Analysis of Student WorkSequences

Leslie RitcheyKim Fong

Amanda Melton

The problem

• You are inviting some friends to a dinner party. As more people arrive, you add tables as shown in the diagram.

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Stage 1

Stage 3

The dinner party problem diagram:

Stage 2

The task

a) Draw the first five stages.b) Write a rule you can use to find the number

of people who can be seated at any number of tables.

c) How many tables will be needed for 100 people?

d) How many people can be seated if you have 37 tables?

How the task was introduced• Reminded students of their previous experience

writing cost equations with the Adventure Zone problem: It costs $5 to get in plus $2 per ride

• Equation: c = 2r + 5

• We showed the repeated addition of $2 per ride as multiplication

• Suggestion: Make a T-chart or table

Questions we asked as they worked

• What does each part of your rule represent?• Can you show me this part of the rule in your

picture?• How did you find your rule?• Have you tested your rule?• How can you make sure this rule works?• What do your variables represent?

Most common misconceptions• Rule as n + 2

• In the rule 2n + 2, the +2 on the end represents the 2 people added with each new table

• Discussion: compare n + 3 with 3n for the sequence 3, 6, 9… Which rule works depends on what n represents.

• Use the diagram to show where each number in the rule is represented

Most common misconceptions, continued

• Not sure what the numbers in the rule represent

• Rule as 2n or 4n without checking for correctness

• Use diagram to connect visual representation to algebraic representation

• Encourage students to test their rule for several stages Encourage students to make and use a table

Categories of student work

• Unusual rules that worked• Typical rules with conceptual explanations• Typical rules with procedural explanations• Typical rules with confused or no explanations• Rules that didn’t work

Unusual rules that worked

Groups that wrote unusual rules all explained them in terms of the diagram. They had difficulty writing verbal explanations for their rules.

“Typical” rules with explanations that were linked to the picture

Procedural at first

Typical rules with procedural explanation only

The “because it works” explanantion

Some groups with procedural explanations arrived at the typical rule very quickly and easily.

Some groups with procedural explanations had difficulty coming up with their rule.

Misrepresentations

Rules that didn’t work

One group could not come up with a rule and used this strategy to answer the questions.