continue at home.

Last week we continued our Shape & Space investigations with a simple (seemingly) question.

What can be measured?

Enthusiastic conversation ensued, and a common response was “A table” or “A book” or “A car.” Students eventually agreed that you can’t measure a table…but you can measure a table’s height, its length, its weight, etc. At some point one or two students offered “how much liquid is in a container” which led to a discussion of volume and capacity. The concepts of mass and weight came up, and since nobody knew the difference, Thea figured it out, offering the class a scenario of one’s weight and density on Earth, compared to one’s weight and density on the moon. 🙂

We then began investigating volume more closely, along with a review of area concepts. Today, in the hopes that students could visualize size/space relationships, we built a cubic meter. How many students could fit in one? Two? What would two cubic meters look like? Three? And finally, I offered the following conjecture:

If you double the dimensions of a solid, the volume will double.

The images below are from the students’ attempts to prove or disprove that conjecture, as well as our initial cube constructions. At no point did we devise a formula for volume, though I could see that a few students were starting to make the connection.

Parents, at home, throw out a challenge at the dinner table. What is the area of this room? What is the volume? Can you find an object in the house that is one cubic meter? Can you prove that it’s one cubic meter? Does it actually have to be a cube?

Mathematical Shift

For the last few weeks our math focus has been computation, particularly multiplication and division, with an emphasis, as always, on conceptual understanding rather than “Here’s an algorithm…memorize it…practice it fifty times.” Or at least, that’s been the goal. Having said that, at this point, our students should be comfortable with at least one algorithm for all operations. Check individual student blogs over the next week to see demonstrations of these algorithms (and hopefully understanding!).

Today’s math investigation represented a shift from number to shape and space concepts. Student approval was immediate, as was engagement.

Challenge: using one sheet of chart paper per group (4 students in each), design four nets (I avoided that term as long as I could during the activity, mostly because I want students to focus less on terminology and more on actual problem solving) that, when cut out in individual pieces, will wrap perfectly around your chosen solid (cubes, triangular prisms, cones, etc.).

The best part of these challenges are the resulting conversations, the debates, the conjectures. And as students complete the challenge, they are encouraged to think mathematically. As opposed to “I finished, now what?” I hope they are starting to say, “I finished, now I’m wondering…” We’re not all there yet. In the video below you may hear me ask, “Is that the only possible plan (net) for that solid?” And perhaps I’ll pose that question to our families. Possible dinner conversation! How many possible nets are there for a cube? Let us know what you come up with!

Math Inquiry

As I become increasingly comfortable with student-led inquiries guiding our learning, our math lessons are becoming more dynamic, unpredictable, and productive. There is also less emphasis on term/procedure memorization in favor of higher level, cognitive investigations and problem solving (and problem posing!).

Yesterday, in an attempt to explore properties of lines and angles, I asked the students to simply construct lines or line segments in their notebooks. Shortly after, I told them to limit the number of lines to five. The ability to actually use a ruler and pencil to construct a line, without ripping through the paper or breaking the pencil or letting the ruler slip, was more challenging for some students than I anticipated. But eventually, we all had five lines. Then questions guided the learning. The first, from me, was intentionally open, as opposed to a specific, restrictive plan or set of questions designed to force kids to see what I want them to see.

“Mathematicians, what do you notice?”

Students observed shapes and angles and intersections, as expected. Then Seth said, “I wonder if there’s a maximum number of shapes, just like that ‘how many squares on a chessboard‘ problem we had.” Oooh…good one. Seth was obviously on to something, and I, of course, hadn’t thought of it. As he explained his thinking, several students were already counting and exploring. Seth’s question obviously intrigued his classmates, but it also inspired me to wonder about intersections, so I posed the following: “If we draw five lines, is there a maximum number of intersections that will occur?”

This one question led to endless conjectures, debates, tests, and hypotheses. At some point I interrupted the busy hum of mathematical thinking in the room and suggested students start gathering data. They were doing this anyway, but I don’t think anyone had a systematic, written approach. Students eventually created a table like this one (Brianna’s):

lines 1

And you can guess what happened next. Patterns! Predictions! Tests! Proofs! None of which, most likely, would have happened had I imposed a traditional “This is exactly what you’re going to learn today” approach. The trick is coming up with those generative questions, the ones Seth and so many of the students are good at asking.

Audience, what questions do you have? Looking at the images in this post (more student work/process below), thinking about the original question, what else could we explore?

lines3 lines2 lines4 lines5