Return to Mathematics IndexINTRODUCTION TO SOLVING EQUATIONSMary Lynn Bochenek Central Junior High 17248 S. 67th Ave. Tinley Park, IL 60477 1-708-532-1771OBJECTIVE:To arrive at an equation of the form X = [ ] in which the variable is isolated and the solution is then obvious through use of equivalent equations.MATERIALS:-film cans (with appropriate weights inside and labeled) -balance scales (at least 2) -cards with equations written on them -handouts (equations to solve, twelve ball problem, the hundred artisans, and various others) -overhead projector -card with equal symbol and not equal symbolSTRATEGIES:Each film can weighs approximately 5 grams so I let that be my 1. A individual film can with cover is approximately 7 grams so I filled each can with sand, nails, money, etc. to have 10 grams be a 2, 15 grams be a 3, etc. I let x = 4, y =3, z =8, w = 15 and v = 2 for my equations. You also need cans labeled 3x-2, 2y, x-1, z-2, 3x, 5y, z/2, w/3 and 2y+1. Equations used v=v, 2+3=5, y+4=7, 6=2+x, 4-1=3, x-1<>x, x-1=3, 6=z-2, 2*3=6, 3x=12, 15=5y, 6/3=2, z/2=4, 5=w/3, 2y+1=7 and 3x-2=10. Prior knowledge: variables and expressions, order of operations, evaluating expressions, properties of operations and inverse operations. Discussion at beginning: Solving equations: "The basic idea is to find out what x is. The catch is, that as soon as you do, they change it to something else!" Equal symbol states that two expressions name exactly the same number. An equation is a number sentence which states that two expressions are equal. Therefore, an equation must have an equal symbol. We discussed a not equal to symbol and defined a variable as a symbol, usually a letter, that can represent any number. Remember when the solution of an equation is found, it must be checked. Symmetric property states that x-4=9 is the same as 9=x-4 and commutative property states that 113 + x is the same as x+ 113. I will use the pan-balance scale to demonstrate how to solve simple equations incorporating the concepts of our properties and inverse operations. Each step is an equivalent equation that is easier to solve. Although the equation changes in the process, the solution remains the same. In your last step, the equation is so simple that it tells you the solution. Okay, let's start with our first equation v=v. I demonstrated on a balance scale how any number equals itself. Also that if I add the same number to both sides of the balance scale, I will have an equivalent equation. Then we proved 2+3=5 on the scale. Now two equations with variables are solved and checked. Discussion of what we did should lead to the Subtraction Property of Equality which states that subtracting the same number from both sides of an equation does not change the equality. For all real nos. a, b, and c, if a=b, then a-c=b-c. Our fourth equation 4-1=3 lead to a discussion of subtraction on the balance scale. We must break 4 down to 3+1 and then take 1 away or we can add 1 to both sides. On the scale I showed that x-1 is not equal to x. In order to equate them I must add 1 to x-1. Therefore, I know that x-1+1=x. Our next equation x-1=3 2 uses the concept from the last equation that we must add 1 to both sides, then substitute x for x-1+1 to reach our goal x=4. Have a student volunteer to demonstrate that 6=z-2. This leads to a discussion of the Addition Property of Equality which states that adding the same number to both sides of an equation does not change the equality. For all real nos. a, b, and c, if a=b, then a+c=b+c. On to multiplication: 2*3=6 means 2 groups of 3 or 3 groups of 2. Both of these need to be demonstrated. Therefore, 3x=12 means 3 groups of x = 12 so 3 x's need to be shown equal to 3x and then substituted for 3x can. Now take away equals and you are left with x=4. Make sure you check your answer. Have a student demonstrate the next equation. This leads to the Division Property of Equality: dividing both sides of an equation by the same nonzero number does not change the equality. For all real nos. a, b, and c, with c<>0, if a=b, then a/c=b/c. Now for division: 6/3=2 is our first equation. If we break 6 up into 3 groups, each one will be a 2. For z/2=4, we need to show they are equal and then show that z/2 + z/2 = z but z/2 + z/2 = 4 + 4. Showing this on the scale leads to z=8. Have a student demonstrate 5=w/3. This leads to the Multiplication Property of Equality: Multiplying both sides of an equation by the same nonzero number does not change the equality. For all real nos. a, b, and c, with c<>o, if a=b, then ca=cb. Finally we tried to solve 2y+1=7 on the scale using the concepts that we had learned in previous problems. Break 7 into 6+1, then subtract 1 from both sides; break 2y into y+y and 6 into 3+3, then take away equals to have y=3. Lastly, 3x- 2=10. Students should now be given the handouts and solve the equations using the methods learned and the scales if needed. When finished with their equations they need to try the Twelve Balls problem from the handout. This can be tried and demonstrated on the balance scale. The other problems from the handouts should be discussed in later sessions.