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-Algebra Tiles

Algebra tiles are not a new technology to classrooms by any means, however it is not infrequent to find that tiles are missing or damaged. This tech tool can replace traditional algebra tiles in the classroom for activities related to substituting, solving, factoring and expanding algebraic expressions in the classroom. This concept is essential to progressing in mathematics and is difficult to fully and adequately expose students to in a PBL or PrBL classroom. This is a skill that requires a lot of procedural practice, which this tool provides.

My Original CEP Post

Overview

Name of Tech Tool: Illuminations - Algebra Tiles

Brief Description of Tech Tool: Four different exercises can be practiced with this technology: solving algebraic expressions, substitution, multiplication of algebraic expressions, and factoring. Manipulation of algebraic expressions are reinforced and practiced with this activity with the use of algebra tiles. This activity helps make these abstract concepts more concrete.

Link to Tech Tool or Tool homepage: http://illuminations.nctm.org/ActivityDetail.aspx?ID=216

Evaluation

Description of Learning Activity

Students have the ability to practice their algebraic skills in four areas of equations in algebra: solving, substituting, expanding and factoring. Students use manipulatives to “write” the expressions that they are given and are provided with a variety of tools to manipulate these expressions to either solve, substitute, expand or factor. This is an online version of the familiar algebra tiles that many teachers already use in their classroom, however this goes a step further by alerting students if they are or are not correct, either in their initial set-up or in their figuring on the problems.

1. Learning Activity Types

Students are able to not only practice solving, substituting, expanding and factoring algebraic equations, but they are able to do so visually, providing them with multiple representations of algebra and a more concrete understanding of why X’s and 1’s can’t be combine with one another. Teachers can use this tool with students to explore the four ideas in a conceptual manner to solidify these ideas with strong learners, and to give an alternative approach to these ideas for struggling with the concept.

2. What mathematics is being learned?

Teachers can very easily use this tool to help students represent, understand and interpret the structure of expressions. All four components of this tool have students identifying terms, factors and coefficients. Additionally, students are also rewriting equivalent expressions in various representations through the expand and factor categories.

NCTM Standards

NCTM-Alg-symbol Represent and analyze mathematical situations and structures using algebraic symbols

Proficiency Strands

• PS-conceptual understanding
• In “Adding it Up”, a key component of conceptual understanding is being able to represent mathematical situations in different ways. With this tool, students are learning how to look at a numerical expression and represent it in a visual/graphic form. This enables students to manipulate the expression and gain a deeper understanding of key ideas, such as what equality truly means in that they must continually balance their equations.
• PS-procedural fluency
• Students are also working on their basic math with arithmetic and understanding the inverse operations. Students are expanding this knowledge into a variable domain and experimenting with how those operations work with variables. In addition to practicing their arithmetic fluency on these expressions, students are also practicing their substituting and solving skills.

3. How is the mathematics represented?

This tool is a virtual manipulative that helps students to tinker with various algebraic expressions. Students are working on solving, substituting, expanding or factoring these expressions while using a technology version of algebra tiles. Students are able to check their work instantaneously with this method, as opposed to using traditional algebra tiles, which provide no feedback. This tool then, helps students and teachers alike to identify where a student is struggling: with the set up, or with the solution of a problem. It also permits them to see how well a student understands the concept conceptually, as the tool provides a secondary means of representing algebraic expressions, in addition to student’s already understood written-out way.

With this tool, students are also able to see the differences in components of algebraic expressions (terms, coefficients, etc), which can help them to gain a better understanding and appreciation of what terms can be combined with others, in that they are able to visualize the like expressions. However, students may use this tool without considering the written mathematical expression, and may think of this more visually (one red cancels out one yellow), instead of a negative number and a positive number of the same value equal zero. This is an aspect that teachers should be wary of when using this tool in the classroom.

4. What role does technology play?

• Computing & Automating -
• Representing Ideas & Thinking - This ‍tool is a nice way for teachers to let their students explore multiple representations of algebraic expressions‍, while providing them with checkpoints and immediate feedback that may not be available to students who are using a more traditional algebra tile manipulative.

• Accessing Information -
• Communicating & Collaborating -
• Capturing & Creating -

While the tool does not provide suggestions via feedback, it does indicate clearly A) that you have made a mistake (ding!) and B) where you made the mistake (turns the box orange). With these features (and the sound on, of course) teachers then can quickly scan the room to see which students are struggling on which types of problems to better address problems as they arise.

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

This tool is best used as an individual tool, in my opinion, as it is practicing procedures. However, a student could benefit from having an elbow partner to work with by bouncing questions off of one another or checking one another's work, before soliciting a teacher for support. In this way, students would be practicing explaining procedures to one another, strengthening their conceptual knowledge on the subject.

4. What do teachers and learners need to know?

Teachers and learners will benefit from reading through the instructions on how to use the applet properly, as it is not necessarily intuitive from the get-go. Users do not need extensive computer knowledge, just familiarity with the applet itself.

How this Supports & Supplements PBL/PrBL

This tool is a great supplement to a PBL/PrBL classroom as it provides a means of practicing algebraic symbol manipulation conceptually and procedurally. Symbol manipulation and being able to work with equations of one variable are essential to success in later mathematics, so it is crucial to have a solid foundation. This applet helps provide additional support and practice that may be necessary for students to have, if not otherwise addressed in your classroom.