Lesson: Equations Are Solved by Backtracking
Attached transparencies and worksheets
Projector or White Board
8-12A 1.0 Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable:
A1.1 Students use properties of numbers to demonstrate whether assertions are true or false.
Launch: Present 3 problems that review recently acquired skills, provide 3-5 minutes for solving, solicit answers, guide student-led clarification if needed
Display the attached detective overhead. Ask students what detectives do. Note the ideas they come up with. Explain that when we are solving an equation, it is like solving a mystery, we have to backtrack to find out what was originally true. We can think of the variable as something a thief has hidden, for example. We are going to UNDO what they have done in order to discover the truth- the value of the variable. Stress that we will know when we have to do this because the directions will say “Solve for x.”
On the overhead or board, write an equation everyone can solve in their head, like 2 + x = 3. We know x is 1, but let’s pretend we don’t. In this case, the thief did not hide the variable very well! We are solving for x, so we need to end up with x = 1. Place this below, leaving space to show work. If we are going to get there, we need to undo the things that are hiding the x. We need to get the x alone. What do we need to get rid of? (2) If I gave you 2 dollars and you didn’t want it, you’d get rid of it, you’d subtract it. Let’s subtract it. But if that’s all we do, we’ll end up with x= 3, which can’t be true because 2 + 3 = 3 is false. Remember when we used the scale to find equal weights? The scale had to balance. We had to do the same to both sides, whether we added or subtracted, we had to do it to both sides in order to stay balanced. So, we subtracted 2 from that side, let’s also do it from the other side. (Show.) Did we end up with x =1 like we wanted? Yes! So, what we did was:
- Identify the variable we want to solve for (get alone)
- Undo whatever was on the same side of the = with the variable
- Do the same to both sides
- Write what’s left on both sides of the = sign
Try again, with the same language, for several examples.
x + 4 = 5 2x = 4 x – 3 = 0 1x = 4 x/2 = 3 10x = 100
What do we want to end up with?
What is on the same side as the variable?
How do we get rid of that?
Let’s do it to both sides.
What’s left on each side?
Is that possibly the answer? (Check by substituting when examples are less obvious.)
Model thinking, writing, doing using the sentence-starter worksheet attached. Encourage student input and gradually increase independence.
Students work on think, write, do scaffolded One Operation Worksheet, attached.
In subsequent days, repeat the structure of the lesson and introduce more complex equations. Guide students in creating Genius Cards listing steps they should follow. Example attached.
Solve for x.
7x + 3y – x + 4 - 2
Scale 1 - 5