Algebra Tech Tool: Interactivate-Algebra Quiz

Differentiation is essential in any classroom, but can be particularly beneficial in a PBL classroom. The following tool, reviewed by my classmate Hilary P again, provides a quick and efficient way of checking students' knowledge on basic algebraic manipulations covering a variety of topics: quadratics, distributive property, variable on both sides of the equation, etc. At the time of our initial review, we only had 4 questions to do, so parts 5 and 6 were evaluated by me after experiences with the tool.

Original CEP Post

Tech Tool Overview

Curator - Hilary P

Name of Tech Tool: Interactivate - Algebra Quiz

Brief Description of Tech Tool: A quiz is given - the red line counts down time, and the user can choose the type of problems to display.

Link to Tech Tool or Tool homepage: http://www.shodor.org/interactivate/activities/AlgebraQuiz/

Evaluation

Description of Learning Activity

This tech tool provides students with practice solving algebraic equations. Students can select whether to practice solving equations that require one or two steps, or whether to practice more complex equations which involve moving variables to one side of the equation or solving by first using the distributive property. When implementing this task in my classroom, I would use this tech tool as independent practice. Most likely, I would ask students to increase the time allotted to solve each problem (from the default of 20 seconds), and require them to write down their steps used to solve each problem. Additionally, I might use this activity as a differentiated practice. If this applet was used after I had explicitly taught how to solve the various equation types, I might do a quick mastery check prior to the practice, and students that were unable to solve one and two step equations would practice that, while students that have mastered each type of solving would practice all types, but increase the level. From what I can tell, Level 1 just involves more negative coefficients, and Level 2 has fractional answers, and Level 3 has fractional coefficients. While all students need to be able to solve all types of equations, if a student has not yet mastered solving one and two step equations, it will be very difficult for them to solve more complex equations.

1. Learning Activity Types

• LA-Practice - practicing for fluency

This algebra quiz applet provides students with the opportunity to practice their solving skills. Additionally, the timer (red scroll bar) pushes students to be able to solve equations quickly.

2. What mathematics is being learned?

NCTM Standards

• NCTM-N&0-understand operations - understand meanings of operations and how they relate to one another;
• NCTM-N&0-compute fluently - compute fluently and make reasonable estimates
• NCTM-Alg-patterns - understand patterns, relations, and functions;

Proficiency Strands

• PS-procedural fluency

This applet simply provides students with the opportunity to solve algebraic equations. Students are not required to write equations from a mathematics situation, but given an equation and asked to solve.

3. How is the mathematics represented?

Equations of various type are given in this tech tool. Students are asked to solve for variable. The tech tool gives students immediate feedback by marking the answer as correct, or incorrect and then displaying the correct answer.

4. What role does technology play?

Make boldface the affordances that play a significant role in this technology use. For each affordance that you select, comment briefly on why.

• Computing & Automating - This applet requires students to quickly solve for the variable, x, and provides students with immediate feedback on whether or not their answer is correct. This is different than providing students with the correct answers after they have done 10 problems, but instead allows students to see their progress as they go.
• Representing Ideas & Thinking -
• Accessing Information - The only reason I think this applet allows students/teachers to access information is because it provides a detailed report of their mastery of each equation type and level. Students can view this report to monitor their own progress, and teachers can monitor the classes progress through this report as well.
• Communicating & Collaborating -
• Capturing & Creating -

What advantages and/or‍ disadvantages‍ does the technology offer for facilitating learning?

This technology tool allows students to practice their solving skills. The app gives the students ‍immediate feedback‍ on how well they are understanding solving equations. Additionally, this tool can provide teachers with the ability to quickly differentiate the practice by assigning students different types of equations to practice or different levels of practice. This technology tool does not provide students with any conceptual understanding of solving equations. Students are simply following a procedure to solve for the given variable, and then typing in the correct answer for x.

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

This applet is most useful to teachers as an individual activity for the students as it provides detailed information about the students' mastery level through the quiz. This kind of information can then be used by the teacher to further differentiate their instruction. Teachers also have the ability to differentiate within the applet by selecting various different inputs for the quiz from the content covered to the type of numbers used to the amount of time each student gets for each problem.

6. What do teachers and learners need to know?

Teachers should be prepared to set up the quiz for students or to provide students with information about the type of quiz that they should be taking based on their skill level. Teachers should also know that the answer does not provide corrective feedback on the problems if a student gets the problem wrong. Rather a pop-up tells you the correct answer. Students may need to write down their problems/work so that you can review wrong answers with them.

How this tool Supports & Supplements PBL/PrBL

Teachers can use this tool as a means to deliver quizzes quickly to students to check their understanding at any point during the PBL or PrBL process. This helps students to practice their procedural skills with algebraic manipulations and solving equations which is not always available through PBL/PrBL instances, and it also provides students with practice answering questions in a timed atmosphere, which they will need in anticipation of high-stakes testing. The information obtained from the applet can help teachers to plan for differentiation within their projects or problems.

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.

Algebra Tech Tool: GeoGebraTube-Polynomial Exploration

Polynomials are a concept that I always have a difficult time working on with my students, largely because it is difficult to find a real-world context relevant to them that is polynomial based. Students have an easy enough time memorizing polynomial rules, but they often have a difficult time conceptualizing these rules or understanding where they came from. While my Algebra 1 students aren't going to be manipulating the function form of the polynomial expression anytime soon, I really like this tool because it helps them to visualize the changes that occur within a polynomial expression and how that change affects the graphic version of the polynomial. 

Overview

Name & Link to Tech Tool or Tool homepage: GeoGebraTube: Polynomial Explorations

Brief Description of Tech Tool: With this tool, students use sliders to manipulate coefficient values for constant through quintic polynomial functions to analyze the graphic changes. Students are prompted with a series of questions to guide their explorations.

Technical & Cost considerations: This tool is a free web-based tool that requires internet connection.

Evaluation

Description of Learning Activity

This activity would be best used for an individual or pair to explore various changes that can occur within a polynomial function graph for polynomials of various degrees (Constant through quintic). Students would take notes on the graphic changes for the various shifts in coefficient values to come up with generalized rules for graphing polynomial functions.

1. Learning Activity Type:

• LA-Explore - This tool is particularly useful for helping students to explore graphic changes in polynomial functions of various degrees (constant through quintic) as different coefficient values are manipulated.

2. What mathematics is being learned?

NCTM Standards

NCTM-Alg-patterns- Understand patterns, relations, and functions

Proficiency Strands

• PS-conceptual understanding Students develop a conceptual understanding of changes in a graphical context to develop generalized rules for these changes.

Additional comments on what is being learned

This tool allows students to view multiple function graphs at the same time allowing them to compare/contrast between even and odd degree functions or between various functions of the same type of degree. Students can make generalizations based on the various coefficient changes for all functions using this feature as well.

3. How is the mathematics represented?

This tool is a virtual manipulative that relies on slide values to alter the coefficient for each of the terms within a polynomial function. Students can slide these values to view the changes in the graphic form of the function.

4. What role does technology play?

Technology enables the users to view graphic changes automatically, an activity that would eat up a lot of class time if done by hand, or even if done by graphing calculator. The slide values are essential to allowing the student to work on developing their conceptual understanding without being bogged down by the procedures of choosing specific values or graphing the function.

Affordances of Technology for Supporting Learning

• Computing & Automating - This tool enables the user to manipulate the coefficient values within the terms of the polynomial and automatically changes the graph accordingly.

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

This tool can be used by students independently or in pairs, but I would not suggest using it for anything larger than a group of two. Students exploring individually may need to be assigned a talking partner that they can bounce their ideas off of after having experimented with the tool for a while. This use of this tool to develop a conceptual understanding of graphic changes could foster good discussion between a pairing of students to compare and contrast their ideas and ultimately come to a conclusion or generalization that is more sound than if they were working solely independently.

6. What do teachers and learners need to know?

Users do not need much technical prowess to successfully use this tool in the classroom. Teachers may want to demonstrate the type of changes that occur while using one of the sliders, but would likely wish to leave the actual exploration to the individuals.

How it Supports & Supplements PBL/PrBL

This tool is very helpful for me in the classroom as it allows for my students to work on a more inquiry driven exploration, similar to the style they use when working on a problem or project. This type of exploration allows for the student to reach their own conclusions about the concept and then check with others for validation. I particularly like this tool because of the automatic changes that it produces within each graph as the coefficient values are being changed, as it frees up the students to think solely about the shifts that are occurring and not about what value they should try next. This also helps students to better analyze graphs of data in later projects as they are more familiar with the type of graphic changes that occur within polynomial functions. 

Algebra Tech Tool: Illuminations-Line of Best Fit

In Project-Based Learning, students are constantly gathering data about their project and making decisions based off of it. Linear data is a fairly common type of data that is extracted from projects, either by design or by chance. Once you know that a set of data is or should be fairly linear, it is helpful for students to look at and analyze the data to make inferences from it. Think of the Barbie Bungee project some students do, where they are trying to figure out how many rubber bands should be on Barbie's bungee to make sure she has a great thrill but doesn't die. Students could run this experiment and gather their data and use a tool, like the one presented here, to make inferences about how many rubber bands would be needed for different drop heights. Without further ado, here is an excellent source for finding the Line of Best Fit!

Overview

Name & Link to Tech Tool or Tool homepage: Illuminations-Line of Best Fit

Brief Description of Tech Tool: This tech tool allows students to explore creating a line of best fit. Students are able to input their own data, and can then guess their own line. Students then check their line with the actual line of best fit created by the program.

Evaluation

Description of Learning Activity

Students have the ability to practice their critical thinking about data with this interactive applet provided by Illuminations. The "Line of Best Fit" activity enables students to work with their own sets of data to plot their coordinates. Students then create a "guess" line which they can manipulate by either changing the numbers in the linear equation or dragging the line to come up with their own line of best fit. From there, they can ask the computer to show them the true line of best fit.

1. Learning Activity Types

Using this activity, students are given a more practical method of "guess and check" problem solving for the line of best fit. Students are able to practice creating their own line of best fit and are also able to explore the concept. Students could use poorly aligned coordinates and really well aligned coordinates to better understand an "r" value.

2. What mathematics is being learned?

NCTM Standards

NCTM-DA&P-Select methods

• Select and use appropriate statistical methods to analyze data
• Develop and evaluate inferences and predictions that are based on data

Proficiency Strands

• PS-conceptual understanding
• Students are working on representing various information in different ways and seeing the usefulness behind each of the representations. Students are practicing with data as a coordinate point, a table, and a graph. Students are also practicing with lines as equations and graphs and are interpreting the various components of those lines as they change one or the other.
• PS-procedural fluency
• Students are becoming more adapt at creating graphs as well as creating equations for lines to best fit the graph.

3. How is the mathematics represented?

Students are working with graphical and numerical representations of numbers in a virtual manipulative that allows them to change the scenario and what they are seeing at any given time. Without technology, students wouldn't be able to test out as many ideas as to the "line of best fit" as easily, and would not see the effect of small changes as readily. However, with the technology, students do not get the same practice of plotting points and understanding the coordinate plane that they would if they were to do a similar activity by hand.

4. What role does technology play?

• Computing & Automating - This tool is very handy for quickly plotting data and creating lines of best fit, tasks that can take up quite a bit of time in the classroom when done by hand.
• Representing Ideas & Thinking -
• Accessing Information -
• Communicating & Collaborating -
• Capturing & Creating - Students can use this tool to create a graph of any data of their choosing and can then create a line of best fit for the data.

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

For this tool, students could be working individually or in pairs analyzing the same data set. Students could talk together to discuss what they believe would be the best line of fit and why, and then could check it against the "real" best line of fit, as given by the computer. However, this technology would not work well for groups larger than two, as there would not be enough work to do or exploration to support large-group collaboration or discussion on an activity.

6. What do teachers and learners need to know?

Students should be familiar with the idea of "line of best fit", as well as the various aspects of an equation for a line. Students also need to be familiar with a coordinate plane and how to input and graph points of data. Students should feel comfortable with basic computer skills: mouse control and typing, but do not need to be "tech savvy" to be successful at using this applet. Teachers should be walking around and facilitating students when using this applet, as well as monitoring their use and discussions, and should expect to model its use with a projector before having students explore.

How this Supports & Supplements PBL/PrBL

As I mentioned at the beginning of this post, a lot of math projects that are designed contain some sort of linear data. This could be because linear data is very prevalent in the world and is easily relatable to students. I plan on using this tool to help with the same project I mentioned above, the Barbie Bungee. My goal is for students to be able to compare their results with one another and make inferences as to why their results were different and what their results would be if they wanted to drop Barbie from a height 2x or 3x greater than the height we dropped (would it change the equation proportionately?).