This topic was addressed in an earlier post for seventh grade (Pct 1) The reader is referred to that chapter for the specifics of the instruction. The only difference is that we now have equations with a distribution in it and which contain fractions. This chapter will provide examples of such problems and will also include word problems that results in variables on both sides of the equation.
At this point, common mistakes and misconceptions will surface. These should be addressed, and students who make such errors should be monitored to ensure that they don’t repeat the mistakes. Mistakes include the following:
Subtracting instead of dividing. Rather than dividing a term like 6x by 6 to isolate the variable, students will subtract the 6. I show students that 5 × 2 – 5 does not equal 2, but the expression divided by 5 does. Similarly 5x – 5 does not equal 5, but 5x/5 does.
Combining unlike terms. Students will try to combine terms that cannot be combined. The expression 6x + 3 does not equal 9x, but students will look only at the coefficient and the number and think that the numbers are like terms, and can therefore be combined. To correct the mistake, students should be given problems requiring them to combine terms which contain uncombinable terms so that they learn to not feel obligated to merge everything together.
Dividing before its time. For an equation like 6x + 3 = 9 students will divide both sides by 6 to get x +3 = 9, or maybe even x + 3 = 9/6 at which point they sit and stare at the mess they created. One way around this is to teach that solving equations like that is reverse PEMDAS. Rather than multiplication and division having precedence over addition and subtraction, the opposite is true: addition and subtraction steps are carried out first. Of course they could divide each term by 6, but I try to keep things simple at this point.
Not defining variables. While not an outright mistake, this is a bad habit. When solving word problems, students should write out what the variables represent. For the problem “John weighs 30 lbs more than Alice and their combined weights are 130 lbs. Find each person’s weight” students should define what x represents. If it represents Alice’s weight, then say so, and then write x + 30 = John’s weight.
Warm-Ups.
1. Write an equation and solve. No credit without equation. Rich is 3 years older than Carla. Ruth is twice as old as Rich. Their ages total 33 years. How old is each person? Answer: Let x = Carla’s age, then x + 3 = Rich’s age and 2(x+3) = Ruth’s age. x + (x+3) + 2(x+3) = 33; x + x + 3 + 2x + 6 = 33; 4x + 9 = 33; 4x = 24; x = 6, x + 3 = 9, and 2(x + 3) = 18.
2. Solve. 2(t+4) -3 = 11 Answer:2t+ 8 – 3 = 11; 2t+5 = 11; 2t = 16; t = 8
3. Simplify. –x + y + 3x -4y + 6 Answer:2x -3y + 6
4. Solve. 69 = 4(h + 5) – (h-1) Answer:69 = 4h + 20 –h+1; 69 = 3h+21; 3h=48; h = 16
5. Solve. 0 = 17h – 102 Answer: 17h= 102; h = 6
Students will likely ask for help on Problem 1. Prompts may include, “What should we let x equal?” “All the ages are in terms of one person. Who is that person?” “If you let x equal that person’s age, how would you represent the others in terms of x?” Problem 3 may cause students difficulty; they may need reminding that –x is the same as -1x. In Problem 4, students may not subtract (h-1) correctly. They may need reminding that –(h-1) can be written as –1(h-1), and that –1 is then multiplied by each number and variable in the parentheses. In Problem 5, students may find this form different than what they’ve seen. A prompt may be “How do we isolate 17h?”
Examples of problems for today’s lesson. (As mentioned above, this is essentially the same lesson as that in Pct 1. The examples contain equations with distributions, as well as fractions. Also some may require combining terms as a first step.)
1. 3t = 2t +16 Answer: t = 16
2. x + 4x – 8=6 + 2x + 1 Answer: 5x – 8 = 7 + 2x; 3x = 15; x = 5
3. 19r + 4 = 19 + 14r Answer: 5r = 15; r=3
4. 5-b = b +5 Answer: 2b = 0; b = 0
5. 3(t +4) – 3 = ½ (10 + 4t) Answer: 3t + 12 – 3 = 5 + 2t; 3t + 9 = 5 + 2t; t = -4
Students often forget how to multiply a fraction by a whole number. Possible prompts: “What is ½ of 10? What is ½ of 4? So then what is ½ of 4t?
Homework. The homework should be a mix of all the different types of equations students have had, with the focus on equations with variables on both sides.