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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.
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