Intent This activity introduces the mathematical symbol for summation notation. Students begin to understand the utility of this notation by working with both numeric and geometric examples.

Mathematics One of the challenges of secondary mathematics teaching is helping students to understand the notational systems used to express complex ideas in a compact form. This activity introduces one such system, summation notation, and offers students opportunities to start to make sense of it.

Progression This activity serves as a useful way to break up student work on Consecutive Sums. Students may elect to utilize summation notation in their posters for Consecutive Sums and in POW 2: Checkerboard Squares.

Approximate Time 10 minutes for introduction 20 minutes for activity (at home or in class) 10 minutes for discussion

Classroom Organization Individuals, followed by whole-class discussion

Doing the Activity Introduce the activity with a multiterm example of a consecutive sum, such as 3 + 4 + 5 + 6 + 7 + 8 + 9. Demonstrate that there is a shorthand way for writing such sums: ∑i=39i size 12{ Sum cSub { size 8{i=1} } cSup { size 8{"10"} } {i="55"} } {} . Explain that this symbol is an uppercase letter in the Greek alphabet, called sigma, and that the expression is read, “The summation, from i equals 3 to 9, of i.” Invite students to articulate the connection between the shorthand and the full expression. [link to math maps] Explain that the letter i is called a dummy variable and that any letter would work. The expression ∑t=39t size 12{ Sum cSub { size 8{i=1} } cSup { size 8{"10"} } {i="55"} } {} means exactly the same thing as ∑i=39i size 12{ Sum cSub { size 8{i=1} } cSup { size 8{"10"} } {i="55"} } {} . Use a more complex example to illustrate in detail how this notation works. For example, ask students what they think this expression means. ∑ w = 3 7 ( w 2 + 2 ) size 12{ Sum cSub { size 8{w=3} } cSup { size 8{7} } { \( w rSup { size 8{2} } +2 \) } } {} How can you “act out” the process described by this summation expression? Help students act out the process. First, w is 3, so the first term is 3²+2. Then, w is 4, so the next term is 4²+2. Then, w is 5, so the next term is 5²+2. Then, w is 6, so the next term is 6²+2. Finally, w is 7, so the next term is 7²+2. Since the symbol Σ indicates summation, these terms must be added together. In other words, the notation represents the expression. (3²+2)+(4²+2)+(5²+2)+(6²+2)+(7²+2) Point out that although this example does not give a consecutive sum, the values for w are a sequence of consecutive numbers. The mechanics of summation notation are summarized in the student activity. Students will work with this notation in geometric as well as in purely numeric contexts. Don’t get bogged down on mastery of the notation; it is intended only as a tool to help students express their ideas. You may want to introduce the use of ellipsis notation, such as writing 1+2+...+100 for the sum of the whole numbers from 1 to 100

Discussing and Debriefing the Activity Give students an opportunity to share responses and ask questions of one another. For Question 2, students will likely see the picture in terms of the sum 1+2+3+4 and produce an expression like ∑i=14i size 12{ Sum cSub { size 8{i=1} } cSup { size 8{"10"} } {i="55"} } {} . The expressions for Question 3 can be written in various ways. Question 3c is especially likely to lead to different answers, such as ∑ t = 2 6 ( 3t + 2 ) size 12{ Sum cSub { size 8{w=3} } cSup { size 8{7} } { \( w rSup { size 8{2} } +2 \) } } {} and ∑ j = 3 7 ( 3j - 1 ) size 12{ Sum cSub { size 8{w=3} } cSup { size 8{7} } { \( w rSup { size 8{2} } +2 \) } } {} . You can leave this question open if students cannot find a way to write the expression using summation notation. For Question 4, the diagram suggests the idea of a sum of squares and can be expressed as ∑ n = 1 5 n 2 size 12{ Sum cSub { size 8{w=3} } cSup { size 8{7} } { \( w rSup { size 8{2} } +2 \) } } {} . Students’ facility with summation notation will increase as they find situations where it is useful.

Key Question How can you “act out” the process described by this summation expression?