Tips and Techniques: Transposing

Transposing is an important technique for students to master as they progress from Algebra I to Algebra II and beyond. It’s the process of moving quantities across the equal sign in an equation to make the problem solving process more efficient. Here’s how it works.

 Solving an equation means isolating the variable on one side of the equation, either the right side or the left. Generally it is better to isolate the variable on the left side of the equal sign because we read from left to right.

Beginning algebra students are taught the Law of Equations:  whatever you do on one side of an equation, you must do on the other side as well. 

Using the Law of Equations, to solve 2x – 3 = 13, we can add 3 to both sides, giving us

2x – 3 + 3 = 13 + 3 ===> 2x = 16

Then, we divide both sides by 2 to find that x = 8.

To solve 2x – 3 = 13 by transposing, we do the following:

Instead of adding 3 to both sides, we can transpose the 3 that’s already there by

moving it from the left side, over the equal sign, to the right side, and changing its sign from “–” to “+.”

Now we have  2x = 13 + 3, which becomes 2x = 16, the same answer we got using the Law of Equations. Again, we find that x = 8.

The beauty of this process is that it is visually easier to see how the parts of the equation change as we solve it. Instead of writing the transposed quantity twice (once on the left and once on the right), we literally pick up the quantity and move it to the other side. The price we have to pay for doing this is changing the sign of the quantity being moved. 

Let’s solve 3y + 2x – 3 = 7 for y.

Since we want to isolate y, we can transpose 2x and – 3.

This gives us y = –2x + 7 + 3. Simplifying, we get y = –2x + 10.

If we used the Law of Equations, we would have to write out

3y + 2x – 3 = 7 ===> 3y + 2x – 2x – 3 + 3 = 7 – 2x + 3

While simplifying the above does give us y = –2x + 10, it comes with a lot more work.

Note that most classroom teachers will not allow transposing until students learn to use the Law of Equations.

Try these:

1. Solve for x: 4x + 9 = 49
2. Solve for y: 2y + 7 – 3 = 24
3. Solve for y: 3x + 9y – 7 = 49
4. Solve for a: 3b – 4a – 6 + 4 = 49
5. Solve for a: 2b + 4a + 5 – 2 = 42 – 2a

Master Fractions and Set the Stage for Success, As Featured in The Wall Street Journal

Algebra Readiness begins in elementary school. Three essential topics in elementary school mathematics are 1) numerical fluency, which includes the effortless recall of addition facts and the times tables stemming from mental math strategies for computation, 2) foundational understanding of the nature fractions, and 3) the basics of problem solving. Mathnasium was recently featured in The Wall Street Journal and highlights our unique and effective approach to teaching fractions. Read the full article here.

As students move into middle school, work with fractions expands to embrace rational numbers (common fractions, decimal fractions, percents, ratio and proportion, and negative numbers). At the same time, student’s problem solving horizons broaden to include multi–step word problems, both routine and non–routine.

Successful mastery of these elements sets the stage for success in Algebra, the rest of high school math, and college. If any of these areas are under developed, the road ahead is fraught with danger.

Visit a Mathnasium Learning Center and see where your child stands.

- Larry Martinek

Is your child being left behind in Math?

When students can do these questions (at their grade level and below), mentally — without using pencil and paper — they are likely doing well. For students who can't handle these questions, this is a warning sign. Very often they need help outside of the classroom. Students who can do the questions at and above their grade level may need a more challenging experience. 

See how well your child answers these questions. The results may surprise you!

First Grade:	11 + 12 = ___
Second Grade:	1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = ___
Third Grade:	How much is 99 plus 99 plus 99?
Fourth Grade:	Count by 1¾ from 0 to 7.
Fifth Grade:	Which is greatest: 17/18, 23/30, or 18/19? Explain how you got your answer.
Sixth Grade:	Halfway through the second quarter, how much of the game is left?
Seventh Grade:	How much is 6½% of 250?
Pre-Algebra:	On a certain map, 6 inches represent 25 miles. How many miles do 15 inches represent?
Algebra:	When you take 3 away from twice a number, the answer is 8. What is the number?
Geometry:	What is the Absolute Value of the point (3, 4)?

For the answers, click here or visit our site!