A preservice teacher leading a review for a quiz on rational number computation invited me to watch the lesson and work with her to improve on it. The objectives addressed in the review activity were from the Michigan Grade Level Content Expectations (GLCE):

**N. FL. 07.08**(GLCE): Add, subtract, multiply and divide positive and negative rational numbers fluently**B. N. FL. 07.09**(GLCE): Estimate results of computations with rational numbers

But the preservice teacher was also interested in developing conceptual understanding - especially around the idea of how multiplying and dividing by numbers between 0 and 1 impact the result. A good activity from NCTM (pdf) was found and modified in an effort to achieve these goals.

Move down or sideways (never up) through the maze fromStarttoFinish. You may not retrace any steps. Begin with 10 and as you move along a segment do the indicated computation. Record your steps on the scorecard. Your goal is to find the path that results in the largest (or smallest) value when you reach theFinish.

After the lesson, we discussed ways of using the activity more effectively. The first idea was to model what a path looks like. Because this kind of activity was new to the students, it took them a while to understand what was expected of them. For example:

What if we just followed along the left-most edge?

- 10 x 0.9 = 9;
- 9 x 1.75 = 15.75
- 15.75 + 5 = 20.75

Next, without doing any computation, we would ask the students to predict the path that would result in the greatest result or the least result. Making predictions is a great way to develop interest in a task. Student would share their predicted path and the rationale for their choice. This would provide some insight into the students' number sense related to multiplication and division of rational numbers.

Then the students would estimate the results of several paths. This would allow them to check out their predictions and refine our list of which paths might represent the largest (or smallest) value. Also, this would address objective

**B. N. FL. 07.09**from above.
Finally, the students would be asked to compute the path they believed would result in the largest (or smallest) value [

**N. FL. 07.08**]. Calculators are not allowed in this classroom but we decided that we might allow students to use calculators on up to half of the calculations. That way they would be exposed to the idea of using calculators strategically instead of with an "all-or-nothing" mindset.
As an extension, we might ask the students to find the easiest or hardest path to follow without using a calculator and why. We thought this would offer the students an opportunity to be metacognitive. It would also provide us with information on areas where students could improve on their fluency.

The preservice teacher was able to apply some of these subtle shifts to her later class with success. She writes:

...they did much better! They were excited to do something "more fun than boring problems." I was really happy with the responses I got...

Here are some thoughts off the top of my head:

ReplyDeleteI can see the estimation helping students quickly check results, but at the same time it feels like it bogs down the activity because one good choice does not necessarily lead to another equally good choice (or does it?). For example, starting at 10, I can multiply by by a number close to 1 (so my product will be a little less than 10), subtract a number close to 0 (so my difference will be nearly 10), divide by 1/3 (so my difference is a little more than 3), or add 7/10 (so my sum is approaching 11).

From these choices I'm drawn to adding 7/10 but is that the route I should actually follow? Is the puzzle designed so that I am rewarded for choosing the greatest answer at every decision point? Or are there times that taking a hit at one intersection will open up a path that ultimately results in a greater result in the end?

In my mind it might be better to have the students work with this puzzle for a time, and then talk about strategies for how students chose their paths, recorded their work, etc. Estimation may come up as a student explains that they didn't want to do all the calculations precisely right away.

If it doesn't come up, then the teacher may ask the class to think about how estimation could help them work more efficiently with the puzzle. At that point it feels the students are more primed to want to use it. Starting out with it just makes it look to the students like the teacher is making the task even more challenging than it really is. Adults generally see the value in estimation more quickly than students.

I don't know enough about where this teacher's students are with regards to their ability to operate with rational numbers, but is a string of 10 or so computations something the students can do successfully/reliably? They might be able to solve 10 distinct problems, but in this case the end result is dependent on success at every step. I could see an additional goal of this lesson to be focusing on how students are attending to precision. How do they accurately and clearly record their work - their calculations and their path?