Sunday, May 21, 2017

Area of Polygons Discovery Lesson on Dot Paper

I created these pictures to give my students the opportunity to explore the areas of triangles, parallelograms and trapezoids in reference to the area of rectangles. I made copies of these two slides from my lesson presentation for the students to write on and they explored the areas of the shapes by drawing on the dot paper, counting square units, cutting off parts of the polygons and moving them to different places, and making conjectures about how the areas of the figures were related and why. This was a very engaging activity and the students used many different strategies to find the areas of these figures and these explorations formed the foundation for understanding the formulas for finding the areas of these figures. I was very pleased to see that students continued to use ideas from these pictorial models to help them with areas of polygons that were not displayed on dot paper (for example, triangle from left side of parallelogram is moved to the right side to form a rectangle with the same area).

The students had lots of interesting ideas for finding the area of Triangle F. I was amazed and thrilled when several students, connected the area of the obtuse triangle to the area of a paralleogram, instead of a rectangle, which was much easier for students in the class to see.

I use this slide to summarize the exploration. I highlighted part of each of the formulas in red to help students see the connection between all of the formulas:  base x height. Even the formula for trapezoid can be understood in this way if we talk about finding the average base x height.

 Here is a link to the google presentation, Area of Polygons Exploration, that contains these slides and some others I used to discuss and summarize the formulas.

Thursday, February 9, 2017

Cliff Jumping and Absolute Value

I love to show this video of Laso Schaller's world record jump of 58.8 meters in Maggia, Switzerland.
I use it when we start our unit on absolute value and it sparks a nice discussion about the distance above and below zero. I show the video and then ask the students what they would want to know before they jumped off the cliff. They usually come up with some very interesting comments including how deep the water is and how far down you will go after you hit the water. Using these questions I guide the discussion towards the total distance traveled by drawing a diagram on the board with a vertical number line showing positive numbers above zero and negative numbers below zero. I use 192 ft (58.8 m) for the top of the cliff and -48 ft. for the depth of the water needed for the jump. I got these numbers by using the following information about the depth of water needed for a safe jump based on the height of the jump.

(Depending on time constraints, I could leave off the numeric expression and ask the students to figure out what the depth of the water must be for this jump.)

This discussion helps provide the mathematical motivation for absolute value and why positive numbers remain positive and negative numbers become positive when we are measuring distance. And then at this point I offer the mathematical notation and definition of absolute value.

Here are some still shots from the video:

The video replays the jump several times and shows different angles and information with each replay. For some of the replays, additional information about the jump overlays the video including the height of the top of the cliff as it is compared to the Leaning Tower of Pisa and also a time lapse of the speed and time of the Laso Schaller throughout the jump.

Sunday, January 29, 2017

A good problem is worth a thousand worksheets

I got this problem from

All of my students worked on this problem very enthusiastically for most of the class period. It was truly a low floor  high ceiling task because every student in the class was dividing fractions (practicing that skill) and noticing that the quotient was either smaller or larger than previous attempts. Some students used guess and check and were thrilled when they found larger quotients. Other students were more systematice and tried the digits 1,2,3 and 4 and found the largest quotient with these numbers. Others tried 6, 7, 8 and 9 and found the largest quotient with these numbers. And other students noticed that they should use 1, 2, 8 and 9. There was a lot of discussion in the room and students were comparing and discussing their results. Some students noticed that there was more than one way to get a quotient of 36 (or 1/36). And many students were starting to notice and generalize that larger quotients result from larger dividends and/or smaller divisors.

Low floor high ceiling tasks are also sometimes called low threshhold high ceiling or low entry high ceiling tasks. NRICH describes a low threshhold high ceiling task like this:
"A LTHC mathematical activity is one which pretty well everyone in the group can begin, and then work on at their own level of engagement, but which has lots of possibilities for the participatns to do much more challenging mathematics."

I am looking forward to trying this problem with my students now that we are working on ratios: