By Jane M. Wilburne, Posted October 27, 2014 –   

Welcome back! I hope you and your students had the opportunity to explore finding the sum of a series of consecutive numbers. This problem can easily be adapted for any grade level and can offer opportunities for good classroom discourse. At the first-grade level, students in one classroom were asked to find the sum of the first five numbers: 1 + 2 + 3 + 4 + 5.

One first grader rearranged the order of the numbers and added to find a sum of 15:

(1 + 5) + (2 + 4) + 3

Another student added the numbers in the order they were given:

    (1 + 2) =         3

    (3 + 3) =         6

    (6 + 4) =       10

  (10 + 5) =       15

The teacher asked the class to compare the two different approaches that their classmates had used. The discussion provided a great opportunity for students to explore the associative (grouping) property. The teacher demonstrated the problem by showing the class one red Unifix^® cube, two blue Unifix cubes, three green Unifix cubes, four yellow Unifix cubes, and five white Unifix cubes. She asked students to think about how they could put the cubes together to find the total sum of cubes. As students shared such strategies as putting the red and white cubes together, grouping the blue and yellow cubes, and then adding the green cube, their teacher helped them make the visual connection to the two different student strategies.

At the third-grade level, some students used a hundred chart to find the sum of the first fifty numbers. First, they found the sum of each row:

      
      1  + 2 + …    + 9 + 10  =   55,

     11 + 12 + …  + 19 + 20  = 155,

     21 + 22 + …  + 19  + 30  = 255,

     31 + 32 + …  + 19 + 40 = 355,

     41 + 42 + …  + 49  + 50 = 455.

Then, they added 55 + 155 + 255 + 355 + 455 = 1275. Students noticed the pattern for the sum of each row increasing by 100 (a fine example of the eighth of the Common Core State Standards for Mathematical Practice, SMP 8: Look for and express regularity in repeated reasoning). When students were asked to justify why the sum of each row increased by 100, several students stated that the numbers in each row had a value that was ten more than the number in the row above it. Thus, 10 × 10 = 100. This showed the teacher that they were connecting their knowledge of place value to the hundred chart.

When the teacher asked students to find the sum of the first 100 numbers, some of them used a calculator and added (500 + 600 + 700 + 800 + 900) + (55 × 5) = 3775 to the 1275 (the sum of the first fifty numbers) for a total sum of 5050. Other students added 555 + 655 + 755 + 855 + 955 = 3775 to the sum of the first fifty numbers (1275) for a total of 5050. The use of the hundred chart to look for patterns with the sums of each row helped students reason quantitatively and make mathematical connections with patterns and groupings of numbers.

In both examples, we see how students from different grade levels used their knowledge of mathematical properties to find the sum of a series of the first n consecutive integers. Problems and tasks such as these give students opportunities to use their knowledge of basic operations, the commutative and associative properties, and different strategies to solve problems as well as to share and defend their approaches. They are engaged in such Standards for Mathematical Practices as SMP 1: Make sense of problems and persevere; SMP 2: Reason abstractly and quantitatively; SMP 3: Construct viable arguments and critique the reasoning of others; SMP 4: Modeling with mathematics; SMP 5: Use appropriate tools strategically; SMP 7: Look for and make use of structure; and SMP 8: Look for and express regularity in repeated reasoning.

The historical connection to Carl Friedrich Gauss (1777–1855) offers students an opportunity to see the richness of mathematics over time. Having students solve the summation of the series of integers from one to n before sharing Gauss’s approach may help students appreciate his contribution to mathematics. Students can see how Gauss engaged in SMP 7: Make use of the structure of the problem to find a shortcut to a solution.

Jane M. Wilburne is an associate professor of mathematics education at Penn State Harrisburg. She teaches content and methods courses for both elementary and secondary mathematics teachers as well as graduate mathematics education courses. She is a co-author of Cowboys Count, Monkeys Measure, and Princesses Problem Solve: Building Early Math Skills Through Storybooks (Brookes Publishing 2011) and has published numerous manuscripts in Teaching Children Mathematics, among other journals. Jane began serving as a member of the Teaching Children Mathematics Editorial Panel in May 2014, and her term will continue through April 2017.

Archived Comments

Wow, this is very interesting. I learnt series and sequence at my 11th grade years back. if this concept can be taught like this to these young ones, they will appreciate mathematics, see its beauty and know that it is fun.
Posted by: OLUSOLAA_45717 at 11/1/2014 4:48 PM