I’m pleased Mathmistakes.org thought my area of walls problem was worth posting and I’m enjoying reading the comments. I submitted the mistake after writing about it here. Presenting this word problem to my 6th grade advanced math students was my aha moment where I learned that kids need to do much more reading, plus apply reading strategies, in math. The problem wasn’t an exercise in manual multiplication of decimals. It was intended for students to apply area in a different context.
After I wrote about the shaded rectangles task I did some additional reflection and I updated that post to include specific reading strategies when giving this task. In my original post I inferred. The update is more direct.
I would love to hear from teachers who teach reading in math. What are your success stories?
Want students to problem solve? Check. Want to challenge their reasoning skills? Check. Need to focus on reading and writing in math? Then check out the Shaded Rectangle task. I found it in the Sept. 2012 issue of Mathematics Teaching in the Middle School and couldn’t be more delighted.
It’s the only geometry task I have ever given where a geometric shape, or visual, is not provided. Students must construct a specific rectangle and 11 triangles within it, shade the alternating triangles, then determine the area of the shaded region–all from written directions. Way cool!!!
Here’s the task:
Draw a rectangle based on the following information
Divide the height into 3 congruent line segments by placing points on the line segment.
Divide the base into 4 congruent line segments by placing points on the base.
Construct triangles and determine the area
From the point just below the top-left corner, draw a line segment to each point on the rectangle.
Shade the triangle at the upper-left corner and then shade every other triangular region.
Before I gave the task, I wanted my department chair’s opinion. “Hmm. I’m visual. Will the kids have a hard time with this?” For me that was exactly the point. Would they think to, at a minimum, just sketch a rectangle as a starting point. Then place points somewhere on the base and height; think about what congruent means; think about a unit of measure and, as one student said, “The points need to be evenly spaced.”
This group began shading the triangles from a different point than what was listed in the directions.The sketch was accurately interpreted by this group of students. Are the bases and heights correct?
As with writing, beginning from a blank page is the most difficult part. I wanted the students to struggle but not feel defeated, so I tweaked the original lesson to include three resource cards. If students were stuck they received a hint.
I am adding this addendum to include additional reflections as it relates to reading. Many students spent nearly 40 minutes persevering on constructing the rectangle. The next time I do this task I am going to add two resource cards that focus in reading strategies. 1) Visualize. Sketch the rectangle and place the points where you think they should be. 2) Re-read. If you don’t understand the passage, re-read it. What context clues can help you with this task?
I talked with a colleague afterwards and we brainstormed the possibility of adding step-by-step directions. We both thought this wouldn’t help students to think for themselves, so we came up with the idea of adding the reading strategies as resource cards for students who needed additional support.
I also modified the task by dividing it into two phases, creating the rectangle and determining the area of the shaded region. After each phase the students had to explain their mathematical thinking in writing. This proved to be difficult for many students. So it’s something I need to do more in class.
Please let me know if you use the task, think of ways to modify it further, or to create more extensions. Speaking of extensions, some teachers may want to omit any reference to the dimensions listed in extension 1.
Again, let me know if you use it and how you may change it.
This is was the mathematical thinking I was hoping for.
Another example of mathematical thinking.Emerging mathematical thinking. I need to help students develop this skill.
My sixth graders and I dig GeoGebra. The students have spent the better part of the quarter diving deep into geometry and GeoGebra has been my anchor activity on many occasions. I worked on curriculum writing over the summer–writing targets, common assessments, etc. Yet the fun, creative part for me was gathering GeoGebra resources and organizing them into a 6th grade geometry wiki. With GeoGebra, the kids discovered area of triangles, parallelograms, and soon, trapezoids. Next they’ll examine volume and surface area with the GeoGebra applets.
The kids are incredibly engaged. Not because it’s technology. The applets are well designed learning tools. All they need to do is explore and follow some directions.
What’s even better is that GeoGebra isn’t limited to teaching geometry. I also collected several GeoGebra files for a 6th grade ratios and proportions wiki. Check out the tape diagrams, solving percents with bar models, comparing ratios, car race simulator, and more.
I am always looking for more middle school GeoGebra resources. If you can recommend any, please share.
It’s apparent that I need to do more reading strategies in math. I took for granted that my sixth grade advanced math students would be able to tackle this problem, yet only 3 or 4 kids were successful. When I gave them the problem most looked at me as if I had six heads. Here’s one student’s solution.
Many students had no idea where to start with this problem. (NY means “Not Yet”.)
This student either thought, “Hey, this is decimal multiplication practice, so all I have to do is multiply the decimals.” Or they are thinking this is a volume problem. Others found the area of the floor then multiply by four to get the area of the walls.
They obviously didn’t comprehend what the problem was asking.
We spent nearly 20 minutes sharing “My Favorite Not Yets” where students presented their work under the document camera and we analyzed their thought process. The class period turned into a reading lesson. We talked about context clues, what could be inferred from the problem, what do we already know, etc.
That afternoon during our team meeting our reading/literature teacher presented this reading strategy list to address reading informational text.
Next week, I’ll hand it out to students, model with a think aloud, then present them with a new problem.
On a side but related note, our team plans to use this with “fidelity” and monitor student progress using the informational text portion of the MAP reading test. We’re using the fall MAP as a baseline. Hopefully, their reading comprehension will improve.
Dan Meyer’s 3 Act Math tasks have inspired me to think about how this approach to learning could be adapted to other disciplines. So later this week I’m presenting Creative Use of Media in the Classroom to my colleagues during our Early Release time. Hopefully our Encore teachers, who sometimes feel left out when it comes to an inservice, will think they have been included.
I’m one of those teachers who needs to create “go to” lesson plans for when I am unexpectedly absent. Up until recently I was of the mindset of relying on the sub to introduce new concepts. I so wanted learning to continue that I even went as far as to create an instructional video on zero pairs for the sub to show in class. Unfortunately the sub did not have the math background to answer questions or to verify the students were using zero pairs correctly.
That’s not fair to the sub and it’s not fair to the students. So now I am one step closer to creating “go to” lesson plans. They will be plans that emphasize review and not introduce new concepts. Here are two that have been inspired by fellow bloggers.
Dee Chadwick reminded me of a site a stumbled upon over the summer, Tarsia, a free, downloadable jigsaw and dominoes generator. Mr. Barton has posted numerous pre-made files that range from elementary to high school.
For adding and subtracting integers my 6th grade Advanced Math students absolutely loved this activity I left for the sub.
Curiouser and Curiouser 