Virtual math

Objective: This virtual activity is designed to be used in math lessons in the next chapter: Theoretical part Definition and representation Examples: Operations on positive and negative numbers Addition: For example, -12+(-3)=-15; -7+9=2. Subtraction: For example, 18-(-23)=41. Virtual Experiment In this virtual experiment, students learn how to reduce an expression. Create complex expressions that may involve subtraction and negative variable values. Creates expressions that match the tasks in the Games section.  Plays with variables in Levels 5-8. Levels 7-8 include division problems. Workflow: Step 1. Launch the simulation. There are 4 different modes. “Basics”, “Explore”, “Negatives” and “Game”. You will work in the “Negatives” and “Game” sections. Open the “Negatives” section. Step 2. In the workspace you are provided with: Step 3. Create the expression x2 – 2×2 + 3y in the workspace. If you place the second -x2 on top of the -x2 variable, you will see a yellow circle and the variables will be joined together to get – 2×2. Assemble each element in this way.  Step 4. If you keep the elements in the same row, the part of the screen where the pattern is will be light, making a sum. You can see the sum on the board where the value of the expression is displayed.  Step 5. Change the x and y values in the Variables panel. Try to calculate the expression yourself and compare it with the value of the expression shown on the board. Step 6. To display the values and coefficients of the variables, run the simplify operation button (-) and calculate the expression.    Step 7. Build the next expression. Create several expressions using variables. Step 8. Open the “Game” section. You are given 8 levels. In this lesson, you will work on levels 5-8. Step 9. Open Level 5. In the work area, you are given: Step 10. Build the templates by looking at the expression bar provided for assembly and drag them into the empty space below the expression.  Step 11. Once you have completed all the expressions on the board, you can move to the next level by clicking the “Next” button. Conclusion Students, working on the simulator, performed calculations on the topic of summing expressions by similar terms, addition, subtraction of rational numbers.

Objective: This virtual work is intended for use in math lessons in the following chapter: Virtual Experiment The virtual work “Efficient Calculations” is intended for students to learn how to make calculations with different coins and reduce expressions. Workflow: Step 1. Run the simulation. You have 4 different modes. “Basics”, “Explore”, “Negatives”, “Game”. You will work in the “Explore” and “Game” sections. Open the “Explore” section. Step 2. In the work area, you are provided with: Step 3. Pull out some coins to the work area. When you hold the coins in one row, part of the screen turns whitish, forming a sum. You can see the sum on the board, where the value of the expression is shown. Step 4. Activate the buttons to display the values ​​​​and coefficients of the coin. Step 5. Build the next expression. Build several expressions using coins. Step 6. You can replace the coin with the variables x, y, z. Step 7. Open the “Game” section. You are given 8 levels. In this lesson, you will work on the first 4 levels. Step 8. Open the first level. In the work area you are provided with: Step 9. Build expressions by looking at the expression panel given for assembly and drag them to the empty space under the expression. Step 10. After you have completed the entire expression on the board, you can move to the next level by clicking the “Next” button. Conclusion Students, working on the simulator, made calculations on the topics: generalization of expressions by similar members, coefficient.

Objective: This virtual work is intended for use in algebra lessons on the following topics: Theoretical part Definition of linear function A linear function is a function that can be written in the form: The graph of a linear function is always a straight line. Graph of the function y = kx The coefficient k determines the slope of the straight line: The graph of the function y = kx always passes through the origin (point (0; 0)). Graph of the function y = kx + b Finding points of intersection of the graph with the coordinate axes The point of intersection with the Oy axis: Substitute x = 0 into the equation of the function and find the value of y. Coordinates of the intersection point: (0; b). Point of intersection with the Ox axis: Substitute y = 0 into the equation of the function and solve the equation kx + b = 0 with respect to x. Coordinates of the intersection point: (-b/k; 0). Determining the coefficients k and b from the graph Coefficient b: We find the point of intersection of the graph with the Oy axis. Its ordinate is the value of b. Coefficient k: Choose any two points A(x1; y1) and B(x2; y2) on the graph. We use the formula: k = (y2 – y1) / (x2 – x1). Virtual experiment In the Graphing Slope-Intercept simulation, students explore a line at an angle of slope. Draws a sloping line through the equation of a given graph. Predicts how changing the values in the linear equation will affect the line shown on the graph. Predicts how changing the line shown on the graph will affect the equation. Course of Work: Section 1. Slope-Intercept Part Step 1. You will be given 2 different modes, “Slope-Intercept” and “Line game”. Start the “Slope-Intercept” mode. Step 2. You are given:  Step 3. Determine the points of the graph of y = ⅔x + 1 using the tools that show the values of the points (x, y) in the graph coordinate. add buttons to show graphs of y = x and y = – x. Examine the graph. Step 4. Change the values of m, b of the function y = mx + b and study the graph.  Step 5. Examine the graph by changing the m, b values of the function y = mx + b. You can use the buttons and tools provided on the screen. Make some calculations. Section 2. Game Part Step 6. Activate the “Line Game” mode. You are given 4 levels. Select the first level. Step 7. You are given:  Step 8. You have an equation or graph in green. This is the task you must complete:   Complete and review the assigned problem.  Step 9. Complete the assignment and move on to the next level.  Conclusion The virtual lab is a valuable tool for students to learn the graphs of linear functions. The simulator has become interactive and visually useful by providing various tools for learning graphs, such as displaying points, displaying equations, and saving graphs.

Objective: This virtual activity is designed to be used in a 9th grade geometry class. Theoretical Part The motion of a projectile is a classical problem in mechanics, often considered in a simplified model where air resistance is not taken into account. However, under realistic conditions, air resistance has a significant effect on the trajectory of the projectile, especially at high speeds and significant altitudes. Trajectory Without considering drag, the trajectory of the projectile is a parabola. Air resistance deforms this parabola, reducing the range and maximum height of lift. Vectors and Forces Velocity: A vector quantity characterizing the speed and direction of motion. Acceleration: A vector quantity that characterizes the change in velocity in magnitude and direction. Gravity: A constant force directed vertically downward. Drag force: Depends on velocity, air density, cross-sectional area, and drag coefficient. Motion in the X and Y Axes The motion of a projectile can be decomposed into two independent components: horizontal and vertical. Only the horizontal component of the initial velocity acts on the x-axis, while gravity and air resistance act on the y-axis. Virtual Experiment In Modeling Projectile Motion, students study factors that affect the trajectory of a projectile, such as angle, altitude, initial velocity, and drag. By combining their math and physics knowledge, students can perform a variety of projectile experiments. Workflow: Step 1. Start the simulation: you will be presented with 4 different modes: “Intro”, “Vectors”, “Drag” and “Lab”. In this paper you will work in the “Intro” mode. Open the “Intro” section. Step 2. In the workspace you are given: Step 3. Shoot the projectile. Object Pumpkin. Using the Explorers tools, measure time, path, height, distance. Step 4. Add air effects, velocity vectors and acceleration vectors. Use the special tool to estimate where the object will land.    Step 5. Fire and examine the projectile. Step 6. Change the initial velocity. Fire and examine the projectile. Step 7. Change the height of the projectile, the angle. Shoot and check the projectile. Step 8. Perform several experiments by changing the settings. Notice how each setting (initial height, initial angle, initial velocity) affects the trajectory of the object with and without air resistance. Step 9. The Object Data panel lists several object names: cannonball, tank shell, golf ball, baseball, soccer ball, pumpkin, human, piano, car. Select one of the objects. Snapping on the selected object changes its mass, diameter.  Step 10. Do some experiments and research by changing the settings of the selected object.  Conclusion The study has allowed a deeper understanding of the physical processes occurring during the movement of the projectile in the atmosphere. The main parameters influencing the trajectory were taken into account: initial velocity, flight angle, mass, diameter and height. The results obtained can be used to solve various practical problems.

Objective: This virtual activity is designed for use in 7th grade algebra lessons on the following chapter: Theoretical Part Imagine a machine that transforms some numbers into other numbers. You give it one number as an input and it gives you another number as an output. This machine is an example of a function! A function is a rule that matches every number from one set with only one number from another set. How do you write a function? A function is usually written as a formula: y = f(x) Here: For example: Function y = 2x + 1. If we replace x with the number 3, we get y = 2 * 3 + 1 = 7. The function has matched the number 7 with the number 3. Examples of functions from life Virtual Experiment This virtual activity introduces students to the concept of function. In the simulation, they will calculate and investigate a hidden function based on the given arguments and the value of the function.  Procedure: Step 1. Launch the simulation. You have 4 different modes. “Patterns”, “Numbers”, “Equations” and “Mystery”. In this paper you will work in the “Mystery” section. Open “Mystery”. Step 2. In the workspace you will be given: Step 3. Enter several arguments into the function machine. Examine the results, try to find the hidden function.  Step 4. Test your guess by clicking the “show hidden function” button hidden on the function engine.  Step 5. Refresh the hidden function. Enter the arguments and try to find the hidden function. Check your guess by clicking the “Show hidden function” button. Step 6. Do some research on the function machine by increasing the number of operations by 2 and entering the arguments.  Step 7. Check your guess by clicking the “Show hidden function” button hidden on the function machine.  Step 8. Do some research by using the buttons presented on the screen on the function machine. Conclusion Students were introduced to the concept of function in a virtual activity. The function engine was an effective tool for visualizing and analyzing mathematical relationships. The variety of controls (panels, graphs, tables) gave students the flexibility to explore the function from different angles.

Objective: This virtual activity is designed for use in geometry lessons on the following topic: Theoretical part Arithmetic mean The arithmetic mean is a number that shows the average of all the numbers in a group. To find the arithmetic mean, you need to add up all the numbers and divide the resulting sum by the number of those numbers. Formula: Average = (Sum of all values) / (Number of values) Example: Suppose we have a data set of 5 measurements of people’s height: 170 cm, 175 cm, 180 cm, 185 cm, 190 cm. Average: (170 + 175 + 180 + 185 + 185 + 190) / 5 = 180 cm. Virtual Experiment Modeling “mean fraction and equilibrium” allows students to understand mean in a variety of contexts. Students observe how gravity balances the height of water in two-dimensional beakers when the valves are opened. The flattened height is the average level of water in the beakers. Procedure: Step 1. Launch the simulation. In the workspace you will find Step 2. Activate the Predicted Average, Marks and Water Level buttons. Looking at the marks in the beaker, if it balances the water in the beaker, guess where the water level reaches the beaker by moving the knob there. Step 3. Press the middle button. Click “Restore Balance”. Compare the actual average water level with the estimated average water level.  Step 4. Press the “Restart” button. Increase the number of jars as desired.  Step 5. Repeat the process for 2 cups. Step 6. Experiment by changing the number of cups to your liking.  Conclusion Students were introduced to the concept of average through this virtual activity. They performed different manipulations of different levels of water in a beaker and balanced the water.

Objective: This virtual activity is designed to be used in math lessons in the next chapter: Theoretical part A ratio is a comparison of two quantities. It shows how many times one quantity is greater or less than the other. A ratio is written as a fraction or with a colon. For example, the ratio of the number of apples to the number of pears is 3:5, which means that there are 1.67 times fewer apples than pears. Proportion is the equality of two ratios. It is written as two fractions separated by an equal sign. For example, 3/5 = 6/10 is a proportion. Proportions are widely used to solve various problems related to the proportional dependence of quantities. Basic properties of proportions: Virtual experiment In the Ratio and Proportions game, students explore the concepts of ratio and proportional reasoning by changing hand positions and supporting ratios with movement to find complex relationships. Move your hands to find a complex ratio and try to maintain the ratio by moving your hands together. Workflow: Step 1. You have 2 different modes, “Discover” and “Create”. Open the “Discover” section.  Step 2. Start the simulation. In the workspace you will see: Step 3. Press the “Display with line scale” button. You will see that the pointer is in row 2 and 4 and the screen is green. Step 4. If you change the position of one of the hands, you will notice that the screen color changes. This is because the ratio between the two numbers is broken. The ratio between the hands is 2 multiples of 2. Step 5. Place one hand on the number 3. Place the value of the other hand on the number 6, which is 2 multiples of 2. The ratio of 2 is maintained and the screen turns green. Step 6. Construct several proportions that maintain this ratio of numbers. For example, 3.5 and 7; 5 and 10, and so on.  Step 7. Select Challenge 2. Here you construct a ratio of 3 times the numbers.  For example, 1 and 3; 2 and 6, etc. Step 8. Select Challenge-3. This is where you create a relationship to 1.33 times the numbers.  For example, 2 and 2.66; 6 and 8, etc. Step 9. Open the Create section. In the workspace you will see: Step 10. Press the button “Display with line scale”. Choose the size of the line. Step 11. Move your hand over the specified multiple. If it is correct, the screen will turn green.  Step 12. Activate the lock button. Create different proportions of numbers by moving your hands. Step 13. Give other types of proportional relationships and do experiments. Conclusion This virtual activity can help students master the topic of proportions. Creates different relationships, and realizes the topic more deeply.

Objective: This virtual work is intended for use in 6th grade mathematics classes. Theoretical Part Imagine a machine that transforms some numbers into other numbers. You give it one number as an input and it gives you another number as an output. This machine is an example of a function! A function is a rule that matches every number from one set with only one number from another set. How do you write a function? A function is usually written as a formula: y = f(x) Here: For example: Function y = 2x + 1. If we replace x with the number 3, we get y = 2 * 3 + 1 = 7. The function has matched the number 7 with the number 3. Examples of functions from life Virtual Experiment This virtual activity introduces students to the concept of a function. The simulator is not designed to calculate with numbers, but with images as arguments; a function is designed to transform these images. On the Expression screen, students explore different functions and make predictions. They can play detective to find hidden functions on the puzzle screen. Workflow: Step 1. Start the simulation. There are 2 different modes. “Patterns” and “Mystery”. Open the “Patterns” section. Step 2. In the workspace you are provided with: Step 3. Place an operation on the function machine. There are 12 different operations, choose one. Step 4. Paste the image from the argument panel into the machine. How did the function transform the image?  Step 5. Insert some more pictures into the machine and study the function.  Step 6. Add three operations to the machine. Add three different operations to the machine.  Step 7. Paste an image from the argument table into the machine. How did the function transform the picture?  Step 8. After each operation performed on the input number in the function machine, activate the button that displays the value of the expression. Run a few more images and examine how the image changes after each operation. Step 9. Open “Mystery”. In the workspace provided: Step 10. Enter several images into the function engine. Examine the results, try to find the hidden function.  Step 11. Test your assumption by clicking on the hidden “show function” button on the function machine.  Step 12. Do a little research by increasing the number of operations on the function machine and inserting images.  Conclusion Students have become familiar with the concept of function in the virtual lab. They have seen that its role in mathematics is important. This work serves as an introduction to the topic of function in high school. They realized that function is not just a formula, but a way of describing the relationship between quantities in the world around us.

Objective: This virtual activity is designed for use in geometry lessons on the following topic: Grade 9. “Sum of Vectors”. Theoretical Part A vector is a directed line segment that is characterized by The Vector Addition Operation Vector addition is a geometric operation to find a new vector by adding two or more vectors. Properties of vector addition Virtual Experiment In this virtual activity, students explore vectors in one-dimensional space and investigate how vectors add up. By placing vectors in Cartesian coordinates and examining the magnitude, angle, and components of each vector. They can use different representations to accomplish different tasks.  Workflow: Step 1. Start the simulation: You will be presented with 4 different modes: “Explore 1D”, “Explore 2D”, “Lab” and “Equations”. In this paper you will work in the first 2 modes. Open the “Explore 1D” section.  “Explore 1D” Section One-dimensional space: This is a line that extends infinitely in two opposite directions. It is described by a single coordinate (for example, the X axis). Step 2. In the workspace provided to you: Step 3. Place the vector a on the coordinate plane. The board above shows the vector data. Step 4. Execute the buttons Sum, Values. Examine the vector  a. |a| -values, |s| -sum. Step 5. Place vector b in the plane. |b| – indicates the length of vector b, |s| -sum indicates the sum value of vectors a and b. Since |s| is the sum value, it does not depend on the location of vectors a and b in the plane, so the value of |s| does not change. Step 6.Place the vector c in the plane. |c| is the length of the vector c, |s| is the sum value of the vectors sum a, b, c. Step 7. You can move the coordinate plane from horizontal to vertical and perform calculations for vectors d, e, f. “Explore 2D” Section Two-dimensional space: used to graph functions and equations. The X-axis is horizontal and the Y-axis is vertical. All the shapes we learn in school geometry (triangles, squares, circles, etc.) are two-dimensional. Step 8. In the workspace, you are given Step 9. Place the vector a on the coordinate plane. The table above shows the data of the vector. |a| – length of the vector, – angle, ax, ay – length along the x,y axes.  Step 10. Activate the Sum, Value, Angle buttons. Explore the different layouts of the vector a for ax, ay by clicking each button on the Component panel.  Step 11. Place vector b in the plane. Examine the value of the sum |s|.  Step 12. Place vector c in the plane. Examine the value of the sum |s|.  Step 13. Choose the type of vector by angle and perform calculations for vectors d, e, f.  Conclusion By working on this simulation, students became more familiar with the concept of vectors and sharpened their skills in adding vectors. Verified that the sum of vectors depends on their dimensions and angles. Studied the arrangement of vectors in one- and two-dimensional space and visualized the arrangement of objects in space.

Objective: This virtual activity is designed for use in geometry lessons on the following topics: Theoretical part A vector is a directed line segment characterized by Multiplying a vector by a scalar Multiplying a vector by a scalar changes the length of the vector. If the scalar is positive, the direction of the vector is preserved; if it is negative, the direction is reversed. Addition and Subtraction of Vectors A line equation is a mathematical expression that defines all the points that lie on a given line in the plane. One of the most common ways to write the equation of a line is the general equation: Ax + B + C = 0 where A, B, and C are some constant numbers, and at least one of A or B is not zero. The geometric significance of the coefficients Virtual Experiment In the “Vector Addition: Equations” virtual activity, students experiment with vector equations and compare vector sums and differences. Students learn about scalar multiplication by performing calculations with vectors and changing the coefficients in an equation. Workflow: Step 1. Launch the simulation. In the workspace provided: Step 2. Explore the information of the vector data panel by clicking vectors a, b, c.  Step 3. Activate the “Values” and “Angle” buttons. Explore the different layouts of the vectors by clicking on each button in the Components panel. Step 4. Perform a vector calculation using the equation a+b=c. Change the coefficients of the vectors a and b and solve the problems.  Step 5. Modify the vectors a and b by changing the values ax, ay for the vector a, bx, by for the vector b from the vector coordinates panel. Since a+b=c, the vector c is changed automatically.  Step 6. Click the Reload button. Select the equation a-b=c.  Step 7. Repeat the operations created for the equation a+b=c and examine the vectors.  Step 8. Click the Reload button. Select the equation a+b+c=0. Step 9. Repeat the operations created for the equation a+b=c and examine the vectors.  Step 10. You can select the vector type by angle and perform calculations for the vectors d, e, f.  Conclusion Students learned more about the concept of a vector by working on this simulation. They studied and compared the changes that occur when a vector is multiplied by a scalar, and the results of adding and subtracting vectors. Mastered techniques for adding vectors using the parallelogram rule and the triangle rule.