Teaching is really only as valuable as the learning experiences and opportunities created for the students.  Crafting and choosing which experiences and opportunities are available to students should be at the forefront of any teacher’s mindset when planning lessons or units.  So let’s delve a little deeper into crafting and choosing learning experiences.

One of my classes has been using the geometric study of area in order to practice and apply multiplicative thinking.  Crafting and choosing specifically rich tasks that engage the students has been one of my major goals for this unit.  So I’d like to just investigate different iterations of reasonably similar tasks that apply multiplicative thinking in the geometric context of area.

You could just start and end your search with:

Draw a rectangle with an area of 56 units.

This certainly gets the job done.  Students have to think about factors of 56 and which make the most sense.  The great thing about this task and others like it (check out Open Middle for more) is that there are a multitude of correct answers.  Having many “right answers” leads to deeper group work and richer class discussions.  The drawbacks to this task are the lack of visual support and the separation from a concrete representation or real world context.  Also the language processing comprehension of this task is high, so students who struggle in this area would require additional supports.

You could add in some visual support:

The extra visual modeling puts students who struggle with language processing on the right track.  One major drawback to this support is that you’ve essentially done the first step for the students.  The direct model approach could be good or less good, depending on the specific group of students.  Also this task requires a bit of comfort with multiplication and does not necessarily support emerging multiplicative thinkers.

To better support emerging multiplicative thinkers you could use this from Illustrative Mathematics:

India is remodeling her bathroom.  She plans to cover the bathroom floor with tiles that are each 1 square foot.  Her bathroom is 5 feet wide and 8 feet long.  India needs to stay within a strict budget and must purchase the exact number of tiles needed.

How many tiles should India buy?

In this task we have taken away the visual support, but added in the real world context and raised the level of language processing again.  This task does support emerging multiplicative thinkers because it relates the area of India’s bathroom to the idea of individual square feet.  The students can construct their own meaning of the area of India’s bathroom using drawings, manipulatives, or graph paper.

In order to keep both the visual support and the real world context, we could use this task from TERC Investigations:

This task combines the visual support with the real world context of tile floors and the rugs that partially cover them.  As if India needed to find the area of her bathroom with a rug halfway out the door.  This also supports emerging multiplicative thinkers because it directly models the 1 square foot idea.  However, the cognitive demand of this task is somewhat lower than the tasks discussed previously.

In a effort to get the best of all worlds I used this task with my students today.

Multiplicative Strategies: Repeated Addition and Area Model

Multiplicative Strategies: Area Model and Derived Facts

If you look at the task PDF above and these work samples, you’ll notice I added in an extension question, because I realized the cognitive demand of this task could be raised.  However, my students were sufficiently challenged solely by this task today and the extension question was saved for another day!  This task combines the visual models, real world context, and cognitive demand of most of the other tasks above.  The language processing demands are also quite low here as well.

As you can see in the work samples, both students (and really all of them) began trying to count all the squares inside the shapes.  They virtually ignored the obstructions to their counting method.  Though all it took was a little prompting to acknowledge this obstacle and then have a quick class discussion to brainstorm alternative strategies.  I have recorded the multiplicative strategies I found in each of these work samples, but if you find anymore let me know in the comments!  My students would benefit from your feedback.

So the next time you’re planning a lesson or a unit, think about the different ways to present a task.  The way you present the task should come from deep thought about the mathematical strengths, goals and cognitive pathways your students use to access the content.  You could use all of these tasks, some of them, or none of them, but any of them is better than worksheets with loads of rectangles and meaningless calculations!

What tasks or units have you presented in different ways?

Share your experiences in the comments!

UPDATE (2/23/15):

I tweeted about some of these tasks earlier in the day and received this from Lisa Noble, a teacher in Ontario, Canada.  She used my tweet in a discussion with her students to compare the tasks.  Here are their thoughts:

@bkdidact is this easier? pic.twitter.com/L4IwKuU6pK

— Lisa Noble (@nobleknits2) February 24, 2015

Reason #842 that twitter is awesome!