Math Background

Lesson: One-Step Linear Equations
Developing the Concept

Now that students have reviewed how to solve one-step linear equations in one variable, they can learn how to write these equations from written word problems. Knowing how to solve word problems makes mathematics useful in real-world situations.

Materials: the poster with the three important ideas for solving equations; another poster of the four-step problem-solving process

Preparation: Make a poster of the four-step problem-solving process found in the students' text.

Write the following problem on the board:

Three soccer players want to buy their coach a present for $19.50. The cost will be divided equally among the three players. How much is each player's share?
  • Say: To help us understand this problem, we need to ask ourselves some questions. What is this problem asking us to find?
    Students will say they are being asked to find each player's share of the cost of the present.
  • Say: Our variable will represent one player's share of the cost of the present. Let's label the variable s.
  • Ask: What information do we have that can help us solve this problem?
    Students should say that there are three soccer players and that the present costs $19.50.
  • Ask: A method that is often used to solve problems in mathematics is to write an equation and then solve the equation. Is there an equation we could write that would help us solve this problem?
    Students should suggest the equation 3 x s = $19.50.
  • Ask: Who would like to volunteer to come to the board and solve the equation using our three important ideas from yesterday?
  • Have a student come to the board and solve the equation, emphasizing isolating the variable, undoing the multiplication by dividing, and dividing both sides of the equation by 3 to keep the equation in balance. Make sure the student properly labels the answer.
  • Say: Let's check the answer by substituting it into the original equation. Does 3 times $6.50 equal $19.50? (yes)
  • Say: Let's look back at our answer. Does it make sense? Does the answer $6.50 seem reasonable in the context of the original problem?
    Students should say that it does make sense because 3 times $6 equals $18, 3 times $7 equals $21, and $19.50 is between $18 and $21.
  • Say: Let's solve another word problem.
    Write the following problem on the board:

    Randy bought a sweatshirt and a T-shirt for a total of $32.85. If the sweatshirt cost $23.95, how much did he pay for the T-shirt?

  • Ask: What are we trying to find in this problem?
    Students should say they are trying to find the cost of the T-shirt.
  • Ask: What do we know that will help us solve the problem?
    Students should say they know that Randy bought a sweatshirt and a T-shirt and paid a total of $32.85. They should also say they know that the sweatshirt cost $23.95.
  • Say: That's right. Could someone tell us an equation we could use to solve the problem?
    Students may suggest a couple of equations. One would be $23.95 + x = $32.85. They may also suggest $32.85 − $23.95 = x.
  • Ask: Who would like to come to the board and solve the equation?
    Have a student come to the board and solve one of the equations. Be sure to stress the three important ideas shown on the poster. Also be sure students label the answer appropriately.
  • Ask: What would I do to check to see if the answer $8.90 is correct?
    Students should suggest substituting $8.90 into the equation.
  • Ask: Does the answer make sense?
    Students should say that it does, since adding $10.00 to $23.95 would give a value of $33.95, which is close to the actual sum.

Wrap-Up and Assessment Hints
There is a big difference between knowing how to do something and understanding why you do the things you do. For example, knowing that you must divide both sides of the equation 3 x n = 15 by 3 in order to solve it is different from understanding that when you divide the left side of the equation by 3, it makes the quantity equal to one-third of its original value. So to have an equation that is a true statement, we need to do the same operation on the right side of the equation. Then the quantities remaining on both sides of the equation will be equal. When assessing students, ask them why they do the things they do, not just what they do. Ask questions which ensure an understanding of the three important ideas for solving linear equations. This will make the students' knowledge much richer and will extend to solving more difficult equations.

Houghton Mifflin Math Grade 6