This worksheet is a great way to introduce solving two step equations. It helps the students use inverse operations in the correct order by first looking at what has been done to the variable (the Do part of the chart), and then how to "undo" those steps. The second page has more practice problems, but no chart, so the students have to practice using inverse operations more independently. This is a pdf; if you'd like it in Microsoft Word (for revising), let me know.
Here is an idea of how to use it:
When students are solving two-step equations for the first time, they are often unclear on which number to “undo” first. I’ve tried a variety of tricks, but the do/undo chart works very well for many of the kids. They use the scaffold for a couple days, and then they are able to do the same steps in their heads. What I like about the do/undo chart is that it connects solving equations to the Order of Operations.
This is one way to present it:
1. Start off by reviewing order of operations with various problems. Then, write a problem on the board, like 15(3) + 9 = 54. Ask if it is true or false, and to explain the steps of proving that it is true (multiply 15 times three first, then add 9, which does in fact equal 54).
2. Then, write the equation 12x + 8 = 92. Highlight the variable. Discuss whether or not they can use order of operations to solve it (they can’t because they can’t multiply first, because they don’t know the value of x).
3. I explain that when there is a variable in an equation, you can’t follow the order of operations, so we have to treat it more like a mystery. I emphasize that we have to “undo” everything that should be done (or has been done to the variable). It’s a mystery…we’re finding an unknown, so we have to work backwards to discover what the variable’s solution is.
4. I introduce the DO/UNDO T-chart. I say that it is just a tool that helps us keep track of how to undo the equation which solves the mystery. I tell them that we’ll only use it for a day or two, and that after that it’s optional. We’re still using the 12x + 8 = 92 equation.
5. Starting on the DO side, I ask, “If we could, what would we DO first in the equation (using order of operations). “ Or, another way I’ve presented it is “What has been done to the variable?” The answer is “multiply by 12), which is what I write in the chart under the DO part.
6. I ask, “What would be done next?” or… “What has been done to the variable next?” The answer is ADD 8. I write that in the chart under •12.
7. Now that we have written the two steps in the DO side, I explain that we can solve the mystery by UNDOING these steps. I tell them to put their pencils on the Star…because that is where they need to STARt. They look at the DO side, which says +8. I ask how to undo it, and they say -8, which we write down. That is the first step to solve the equation, so I have them do that step on the actual equation. They subtract 8 from both sides (which they already know from solving one-step equations). Then we move to the next DO/UNDO step. Since it says •12 in the DO side, then need to write /12 in the UNDO side, and then divide by 12 on both sides of the equation. They are left with x = 7, which is the solution.
So, the DO/UNDO chart helps students identify what has been done to the variable, and how to undo it. It helps to explain to them that solving an equation is using Order of operations backwards, because we’re undoing a problem…solving a mystery. It also helps if they know that the order of the steps matters. If they had divided by 12 first, and then subtracted 8, they would have gotten an incorrect answer. The chart helps them know which step to do first until it becomes more automatic.
Finally, it’s good to give them equations with simple math, but also harder math. If all the equations are too simple, they hate having to show their steps. One year I gave them really difficult numbers (big numbers or with decimals) but let them use calculators to do the calculations as long as they showed their steps. It worked well…they were so happy that they didn’t have to do the math, they were willing to show their steps. They also couldn’t use mental math to solve the problem before they started.
Here is an advanced activity where students solve two-step equations with integers:
Solving Two Step Equations: A Riddle