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.
This post describes a few assessment strategies several colleagues and I piloted with a group of middle school students. One of our main goals was to provide students an opportunity to learn through the assessment process, to revise their initial work, and to make generalizations that reach beyond the boundaries of a given task. The impetus for this work stems from the Mathematics Assessment Project and their emphasis on students' use of mathematical practices throughout task completion.
MAP Task Security Camera found at http://map.mathshell.org/materials/tasks.php?taskid=273&subpage=expert.
When students reached question 3, we turned to ActivePrompt to collect realtime responses from students and share those responses with the class. We found that students were engaged with this portion of the task from the outset primarily because they were given a chance to use their intuition and make a guess.
Give this link to your students: http://activeprompt.herokuapp.com/SPPFK
Go here to see their responses: http://activeprompt.herokuapp.com/TQPJB
Teams explored the task further using the following GeoGebra exploration tool.
Many thanks to the teachers who were willing to pilot some new assessment practices. Our debrief surfaced several key questions that will help to guide our future implementation:
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.