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: 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?).

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.

Geometry Tech Tool: Math Is Fun-Geometry Constructions

Constructions are largely discussed in the CCSSM, but their practice has been long-gone in many Geometry classrooms. Some teachers don't even know how to create constructions; and I know this because I was one of them. Through the years, I've now had my practice at them, but I am still uneasy about Constructing a Hexagon inscribed in a Circle (I'm not kidding... CCSSM HSG-CO.D.13).  I've come to rely on technology for helping create constructions in my classrooms, as it is easier to start over and see mistakes as you make them than it is when you make hand-drawn constructions. This tool is one of the "go to" resources for constructions since it shows students how to create the constructions. They can then apply these in any of our online workspaces or by hand!

Overview

Name & Link to Tech Tool or Tool homepage: Math Is Fun-Geometry Constructions

Brief Description of Tech Tool: This tool allows for students to build up a procedural fluency in creating constructions. The website offers instructions in creating geometric constructions using the basic tools: straight edge, compass. and pencil.

Technical & Cost considerations: This is a free tool provided from Math is Fun, a common website for math games. You may need to check your firewall at work to make sure your students can access this site.


Evaluation

Description of Learning Activity

This tool can be used to assist an in-class or at-home practice of creating geometric constructions. This tool is useful for helping students to learn the rules of constructions and to practice making the various constructions. Students will be able to apply these constructions outside of the applet either via paper and pencil or with other technology such as GeoGebra or the Geometer's SketchPad.


1. Learning Activity Types
LA-Present-Demo -Students are watching demonstrations of how to create various geometric constructions using simple tools.


2. What mathematics is being learned?

NCTM Standards

NCTM-Geo-Visualization - draw and construct representations of two- and three-dimensional geometric objects using a variety of tools;


Proficiency Strands

PS-procedural fluency Students work on developing their fluency with the procedures necessary to create various constructions using simple tools as they watch the videos. 

Additional comments on what is being learned

Students will be practicing making geometric constructions using simple tools, however since the tools in the demonstrations are virtual, students will likely struggle with them if they are able to use a hands-on version of the tool. They may be used to the tools staying in precise places and moving fluidly, but will find that the real-world version is not as easy to maneuver. Keep this in mind while planning lessons that involve hands-on construction, as even though students will be prepared with the procedure, they will still be sticky with the enactment.

3. How is the mathematics represented?

The tool provides videos of virtual geometric tools uses to create constructions of various Euclidean geometry components: lines, angles and figures. The videos also support learning key geometric vocabulary such as bisector and midpoint.

4. What role does technology play?

Technology is helpful in automating the process for creating constructions. If students have the tools readily available to them, they are able to watch the video and create their own constructions alongside the instructions. Technology enables students to view and practice this independently or with a partner. Technology assists the teacher here in providing clear and accurate descriptions of constructions along with showing the tools moving fluidly and working together, which is sometimes difficult with the real-world tools.


Affordances of Technology for Supporting Learning

Communicating and Collaborating This tool is helpful for teachers to communicate with students about how to create various constructions. 

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

This tool could be quite helpful for teachers to give to students as individuals or pairs or even small-groups to watch the videos and practice their constructions. Students can pause the video at any time and can even read the instructions alongside the video to make sure they are creating the correct construction.


6. What do teachers and learners need to know?

Teachers and students only need access to watch the videos for creating constructions. However, it is helpful if students and teachers are familiar with geometry software such as GeoGebra or Geometer's SketchPad so that they can create the constructions while watching the videos. 


How it Supports & Supplements PBL/PrBL

Constructions can prove invaluable in projects where students are designing or creating final products. However, not all students are on the same page when it comes to creating constructions: some have done constructions the old-school way in their elementary classes, and others have no idea that "construction" could be a mathematics term. Either way, this tool is helpful for teachers to share with their students to gain individual practice in the procedures of creating constructions using hands-on or virtual manipulatives.

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).