Math Background

Lesson: One-Step Linear Equations
Introducing the Concept

Stress three important ideas when solving linear equations: (1) isolate the variable; (2) undo what was done to the variable by using the inverse operation; and (3) keep the equation in balance by doing the same thing to both sides of the equation.

Materials: a large poster to hang up in front of the class with the three ideas mentioned above listed on it

Preparation: Make the poster mentioned above. Make a picture on the board or on an overhead transparency of a balance scale to remind your students to keep the equation in balance.

Write the simple equation x − 3 = 4 on the board.

  • Ask: Who can tell me what this equation says? How can you say it in words?
    Students should say that the equation says “Three is subtracted from a number and the answer is 4,” or something similar.
  • Say: Yes, that's right. How can I find the missing number?
    Students may say “The answer is 7, because when 3 is subtracted from 7 the answer is 4.”
  • Say: Good, is there another way I could get the answer?
    Some students may suggest adding 3 and 4 to get 7. If they don't, suggest it to them.
  • Say: Great. In fact, that is how I want us to think about solving equations. Let's look at the poster with three important ideas to help us solve equations.
    Point to the poster with the three important ideas on it. Discuss the three ideas and show how they relate to solving the equation x − 3 = 4.
  • Say: The first idea on this poster tells us that when solving an equation, we want to get the variable on one side of the equation by itself. What is the variable in this equation? (x) What did we do to find its value?
    Students will say that they added the 3 and the 4.
  • Say: That's right, we added 3 because 3 is subtracted from the variable in the equation. In mathematics, we say that addition and subtraction are inverse operations, because one undoes the other one. That is, if we subtract 3 from a number, we can add 3 to the difference to get the number back again.
  • Say: What we are really doing is adding 3 to both sides of this equation.
  • Write the following on the board:
    x − 3 + 3 = 4 + 3
  • Say: When we subtract 3 from a number and then add 3 to the difference, we get that same number. Therefore, we get x on the left side of the equation. When we simplify the right side of the equation by adding 3 to 4, we get 7. Therefore, x = 7.
  • Say: Note that we can think of an equation as a balance. In order to keep a scale balanced, we must do the same operation to both sides. If we had four blocks on each side of a balance and we took away one block from the left side, what would we need to do to the right side to keep it balanced?
    Students should say that you need to take a block away from the right side.
  • Say: That's right. Let's try another one. How would I solve this equation using the three ideas on the poster? What operation would I need to do to the y to get it by itself?
  • Write the equation y ÷ 3 = 12.
    Students should say that in order to get y by itself, you have to “undo” dividing by 3 by multiplying both sides of the equation by 3. They might not mention multiplying both sides by 3, so be sure to stress the need to do that.
  • Say: That's right, I would need to multiply both sides by three. What do I get if I do that?
  • Write the following on the board:
    y ÷ 3 = 12
    (y ÷ 3) x 3 = 12 x 3
    y = 36
  • Have students do the following exercises at their desks and then have volunteers do them on the board.
    x + 14 = 48 (x = 34)
    4 x s = 56 (s = 14)
    y − 19 = 23 (y = 42)
    x ÷ 6 = 14 (x = 84)
Houghton Mifflin Math Grade 6