Compound equations

Objective:

  • To learn how to make and solve an equation
  • To learn how to solve complex equations.

This virtual activity is designed to be used in math lessons on the following topics:

  • Grade 4. “Equations”
  • Grade 4. “Numerical and alphabetic expressions. Conversion of expressions”
  • Grade 4. “Equations. Solving equations by transformation”.
  • Grade 4. “Composition of expressions with variables by problems”.

Theoretical part

Compound equations are equations that contain two or more arithmetic operations.

A variable is a quantity that can take on different values. Expressions with several variables can also be simplified and then find the values at given letter values.

Transformations of expressions that result in a simpler expression are called simplifying expressions. For example: 

x + x + x + x + x 

“4 times x” can be written shorter (simplify): 4 ∙ x or 4x. 

The multiplication sign is often not written in such cases. x + x + x + x + x = 4x

If a transformation can be performed on the left or right side of an equation, simplify the expression first and then solve the equation.

Consider how you solved the equation. 

5x + 2x = 49 

The left side of the equation can be simplified. Let’s do this. 

7x = 49 

Now let’s solve the simple equation by the rule of finding the unknown multiplier.

x = 49:7 

x = 7 

Verification. 

5 ∙ 7 + 2 ∙ 7 = 49 

35 + 14 = 49 

49 = 49

Virtual experiment

The “Equation” simulation allows students to explore the concept of equation, inequalities, and variables. On the “Variables” screen, students explore how different values of a variable affect the state of equality. On the “Calculations” screen, students can construct an inequality or equation and apply universal operations to find out what happens to each term and learn how to undo the operation.

Progression:

Part 1. Variables

Step 1: Start the simulation: you will be presented with 5 different modes: ‘Basics’, ‘Numbers’, ‘Variables’, ‘Operations’ and ‘Solve it’. You will work on this experiment in the “Variables”, “Operations” sections. Open the “Variables” section.

Step 2. You are given: 

  • A board on which to display equation (1);
  • Scales to plot the equation with equating variables and numbers (2);
  • Variables and numbers to construct the equation (3);
  • Eraser to remove the expression standing on the scales (4);
  • Button to adjust the expression above the scales (5);
  • You can set the same numbers in both heads of the scales by pressing the lock button (6);
  • Button to change the value of x (7);
  • You can save the generated equations in the image panel by pressing the camera button (8);
  • Buttons to refresh, garbage can and display x value (9);
  • Reload button (10).

Step 3. Generate the equation in the value of x =1. If necessary, you can use the eraser and button to adjust the items above the scales.

Step 4. Click the camera button and save the equation in the image panel.

Step 5. Change the value of x and make the equation. You can press the lock button and express both sides of the equation with the same numbers. Save in the image panel.

Step 6. Compose the equations by assigning a different value to the variable x. Complete the image panel.

Part 2. Calculations 

Step 7. Open the “operations” section. You are given: 

  • A board on which the equation is displayed (1);
  • A panel to apply the same operation to both sides of the scales (2);
  • Scales to plot the equation with the equation of variables and numbers (3);
  • Eraser to remove the expression standing on the scales (4);
  • Variables and numbers to build the equation (5);
  • You can set the same numbers and variables in both heads of the scales by pressing the lock button (6);
  • A board to change the value of x (7);
  • You can save the generated equations in the image board by pressing the camera button (8);
  • Buttons to refresh, garbage can and display the x value (9);
  • Reload button (10).

Step 8. Construct the first equation: x =1. You can construct the equation using the variables and numbers located at the bottom. You can use the operations panel above. In the operations panel, you can create expressions with numbers from 1 to 10 in the operations ( + ), ( – ), variables 1x-10x, numbers from 1 to 10 in the operations ( * ), ( / ).

Step 9. Click the camera button and save in the image panel.

Step 10. Change the value of x and make an equation. Save to the image panel.

Step 11. Compose equations by assigning a different value to the variable x. You can use all of the tools in the workspace when composing an equation. Fill in the image panel.

Conclusion

The virtual “equation” simulator allowed students to gain a deeper understanding of the topic of equations. The equations that the students have learned to solve so far entered a complex form in this activity and helped them to sharpen their knowledge. The fact that the variable has a different value by balancing it on a scale makes it easier for students to visualize and learn how to make different equations.