Children often confuse the terms equation and expression. Though both are algebraic problems, they have different meanings and steps to find solutions. Where an expression names a number, an equation describes the relationship between two expressions on opposite sides of an equal sign. Begin by clearing up the difference for your child. You can do this by looking and thinking about word families. Have your child think of words that are in the same family as the word equation. He/she should be thinking about words that are related in spelling or sound (i.e. equal, equality). Explain that all of these words come from the word root, equi-, meaning equal. Tell your child that the word equation means a math problem where both sides of the equal sign are equivalent (worth the same amount). Remind them that they can always remember this by looking or listening for the “equa” sound. Next, show your child examples of equations and discuss what both side would need to be in order to be considered an equation. For example, use the problem: 4 + b = 8. Ask questions like, “If this is an equation and both sides must be equal, what will both sides be worth?”. You may want to wrap up your discussion by informing your child that an expression won’t have an = sign, but and equation will always have an = to show the equality between the two sides of the equation.
For the visual learner, it may be helpful to use to see equations. For example, if you have the equation 5 – c = 1, you could have your child draw five circles on the left side of the equal sign and one circle on the right side. Next, ask him/her what c will have to take away from 5 to make both sides equal. Have your child draw a large c around the 4 circles that c needs to take away. You’ve just proven that c = 4.
A similar activity can be done with a hands-on approach by using manipulatives, such as counters (dried beans work well) to represent the numbers. Instead of drawing the circle you place 5 counters on one side of the equal sign and 1 counter on the other side. Then have your child remove 4 of the beans to a separate pile so that both sides of the equal sign are equivalent. Tell him/her that the pile represents the value of c.
If an equation can’t be solved mentally, they can be solved using two different ideas: inverse operations and the properties of equality. To use inverse operations simply means to use the opposite operation. Students do this all the time to check their work. For example, if your child solves the problem 10-7=3. He/she can check his/her answer by adding the difference to the smaller number to get the larger number (3+7=10). Addition and subtraction are inverse operations. They are opposite and undo one another. The same holds true for multiplication and division. This concept can be used to solve equations, as follows. Let’s say you have the problem n x 2 = 24. To use the inverse operation of multiplication, you’d divide. To do this, you’ll need to use the second idea, the property of equality. This property says that if both sides of an equation are added, subtracted, multiplied or divided by the same number, the sides will remain equal. Knowing this, we can find the value of n by moving all numbers to the opposite side of the equal sign using both ideas together. So, to solve n x 2 = 24, you divide both sides of the equation by 2. When you do this x2 is canceled by 2, n is left alone and 24 is divided by 2 (n = 24 2). Now that all numbers are one the same side of the equation you simply solve it by performing the operation (n = 12).
When working with variables, there are ways to note the operation for multiplication other than the “x” symbol. This confuses many children. Here are a list of different ways a multiplication problem can be written as an equation:
4n=16 (A variable next to a number with both no sign always means multiply.
4n=16 (The symbol represents multiplication.)
4(n)=16 (Any time a number is next to parenthesis and there is no operation sign between them, you must multiply.)
There are two ways to note the operation of division. Most children are familiar with the sign. However, please remind your child that a problem can also be written in fraction form to show the same operation. Here are the two ways the same division problem may be represented:
60 n = 6 or 60/n = 6
The last step to solving an equation should be checking the answer. This will prevent both errors in computation and operation choice. Checking the answer can be easy and fun. Tell your child that after solving the problem, it’s time to “plug it in”. Let’s say he/she solves the problem, 30 = n x 2 and gets the answer n = 15. Now, he/she will plug the 15 in where the n used to be (30 = 15 x 2) to decide if the number sentence is true. This is particularly helpful in working with division problems. If the problem 40 b = 4 is presented and a student solves it using the inverse operation, the answer would be 160. When we plug it in, we get the number sentence 40 160 = 4. This number sentence is untrue, 40 160 = .25. If we read the original problem carefully, we see that b is taking the place of the divisor (the number doing the dividing). If a divisor is larger than its dividend (the number being divided) the answer will always be a decimal. Since our answer is positive 4, we know the divisor needs to be smaller than 40. This problem can be solved mentally or by trying different divisors for b until the number sentence is true (b = 10, because 40 10 = 4).
Objective: Player will rescue numbers from variable bullies to solve equations involving addition, subtraction, multiplication and division.
Explanation: A group of variable bullies have moved into Numberland. These bullies torment numbers anytime they get near them. An = sign represents a number safety zone. The player must help get all numbers to a safety zone using inverse operations and the property of equality. For example, if when walking through the town the player sees 8 + m = 42, it is the player’s job to get the 8 safely away from the variable bully, m. To do this the player must click on the 8 and drag it behind the empty box after the 42. Behind the 42 will be an empty box with a pull down menu. This menu will have operation symbol choices (+, -, x,). The player must choose the correct operation to perform the inverse operation. The player will receive points for choosing the correct operation. However, he/she will need to solve the problem to receive bonus points. On the right side of the screen will be a Bully Busters zone. Here, the variable and numbers are learning to live together. It is here that the player inputs what the variable’s numerical value is. So, for the problem above, m = 34 would be typed in.
If x – 5 = 10, which of the following is also true?
x – 5 - 5 = 10 – 5 B) x – 5 + 5 = 10 + 5
8 + m = 9(4)
(m = 28)
16 2 = 4c
(c = 2)
Adriana and James collect marbles. James has 6 times as many marbles as Adriana. If James has 120 marbles, how many does Adriana have? Write an equation with using the variable m for Adriana’s marbles and solve it.
(120 = 6 m; m = 20)
Jaime wants to make a CD of his favorite songs. There are 56 minutes of time on his CD. He records 16 songs. Write an equation that could be used to find the average length of each song and then solve it.
(16n = 56; n = 3.5)
Noah earns $20 for doing chores each week. He gives a set amount to charity and has $14.50 left over. Write an equation that could be used to find how much money Noah gives to his charity and then solve it.
(20 – n = 14.50; n = $5.50)
The Cubs won 40 games this season, this is 4 more games than last season. Solve the equation g + 4 = 40 to find g, the number of games they won last season.
(g = 36)
Shane has $23 in his pocket after spending $12 at the movies. Solve the equation d – 12 = 23 to find d, the amount of dollar he had before the movies.
(d = $35)
Reyanne buys 3 notebooks and one binder. The binder costs $3.20, which is 4 times the cost of the notebooks. How much do the notebooks cost?