case 1

Defining Quantities (Slide 2)
Computing and Explaining the Meaning of Speed (Slides 3-11)
Distance Forumula and Equation of a Circle (Slides 12-14)
The Box Problem: Introduction of the Box Problem (Slides 15-21)
Summary (Slide 22)

Defining Quantities (Slide 2): Click for Purpose

Purpose: Having students speak in terms of quantities will lay the groundwork for the entire semester. It is important that students speak with meaning about quantities so students must be pushed to express their thinking in terms of the identified quantities.

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Worksheet 1
Notes
Click for Summary/Goals
  • Introduce students to notion of quantity
  • Differentiate between quantities, their units of measurement, and variables

 

Click for Implementation Notes
  • Allow students to work on problems in their groups
  • Create a whole class discussion to go over each question as this is an opportunity to begin to introduce the normative way of approaching and discussing problems in the class
  • Be sure to push students to make their thinking clear
  • Reinforce the terminology that will be used during the semester
  • Recommended time: 15 minutes

 

Computing and Explaining the Meaning of Average Speed (Slides 3-12): Click for Purpose

Purpose: Since rate of change and average rate of change will be underlying themes during the course, dealing with average speed provides a context to introduce this notion that is familiar to students. Students should begin to talk about average speed in a conceptual way and be able to explain why it works as opposed to just memorizing a formula.

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Worksheet 1
Notes
Click for Summary/Goals
  • Introduce students to average rate of change in the context of average speed
  • Realize that using smaller interverals will result in more accurate speed
  • Create a discussion about instantaneous speed and its relation to average speed
  • Understand that average speed is determined by coordinating amount of distance with amount of time
  • Contrast constant speed with varying speed
  • Represent these situations graphically

 

Click for Implementation Notes
  • Allow students to work on problems in their groups
  • Create a whole class discussion to go over each question as this is an opportunity to begin to introduce the normative way of approaching and discussing problems in the class
  • Be sure to push students to make their thinking clear
  • Reinforce the terminology that will be used during the semester
  • Be sure to have a discussion about average speed before the formula is formalized
  • Recommended time: 15 minutes

 
Worksheet 2
Notes
Click for Summary/Goals
  • Interpret an algebraic model of a problem context
  • Represent the algebraic model graphically
  • Calculating and interpreting average speed in this context
  • Discuss the difference between arithmetic average and average rate of change
  • Develop and use the distance formula

 

Click for Implementation Notes
  • This worksheet should be given at the beginning of slide 11
  • The teacher should reinforce the students focus on how to calculate and interpret average rate of change
  • Students may use calculators on this problem
  • The teacher should make sure the students describe the significance of the intercepts in terms of the problem
  • The teacher should continue the discussion on average rate of change vs arithmetic average until the students are able to describe the difference
  • Recommended time: 15 minutes

 

Distance Forumula and Equation of a Circle (Slides 13-15): Click for Purpose
Content
Worksheet 2
Notes
Click for Summary/Goals
  • Give meaning to the distance formula
  • Give meaning to the standard form of the equation for a circle

 

Click for Implementation Notes
  • This worksheet should be given at the beginning of slide 11
  • The teacher should allow the students to build the formulas on their own
  • Recommended time: 15 minutes

 

The Box Problem: Introduction of the Box Problem (Slides 16-22): Click for Purpose

Purpose: This is where the students get introduced to the Box Problem activity. The teacher should use this opportunity to give the students the necessary supplies and go over the instructions with them. The students should work in their groups and attend to the quantities involved.

The students will have the opportunity to examine their boxes numerically, graphically, and eventually with a formula. The intent is that students become comfortable in recognizing and working with quantities that covary in this situation. Students will also have an opportunity to connect different representations for the same functional relationship. The box problem will also set up some of the major themes of the subsequent modules: average rate of change, relative extrema, covariation, and multiple representations.

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Worksheet 3
Notes
Click for Summary/Goals
  • Students work in groups to construct boxes, measure dimensions, and determine other physical quantities
  • They identify all quantities both on the net and on the box and determine constraints
  • Students will be asked to determine what quantities vary and what quantities remain constant
  • This leads into a class discussion about quantification, variables and parameters
  • Teachers form a concrete image of the situation.
  • This provides experience in identifying the quantities that change in a situation (variables) and those which are constant for the particular situation (parameters).

 

Click for Implementation Notes
  • This worksheet should be given at the beginning of slide 16
  • The students should spend 10-15 minutes creating their boxes and putting up their results

 
Explore Box Applet
Notes (doc, pdf, html)
Click for Summary/Goals
  • To give students a dynamic image of box problem
  • To help students determine what is happening with the volume as the length of the cutout is increased
  • To help students coordinate between the length of the cutout and the volume of the box

 

Click for Implementation Notes

This applet should be used whenever the students are struggling to create a dynamic image of the situation.

 
Explore Box with Cubes Applet
Notes (doc, pdf, html)
Click for Summary/Goals
  • Help students create a dynamic image of the box problem
  • To help students determine what is happening with the volume as the length of the cutout is increased
  • To help students coordinate between the length of the cutout and the volume of the box
  • The cubes allow the students to conceptualize how the volume is changing

 

Click for Implementation Notes
  • This applet should be used to help students create the graph of the situation
  • The teacher should focus students on how the quantities are changing as the length of the cutout changes

 
Exploring Intervals Applet
Notes (doc, pdf, html)
Click for Summary/Goals
  • Help students create a dynamic image of the box problem
  • Allow students to see how the volume is changing over discrete intervals of change in length of cutout
  • Help students recognize how average rate of change changes in this situation and how that affects the graphical representation

 

Click for Implementation Notes
Content
Graphing Quantities Applet
Notes (doc, pdf, html)
Click for Summary/Goals
  • Help students create a dynamic image of the box problem
  • Allow the various quantities (length of box, width of box, height of box, volume of box) to be plotted as a function of the length of the cutout

 

Click for Implementation Notes
Content
Exploring Algebraic Expression Applet
Notes (doc, pdf, html)
Click for Summary/Goals
  • Help students create a dynamic image of the box problem
  • Help students generalize the situation into relationship between the other variables (height, length, width, and volume) and the length of the cutout for specific lengths
  • Develop the algebraic formulas as generalizations of the arithmetic

 

Click for Implementation Notes
Content

Summary (Slide 23): Click for Purpose

Purpose: This slide gives the teacher an opportunity to reinforce some of the terms that will be used throughout the semester. The notions of quantity and covariation will be a part of each subsequent module so it is imperative that students leave module 1 with some kind of understanding of them.

This is also an opportunity for the teacher to reinforce the connections between different representations. In this case the students will do it for the box problem, but as new applications emerge in the modules, it is important that the students strive to make these kinds of connections. Thus it is important to initiate these kinds of conversations at this early stage of the class.

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