You don’t teach students the problem-solving strategy of Organize the Information: Make a Table by having them “complete the table.”
The activity “That’s Sum Challenge!” from AIMS asks “What sums from one to 25 can by obtained by adding two, three, four, five, or six consecutive numbers?”
One of the student pages looks like this:
I’ve designed this type of thing before. Fortunately, there’s a quick fix: ask the question, allow students time to work on the problem, ask the groups–or regroup and ask the class–”How can we organize this information?”
Likely, students’ tables won’t match the one above. Some students will probably make a table for two consecutive numbers, then three, and so on. To highlight the impossible sums, the helpful folks at AIMS have done the work of merging these tables into one. In their defence, kinda, the teacher pages has this under “Management”:
If you have a class that functions well with open-ended problems, you can explain the problem to them and have them solve it without using the student pages.
Subtracting the table engages more students at more levels. From “two consecutive numbers are always even and odd (or odd and even) and that gives us all the odd sums” to “the sums made by adding three consecutive numbers are all multiples of three” to “powers of two cannot be obtained because…,” each student can contribute to answering the key question “What sums can be obtained by adding consecutive numbers?” (The ellipsis is there because the reason isn’t immediately obvious to me.)
In the past, I had it back-asswards. Take the “How many different possible meal combinations are there on the kids¹ menu?” problem. I’d give ’em tables and tree diagrams up front. A problem became practice. Once I “turned the tables” and allowed students time to get started, I could later ask groups to share their tables or I could step in at just the right time with tree diagrams to help make sense of spaghetti nightmares.
¹Kid’s? Kids’? This is why I’m not a prolific blogger.
Related: The more sides you have, the smarter you are.
Recommended: “You Can Always Add. You Can’t Subtract.” Ctd. by Dan Meyer
Chris Hunter 
Nailed it Chris!
But with that being said, the thought of a table-less task/activity can be overwhelming for student and teacher. Having a possible table on standby is a great idea but most of the time it won’t be needed if the right discussions take place.
In the past I’ve been guilty of providing a table for students for activities such as this and then asking them “how do you know you found all of the possible solutions?”
Their reply ” because there’s no room left on the table and all the squares have something in them!”
Me….”DOH!!!”
Made me realize that my tables were limiting my students’ reasoning and ability to make mathematical conjectures.
Thanks Graham.
Yep. I did the Four 4s the other day. I meant to give students my table. But I didn’t and noticed a couple of things. First, students created lists, leaving space when necessary, not tables. I don’t quite know what to make of that. Two, a couple of groups made conjectures about which target numbers would be easy or difficult to make — multiples of four and prime numbers, respectively — and explored that for awhile rather than “complete the table.” I mean, they still could have gone in this direction, even with the tables there, but this hasn’t been my experience.