December 2014 / January 2015 Newsflash: Tips & Techniques

Upper Elementary and Middle School

Find the perimeter of the shape below. (Hint: there is enough information.)

Solution:

The length on the bottom of the rectangle is the same as the length on the top, although the top has been split into two pieces. Similarly the width on the left is 11 inches and the width on right is also 11 inches, again in two pieces. 

Now, the length of the side is given in inches and the length of the bottom is given in feet. Since we have to add to get the perimeter, the Law of SAMEness requires the units to be the same. Let’s change 2 feet into 24 inches (2 x 12). 

Note that it is not necessary to know the exact measurement of the length and width. All that matters is that we know the total length of each. 

With that, 11 in + 11 in + 24 in + 24 in = 70 inches.

Algebra 

If ax + 2a = c and x + 2 = 3, express a in terms of c. 

Solution: 

If ax + 2a = c, then a(x + 2) = c. Now, substitute “3” for “x + 2” and you get 3a = c, 

so a = c/3.

October/November 2014 Newsflash: Tips & Techniques

From time to time, we get lucky and the computation we have to do can be solved using a specific technique or trick that allows us to do effortless mental math.

Level 1:

12 x 125 = _______

The trick to this question is to see that “12” can be broken down into “3 x 4.”

            Think of “12 x 125” as “3 x (4 x 125).”

Now 4 x 125 = 500. (4 x 100 + 4 x 25)

Then, 3 x 500 = 1,500.

Try this (mentally): 8 x 175 = ________.

Level 2:

1/5 + 1/6 = __________

When you add two fractions, and the numerators of the fractions are both 1, the numerator of the answer is the sum of the denominators, and the denominator of the answer is the product of the denominators. In this problem, the numerator is   (5 + 6), and the denominator is (5 x 6).

So, 1/5 + 1/6 = 11/30. [Note: If the denominators are not relatively prime, the answer will have to be reduced.]

Try this (mentally): 1/12 + 1/13 = _____________.

Level 3:

2,000 ÷ 16 = ____________

The trick here is to recognize that 16 is a power of 2, and that powers of 2 make division into “half-of-a-half-of-a…”

In this case, since 16 is “2 to the fourth power,” we simply cut 2,000 in half 4 times.

So,

            Half of 2,000 is 1,000,

                        Half of 1,000 is 500,

                                    Half of 500 is 250, and finally,

                                                Half of 250 is 125.

So, 2,000 ÷ 16 = 125.

Try this (mentally): 3,000 ÷ 8 = ____________.

Note that there is more than one trick for each of these questions. See if you can find others!

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

Friday Afternoon Cartoon

Want to make change? This cartoon illustrates how making change for a dollar can be an exercise in computation. Try it with your kids, but aim for 5 different ways to make change, not 293!