Guiding Principle
Need a task that is open ended , accessible , and has a high ceiling (Lotan, 2003)
By open ended , we mean the task has one or more of the following features:
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Has Partial Solutions
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Has Multiple Solutions
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Has Multiple Solution routes
By accessible , we mean the task has multiple ways for students to enter the problem, such as:
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Requiring many different skills o For example: someone good at drawing, modeling, describing, imagining, acting, or computing, someone with logic, someone willing to go by trial and error, someone who can think of counter examples)
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Setting the problem in a familiar context
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Using manipulatives or pictures
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Building off previous mathematical concepts
By has a high ceiling , we mean the task is cognitively demanding. This might be inherent to the task, or the teacher might look for quick ways to add on to the task to make it more difficult. For example:
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Pushing students to find more than one way to solve the problem
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Posing sense-making tasks (in y = 5x + 2, why do we multiply by 5? Where do you see repeated addition in the graph?)
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Asking students to find similarities or differences between two mathematical objects (or solutions, or solution routes)
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Other deep questions (Which [insert two linear equations] makes larger y-values?
Where, exactly, are these y-values larger?)
Examples (and Non-Examples) of Discussion-Worthy Tasks
1. Solve the following.
(a.) 3x – 7 = 2
(b.) 1 + 2x – x = x – 5 + x
(c.) 3 – 2x = 2x – 5
(d.) 3 + 2x – (x + 1) = 3x – 6
This task is quite procedural. It is not very open, nor accessible, nor does it have an obvious high ceiling.
2.
With your equation mats, solve the following in at least two ways.
(a.) 3x – 7 = 2

(b.) 1 + 2x – x = x – 5 + x
(c.) 3 – 2x = 2x – 5
(d.) 3 + 2x – (x + 1) = 3x – 6
This task, now, is more open and accessible. Solving in two ways puts an emphasis on the different routes you can take to get to the answer, rather than performing computations to get to an answer. Equation mats are manipulatives that allow children to “see” and “touch” an equation, which may help give students access to thinking about equations. However, we must say that equation mats took lots of work for our students to understand. So, this task is only more accessible to students who have already worked with equation mats.
3.
What can we say about “x” if the left side is larger than the right? Smaller? Equal?
This task has many correct solutions, and partial solutions. A student might say (if the right side is larger than the left) that x can be 6, and this would be true! It is not a complete solution, but it gets kids talking.
4. Using the following pattern, find the shape of figure 1 and 5. What does figure 100 look like?
How do you know?
Block patterns allow students access to the mathematics in many ways. If one focuses on the way students see the block pattern, then these tasks can be quite open ended too, as there are many ways to see the pattern. Additionally, there are many ways to make these tasks more demanding quickly and easily (Which figure will have 444 tiles? Will a figure ever have 337 tiles? What’s so special about multiplication; why are we multiplying the figure number by 4?)
5. Solve for x: 3x + 2 = 62

This task is not very open nor accessible, nor does it have a high ceiling. The next two examples are modification of the previous task to make it more open and accessible.
6. At a swap meet, I bought 3 hammers and a 2 dollar box of nails. Altogether I spent 62 dollars.
When I came home my neighbor wanted to buy one of my hammers from me. What would be a fair price for my neighbor?
7. I want to buy a concert ticket that costs 62 dollars. Right now I have two dollars. I decide to sell action figures that I make out of clay, for 3 dollars per figure. How many action figures do I need to sell?
8. Consider the equation 3x + 2 = 62. Create a word problem that can be modeled with this equation. Then, draw the situation, and solve for “x”.
Story problems can give students access because students can re-enact the problems, draw the situation, or imagine the situation in some other way. Although, we must say that students who have seen traditional instruction for years don’t naturally do this. Teachers might need to push kids to reenact a situation, or draw a picture. Also, we do not imagine example 8 could be accomplished by students who have not worked with word problems very often, but if so, it might be an interesting task!
To see more discussion worthy tasks, check out this website! http://map.mathshell.org/tasks.php
Each task has assessment rubrics as well
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Enjoy!
Reference
Lotan, R. A. (2003). Group-worthy tasks. Educational Leadership , 60 (6), 72-75.