Web-Based+Tools

Gizmos - Web-Based Learning Tools
The tool uses a square of adjustable side-length to demonstrate a geometric representation of how square roots work. You can see how many squares of area are in the are of the square (and this is the number under the square root), and it shows the side lengths (which is the answer to the square root question). I chose this because I think square root problems are fairly abstract, so students might have a difficult time understanding them. This tool should provide a direct link to something students can imagine easily - geometry. I like that this gizmo can displays any number in blocks and it can show its square root. I like the visual aspect and that you can change the size of the block. But I don’t think it is that effective – it kind of just gives the answer. Doesn’t have much depth to it. - Denise Good tool choice for a relatively hard concept. Good idea to highlight that square roots have a visual (geometric) representation. Be careful in assuming that students can easily imagine geometric objects.
 * Tool**: Square Roots (Bryan)
 * Math Strand**: Number Sense and Numeration
 * Concepts**: Square Roots and their geometric representation
 * Evaluation**:
 * Review **:
 * Review #2:**

In this activity, the radius of a circle can be changed. The diameter, area and circumference of the circle can be found in three tables. Each table can be displayed or hidden by clicking on a check box The gizmo was a good idea for showing practical examples of circles in geometry. There is even an additional option which allows users to measure arc lengths (a feature which becomes more useful when you teach measurement in radians). I think the tool is useful for its intended purpose. The evaluation which is given to this tool is detailed in its description, but lacks a justification for why the tool is useful in the classroom.
 * Tool: ** Circumference and Area of Circle (Jason)
 * Math Strand**: Measurement
 * Concepts**: Circumference and Area of Circle
 * Evaluation**:
 * Review ** :

In my experiences tutoring high school students, I have encountered a great sense of fear and confusion on their part towards square roots. I have found that they rarely understand what a square root truly is. I combat this by showing them a picture of a square composed of various smaller squares and explaining that they can remember what a square root is by realizing that the word "square" in the name has significance. I like this tool because it does the same! Students are provided with a visual representation of a square root. By dragging the corner of the square and making it bigger or smaller, the student is able to look at the way square roots increase and decrease. In addition, I really like the way students are able to understand the concept of a perfect square by using a Gizmo. This Gizmo has a useful interactive visual component. The “show grid” option makes it easier to count the number of selected squares. However, I am not really sure if this Gizmo would be useful for students if they do not know or understand the definition of a square root or a perfect square. Can they deduce the meanings of these terms when they are working with this Gizmo?
 * Tool:** Square Roots (Stephanie Romanello)
 * Math Strand:** Grade 7 Number Sense and Numeration
 * Concepts:** Displaying a square root visually while demonstrating its meaning to students
 * Evaluation:**
 * Review: **

This tool is great for students to understand the theoretical and experimental probabilities. Students can pick either 1 or 2 spinners with each having up to 10 sections, and they can pick how they want to compare the outcomes in relation to the sections in the spinners. For instance, students can look into the probabilities of getting either <, <=, =, >, >= in the first spinner as compared to the second. Students can go through different stimulation and compare the theoretical and experimental values. The great thing is that this tool allows you to have over 5000 experimental trials, and all the trials are listed in a table that students can look at and record. With each trial added, the experimental probability changes and you can really see that with the increasing number of trials, the experimental probability gets really close to the theoretical probability. Also, this tool lists the total outcomes in the theoretical probability page. This tool can be used to solve probability questions such as rolling 2 die, flipping a coin and rolling a dice, or anything with 10 max outcomes per object. I agree with Nabgha's views on this gizmo. The fact that students can run multiple trials and follow results really allows them to grasp the relationship between theoretical and experimental probability. I have one small issue with this gizmo. I found the interface a little but uninviting and it took me a while to understand the significance of all the parts of the screen. Also, I found that the visual representations of the spinners were too small. The user could barely see the different sections of the spinner, and there were no numerical labels on each section. In a redesign of this Gizmo, I would enlarge the spinners, add numerical labels to each section, and possibly incorporate a small animation of the spinner at work. Overall though, I think this is a great tool for students to use when trying to grasp the concepts of theoretical and experimental probability.
 * Tool**: Theoretical and Experimental Probability (Nabgha Ejaz)
 * Math Strand**: Data Management and Probability
 * Concepts**: Create stimulation using spinners to understand and compare the theoretical and experimental probabilities.
 * Evaluation**:
 * Review **:

This gizmo is great for the abstract thinking involved in modeling and solving expressions. Using great visual aids provided in this gizmo ('x' cups as variables, unit counters as the integers), students first model an expression. For instance, for 2x+1=3x+2, students are asked to model each side of the equation using the 'x' cups and unit counters. Then, students are walked through the process to isolate variables on one side and integers on the other. Finally, students are explained how to isolate for the variable by using prior knowledge of division. This gizmo enables the math concept of "solving expressions" to become highly visual and hands-on, thus students are more inclined to learn, grasp the concept and transfer the information to their long term memory. In addition, since students must follow a series of steps the gizmo incorporates the importance of literacy/reading comprehension. Ultimately, I believe this gizmo caters to a wide range of learners and simplifies a concept many students struggle with at the elementary level. Great description. I can see how this tool works and helps students however I think this isn't too beneficial for the students. It's time consuming and maybe doesn't really help them learn so much. I don't see how students are actually learning; they can just get the appropriate cups and counters on the gizmo by following the instructions on the checklist and not really learn anything about the concept itself.
 * Tool**: Christine Dabrowski. Modeling and Solving Two-Step Equations
 * Math Strand**: Grades 6-8 Algebra List strands that might be covered by this toll
 * Concepts**: Modeling and solving expressions, equations, inequalities and word problems.
 * Evaluation**:
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 * Tool**: Probability Simulation (Lauren Quinto)
 * Math Strand**: Grade 8: Data Mangement and Probability
 * Concepts**: Compare, though investigation, the theoretical probability of an event with experimental probability, and explain why they may differ.
 * Evaluation**: T his is a great gizmo that will allow students to compare experimental and theoretical probability. In this gizmo, students will be able to create a spinner with up to 10 sections. They will then decide how they want to determine the probability. For example, if the student creates a spinner with 6 sections, they can choose to see what the probability of spinning <, >, =<, or >= of a number in the spinner. So for this example, the student can find the probability of spinning a number that is greater than 4. This gizmo will record each trial in a chart with the total probability at the bottom. Students can also use this gizmo to compare two spinners. Prior to using the gizmo, the teacher can have the student’s estimate what the different probabilities will be. After the activities, the teacher can have a discussion as to why theoretical and experimental probabilities can differ.
 * Review **: Great explanation of the gizmo; the benefits and the capability of the gizmo are clearly and concisely described. Since the gizmo is highly visual and numerous examples can be completed, I believe the gizmo is useful and has the capacity to promote student learning.


 * Tool**: Linear Functions Gizmo (Diana)
 * Math Strand**: Grade 9 academic - Linear relations
 * Concepts**: Plotting points on a graph to create a scatter plot and test the relationship.
 * Evaluation**: This is a great tool to use as a visual representation for linear relationships. The tool lets you drag points to a graph and shows the input and output values. You can then test the relationship to determine if it is a linear function.
 * Review **: I agree that this is a great tool to use as a visual represenation. You do a good job at describing the different things you can do with this gizmo. I would also make mention that each of the points you make from the input and output controls is recorded in a table of values. This will also teach students how to organize different points on a graph. I would also explain how the gizmo tests the relationship to determine if it is a linear function. Great job!

**Evaluation**: This activity simulates randomly throwing 10 (or 100) darts at a dart board. It is useful in showing the relationship between experimental and theoretical probabilities - students can see after a large number of trials that the experimental probability is very close (but not necessarily equal to) the theoretical probability (click "Show area information" under the Control tab to see the theoretical probability using proportion of areas). By comparison, students can see that throwing 100 darts at random and doing 20 trials yields an experimental probability closer to the theoretical probability than throwing 10 darts at random and doing 20 trials (i.e. larger data set = closer to theoretical prediction). This activity is useful for a class demonstration because it is very quick, has a real-life application, and lays out all of the necessary information (don't have to run to the white board to explain everything). ** Review **: This is a nice tool, but I don't think that it indicates the reasoning behind why the probability of hitting the target is higher when there are more random darts thrown. Good review of the tool!
 * Tool ** : Geometric Probability - Activity A - Random Dart Throw (Jake)
 * Math Strand**: Grade 8 Data Management and Probability
 * Concepts**: determine, through investigation, the tendency of experimental probability to approach theoretical probability as the number of trials in an experiment increases, using class-generated data and technology-based simulation models

**Tool:** Circumference and Area of Circles (Peter Vlasenko) **Math Strand:** Grade 6-8 Measurement **Concepts:** This gizmo demonstrates the relationships between the diameter/radius and area and perimeter of a circle. **Evaluation:** I like this tool, first of all because it is relatively simple, and I think that simplicity in these kinds of tools is more likely to appeal to students as opposed to complicated applications that are more difficult to understand. It visually demonstrates how the area and circumference of a circle depend on its radius, and students are able to stretch out the size of the circle themselves (make it bigger or smaller). The tool states the exact dimensions of the circle as the student varies its size. I think this is a good way for students to visualize the concepts of area and circumference of a circle.
 * Review **: I agree that this is a simple tool, but it may not be the greatest tool for teaching circumference and area – it is a useful tool for quickly changing the size of circles (for many examples in a short time), but there’s no explanation as to **why** the formula for circumference/area is what it is; at the very least, it simply spits out the answers like a calculator and offers no explanation. Possibly, an animation or game (i.e. a person walking along the circle and a display of his distance as he walks) would be more beneficial.

**Tool**: Quadratic and Absolute Value Functions (Adele)
 * Math Strand**: Grade 10 Relations of the Form y ax2 = + bx + c
 * Concepts**: Visual manipulations of the quadratic formula
 * Evaluation**: This gizmo allows the user to use sliders to manipulate the variables “a”, “b”, and “c” of the quadratic and absolute value functions, which modifies the graph shown. A student is able to see the effect that modifying these variables in the function has on the graph. The user is also able to view a table of values for the function. This is a fantastic tool to use compared to the graphing calculators that I remember, and I believe would be very effective in teaching quadratic and absolute value functions.
 * Review **:

**Tool**: Comparing and Ordering Fractions (Tiffany Lee)
 * Math Strand**: Grade 6-8 Fractions and Factors
 * Concepts**: List concepts covered using this tool
 * Evaluation**: Very good use of visuals. You can also compare fractions using LCD, so you can always use this tool if you're touching on the topic of LCD and fractions together. I like how you can make two different set of fractions by setting two different denominators. They are shown directly above/below one another so it is easy for visual comparison. The number line is great too, since you can visually see where the fractions are on the number line. Great tool to compare and order fractions. Also, can help introduce adding and subtracting fractions with unlike denominators.
 * Review ** :  I found the user interface for this gizmo unintuitive, and there was not enough labels or instructions on how to use the gizmo within the gizmo. A lot of instruction from a teacher is needed for students to use this tool, but it could be used in addition to a lesson but not as a stand alone.

**Tool**: Finding Factors with Area Models (Warren)
 * Math Strand**: Grade 8- Number Sense and Numeration
 * Concepts**: Finding the factors of a positive integer, writing the prime factorization of a positive integer
 * Evaluation**: This gizmo represents a number as the area of a rectangle with the side lengths as factors. I think this is a great demonstration for students who have trouble with the abstractness of factorization since the area of a shape is a fairly concrete concept. I also like how you can play around with it to break down a number in different ways.
 * Review **: I like how you talked about how this gizmo can help students with abstractness. Maybe you could have talked more about the tree and how you can split up the factors. Emphasize more on the other gadgets you can use with this gizmo.


 * Tool**: Adding and Subtracting Integers (Joe Anne Guerrero)
 * Math Strand**: Number Sense and Numeration
 * Concepts**: Adding and subtracting integers using a number line
 * Evaluation**: This tool shows how to add and subtract integers on a number line using dynamic arrows. It is very straightforward and easy to follow along with the arrows, so I think this would be a great tool for beginners. There is an option to either add or subtract integers, but the tool only works with adding/subtracting two integers. The equation is shown at the top, first number is red, second number is blue, and the answer is purple. You choose the red and blue numbers at the bottom and the red line will extend from zero to the indicated red number, then the blue line will continue in either the positive or negative direction towards the answer that is indicated with a purple dot.
 * Review **: Overall I think this is a useful gizmo for teaching the addition of integers. It would require some explanation before students used it for subtracting integers since there is no explanation within the gizmo (i.e., it seems a bit arbitrary that sliding the slider one way moves the arrow the other way).

** Tool ** : Comparing and Ordering Fractions (Justin) **Math strand**: Number Sense and Numeration
 * Concepts **: least common denominator of two fractions
 * Evaluation **: This gizmo tool allows you to compare and order fractions, as well as find a least common denominator through visual representation. You can change the denominator of each fraction and it will show you where that fraction belongs on the number line and a block representation of the fraction. It finds the least common factor (LCD) of the two fractions you made. With the help of the visual representation of the fractions and then with the LCD gives an understanding of what a LCD is and why it is important for adding or subtracting fractions. With this concept grasped the students can now learn how to add and subtract fractions.
 * Review ** This is a very useful tool but requires some playing around in order to understand how to use it. I like how it compares fractions in block form and on the number line. You also have the option of comparing the fractions further using the LCD, which also leads to creating equivalent fractions and showing which fraction is larger. For students that are being introduced to comparing fractions and LCD, this tool would be useful along with some instruction on how to find the LCD and how to use it to compare fractions. (Joe Anne)

** Tool ** : Reflections (Alisa Magarelli) **Math Strand**: Geometry:
 * Concepts**: Symmetry and Transformations
 * Evaluation**: This Gizmo tool allows you to understand how objects are reflected across a line. As you move the line around you can see how the reflection of the image changes. Visuals such as this make it clear how reflection across a line works. It is an interactive way to see how the location of an image affects how it is reflected. The movement of both the line and the object allows the user to better understand reflections and see how it affects the actual shape of the image being reflected.
 * Review **:

**Tool**: Adding and Subtracting Integers with Chips(Inga)
 * Math Strand**: Number Sense and Numeration
 * Concepts**: Adding and subtracting integers using blue and yellow chips
 * Evaluation**: This tool lets you practice adding and subtracting integers using chips. There are practice questions below the gizmo. This is a good tool for students since it guides through the steps and gives feedback.
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** Tool **** : ** Adding Fractions - (Sylvia Liu) **Math Strand****:** Grade 8: Fraction Operations Evaluation: This is a very useful Gizmo freeware. This Gizmo makes adding fractions fun by allowing students to visualize the concept of fractions and the addition of fraction. This freeware makes it interactive and stimulating for students by allowing them to create fraction tiles, select different color to represent different fractions and express the sum in an improper fraction or mixed numbers. This Gizmo is also very helpful because it provides sample questions to assess students’ understanding of addition fractions. I agree that this is a useful tool that can help illustrate and visualize the relationship between fractions with different denominators. It can also help the user with the addition of fractions. This tool might be a little difficult to use for students. I think that teacher guidance is necessary in order for students to be able to use this tool effectively.
 * Concepts****:** To add fractions with common denominators, add fractions with different denominators, find the lowest common denominators (LCD) then add the set of fractions, express the sums as an improper fractions or mixed numbers, use a model to illustrate the physical size of the fractions and its sum.
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**Tool:** Rational Functions (Sean) It is very powerful and useful tool for students to visualize the cause and effect of the value “a”(horizontal asymptote), value “h”(h vertical asymptote) and value “k”(vertical shift). This Gizmos is well organized, has nice graphic and easy to use, (user friendly). Good choice of Gizmos and an excellent evaluation. (Sylvia)
 * Math Strand:** Algebra
 * Concepts:** Rational Functions – Graphing from formula (transformations)
 * Evaluation:** This gizmo allows you to graph rational functions with all different types of transformations. You can visually see the effect of the different numbers on the shape of the graph, and also move the graph yourself and see how it affects the formula. I think this could be a useful tool if used by students in grade 11 math since that is one of the first times they may be seeing this type of function. It allows them to familiarize themselves with it and get used to its peculiar shape (having both a vertical and horizontal asymptote). Being able to access the table of values adds another level of depth, and may help deepen the understanding for some learners.
 * Review: **I totally agree with your evaluation. It is a good tool to graph rational functions in the form

** Tool ** : Comparing and Ordering Fractions (Lukas)  **Math Strand**: Grade 7 Number Sense and Numeration
 * Concepts**: Number line, fractions, lowest common denominator
 * Evaluation**: This tool uses a visual model of two fractions and allows you to compare them side-by-side and order them on a number line. It also allows you to compare them using their LCD. Many students have trouble ordering fractions with different denominators. This tool not only makes the problem easier by finding a LCD, but also shows how the initial fractions look using visuals (blocks) and shows where each one lies on a number line. It’s a multi-dimensional approach to understanding fractions.
 * Review ** :   This Gizmo is definitely good for comparing two fractions as you mentioned. It uses both pictures and numbers to illustrate the big picture, and the LCD comparison is a bonus. The only thing you didn't mention was the assessment questions below it. I think these should be used together with the Gizmo to help the students learn the concept and how to use the Gizmo. Also depending on what grade level you’re teaching they may not understand the “dot” as a multiplication sign, or have a firm understanding of a greater than sign, so you’d have to prepare accordingly.

**Tool**: Parabola (Shima)
 * Math Strand**:Grade 10 : determine the basic properties of quadratic relations;
 * Concepts**: The vertex, orientation of a parabola
 * Evaluation**: This tools allow you to graph different parabolas. You can see that by using different number for a,h, and k you will have different shapes of parabolas.
 * Review **: I agree with Shima; this tool is really good at allowing students to see the effects of a, h, and k on the shape of your parabola. Thanks! (Caroline)

**Tool: ** Addition of Polynomials (Katie) A great way to visualize the concept. I think the questions about removing zero pairs in the exploration guide will help students to understand the subject better. I found the idea of dragging a box around zero pairs is very useful. (shima)
 * Math Strand **:Grade 9 Number Sense and Numeration
 * Concepts ** :Adding and Subtracting Polynomials
 * Evaluation **:I really enjoyed using this gizmo because it provides a great use of visuals for students to use. It allows the students to add polynomials using step-by-step feedback when they arrange their tiles. This is a great manipulative for the students to use on their own as well as in a group setting.
 * Review ** :

**Tool ** : Roots of a Quadratic (Taylor) I really like this gizmo and agree with what Taylor has said. I think that students would find it very beneficial because they are able to manipulate the values of all variables on their own. I think that it is also very useful to students because it shows the step by step calculations on how to solve for the roots as well as showing the x and y components. I also really like that there are assessment questions under the gizmo so the students can test their knowledge. (Katie)
 * Math Strand **: Grade 11 Characteristics of a Function
 * Concepts **: Manipulating quadratic equations and solving for its roots
 * Evaluation **: I really like this gizmo because not only does it show you the roots of the quadratics graphically, but there are also tabs that allow you to see it mathematically and also in a chart of x and y components. This gizmo also allows you to manipulate the quadratic, for example you can change the ‘a’, ‘b’, and ‘c’ values of the quadratic and see how the graph changes and also how the roots change. SO this is a great way of visualizing BOTH manipulations AND roots.
 * <span style="color: #800000; font-family: Arial,sans-serif; font-size: 10pt;">Review **<span style="font-family: Arial,sans-serif; font-size: 10pt;">:

**<span style="font-family: Arial,sans-serif; font-size: 10pt;">Tool **<span style="font-family: Arial,sans-serif; font-size: 10pt;">: [|Classifying Triangles] (Denise)
 * <span style="font-family: Arial,sans-serif; font-size: 10pt;">Math Strand **<span style="font-family: Arial,sans-serif; font-size: 10pt;">: Geometry
 * <span style="font-family: Arial,sans-serif; font-size: 10pt;">Concepts **<span style="font-family: Arial,sans-serif; font-size: 10pt;">: Classifying triangles based on sides and angles.
 * <span style="font-family: Arial,sans-serif; font-size: 10pt;">Evaluation **<span style="font-family: Arial,sans-serif; font-size: 10pt;">: I like that this tool can classify the triangles based on angles and sides. I like how students can change the triangle and see how it will then be classified. I’m not entirely sure what the click boxes on the bottom do – doesn’t explain that.
 * <span style="color: #800000; font-family: Arial,sans-serif; font-size: 10pt;">Review **<span style="font-family: Arial,sans-serif; font-size: 10pt;">:

** Tool ** : Fractions with unlike denominators ( ** Ted Baarda) ** **Math Strand**: Grade 7/8 Number Sense and Numeration
 * Concepts**: Add and subtract simple fractions with like and unlike denominators.
 * Evaluation**: This gizmo is great for teaching addition and subtraction of fractions. You can see both fractions that you are working with represented as bars being added or subtracted and visually see the importance of finding a common denominator. The gizmo can also give you the answer with the lowest denominator. I really like the visual aspect that this gizmo provides, since it makes fractions (which is a difficult subject for many students) more concrete.
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** Tool ** : Exponents and Power Rules (Kelly) **Math Strand**: Grade 9 Academic, Number Sense and Numeration **Concepts**: Exponents and Power Rules This Gizmo, I personally felt, was only good for students who could skip a couple of steps. When finding the next step for multiple variables, the exponents were evaluated for each variable at the same time. For example: (4xy^2)^6 = (4^6)*(x^6)*(y^12) Although this is a Grade 9 Academic class, not all students will be able to compute all this so quickly, especially by observing and not being able to write rough work. This could cause more frustrations than anything. Also, in terms of the multiple choice, it was annoying to DRAG the answer into the box, rather than just CLICK the answer. The font is pretty small, which could make it harder to read.
 * Evaluation**: This is a good tool for learning which power rule is appropriate for an expression with an exponent in it. This gizmo gives the students an expression and gives them a choice of four possible answers to simplify the expression. The choices are the four different power rules using the numbers in the expression and the students are asked to choose one. If they choose the wrong the answer, it explains to the students why they are incorrect and asks them to try again. This process can continue for as a long as the students need.
 * Review: ** (K.V.)


 * Tool**: Theoretical and Experimental Probability (K.V.)
 * Math Strand**: Grade 8, Probability
 * Concepts**: Theoretical Probability
 * Evaluation**: Strong tool to learn about theoretical probability using 1 or 2 spinners. With the 1 spinner, you had up to 10 sections, and you could determine the probability of getting the one spot, or getting a spot with a number greater than, less than, or equal to. It also had the option of NOT equal to (so P not). With the 2 spinners, it's actually comparing with each other. This was confusing at first, as it only shows you're comparing near the top with a small diagram. It can answer questions such as "what is the probability of getting a 1 on both wheels?" and "what is the probability of getting a number on wheel 1 greater than wheel 2?", which is still good.
 * Review **:

The tool allows the student to shade a portion of a square grid and then creates the corresponding fraction, reduced fraction, decimal, and percentage. The calculation is completed for both the shaded and un-shaded regions. There is also a number line to compare which is the larger fraction, percentage and decimal of the shaded or un-shaded. There are two short comings when it comes to this gizmo. First, the un-shaded area is white in the grid but represented with aqua coloured numbers. There is no point to colour coding the squares if the colours do not completely connect to the numbers. The second short coming is only a slight possibility. It could be too abstract for students at this age to understand that the shaded areas can be disconnected. A better way to explain this: you can colour a 3x3 and a separate 4x4 section of the 10x10 grid. Meaning you have coloured 25 of the 100 squares without drawing over a portion of the grid that actually looks like 1/4 of the grid. I liked this tool because it allows students to change the area of a target on a dartboard, and observe the r elationship between target area and the proportion of darts that strike it. The Gizmo calculates the percent area of that target in relation to the rest of the dartboard in order to determine the theoretical probability of throwing a dart that lands in the target area. The user can then randomly throw varying numbers of darts at a target and see what percent are "hits." The user can also vary the number of darts that are thrown to show that as sample number increases, the accuracy of the experiment increases, as it more closely resembles the theoretical probability.
 * Tool ** : Percents, Fractions and Decimals (Andrew)
 * Math Strand ** : Grade 8 Number Sense and Numeration
 * Concepts ** : Quantity Relationships
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 * Tool**: Geometric Probability (Caroline)
 * Math Strand**: Grade 8 Probability
 * Concepts**: Theoretical Probability and experimental probability
 * Evaluation ** :
 * Review **: