#GETMath: Launching the Problem for Sums of Consecutive Numbers

Have you had a chance to try the Sums of Consecutive Numbers problem with your students?  If not, you might be wondering how you might launch the problem.  I did not give specific ways to launch the problem in the video because of the wide range of learners that might be participating.  After further reflection, I thought it might be worthwhile to share how I involved students in the problem.

I started the problem with a grade 3 class late last week.  We began by talking about the Global Read Aloud project and what they enjoyed about it.  Students said what they enjoyed most was talking with others about the book.  After explaining that I wanted to do something similar with a math problem, the class was eager to get started!  I shared the title of the problem "Sums of Consecutive Numbers" and asked students to tell me what they knew about "sums".  The class quickly agreed that sums were answers to addition problems.  Defining "consecutive" was a little more challenging but with the help of a dictionary, students soon agreed that consecutive numbers would be numbers that follow one another in sequence, such as on a number line.

 After drawing a number line to use as a reference, we began to brainstorm possible answers.  On a large chart, students were able to write their answers once they shared their idea and the class was in agreement that the example fit the rules of the problem.

After we brainstormed a few examples and I felt that students had a strong understanding of the problem, I encouraged students to explore the numbers on their own.  Students received a chart for them to record their thinking individually.  To begin, students were encouraged to copy the examples from the class chart.

While students were working, I circulated the room and asked questions about their strategies.  Some students immediately saw a pattern with the odd numbers - noticing that any odd number could be made as the sum of two consecutive numbers.  Other students tried using three or four addends and writing down the sum they discovered.

Students worked for about 15 minutes and then I stopped the class.  We discussed our observations about the patterns and wrote them on a chart.  I encouraged students to continue working on the problem and told them that they should not be limited to the 50 numbers on the chart.  I asked students to continue adding to the chart as they noticed other patterns.  I can't wait to check in this week to see what else they have added to the chart!

If you have been waiting for the moment to introduce this problem, I hope this helps you get started!  The students were so engaged with this problem that they continued working during their indoor recess!  Please ask questions or share how things are going on Twitter at #GETMath!  I look forward to hearing from you!

Using #GETMath to teach the Standards for Mathematical Practice

Some of you may have thought about using the Sums of Consecutive Numbers and joining in the #GETMath conversation but you're feeling stuck because you don't feel like the problem clearly connects to the content for your grade level.  (If you haven't heard about #GETMath, learn more here: http://bit.ly/1eybFKx)

This problem does not need to be (nor should it necessarily be) completed in one or two class periods.  Students can play with the problem during the month of February and share their observations once or twice each week.

This is an opportunity emphasize the Standards for Mathematical Practice!  

1. Make sense of problems and persevere in solving them.
• Sums of Consecutive Numbers has many different aspects to explore.  In introducing the problem to a 4th grade class, some students immediately thought to use more than two addends.  Another student made a conjecture that no even number could be made by only two addends.  The students were ready to get started and excited to look for patterns!  We created a chart where they can confirm or revise their findings and add additional observations.
2. Reason abstractly and quantitatively.
• Because this standard focuses on connecting real situations with symbolic representations, it is not really a standard that could be emphasized during this exploration.  
3. Construct viable arguments and critique the reasoning of others.
• As noted above, students were making conjectures about the patterns and working to prove or disprove their thinking.  As students continue working through the problem, they can add to the class chart.  We even plan to video chat with another 4th grade class - what a great opportunity to share our arguments and discuss our thinking about the problem!
4. Model with mathematics.
• As with SMP #2, this would not be the best standard to apply with the Sums of Consecutive Numbers problem.  Students might use a table to organize their thinking but they do not apply math to solve a real-world problem.
5. Use appropriate tools strategically.
• As with SMP #2 and #4, this would not be a strong standard to emphasize with this particular problem.  While the table or organizer used by students might be considered a tool, it would be a weak connection to this standard.
6. Attend to precision.
• In the Sums of Consecutive Numbers problem, students need to be sure that the numbers they use are indeed consecutive numbers and check to be sure the sum they find is accurate.  They also need to be able to communicate their thinking in a way that is clearly understood by others.  This would be a strong standard to emphasize with this problem!
7. Look for and make use of structure.
• The Sums of Consecutive Numbers problem provides students with an opportunity to notice and discuss the structure of mathematics.  For example, why can the odd numbers always be made by the sums of two consecutive numbers? 
8. Look for and express regularity in repeated reasoning.
• The Sums of Consecutive Numbers problem provides students with an opportunity to observe patterns and reason about generalizations.  For example, what numbers cannot be made by the sums of consecutive numbers?  Could you predict the next number that cannot be made as a sum of consecutive numbers?  Can you write a rule that will help you find all of the numbers that cannot be made by the sum of consecutive numbers?

The Sums of Consecutive Numbers problem has clear connections to five of the eight Standards for Mathematical Practice.  The CCSS emphasizes that the Standards for Mathematical Practice are as essential as the mathematical content.  While it would be difficult to emphasize five Mathematical Practices within your lesson, there are many opportunities for you to engage students in the Standards for Mathematical Practice through this problem. 

I hope you will join us and share your thinking at #GETMath!