Using data to generate conversations about strategies

Recently, as a part of my evaluation, I gave students in a 5th grade class the following two problems to solve:

1. OfficeMax sells pencils in large boxes.  Mrs. Bailey bought some pencils for the students at school.  There were 150 pencils in each box.  She bought 8 boxes.  How many pencils did Mrs. Bailey buy?
2. 86 x 23 = ____

Students were able to solve the problem in any way they wanted during this pre-assessment.  In past years, I felt like I had involved students in discussing their strategies and analyzing both their own work and the work of others.  Because I am not a classroom teacher this year, I was more explicit in presenting the data to students and in this process we noticed some interesting patterns about strategies.  Here is a chart of the accuracy and strategies students used to solve the problems above:

For the first problem:  OfficeMax sells pencils in large boxes.  Mrs. Bailey bought some pencils for the students at school.  There were 150 pencils in each box.  She bought 8 boxes.  How many pencils did Mrs. Bailey buy?

Strategies                                                       Accuracy

Repeated Addition	10/11 students answered correctly	91%
Standard/Traditional Algorithm	6/9 students answer correctly	67%
Doubling/Halving	1/1 students answered correctly	100%

For the second problem: 86 x 23 =

Strategies                                                       Accuracy

Repeated Addition	0/1 students answered correctly	0%
Standard/Traditional Algorithm	6/15 students answered correctly	40%
Partial Products with Boxes	4/4 students answered correctly	100%
Partial Products	0/1 students answered correctly	0%

Students were engaged in the conversation and shared great thinking!  For example, students explained why someone might use repeated addition in the first problem but not in the second problem since the numbers became too large to easily keep track of the addition. 

We WANT for students to have multiple strategies for solving problems and to know WHEN and WHY to use them.  It might make more sense to use repeated addition when solving 175 x 3 because the numbers are reasonable for addition.  Students should recognize, however, that when solving 175 x 83 that repeated addition is not a reasonable strategy to use.  This activity also led to a great conversation about efficiency.  Which algorithms are most efficient?  In what situations?  I'm sure the students in your classrooms have great ideas to share!

Applying Powers of 10

Our district has a new superintendent this year.  In recognition of his 100th day since joining the district on July 1, 2013, our 5th grade students created a sign that read "The Power of 10".  We took a photo as students stood in front of the sign holding white boards with various representations of 10 x 10, including 10^2 and 100.  We then did a second photo with students standing in front of a sign that read, "We can't wait to see what happens by the next power of 10" with students again holding white boards with representations of 10 x 10 x 10, including 10^3 and 1,000.  We e-mailed these pictures to the new superintendent and he was appreciative of the message.

This was a great activity that allowed students to apply their understanding of powers of 10.  They were finishing a unit about powers of 10 and were preparing to take their unit test.  I challenged the students to figure out on what date that 1000th day would fall.  The classroom teacher said the students were engaged and talked about the problem for at least two days.

One student had an elegant solution.  He started with October 8, 2013 being the 100th day.  The student added 365 days to find October 8, 2014 and then an additional 365 days to find October 8, 2015.  From this date, he added by months until he reached 1,000 days on March 26, 2016.

When the superintendent next visited our building, he stopped in the classroom of this student and asked where the student would be on March 26, 2016.  Our superintendent then put a reminder on his phone to take this student out to lunch on March 26, 2016 to celebrate his 1,000th day.  What a great way to acknowledge the great mathematical thinking that occurred in this classroom!  Thank you to our new superintendent for encouraging such excitement about mathematics!