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

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!

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__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.