Intermediate Steps Matter

So I asked a student the following question:  If five notebooks cost $7, how much does 40 notebooks cost? This is the student work that she produced:

Is this correct? How would you grade the process? I would argue that some of the intermediate steps would definitely need attention.

A Toy Story

Finding the Average Speed of Moving Toys in Mrs. Tao's Algebra II Class...

Students were instructed to choose two wind up toys and find their average speed after winding them to their maximum capacity. Students were given a variety of tools to accomplish this. I also reminded them of the definition of speed. Students began by formulating different strategies and plans based upon the materials provided. Critical thinking began when the students raced to the table to select their toys. They immediately began winding them and observing their movements. Many students made the assumption that the simpler moving toys would be easier to work with. Once the students chose their toys, they began to formulate questions, such as, “If the toy doesn’t move in a linear path, how will I calculate the distance?” and “How do I address the fact that the toy slows down the longer it travels?”.

Ultimately, students marked the toys’ path on grid chart paper, then used one of two popular strategies. The first was to measure the toys’ linear paths with rulers and circular paths with compasses, using circumference. The second, was to trace the toys path with a piece of string and then measure the string of the sheet of paper was what we were looking for.  

What are some other ways that students could have found the speed of their toys?

(Sarah Tao teaches Pre-AP Algebra II at Galena Park High School in Galena Park ISD.)

Hey, I Noticed... (Parking Meter)

Hey, I noticed this while looking a video from a couple of months back.

As I was paying for street parking, I noticed what appeared to be a function that compared the amount I was paying with the time I would be allowed to park here.  But was this a function?

• How much was I paying per minute?
• Could I figure out how much I would have to pay for 3 hours? 30 hours?

What are some good inquiry questions that you can pose for your students with this inquiry starter?

Since it's 03/14....

With it being Pi Day and all, I decided to do some research on early mathematicians (bearded or not) deriving a value for pi.  I always thought it would be somewhat hard to do without having Calculus at your disposal.  I found this good article that explains exactly what I was looking for:

https://betterexplained.com/articles/prehistoric-calculus-discovering-pi/

It got me to thinking, how could you use the concept of inscribed and circumscribed circles to get an estimate for pi in math classes before Calculus?  I love the algebraic thinking that can come about with this type of task!

Folding Up!

Deriving Surface Area Formulas in Mrs. Gutierrez’ Math 8 Class...

Students were given a sheet of paper and asked to fold it as many times as wanted in the same direction.  After folding the sheets, they labeled the base and height of each resulting rectangle using variables and then taped the figure closed and stood it up to create a prism.

“When I asked my students to find the lateral surface area of the paper, most students immediately noticed that the area of the sheet of paper was what we were looking for.  BUT what I found interesting was that some students, when I let them struggle, wanted to find the area of each section of their paper and then add them up.  Another group of students found it more efficient to add all of the bases together and then multiply by the height.  This allowed me to have a rich, authentic discussion about using the distributive property and factoring.  Once students realized they were all doing the same thing, they decided that the best method was to all up the bases and multiply by the height which was the same thing as the perimeter of the base times the height (Ph).  Students walked away with a better understanding of the lateral surface area formula for prisms because they came up with it on their own.”

 When have you had those teachable math moments that you couldn't pass up?

(Rosa Gutierrez teaches 8^th Grade Mathematics at YES Prep Southwest in YES Prep Schools.)