Virtual math

Objective: This virtual activity is designed for use in math lessons on the following topics: Theoretical part Median The median is the number that is in the middle of an ascending-ordered list of numbers. If the number of numbers is even, the median is the arithmetic mean of the two center numbers. How to find the median: Example: Suppose we have a dataset of 5 height measurements of people: 170 cm, 175 cm, 180 cm, 185 cm, 190 cm. In a data set of 5 measurements of people’s height, the median is 180 cm. 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 This simulation allows students to explore the mean, the median, by working with small data sets with points far apart. Using the median screen, students can see how a data point affects the value of the median. In the mean and median screen, students compare the mean and median. They can see how they are affected by new points or the movement of individual points. Progression: Section 1: Median Step 1. Start the simulation: you will be given 3 different modes: ‘Median’, ‘Mean&Median’ and ‘Variability’. In this work, you will be working in the first two sections. Open the “Median” section. Step 2. In the work area you are provided with: Step 3. Add data sort, median, predict median and median buttons. Step 4. Press the hit the ball button 1 time. Examine the data.  Step 5. Hit the ball 2 more times. Examine the data. Step 6. Delete the data. Step 7. Press the button to hit the ball 5 times. Examine the data.  Section 2: Arithmetic Mean & Median Step 8. Open the Mean&Median section. In the work area you are provided with: Step 9. Enable the buttons in the area displaying the data and the predict median, predict mean, median, mean buttons. Step 10. Press the hit the ball button 1 time. Examine the data.  Step 11. Hit the ball 2-3 more times. Examine the data. Step 12. Delete the data. Step 13. Press the button to hit the ball 5 times. Examine the data. Conclusion Two ways of evaluating data were explored in the simulation: the median and the arithmetic mean. Comparison of the median and the mean:

Objective: This virtual activity is designed for use in math lessons on the following topics: Theoretical part Median The median is the number that is in the middle of an ascending-ordered list of numbers. If the number of numbers is even, the median is the arithmetic mean of the two center numbers. How to find the median: Example: Suppose we have a dataset of 5 height measurements of people: 170 cm, 175 cm, 180 cm, 185 cm, 190 cm. In a data set of 5 measurements of people’s height, the median is 180 cm. 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. The average: (170 + 175 + 180 + 185 + 190) / 5 = 180 cm. Variation Range is the distance between the minimum and maximum data points. Example: Suppose we have a data set consisting of 5 measurements of people’s height: 170 cm, 175 cm, 180 cm, 185 cm, 190 cm. Range: 190-170=20. The interquartile range (IQR) is the range from the 25th percentile to the 75th percentile, or the average of the 50th percentile of a set of numbers. It is often considered a means of determining what the average range should be. Example: Suppose we have a data set consisting of 5 measurements of people’s height: 170 cm, 175 cm, 180 cm, 185 cm, 190 cm. Q1=(x1+x2)/2=(175+170)/2=172.5 Q3=(x4+x5)/2=(185+190)/2=187.5 IQR=Q3-Q1=187.5-172.5=15 The mean absolute deviation is the average distance from each data point to the mean. Example: Suppose we have a data set consisting of 5 measurements of people’s height: 170 cm, 175 cm, 180 cm, 185 cm, 190 cm. ⅕(170+175+180+185+190)=180 MAD=⅕(∣180−170∣+∣180−175∣+∣180−180∣+∣180−185∣+∣180−190∣)=⅕(10+5+0+5+10)=6. Virtual experiment This simulation allows students to explore the mean, the median, by working with small data sets with points far apart. Using the median screen, students can see how a data point affects the value of the median. In the mean and median screen, students compare the mean and median. They can see how they are affected by new points or the movement of individual points. Progression: Section 1. Arithmetic mean and median Step 1. Start the simulation: you will be given 3 different modes: ‘Median’, ‘Mean&Median’ and ‘Variability’. In this paper, you will be working in the last two sections. Open the “Mean&Median” section. Step 2. In the work area you are provided with: Step 3. Enable the buttons in the area displaying the data and the buttons predict median, predict mean, median, mean. Step 4. Press the hit the ball button 1 time. Examine the data. Step 5. Hit the ball 2-3 more times. Examine the data. Step 6. Delete the data. Step 7. Press the button to hit the ball 5 times. Examine the data. Part 2. Variation Step 8. Open the “Variability” section. In the work area you are provided with: Step 9. Add the median, arithmetic mean, pointer, and interval buttons. Step 10. Hit the ball 3 times. Start the range button (range). Examine the data. Step 11. Delete the data. Step 12. Press the button to hit the ball 5 times. Examine the data. Step 13. Go to the “interquartile range” chapter. Step 14. Start the interquartile range (IQR) button. Examine the data. Step 15. Go to the mean absolute deviation chapter. Step 16. Run the mean absolute deviation (MAD) button. Examine the data. Step 17. You can try the experiments several times by changing the data. Conclusion This virtual activity is a valuable tool for learning basic statistical concepts such as mean, median, variance, standard deviation, interquartile range and mean absolute deviation. It allows students to not only learn the definitions and formulas, but also visualize how these concepts are applied to real data. Visualizing data with graphs and charts makes learning more understandable and engaging.

Objective: This virtual activity is designed for use in math lessons on the following topics: Theoretical part 1. Equality of fractions: Two fractions are considered equal if they denote the same part of a whole. Example: 1/2 = 2/4 Explanation: 2. Comparing Fractions: 2.1 Compare denominators. Example: 1/3 > 1/4 Explanation: 2.2 Bring fractions to a common denominator. Example: 1/2 > 3/8 Explanation: 2.3 Compare the numerators. Example: 2/3 > 1/3 Explanation: 3. Equivalent fractions: Equivalent fractions are fractions that are equal to each other. Example: 1/2, 2/4, 3/6-equivalent fractions. Explanation: Virtual experiment The simulation game “Fraction: Equality” allows students to explore equivalent fractions with different denominators and then test their concepts on the game screen. The figure below shows the functions each button performs. Progression: Part 1. Equality Lab Step 1. Start the simulation: you will be presented with 2 different modes: “Equality Lab screen” and “Game”. Open the “Equality Lab screen”. Step 2. In the working area you are presented with: Step 3. Place the mold on the empty skeleton located on the left side of the screen. You will automatically have the empty skeleton on the right side replenished. You will see from the equality below the shapes that 1/1=2/2. Step 4. Fill the empty skeletons on the left with more shapes. Make it a point to change the equality each time you fill. Step 5. Click the “Reload” button. Step 6. Increase the number of parts of the empty frame of the fraction model by 3. Step 7. Place the shape on the empty frame on the left. You will automatically have the empty skeleton on the right automatically replenished. You will see from the equality under the shapes that 1/1=3/3. Step 8. Fill the empty skeletons on the left with more shapes. Attribute the change in equality to the change in equality each time you fill. Step 9. click the Reload button. Increase the denominator of the fraction by 2. Step 10. Examine the changes to the equality by placing the shapes into empty frames. Step 11. Explore the changes by replacing the fraction model shapes. Step 12. Explore multiple types of equality by changing the denominator of the fraction by starting over. Part 2. A game for learning to balance fractions Step 13. Open the “Game” mode in the simulator. You are given 8 levels. Select the first level. Step 14. In the work area you are given: Step 15. Select one of the fractions to be balanced and place it on the scales. Step 16. Among other fractions, find a fraction equal to it and place it on the second scale. Step 17. Check for correctness using the “check” button. If correct, place it in the empty cell above by clicking “ok”. And if there is an error, click the “try again” button and do the equality again. Step 18. Complete the tasks given by balancing the other types of fractions. Step 19. Once you have done everything correctly, you can move to the next level by clicking “continue”. Step 20. You can continue and complete the tasks of each level as you wish to sharpen your knowledge. Conclusion In completing this virtual lab activity, students explored different aspects of equality in the context of fractions. In Part 1 of the experiment, students worked with models of fractions and explored their equality. Part 2 introduced a new game format for learning how to equalize fractions. Here, learners practiced finding correspondence between different parts. This allowed students to put their knowledge and skills into practice, as well as develop their logical thinking skills and work with parts in a playful way.

Objective: This virtual activity is designed for use in math lessons on the following topics: Theoretical part Ordinary fractions are numbers represented as fractions where the numerator and denominator are integers and the denominator is not zero. Writing a fraction: Example: Meaning of the fraction: Example: Types of fractions: Example of a mixed number: Writing a mixed number: Converting fractions to mixed numbers: Virtual experiment The “Fraction: modeling mixed numbers” simulation allows students to become familiar with and compare several representations of fractions, including mixed numbers. The correspondence between numbers and representations allows for flexible exploration of fractions. The figure below shows the functions each button performs. Progress: Part 1. Learning to build simple fractions Step 1. Start the simulation: you will be offered 3 different modes: “Intro”, “Game” and “Lab”. Open the “Intro” section. Step 2. In the working area you will be presented with: Step 3. Start the mixed numbers button. Select the appearance of the fraction to your liking. Step 4. On the empty frame of the fraction model, hover the mouse over the shape that makes up the fractions. You will have both the numerator and denominator equal to 1. That is, the fraction 1/1 – forms a complete shape. Step 5. Increase the fraction section by 2. You have gotten ½ – a semicircle. Hence, it turns out that ½ represents half of the shape. Step 6. Click the button that increases the amount of empty frame of the fraction model. You will have 2 wireframes. Step 7. Fill the first skeleton completely and the second one halfway. You will have a mixed number. Step 8. Increase the section of the fraction by 3. You will have a fraction of 3/3. Step 9. Create another mixed number by completing the skeletons. Step 10. Click the button that increments the empty frame of the fraction model and pull the third empty frame onto the screen. Step 11. Form mixed numbers by collecting and filling the frames with different fractions from the shapes. Step 12. You can also repeat the experiments for other types of shapes. Part 2. A game for learning fractions Step 13. Select the “Game” section of the simulator. Step 14. You will be presented with different levels. Drawing figures on the given levels fractions are represented as numbers, by which you will construct fractions from the figures. And in levels given by numbers, fractions are represented as figures by which you will write fractions by numbers. Choose the type of level to your liking. Step 15. On the right side of the screen, you will find a list of fractions to collect. You can collect these fractions in the center area. At the bottom of the screen are the items from which you need to collect the fractions. On the left side are the back and restart buttons. Step 16. Collect the first fraction. If you have selected a level that has an image of shapes, you can add as many more empty wireframes as needed using the “Add” button to express a mixed number. Step 17. To check for correctness, move the cursor over the empty space next to the given fraction, left click and drag the fraction into it. If the fraction is correct, it will occupy the space. If you made a mistake, however, the fraction will return to its place of assembly. Step 18. Assemble the other types of fractions as well. Step 19. Once you have correctly assembled all the fractions, you can move on to the next level and complete other tasks. Step 20. You can complete other levels by clicking the “Back” button above left and returning to the “levels” section. Part 3. Lab part Step 21. Select the “Lab” section. You are given shapes to create fractions. You can choose a shape: round or rectangular. Here you see a blank space to write the fraction and an empty frame. Underneath the template are the numbers to use for the numerator and denominator of the fraction. Step 22. Construct a mixed type of number from the numbers. Step 23. Place the shapes on the empty frame following the given fraction. You can add more empty skeletons as needed using the “Add” button. Step 24. Move the fraction and shape to the left side of the screen by holding down the left mouse button. Step 25. Pull a copy into the workspace by dragging the empty space next to the numbers. And in the same way, move the empty wireframe next to the shapes to the same workspace. Step 26. Build the correct type of fractions from the numbers and form the corresponding shape. Step 27. Try to build some more mixed and proper fractions. Conclusion In the course of the work, students not only learned theoretical information about fractions, but also acquired practical skills of working with them. They learned how to create, analyze and work with ordinary fractions using virtual tools and modeling. This experience enriched the knowledge in the field of mathematics and will come in handy for further learning.

Objective: This virtual activity is designed to be used in math lessons on the following topics: Theoretical part Fractions are numbers represented as fractions where the numerator and denominator are integers and the denominator is not equal to zero. In mathematics, ordinary fractions play an important role, used to describe fractions, parts of a whole, and to perform various arithmetic operations. Writing a fraction: Example: Meaning of the fraction: Example: Reading fractions: Types of fractions: A mixed number is a number that consists of a whole part and a fractional part. Example: Constructing fractions: An integer must be divided into equal parts. For example, divide a circle into 4 equal parts. If we color 2 parts of the 4, we get 2/4. Virtual experiment The Build a Fraction virtual activity allows students to create fractions from differently shaped parts by playing games and conducting lab experiments. Modeling the construction of a fraction allows them to predict and understand how changing the numerator of a fraction will affect its value, and how changing the denominator of a fraction will affect its value. The figure below shows the functions each button performs. Progression: Part 1. Learning how to construct proper fractions Step 1. Start the simulation: you will be given 3 different modes: “Build a fraction”, “Mixed numbers”, “Lab”. Select the “Build a fraction” section. Step 2. You will be given different levels. Drawing figures on the levels given fractions are presented as numbers, by which you will construct fractions from the figures. And on levels given by numbers, fractions are represented as shapes by which you will write fractions with numbers. Select level 1, represented by a picture of shapes. Step 3. On the right side of the screen, you will find a list of fractions that you need to collect. You can collect these fractions in the center area. At the bottom of the screen, you will find the elements from which you need to collect the fractions. On the left side are the back and restart buttons. Step 4. Collect the first fraction. Step 5. To check if it is correct, hover over the empty space next to the given fraction, left click and drag the fraction into it. If the fraction is correct, it will occupy the space. If you made a mistake, however, the fraction will return to the assembly space. Step 6. Assemble the other types of fractions as well. Step 7. Once you have correctly assembled all the fractions, you can move on to the next level and complete the other tasks. Step 8. Go back to the levels section by clicking the back button at the top left. Step 9. Choose the first of the levels given by the numbers and complete the tasks. Part 2. Learning to build mixed fractions Step 10. Select the “Mixed numbers” section. Step 11. You will be given different levels as in the “Build a fraction” section. Select level 1, represented by a picture of shapes. Step 12. Build the first type of fraction. Because of the mixed number, the shapes do not correspond to a single empty skeleton. Therefore, click the “Add” button next to the empty skeleton to display the number of skeletons you will need. Step 13. Collect the fraction and place it by dragging it to the empty space next to the given fraction. In case of an error, the fraction will be returned to the place where it was assembled. Step 14. Assemble the other types of fractions as well. Step 15. Once you have done everything correctly, you can move on to the next level and complete the other tasks. Step 16. You can also complete the levels given by the numbers if you want. To do this, go back to the levels section by clicking the back button at the top left. Select the first of the levels given by numbers and complete the tasks. Part 3. Laboratory part Step 17. Select the “Lab” section. You are given shapes to create fractions. You can choose a shape: round or rectangular. Here you see a blank space to write the fraction and an empty frame. Underneath the template are the numbers to use for the numerator and denominator of the fraction. Step 18. Construct a mixed type of number from the numbers. Step 19. Place the shapes on the empty frame following the given fraction. You can add more empty skeletons as needed using the “Add” button. Step 20. Move the fraction and shape to the left side of the screen by holding down the left mouse button. Step 21. Pull a copy into the workspace by dragging the empty space next to the numbers. And in the same way, move the empty wireframe next to the shapes to the same workspace. Step 22. Build the correct type of fractions from the numbers and form the corresponding shape. Step 23. Try to build some more mixed and proper fractions. Conclusion Using a virtual experiment is an effective and interactive way to learn how to create different types of fractions. This tool allows students not only to visualize the process of building fractions from different shapes, but also to conduct laboratory experiments to better understand their properties. With the ability to check the correctness of assignments in real time and instant feedback, students can effectively progress in mastering this math topic.

Objective: This virtual activity is designed for use in math lessons on the following topics: Theoretical part Fractions are numbers represented as fractions where the numerator and denominator are integers and the denominator is not equal to zero. In mathematics, ordinary fractions play an important role, used to describe fractions, parts of a whole, and to perform various arithmetic operations. Writing a fraction: Example: Meaning of the fraction: Example: Reading fractions: Types of fractions: Virtual experiment Modeling the construction of a fraction allows students to predict and understand how changing the numerator of a fraction affects its value, and how changing the denominator of a fraction affects its value. The correspondence between numbers and pictures allows for flexible exploration of fractions. The figure below shows the functions each button performs. Progression: Learning to build simple fractions Step 1. Start the simulation: you will be offered 3 different modes: “Intro”, “Game” and “Lab”. Open the “Intro” section. Step 2. In the working area you will be presented with: Step 3. On the empty frame of the fraction model, hover the mouse over the shape that makes up the fractions. You will have both the numerator and denominator equal to 1. That is, the fraction 1/1 – forms a complete shape. Step 4. Increase the fraction section by 2. You have gotten ½ – a semicircle. Hence, it turns out that ½ represents half of the shape. Step 5. Increase the fraction section by 3. You will get ⅓ – one third of the shape. Step 6. Increase the numerator of the fraction by 2. You will get 2/3 – two-thirds of the shape. Step 7. So you can see how the fraction looks on the shape by changing the numerator and denominator. Try to construct several types of fractions. Step 8. Click the button that increases the number of empty skeletons of the fraction model and display 3-4 empty skeletons. Match the fraction with the number of empty skeletons. Step 9. Fill the empty skeletons using different shapes of fractions. Step 10. You can repeat these experiments with other shapes. A game for learning fractions Step 11. Select the “Game” section. Different levels will be available to you. Choose one level to your liking. Step 12. On the right side of the screen, you will find a list of fractions that you need to collect. You can collect these fractions in the center area. At the bottom of the screen are the items from which you need to collect the fractions. Step 13. Collect the first fraction. Step 14. To check for correctness, move the cursor over an empty space next to the given fraction, left-click and drag the fraction into it. If the fraction is assembled correctly, it will occupy the space. If you made a mistake, the fraction will return to the place where it was assembled. Step 15. Assemble the other types of fractions as well. Step 16. Once you have correctly assembled all the fractions, you can move on to the next level and complete other tasks. Laboratory part Step 17. Select the “Lab” section. You are given shapes to create fractions. You can choose a shape: round or rectangular. Here you see an empty space to write the fraction and an empty frame. Underneath the pattern are the numbers to use for the numerator and denominator of the fraction. Step 18. Form the fraction using the numbers. Step 19. Place the shapes on the empty frame, following the given fraction. Step 20. Move the fraction and shape to the left side of the screen by holding down the left mouse button. Step 21. Pull a copy into the workspace by dragging the mouse over the empty space next to the numbers. And in the same way, move the empty wireframe next to the shapes to the same workspace. Step 22. Build the next fraction from the numbers and form the corresponding shape. Step 23. Try building a few more fractions. In this lab section, you can improve your fraction skills by creating different fractions. Conclusion By creating this virtual activity, students will reinforce their knowledge by experimenting with the topic of simple fractions that they have been studying in math.The simulation demonstrates how fractions relate to shapes, which promotes a deeper understanding of the concept of fractions.