Technology Review: Activity
Shodor
Activity
Goal: Use the Sequencer applet, to explore how various "multipliers" and "add-ons" create sequences that resemble linear and exponential equations.
Warm up: Have students explore (in partners or groups) how different start values, multipliers, and add-ons effect the sequence. Students should record what they observe on the Sequencer Exploration Questions Worksheet. As a class, discuss their findings.
Lesson: Now that students are comfortable with the applet and have thoroughly explored the different input values, we want to focus specifically on arithmetic and geometric sequences. Use the following questions to guide student exploration.
- What do students notice about how the starting point relates to the linear model?
Select specific values for the various quantities and have students use the points on the graph to determine the equation of the linear or exponential function. I suggest using specific values so that you can check to see students reasoning. This will also avoid students choosing to have a multiplier and an add-on.
After students have gotten the hang of making several linear and exponential functions, have them analyze how the functions relate to the values selected. From these generalizations, we should be able to derive the explicit formula of an arithmetic and geometric sequence.
Example:
Starting point: 2 Multiplier: 3 Add-On: 0 Equation: y=2(3)^x Generally: y=StartingPoint*(Multiplier)^x
Starting point: 4 Multiplier: 1 Add-On: 5 Equations: y=5x+4 Generally: y=AddOn*x + StartingPoint
Warm up: Have students explore (in partners or groups) how different start values, multipliers, and add-ons effect the sequence. Students should record what they observe on the Sequencer Exploration Questions Worksheet. As a class, discuss their findings.
Lesson: Now that students are comfortable with the applet and have thoroughly explored the different input values, we want to focus specifically on arithmetic and geometric sequences. Use the following questions to guide student exploration.
- What values create a sequence that could be modeled by a linear function?
- What do students notice about how the starting point relates to the linear model?
- What values create a sequence that could be modeled by an exponential function?
Select specific values for the various quantities and have students use the points on the graph to determine the equation of the linear or exponential function. I suggest using specific values so that you can check to see students reasoning. This will also avoid students choosing to have a multiplier and an add-on.
After students have gotten the hang of making several linear and exponential functions, have them analyze how the functions relate to the values selected. From these generalizations, we should be able to derive the explicit formula of an arithmetic and geometric sequence.
Example:
Starting point: 2 Multiplier: 3 Add-On: 0 Equation: y=2(3)^x Generally: y=StartingPoint*(Multiplier)^x
Starting point: 4 Multiplier: 1 Add-On: 5 Equations: y=5x+4 Generally: y=AddOn*x + StartingPoint
Standards
CCGPS Coordinate Algebra Standards
- MCC9-12.F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. (Draw connection to F.BF.2, which requires students to write arithmetic and geometric sequences.)
- MCC9-12.F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
- MCC9-12.F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
National Council of Teachers of Mathematics Standards
- Grades 6-8 (Numbers and Operations): Understand meanings of operations and how they relate to one another
- Grades 9-12 (Algebra): Understand patterns, relations, and functions; Use mathematical models to represent and understand quantitative relationships