Intent

The core for a series of activities that open Investigations, Consecutive Sums poses an open-ended situation in which students are encouraged to make and test conjectures, construct proofs, and find counterexamples. This activity helps to establish a classroom environment of student-student interaction during the exploration of a challenging mathematical investigation.

Mathematics

Students examine complex, open-ended mathematical questions, develop and test ideas, write proofs using logical reasoning and algebraic notation, and disprove conjectures using counterexamples.

Students are also introduced to summation notation. Consecutive sums are defined as sums of consecutive natural numbers, such as 6+7+8+9=30, 35+36=71, and 1+2+3+...+10=55. The third example can be written, using summation notation, as ∑i=110i=55∑i=110i=55 size 12{ Sum cSub { size 8{i=1} } cSup { size 8{"10"} } {i="55"} } {} .

Progression

This activity requires a brief introduction, followed by a significant amount of time working in small groups to investigate, prepare posters, and present findings to the class. Split the activity over at least two days, assigning the individual activity Add It Up in the break. In Group Reflection, students reflect on the nature of the activity and the ways in which they worked together during the investigation.

Approximate Time

85 minutes

Classroom Organization

Groups

Doing the Activity

Students’ task is to explore patterns in consecutive sums and to create a poster summarizing their work, including descriptions of confirmed, disproved, and still-open conjectures.

Once you ensure that students understand what consecutive sums are and what they are being asked to do, the best way for students to begin is to “try stuff.” For the purpose of this activity, a conjecture might be defined as a “guess based on some evidence.”

To introduce the activity, you might spark student interest by offering a few examples of consecutive sums and posing a challenging question, such as one of those given below.

This activity will be explored over at least two days. At the end of the first day’s work is a good time to refocus groups on the products they are to create: posters that summarize their results and contain summary statements of the patterns they have observed. Remind students that they will classify patterns as conjectures, certainties (statements that are always true), and false conjectures.

Encourage groups to record clearly worded summary statements about what they think the pattern is. Offer an example of a clear summary statement, such as “Every number can be written as a consecutive sum.” Tell students that while this statement may or may not be true, it is the type of statement you are looking for.

The following are some of the questions that groups might investigate.

What numbers can be written as consecutive sums?

What numbers can be written as more than one consecutive sum?

Are there patterns to the answers to consecutive sums that are two terms long (such as 4 + 5), three terms long, or four terms long?

If groups have gathered some information but are not seeing any patterns, suggest that they try to reorganize the information in a way that might make patterns more visible.

As the exploration draws to a close, circulate to help groups focus on their summary statements. Following are some possible summary statements.

Discussing and Debriefing the Activity

Once groups have displayed their posters, review and discuss this collection of conjectures and summary statements. Ask a member of each group to state one of the patterns that the group found that hasn’t yet been mentioned. Continue until no group has summary statements that haven’t already been mentioned.

It may work best to have all the statements read before getting into discussion of or challenges to any of them. When ready, invite students to comment on the summary statements of other groups. They may have facts that contradict a given statement, or they may simply question whether a given generalization is valid.

Introduce the word counterexample in the context of these summary statements by asking whether there are any cases in which a generalization doesn’t hold. (If no one offers one, suggest one yourself.) For example, the summary statement “If a number can be written in three or more ways as a consecutive sum, then it must be odd” is false, and 30 is a counterexample. Although 30 fits the condition that “it can be written in three or more ways as a consecutive sum,” it doesn’t have the property “it must be odd.”

On the basis of this discussion, the class may eliminate or confirm some of the summary statements, while others will remain conjectures. For example, the statement “Powers of 2 cannot be written as consecutive sums of positive whole numbers,” though a true statement, will probably remain unproven at this time.

Key Questions

What numbers can be written as consecutive sums?

What numbers can be written as more than one consecutive sum?

If a number can be written as a consecutive sum, is that consecutive sum unique?

What numbers are not answers to some consecutive sum? Are there patterns in these numbers?

Are there patterns to the answers to consecutive sums that are two terms long (such as 4 + 5), three terms long, or four terms long?

Supplemental Activities

Three in a Row (extension) offers students an opportunity to explore sums of three consecutive numbers as well as sums of other lengths.

Any Old Sum (extension) asks students to examine sums that are not consecutive.