Overview
Curator: Marissa Hoffman
Technical & Cost considerations:
There are not a lot of obstacles standing in a teacher's way if he or she wants to implement this activity into her lesson plans. Java and or Flash Player may need to be downloaded and updated on whichever wifi enabled device the teacher has available, and a high-speed interconnection coupled with Google Chrome is recommended. But other than that, this engaging and individualized tool will work on a SmartBoard for whole class instruction or on a laptop/tablet for students who want to work in small groups or individually. The best part is that this tool is free and allows students to use it an unlimited number of times.
Evaluation
Description of Learning Activity
The goal of this learning activity depends on the goal of the teacher's lesson. This independent, clear-cut technology tool allows teachers to use this tool as the center-point of a lesson as students construct their own knowledge about Riemann Sums. Or, the teacher could use the tool as a way for students to experiment which type of Riemann Sum will give them the most accurate approximation of the area between two curves. The features this tool provides gives the students the chance to change functions, or parts of functions, change the type of Riemann Sum and the number of partitions. In turn, this saves time and mental energy when it comes to conceptually understanding Riemann Sums which should also increase student motivation since time is not spent on manual tasks.
1. Learning Activity Types
- LA-Present - This tech tool presents to students the behavior and relationships of two functions and the use of successive rectangles (of equal width) to approximate the irregular area between the two functions.
- LA-Present-Demo - As students change parameters and functions for this activity the changes are automatically demonstrated graphically and numerically in terms of the behavior of each function in a given domain and range window, the approximated area, and the number of rectangles.
- LA-Explore - Students can explore and establish rules of Riemann Sums as the technology gives them the freedom and ease to change entire functions, parts of them, and certain properties of the Riemann Sums.
2. What mathematics is being learned?
Standards
Principles and Standards of School Mathematics (PSSM)
- Analyze properties and determine attributes of two- and three-dimensional objects
- Draw and construct representations of two- and three-dimensional geometric objects using a variety of tools
- Use vertex-edge graphs to model and solve problems
- Use geometric models to gain insights into, and answer questions in, other areas of mathematics
Common Core State Standards Mathematics (CCSSM)
- Make geometric constructions
- Use coordinates to prove simple geometric theorems algebraically
- Visualize relationships between two-dimensional and three-dimensional objects
- Apply geometric concepts in modeling situations
Mathematical Practices
- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.
Proficiency Strands
- conceptual understanding - Students are conceptually learning how the behavior of two graphs and the type (and number) of Riemann Sums being used changes an approximated area between two curves. Students are not taught how to calculate this by using this tool but are able to draw conclusions and pose rules for the properties of Riemann Sums.
- strategic competence - Predictive strategies are used as students master the concept of Riemann Sums and must determine which kind of sum and how many rectangles, will give the most accurate approximate areas between two curves. This strategy will develop over time as students familiarize themselves with the properties of Riemann Sums from what is observed from the Finding the Area Between Two Curves activity.
- adaptive reasoning - Students will slowly develop rules for using Riemann Sums based on initial use of this technology. However, students will have to adapt their reasoning and conclusions when a counterexample is used to expand the conceptual understanding students will gain from this activity.
This activity focuses heavily on the geometric properties of rectangles and the concept of change in the algebra standards. Using Riemann Sums to calculate the approximate area between two curves is dependent on a students ability and understanding to construct a series of equivalent rectangles, determine their height based on the difference between two functions, and then summing it all together. The number of rectangles used will drastically change the approximated area and students will see this as they use the slider tool to increase and decrease the number of rectangular partitions. This method of integration is an intersection of geometry and algebra that students can only master if they are proficient in both strands of mathematics.
3. How is the mathematics represented?
This tech tool works for a variety of learners because it is hands-on and visual by allowing the mathematics to be represented graphically, algebraically, and as a virtual manipulative. The visuals and inputs into this technology are dynamic and unlimited. Students can study any type of function using any kind of Riemann Sum to study the changes in the approximated area. However, this tool is lacking the procedural knowledge students may need to fully understand how the answer is determined. Students can see the rectangles (and how they change) depending on different values and this altogether changes the approximate area which is always compared to the actual area, but the numeric mathematical work is not shown or explained.
4. What role does technology play?
Like with any other calculator, Finding the Area Between Two Curves can become a crutch for students. Although the intent is for students to study and critically think of the relationship between Riemann Sums and the different areas under a curve, students can choose to remain with an instrumental understanding of the material if they do not put in the effort required of logical reasoning and for drawing conclusions. However, the features of this activity - being able to focus on the connections between the behavior of two functions and the different calculated areas allows students to develop a relational understanding of the material that requires less menial work by hand that is prone to errors. Find the Area Between Two Curves helps facilitate student learning by answering all of the "what if' questions students may have. "What if my function is decreasing? What if I use the inverse? What if I use more rectangles?" can quickly be answered as the students play around with the function they put in to the program and the features this tool provides.
Affordances of Technology for Supporting Learning
- Computing & Automating - Along with graphing functions (with the ability to adjust the viewing window) this Geogebra activity includes two area calculators: one for the Riemann Sum and the other for the actual area under a curve. By automatically calculating the area, this technological resource allows students to focus on comparing the changes in the found area rather than the manual work of finding it.
- Representing Ideas & Thinking - The technology allows for functions, their intersection, and the various Riemann Sums to be represented and changed with the slider tool. This allows students to quickly compare the differences in the area under a curve depending on which Riemann Sum is being used.
- Capturing & Creating - Students are given the ability to create their own functions and quickly see how two interact with each other. Students can create different functions and automatically see how their behavior changes. This is beneficial if the student created functions need to meet specific requirements.
5. How does the technology fit or interact with the social context of learning?
This tech tool is versatile enough to be used as with a whole class, groups, or individually, lecture method instruction or exploratory too. Because of this, this technology allows students to work collaboratively or independently depending on their learning preferences. Although it is my belief that mathematical concepts are best learned through communication, Finding the Area Between Two Curves is interactive and clear enough to allow students to work individually if they would like to.
6. Additional Comments
Finally a technological tool that can help students develop and challenge their relational understanding of a concept with automatic feedback!