Thank you so much for introducing this fantastic resource.

I really love how the Open Middle problems engage students, especially in a mixed-level math class.

Consider an alternative to the subtraction example you wrote about, where a teacher gives their students a handout with a bunch of problems requiring the subtraction of three digit numbers.

Faster students who are comfortable with the procedure will race through the handout, some completing it before you’ve even finished handing out the sheet. And then you have to give them something else to do. This problem gives students a lot of practice with calculation (think about how many different subtraction problems they’ll do on their way to solving this problem) but it requires more than that. Students have an opportunity to reason, look for patterns and structures, which deepens their learning and extends it beyond just this one problem. There is a puzzle-like quality to the problems on Open Middle. Students persevere because they are not just doing all of these one-off calculations. They are doing a series of calculations that connect to each other and are moving towards the overall goal of finding the smallest difference. It is also very easy to extend the problems here for students who do finish early. If a students finds what they think is the smallest difference you can ask them to prove that their difference is the smallest. You could also ask them for the largest possible difference between two 3-digit numbers, given the same conditions.

I love that this site has challenging problems for exploring number sense and operations and challenging problems in functions, algebra and geometry. One thing I really appreciate about the algebra problems is that because they are “open middle” – meaning they have more than one way to solve them – they are accessible to a wider range of students. When students work on problems where there is only one solution method, it often curtails their sense-making and perseverance because for those kinds of problems, you either know how to solve them or you might as well put your pencil down. These open middle problems keep students in the game and give them an opportunity to do a lot of great mathematical reasoning.

Also, I had a lot of fun working on this great Open Middle function problem posted on Twitter by Graham Fletcher (@gfletchy) – https://pbs.twimg.com/media/C9VkpFwVwAAV5BH.jpg:large

]]>What is your best guess for what the answer will be?

What is a number you know is too big?

What is a number you know is too small?

The big wow in this problem for me was the Support and Push questions which worked extremely well in my multi-level classroom, because everyone could work at their own pace. As students worked on the problem, I walked about the room to see where students were taking the problem, where they were stuck, or where they could extend their thinking. Then I gave students questions based on what I saw the students doing or struggling with, which kept them moving forward and engaged. I like the fact that the Support and Push questions are visual – something that they can hold onto and keep mulling over.

I’m including a link to a picture of the work of one student, along with her support and push questions. You can see that she started with the problem by trying to figure out the hourly rate and coming up with something different for each week, as did most students. As soon as I gave her the slip of paper with that puts the hours in sequential order, though, it was all she needed to move forward to the “push” questions.

Also available with the link is the work of one student’s work that I found fascinating. Before I gave him any support or push questions, he had put the hours in sequential order and discovered a sort of pattern. I gave him the support question “How much money would Kareem make if he worked 28 hours?” and was surprised to see that he had correctly written $465 on the slip of paper (even though he hadn’t got the the point yet where he knew that the rate of change for every 2 hours was $35). I asked him, “How do you know?”. You can see from the attached work that he had figured out about how much it would be per hour and then rounded it off.

Here’s the link to my students’ work: https://drive.google.com/open?id=0B5AHThRUZrr-RnNBNEF3ZVI3OGc

If you haven’t tried the problem yet with your class, I highly recommend it!

Happy Numeracy Adventures,

Patricia

]]>I especially like the last picture (which I wasn’t sure was a picture at all). It would be interesting to ask students what they thought was the question, and why certain blocks were bigger or smaller. This would be especially interesting as a Think, Pair, Share version of Notice/ Wonder. I love it!

]]>You might pair the video with this article by Parrish on number talks.

http://www.mathsolutions.com/documents/NumberTalks_SParrish.pdf

]]>The students were reluctant to say what they noticed and what they wondered in front of the whole group. We used Think, Pair, Share and many of the responses from the students came from bouncing off ideas from their peers.

Overall it was a very successful lesson in a group of mixed abilities. I will continue to use similar lessons with my class.

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