Here's a salute to one of the oldest manipulatives still in use in today's digital classrooms.
1. Investigating Area by Folding Paper
2. Algebraic Models
4. How many times can you fold a piece of paper?
I used this task from NRICH at a teacher workshop several years ago to give teachers an opportunity to practice making conjectures, reasoning, and justifying. It caught my attention again today as I visited the NRICH website in search of rich tasks. I think it's held up quite well over the years and after reading about Alan Parr's implementation with students, I'm impressed with the multiple opportunities for extension this task provides.
Start with a few key questions:
Provide students with number cards to try out their ideas. As they work together, ask questions about what they are noticing and wondering. You might find that this recording sheet can assist with keeping track of student observations. These resources from Max Ray of the Math Forum can help guide students toward more productive problem solving.
When students have found and shared some methods and solutions, give them an opportunity to make up some of their own problems. Alan Parr shares some student-generated problems that might be a good starting point.
I recently prepared a series of number explorations for a 2nd grade classroom. The students in this classroom have been working with base ten concepts and place value and are now beginning to compare numbers. In this post I'll share a menu of activities that can be used to support deeper understanding of these concepts. The goal here was to help students reason about the relative size of numbers and quantities without relying on an algorithm to compare numbers. I've also tried to help students build some bridges between various representations of numbers.
1. We started with Andrew Stadel's Estimation 180. While many of his explorations might be targeted at an older audience, I found the following extremely well-received in this 2nd grade classroom.
First huge moment: seeing licorice on the big screen at 12:45 in the afternoon.
Second huge moment: getting to guess and seeing those guesses on the board.
Two equally huge final comments: "I was sooo close!" vs "Wow, I was not even close!" These are both satisfying to me.
Students had opportunities to consider the relative size of numbers as each classmate shared a guess. We discussed our smallest and largest guesses and any that appeared more than once. It was quite evident that students were wondering whether they should guess a really big number, and if so, what that number would be in this scenario.
2. We followed this up with some number line estimation. I used the cards below with the students seated on the carpet for a brief number talk. Along with estimating the location of the red dot, students answered the question, "How did you decide?" This line of questioning encouraged students to use the numbers given as landmarks and do some broad thinking about tens and hundreds.
3. Next we pulled from the deep constructivist work of John Van de Walle. I was first introduced to John Van de Walle's work in my teacher credential program back in 1996. These activities are great work for both teacher and student and make mathematical reasoning about number the focus.
4. I pulled this next task from Illustrative Mathematics as a progressive step toward more challenging estimation with number lines. I drew a long number line across the board at the front of the room and labeled every hundred up to one thousand. Students teamed up and together discussed and decided where to place their number.
5. The final selection in this menu of activities is from the National Library of Virtual Manipulatives. I used the Practice mode with one dot in which you are asked to drag the dot onto the number line to match the value shown. If the location you choose is not close enough to the actual value, you are asked to zoom in and be more precise.
In the first lesson, students were invited to create a new game based on the four-square pieces used in TETRIS. Using five identical squares, groups explored possible arrangements and clarified guidelines for piece creation along the way. Part 2 continued as students kept track of possible pieces by cutting them out of a 1-inch grid on cardstock.
As collections grew, quite a few groups had several pieces that were duplicates. I anticipated this, and so prepared a few slides to challenge their spatial reasoning and sharpen their skill at finding these duplicates.
After groups were convinced we had found all 12 unique pentominoes, I reminded students that what made someone really exceptional at TETRIS was an understanding of how the game pieces fit together. I explained how clearing lines was related to making rectangles and challenged them to practice making rectangles by putting together various pentominoes. I used this handout to help structure their exploration.
I suggested that students treat this part of the lesson as a tutorial, similar to the TETRIS tutorials. It’s not the real game, but exploring how the pieces fit together would certainly support their gameplay. We spent about 10 minutes on this task before getting to the real deal.
I asked students, “What do we do with these new game pieces? How does the 5-square game work?” These questions led us to take a closer look at what makes the TETRIS game tick. Together we brainstormed some of the game features:
Students immediately pointed out the constraints we were dealing with that would prevent a TETRIS-like experience. We didn’t have unlimited pentominoes and clearing lines wasn’t possible with physical pieces. I decided to use this opportunity to present them with the challenge of actually putting together a new game. Here’s what I said:
In order to create a working game, you’ll need to come up with precise responses to the four questions we just discussed.
Groups got to it and after a few minutes a whole class discussion yielded these results:
Answers to questions 1-3 were worked out pretty easily, but defining the game board size left us stumped. We ended up creating several sizes and ran some game trials.
Gameplay using these two boards led us to think about how many squares we really needed.
From their work building rectangles during the tutorials earlier in the lesson, students knew it was challenging to create a rectangle using all of the 12 pentominoes. And from their gameplay using the two trial boards, they knew there needed to be some extra squares for those times pieces didn’t fit together perfectly. Thus, a board with exactly 60 squares was ruled out. We settled on trying an 8×9 and an 8×8.
Based on these results, most groups decided to use the 8×8 board for future play. A few explained that the extra row on the 8×9 board made play easier and they preferred it over the 8×8. Overall I found this activity to be a motivating context for student use of mathematical practices and an inexpensive way to get a great game in kids’ hands during the last few days of school.
I started this lesson by telling kids they would be working together to create a new game. I asked them if they’d ever played TETRIS. The response was mixed. I don’t know when this game hit its peak, but I remember wasting tons of time on it and practically having a heart attack as pieces didn’t go where I wanted them to and the screen filled up. I wanted to use that sort of hysteria as classroom fuel, so I went over to tetris.com and pulled up some of the interactive tutorials. These are quick, direction-led tutorials that allow you to get a feel for the game. I randomly picked a few students to give these a shot while the class watched.
The first tutorial involves moving a piece from the right wall to the left.
Piece of cake. Then we rotated a piece and cleared a line. But then we had a few pieces coming one after another and had to move rather quickly.
A student came to the front and got started. When she got to this point, the class was screaming at her.
ROTATE! ROTATE! ROTATE!
It was great to see such a need for rotation.
After we discussed the rules and some observations they made about the pieces, I presented them with the new challenge. What if we used five squares, rather than four, to make these game pieces? How would it change the game? How many unique pieces would we need? Students made some conjectures and most agreed that we’d have more pieces than we used in the TETRIS demos.
I gave each student five squares and asked them to build a piece they thought should be included in the game. After about a minute I asked students to get up, leave their pieces on their desks, and do a quick gallery walk around the room to view the other pieces. After they returned to their seats I asked students if they had seen any pieces they would include in the game. I also asked them if there were any they thought should be excluded.
Several students weren’t sure about this one:
But others pointed out that it made sense because of this one:
Then the attention was drawn to this one:
Most students thought this should be excluded. I told them that if they wanted to exclude it, they would have to come up with a new rule to clarify the guidelines for piece production. Here were their first attempts:
“A square can’t be in the middle of two other squares.”
“A square can’t be on a crack.”
“It has to allow others to fit without gaps.”
“It can’t be over half a square.”
With each of these attempts, I tried to show how these statements eliminated viable pieces they had already created. I pressed them regarding how very imprecise their rules were, and encouraged them to think about what each square had to do, rather than what it couldn’t do. From this discussion, we came up with:
Use five identical squares. Put the squares together so that each square shares an edge with at least one of the other squares. Turns or flips are duplicates and should be excluded.
Satisfied with the guidelines, students set off to explore possible pieces.
In the next lesson students will be cutting these out and exploring ways they fit together. Some will most likely find pieces that slipped past the guidelines, like this one: