Algebra Tech Tool: Math Open Reference- Quadratic Function Explorer

Quadratics are a very popular topic in PBL and PrBL, mainly because of the ease of accessibility with students in terms of projectile motion. Projectile Motion is a topic many students become easily enamored with, from Pumpkin Chunkin to Angry Birds to shooting hoops, there is a tie in for everyone in your classroom. Unfortunately, while the students may get wrapped up in the project for these reasons, they may miss out on some of the key mathematical characteristics of quadratics due to this excitement and engagement. The following is a tool reviewed by a classmate of mine, Hilary P, that explores quadratic functions. It is similar to the Polynomial Exploration I reviewed before, but focuses intently on Quadratics.

Original CEP post

Overview

Curator: Hilary P

Name & Link to Tech Tool or Tool homepage: Quadratic Function Explorer - Math Open Reference

http://www.mathopenref.com/quadraticexplorer.html

Brief Description of Tech Tool: This tech tool allows students to explore how the coefficients (a, b, and c) of y = ax^2+bx+c affect the shape of the graph. The students can use the scroll bars to change the a, b or c value. I believe this would be a great applet to use towards the beginning of a quadratic unit. Students can use this tool to investigate the role of each coefficient. The website also provides some guiding questions and asks students to explore certain aspects of the graph. These questions encourage students to set the other values to 0 to see each coefficients role. I find this especially helpful to help students see the role of a and c, but it is a little trickier to identify that b, which creates the slope of the line, has an impact on the location of the vertex. You might decide to write your own questions to target your specific learning goals. For example, maybe your questions are directed simply at the role of the a value and the role of the c value. Also, the exploration could be targeted to how the a value changes the width of a function. Whatever learning goals you decide to target, this tech tool is a great way to have students explore quadratic functions through graphical representations.

Technical & Cost considerations: This applet does not require any additional platforms to run. After the students investigate this activity in small groups or individually, it could be very helpful for the teacher to project the applet on the SMARTBoard and do several demonstrations as students discuss their findings.

Evaluation

Description of Learning Activity

This activity would be used as an exploration to help students understand the coefficients of a quadratic function in standard form, ax^2+bx+c. After students had some understanding of how a quadratic function differs from a linear function and how these differences can be viewed in a table, graph, equation and description, I would use this applet to help students discover the role of a, b and c in the equation. Just as students have an understanding of m as slope and b as the y-intercept of a linear function in slope-intercept form, students should understand that a affects whether the graph opens up or down, that b value "controls" the location of the vertex, and c is the y-intercept. This activity is designed for students to make conjectures about each of these roles based on their exploration.

1. Learning Activity Types

• LA-Present-Demo - demonstration
• If time is limited in your classroom, you could also use this applet to demonstrate how the a, b, and c value affect the graph of a quadratic function.
• LA-Explore - exploring/investigating mathematical ideas
• This activity allows students to explore using the a, b , and c scroll bars how these coefficients affect the graph of a quadratic function. Based on their observations, students can make conjectures and test their theories.

2. What mathematics is being learned?

NCTM Standards

• NCTM-Alg-patterns - understand patterns, relations, and functions;
• NCTM-Alg-symbols - represent and analyze mathematical situations and structures using algebraic symbols;

Proficiency Strands

• PS-conceptual understanding
• I believe this applet with some additional discussion and guiding from the teacher can lead to a strong conceptual understanding of why the coefficients play the role that they do. For example, students can recognize that if we start with y=x^2 and multiply that by a number greater than 0, then just the width of the graph changes. On the other hand, if we multiply that parent function by a number less than 0, then the graph flips, and opens down in addition to the width changing. Also, the applet in itself helps students discover the roles instead of simply memorizing the information a teacher presents.
• PS-adaptive reasoning
• This applet provides students with the opportunity to investigate the roles of each coefficient. Based on their observations, students must draw conclusions. Students must make conjectures and test their conjectures through their exploration with the applet.

3. How is the mathematics represented?

The tool shows a graphical representation of a quadratic function. As students use the scroll bars to change the a, b and c values, when the quadratic function is in standard from, the parabola is transformed on the coordinate grid. Students can very clearly identify the graphical shifts as they use the scroll bars.

4. What role does technology play?

What advantages or disadvantages does the technology hold for this role? What unique contribution does the technology make in facilitating learning?

Advantages: While it is very important for students to discover the role of the coefficients on their own, it can be very time consuming to ask students to graph multiple quadratic functions. So, this tool allows students to access graphs of "multiple" quadratic functions in a short time period. Students can then work together to identify trends, and can come to conclusions about the role of each coefficient. Also, students can add additional information to the graphs of the parabola by checking certain boxes. Students can display the axis of symmetry and the roots. Also, the technology allows students to only use integer coefficients which may be helpful when exploring the a, b and c value, but using fractional coefficients can also be helpful when exploring only the a value.

Disadvantages: I think it may be fairly difficult for students to use this tool to determine how the b value affects the graph, so students may need some extra guidance. Also, while it is very helpful that this applet graphs several quadratic function for the students, it does graph FOR the students. Students can do a similar activity with pencil and paper, and doing it in this manner would help them solidify their graphing quadratic skills. As mentioned previously, while some of the questions on the website may be beneficial for helping students learn about quadratics, the website provides TONS of information about a quadratic function. If you scroll to the very bottom of the page, you will see that it discusses how to find the vertex and estimate roots, and I think that seems like to much information to learn from this simple applet. So I would be hesitate to use these guiding questions, and would suggest designing your own.

Affordances of Technology for Supporting Learning

• Computing & Automating -
• Representing Ideas & Thinking - This applet provides students with a visual representation of how changing the coefficients impact the graph.
• Accessing Information - This applet allows the students to quickly "graph" many quadratic functions. Students can use the scroll bars to create parabolas and then through their observations they can draw conclusions.
• Communicating & Collaborating -
• Capturing & Creating -

5. How does the technology fit or interact with the social context of learning?

I believe this would be an excellent activity to complete in partners or in small groups. This technology tool encourages collaboration as students will be drawing conclusions based on their observations. I believe the discussion between students in the classroom would be very beneficial in helping students best understand the coefficients in a rule. Additionally, I believe students would really benefit from a class conversation to summarize their investigation findings.

6. What do teachers and learners need to know?

I suggest that teachers create their own guiding questions/worksheet which would lead students through the discovery of the role of each coefficient. I think teachers can write questions that will meet their specific learning goals better than the website has done. As mentioned above this activity could be used to learn about how the a value affects the width of a graph, or how the a value affects whether the graph opens up or down, or can be more broad and investigate each coefficient.

Also, it is very nice that the user can select the "snap to integers" so students can simply observe integers. The range of the scroll bar can also be adjusted. So teachers, make sure to point out those features to your students!

How this tool Supports & Supplements PBL/PrBL

As I mentioned earlier, quadratics are a hot-topic in the PBL world, at least in my experiences. However, if the project is focused on projectile motion then a lot of the intricacies of quadratics can get missed: positive a-values, negative roots, imaginary roots, quadratics with one zero, etc. This tool helps students to explore some of these concepts. This is also an activity that can be completed with partners or small groups, which is a situation that students are familiar working in. This would help foster a sense of learning community amongst teammates.

Geometry Tech Tool: Proofs involving Congruent Triangles

The concept of congruence and proofs is a difficult one to cover exclusively through Projects and Problems. I've mentioned another tech tool before, Illuminations-Congruence Theorems, that I have used to help me help students develop a conceptual understanding. This tech tool comes from another colleague of mine and is helpful with completing the proof-writing process involved with congruent triangles.

Original CEP Post

Overview

Curator: Hilary P.

Name & Link to Tech Tool or Tool homepage: Proofs Involving Congruent Triangles

http://www.regentsprep.org/Regents/math/geometry/GP4/PracCongTri.htm

Brief Description of Tech Tool: This tech tool provides a way for students to practice writing two-column triangle congruence proofs and get feedback about whether or not they are writing proofs correctly (students simply click "proof" to reveal the answer). I find it is sometimes difficult to find resources for writing proofs and it seems different textbooks provide very similar if not the same proofs, so it is great to see some different proofs on this website. The proofs get more difficult as students progress through the practice so it could easily be used as a type of leveled practice. The higher number proofs also use CPCTC to prove corresponding segments/angles are congruent. Finally, this tool should be used after all congruence properties are covered as it uses ASA, SSS, AAS, and SAS.

Technical & Cost considerations: There is no cost to use Regents Prep. Students can easily access the website and use the "proof" button to see the correct answers.

Evaluation

Description of Learning Activity

I would use this activity as an independent practice. This practice must be implemented after students have an understanding of each of the the triangle congruence postulates as well as how to use CPCTC in a proof. This is a great way to change up the flow of a lesson (instead of simply practicing paper/pencil proofs). Students can visit website, and write their proofs on paper and easily check their answers. As we know as teachers, it is very time consuming for us to write answer keys for proof practice, so I like that this comes with a "built-in" answer key.

1. Learning Activity Types

• LA-Practice - practicing for fluency
• This activity is a great way to have students practice writing two column proofs. Students can easily click the "proof" button to check their answers, and monitor how well their proof writing is going. The practice proofs get more difficult and more involved as you progress through the problems, so it can also be used as a leveled practice.

2. What mathematics is being learned

NCTM Standards

NCTM-Geo-analyze - analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships;

Proficiency Strands

• PS-procedural fluency
• This website provides a total of 10 two-column triangle congruence proofs. Students can work at their own pace through these problems, and can check their answers against the key as they go.
• PS-adaptive reasoning
• Students are constructing mathematical arguments as they write two-column proofs. Students make statements and must be able to justify their statements in order to write a cohesive proof. Students are not told which triangle congruence postulate to use, but must be able to analyze the diagrams and givens to determine how to prove the two triangles are congruent.

Additional comments on what is being learned

This website is designed to help students practice writing two-column triangle congruence proofs. Students must use statements and the correct mathematical reasons to write a cohesive proof. Students must use a variety of triangle congruence postulates such as SSS, SAS, AAS, ASA. Additionally, students may also need to use CPCTC in their mathematical proof to prove that corresponding sides and angles are congruent.

3. How is the mathematics represented?

The students are given both a diagram of the two triangles as well as mathematical statements that list what information was given. When students click on "proof" a two-column proof answer key opens in a additional window, and students can use this to check their answers.

4. What role does technology play?

Advantages: Students are provided with different proofs. As mentioned earlier, there are often several different proof problems that repeat from textbook to textbook, so it is nice to find a website with so many practice problems. Students are interacting more with this website much like a textbook, but at a click of a button are told the answers to the two-column proof.

Disadvantages: As the website notes, there is sometimes more than one way to write a mathematical proof and the answers only give one answer. Additionally, sometimes the statements or reasons might be written differently than students were taught from the textbook and may cause some confusion. Also, students are asked to use integrity when interacting with this website. Students should be writing two column proofs on paper and then checking their answer against the key. The teacher will want to closely monitor this interaction. Finally, I wish the website would offer hints for each step before just giving students the answer - this would be a great addition to this tool.

Affordances of Technology for Supporting Learning

• Computing & Automating -
• Representing Ideas & Thinking - This website provides students with proofs of varying levels of difficulty and students have to visualize which triangle congruence postulates can be used to prove congruent triangles.
• Accessing Information - This website provides students with an answer key for each of the proofs they write. This is immediate access to information and can be very helpful formative data for students.
• Communicating & Collaborating -
• Capturing & Creating -

5. How does the technology fit or interact with the social context of learning?

This tech tool would best be used by individuals and does not foster tons of interaction between peers. However, I hope that providing the answer key will spark questions within students' minds about the way they wrote something in their proof, or why the textbook writes it differently than the website. Hopefully, the teacher could answer these questions as he or she circulated the room.

6. What do teachers and learners need to know?

As mentioned, teachers must monitor student use of the website. The answer keys should not be revealed until students have first written the proofs. Again, students must have the background knowledge of all triangle congruence theorems as well as CPCTC in order to fully engage with this website. The proofs get more difficult, and number 8 and 9 in particular, require some properties that may or may not be covered in you course, so you may choose to skip those. Overall, this is a good website, and a nice way to have students use computers to practice proofs which is traditional simply a pencil and paper activity.

How this Supports & Supplements PBL/PrBL
As I mentioned at the beginning of this post, proofs are a difficult concept to address directly through projects and problems. Students can most certainly practice attending to precision and justifying their statements and claims in projects and problems, but create a direct proof using geometry theorems is a tough subject to master through problems alone. (I say just problems, because I have yet to do a project with proofs, though I frequently run activities and seminars that are problem-based).

Geometry Tech Tool: Geometer's Sketchpad Tessellations that use Rotations

You'll see in another post that the Geometer's Sketchpad and it's iPad version: Sketchpad Explorer are wonderful tools to use to complement a PBL/PrBL classroom. Below is a review of a specific tech tool that uses the software, written by a colleague of mine: Terry C.

Original CEP Posting

Overview

Curator: Terry C.

Name & Link to Tech Tool or Resource homepage: Tessellations That Use Rotations

Link to parent Wiki page: Geometer's Sketchpad

Link to Other GSP Activities: Geometer's Sketchpad Activities

Brief Description of Specific Math Activity: Students construct an irregularly shaped tile based on an equilateral triangle, and then use rotation to tessellate the plane with it (Key Curriculum Press, 2009).

Evaluation

Description of Learning Activity

By creating the tessellation and then dynamically changing the original tile to see the effects, students get a deeper understanding of what makes the tessellation work (Key Curriculum Press, 2009).

1. Learning Activity Types

• LA-Present - (read or attend to) presentation of new content/ideas
• LA-Present-Demo - demonstration
• LA-Present-Explain - explanation
• LA-Explore - exploring/investigating mathematical ideas
• LA-Apply - applying mathematics to problems and situations

2. What mathematics is being learned?

The objective of this activity is to use rotation to tessellate and to explore rotational symmetry (Key Curriculum Press, 2009).

NCTM Standards

• NCTM-Geo-analyze - analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships;
• NCTM-Geo-specify locations - specify locations and describe spatial relationships using coordinate geometry and other representational systems;
• NCTM-Geo-visualization - use visualization, spatial reasoning, and geometric modeling to solve problems.

Proficiency Strands

• PS-conceptual understanding - Students work with a specific example but the activity tries to promote the idea that students can wander down a path of "What if I did this?" and still reach similar results.
• PS-adaptive reasoning - Throughout the construction and manipulation, students have to understand what is happening, how its happening, and why its happening.
• PS-productive disposition - Seeing the tessellation animated at the end of the activity is fun and rewarding.

Prerequisite Knowledge Required: Experience with equilateral triangles, translation, tessellation, and rotation.

3. How is the mathematics represented?

Mathematics can be represented symbolically and graphically in this activity. Geometer's Sketchpad allows for both dynamic and static representations of mathematics, and it depends on the inputs made by the creator how the mathematics is represented in the tool. Through the instructions, this activity guides students to creating graphical and symbolic representations of mathematics that act as virtual manipulatives.

4. What role does technology play?

Geometer's Sketchpad makes a unique contribution in that it automates tessellation of a plane through tile rotation, so that students can see the effects more easily and quickly.

Affordances of Technology for Supporting Learning

• Computing & Automating - Sketchpad allows students to construct shapes on a computer more easily than if they were drawing them with pencil and paper. Sketchpad also allows them to manipulate and change the shapes more easily than with pencil, paper, and eraser. The effect of rotating the tile to tessellate the plane is achieved through technology; this cannot be done on paper.
• Representing Ideas & Thinking - Sketchpad allows students to visualize and explore mathematics problems.
• Accessing Information -
• Communicating & Collaborating -
• Capturing & Creating -

5. How does the technology fit or interact with the social context of learning?

According to the instructions, this activity is meant to be performed in student pairs. It can also be modified for a whole class. The technology facilitates the ease in which students can construct tessellations together (or side-by-side on separate computers) and then share the results with each other. Using technology allows students to rotate the original tile dynamically, which certainly is not easily accomplished with pencil and paper.

6. What do teachers and learners need to know?

This activity requires that students and teachers have access to Geometer's Sketchpad, and intermediate knowledge of Sketchpad is required to do this activity. The suggested grade level is 9 to 10; however, advanced seventh or eighth grade students could work on this activity. The activity includes an introduction on the activity for the teacher, notes, instructions, and discussion points for the teacher to guide the class through the activity, and a worksheet for the students that will guide them through the activity in Sketchpad and ask them questions at various intervals. The recommended duration for this activity is 45 minutes.

How this Supports and Supplements PBL/PrBL

 This activity is a great way for students to work on developing a conceptual understanding of tessellations as well as the procedural process of making tessellations, both of which can be daunting and time consuming for students. A lot of great projects and problems can focus on tessellations, or can take a tessellation turn (try turning ANY tiling problem into a tessellation problem!). The exposure students get by using this tool and doing this activity helps familiarize them with the concepts well enough to prepare them for thinking this way in projects and problems.

Geometry Tech Tool: Illuminations-Geometric Solids

Overview

Name & Link to Tech Tool or Tool homepage: Illuminations - Geometric Solids

Brief Description of Tech Tool: From their website:

This tool allows you to learn about various geometric solids and their properties. You can manipulate and color each shape to explore the number of faces, edges, and vertices, and you can also use this tool to investigate the following question:

• For any polyhedron, what is the relationship between the number of faces, vertices, and edges?

This tool could also be used to explore other relationships among 2D and 3D figures while focusing on nets of solids and can be adapted to fit a variety of needs from pre-k through high school. This tool provides a virtual manipulative to help students develop spatial reasoning regarding these figures.

Technical & Cost considerations: As with other Illuminations tools, lessons and activities, the resources are free. This applet is iPad and tablet compatible. It also runs in a variety of web browsers on computers and Macs.

Evaluation

Description of Learning Activity

For lower elementary students, this applet could be used to explore and identify the various 3D figures and the 2D figures that make them up. They could also practice counting the number of 2D shapes and relating that to the name for the 3D figure.

Upper elementary students could practice exploring the nets that are created as they "unfold" each of the 3D shape through the virtual manipulative. Students could then practice designing their own nets to be printed out to see if their net creates a 3D figure. Students could compare and contrast their findings with one another to see if they can come up with any generalizations for nets.

Students in the middle grades could expand on the activities for the upper elementary students and could complete the Geometric Solids Exploration worksheet to try to come up with Euler's formula on their own. They can then try to develop an informal proof for it.

Students in high school could expand upon the middle grades activities to develop Euler's formula and a formalized proof for it. They could also explore and establish their own criteria for creating "working" nets and "non-working" nets and generate their own examples for this using the "My Own Net" feature and printing it out.

1. Learning Activity Types

• LA-Present - (read or attend to) presentation of new content/ideas
• LA-Present-Demo - This tool could be used by a teacher to demonstrate what a net is in relation to a 3D figure.
• LA-Explore - This tool is particularly useful for helping students to explore the various relationships between 2D and 3D figures at any level.

2. What mathematics is being learned?

NCTM Standards

NCTM-Geo-analyze - analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships;

NCTM-Geo-visualization - use visualization, spatial reasoning, and geometric modeling to solve problems.

Proficiency Strands

• 
  PS-conceptual understanding Students develop a conceptual understanding of the 2D relationships to 3D models through experimenting and visualization with this tool
• PS-strategic competence Students build strategic competence through their continual trial and error with the creation of nets that create 3D figures.
• PS-adaptive reasoning Students' reasoning is likely to adapt as they continue to explore various combinations of polygons that create nets of 3D figures.
• PS-productive disposition Students will likely struggle with net creation at first, but as they print out and examine more nets from the program, they will become better at creating nets for 3D figures.

Additional comments on what is being learned

This tool is accessible to students at all levels of learning. Younger students made need assistance in using the tool, but once the students have learned basic mouse control, they should be able to use the manipulative to analyze the characteristics of 3D figures in terms of 2D figures to give them a solid foundation for their conceptual understanding of the relationship between the two.

Older students will have to work strategically and diligently on their ideas as they develop Euler's formula as well as when creating their nets to ensure that they create a solid figure. Both of these ideas may take several attempts for the student, but each time, they should be able to adapt their reasoning and work through until they find success. This may require more teacher encouragement and feedback depending on the students' mindset about mathematics.

These features help students to visualize the relationship between the 2D and 3D figures that is outlined in both the CCSS and the NCTM standards, with the upper grades being able to reach the level of developing argumentation about the relationship and proving their conjectures.

3. How is the mathematics represented?

This tool is a virtual manipulative that can take the place of traditional cut and tape nets in the math classroom. This tool could be particularly beneficial for visual learners in helping them to deepen their conceptual understanding of the relationship between 2D and 3D figures and to improve their visual-spatial reasoning. However, for more tactile and kinesthetic learners, this tool may be more of a starting point for a student to see the process of unfolding a solid to create a net, and then the student can use the "My Own Net" feature to create their own nets to print out and try working with.

4. What role does technology play?

This tool provides a great advantage in initially learning and exploring the relationship between 2D and 3D figures as students can continually fold and unfold their 3D figure, as well as color the various aspects of the figure to see just how the net is a mapping of the 3D figure.The ability to look at the figure from multiple perspectives and as a filled solid or a transparent solid is really unique and provides perspectives you don't get with a hands-on manipulative.

Affordances of Technology for Supporting Learning

• Representing Ideas & Thinking - This tool enables the user to manipulate the way the 3D figure is being represented as they explore the relationships between 2D & 3D figures.
• Capturing & Creating - This tool allows the student to create their own nets in an attempt to create a net of a solid figure.

5. How does the technology fit or interact with the social context of learning?

This tool can be used by individuals or partners to support the students' exploration as they work with the applet. Students can play around with the tool individually and form their own conjectures about the various relationships between the 2D and 3D figures that they are able to create and can then share out these ideas with partners and continue their individual exploration from there. Alternatively, students can begin their explorations in a partnership and continually discuss the math that is unfolding before their eyes as they work with this applet. Partner work may prove beneficial for promoting good discussion when using the "My Own Net" explorations.

6. What do teachers and learners need to know?

Users should be comfortable with basic mouse control and should be familiar with the terms: face, edge and vertices when using this applet. Users should also be provided access to a printer to get the most out of their experience and explorations with the applet.

How it Supports & Supplements PBL/PrBL

The various representations of 2D/3D figures comes up quite often in my projects in the geometry classroom, as students are often sketching buildings, bridges, sculptures and other designs and trying to determine how to best construct them. This applet is useful in the classroom because it allows students to get a good visual representation of the way in which the two models are related to one another. I've given students what I consider simple nets before, and they've had no idea what it would become. This applet helps those students to see the changes that happen in a net to see how it maps onto a 3D figure. After getting a conceptual understanding here, students can practice and implement the relationship in their projects.