August/September 2014 Newsflash: Tips & Techniques

“Thinking backwards”—starting at the end of the problem and using reverse operations—can help you solve certain problems at different grade levels.

Upper Elementary and Middle School:

A certain number is doubled. That answer is tripled. Finally, that answer is quadrupled and the answer is 60. What is the original number?

Answer:

When we quadruple a number, we multiply it by 4. As division is the inverse of multiplication, 60 ÷ 4 = 15. Tripling a number means “multiply by 3,” so 15 ÷ 3 = 5. Finally, doubling a number means “multiply by 2,” thus, 5 ÷ 2 = 2.5, or 2 ½.

Algebra:

A certain number is quadrupled. 3 is added to the answer. That answer is then tripled. Finally, when that answer is cut in half, the answer is 12.

a)     Write an equation that describes this problem.

b)     What is the original number?

Answer:

a)   We’re looking for a certain (unidentified) number, x. “x, quadrupled” means “4x.” Then, we add 3 to the answer, so 4x + 3. We then triple the quantity “4x + 3”: 3(4x + 3). Finally, we split the quantity 3(4x + 3) in half in order to yield the final answer, 12, so  {3(4x + 3)} /2 = 12.          

b)     To find the original number, we solve for x, which involves “canceling out” the numbers by using inverse operations.

So, (2) {3 (4x + 3)} /2 = 12 (2) multiply both sides by 2…

{3 (4x + 3)}/3 = 24/3 divide both sides by 3…

4x + 3 – 3 = 8 – 3  subtract 3 from both sides, and…

4x /4= 5 /4  divide both sides by 4.

Thus, x = 1.25

April/May 2014 Math Muscle Challenge!

Can you solve this month's Math Muscle Challenge?

Grades 1 – 5: A recycling center pays $1.90 for every pound of aluminum recycled and 90 cents for every pound of plastic recycled. Anthony collected 10 1/2 pounds of plastic bottles and 13 pounds of aluminum cans. If he turns the bottles and cans in to the recycling center, how much money will Anthony get?

Grades 6 and up: The volume of a cube is 8 in³. Amy created a new cube by quadrupling the length of each side of the original cube. How many times larger is the volume of the new cube than the volume of the original cube?

Answers to Last Month’s Math Muscle Challenge:

Grades 1 – 5: 3 ½ miles.

Grades 6 and up: 60 regular subs

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