Kiss those Math Headaches GOODBYE!

Archive for September 23, 2010

Answer to Problem of the Week

Answer to the 9/17/2010 Problem of the Week

Flying Flora is traveling an average speed of 76.4 miles per hour, rounded to the nearest tenth of a mile per hour.


Let d = distance between Santa Fe and Las Cruces. So 2d = the distance for the round trip. To get the average speed for a trip with two or more “legs,” add up the distances to get total distance, then divide total distance by total time.

For this trip, we get the time for each “leg” by dividing the distance for the leg by the rate for the leg, using the formula, t = d/r. Traveling from Santa Fe to Las Cruces, Flying Flora’s time was d/105; traveling from Las Cruces to Santa Fe, her time was d/60. So the complete formula for average speed is given by:  (2d) ÷ [(d/105) + (d/60)].  Solving this, the d-terms cancel out, and we are find that the expression simplifies to 76.3, with units of miles per hour. So the answer is
76.4 mph.

The people who got this right —

Chris Mark
Irving Lubliner
Jeanine Rose
Sarah Gopher-Stevens

Congratulations to everyone who worked on the problem.

For anyone seeing this for the first time, the problem is this:

Flying Flora, late as usual for her business meeting, speeds from Santa Fe to Las Cruces at 105 mph. After arriving in Las Cruces, she gets an email alerting her that she was caught by a radar gun and received a speeding ticket (she knows the local DA; otherwise she would have been thrown in jail!). Much chastened, Flora drives back from Las Cruces to Santa Fe at just 60 mph. Your task:  Without using a specific value for the distance between the towns, find Flying Flora’s average rate for the round trip. Please show your work and round off your answer to the nearest tenth of a mile per hour.

Facade of the Cathedral of St. Francis in Sant...

St. Francis Cathedral, Santa Fe, Image via Wikipedia

Times Tables, Learning the Threes

What’s more important for early math than knowing the times tables?

Not much, right?

Since the times table facts are so fundamental, and because many students struggle with them, I’d like to share a strategy I came up with today for
learning the 3s. This technique works particularly well with students
who struggle with memorizing apparently random facts. (We know these
facts are not random, but if learned with nothing more than flash cards,
they can appear random.)

The strategy involves three stages, each stage bringing the child closer
to being able to QUICKLY access the desired multiplication facts. Here are the stages, in order they should be taught.

STAGE ONE:  “Patty-Cake Threes”

What I do here amounts to a “patty-cake” approach to learning the threes, which works like this.

The student and I sit facing each other with our hands up. We hit our right hands together and say “one,” then hit our left hands together and say “two.” Then we
hit BOTH HANDS TOGETHER and say, “THREE.” When saying the “one” and “two,” we utter the numbers quietly. But when we say “THREE” and all successive multiples of three, we say these numbers loudly, almost (but
not quite) shouting.

After three, we continue:  “four, five, SIX … seven, eight, NINE, ten,
eleven, TWELVE … ” and so on. So this gives children a fun way to
hear — and get a feel in their body for — the multiples of three, in the proper

Patty Cake

Image by davie_the_amazing via Flickr

STAGE TWO:  “Finger-Drumming”.

After the child has the rhythm of the number three, from the “patty-cake” approach, we do “finger-drumming.” To “finger-drum” the multiples of 3, the child makes a fist with one hand, and shakes it, saying with each shake, “one, two, THREE!” And when saying “THREE,” the child extends one finger from the fist. The child continues: “four, five, SIX,” and at “SIX,” he extends another finger, so he has two fingers out.

Then you ask the child, for example, “What is three times two?” Answer: the number he just said, “six.”

In this way, the child can “finger-drum” out all of the multiples of three. To
reinforce the times tables as you go, ask questions like:  “What is 3 x 4? What is
3 x 5? etc.” Each time you ask, the child must “finger-drum” till s/he gets the
correct answer. This flows very nicely from the “patty-cake” approach as it
builds on the rhythmic feel for counting in threes.

STAGE THREE:   “Finger-Skip-Counting”   The third stage follows “finger-
drumming.” To begin finger-skip-counting, the child must have done enough “finger-drumming” so s/he is quite familiar with the multiples in the correct order.

To “finger-skip-count,” 3 x 4, for example, the child holds out a fist and
runs through the multiples of 3, like this:  “Three (extending one finger), Six (extending two fingers), Nine (extending three), Twelve (extending four fingers).”  You ask, “So what is 3 x 4?” And the child answers:  “3 x 4 equals 12.”

I found it helpful to first just challenge the child with the multiples from 3 x 1 through 3 x 5. Once s/he develops competence there, proceed to “finger-skip-counting” the multiples from 3 x 6 through 3 x 10. Finally do 3 x 11 and 3 x 12.

Put all together, these three stages offer a fun and rhythmic way for children
to learn their multiples of three. I’m curious to find out if I can use a similar
approach for the 4s, and I’ll find out soon.

I can’t be sure, but it seems like children could probably learn their 4s
by jumping rope, or doing other activities with a rhythmic nature.

If any of you have used an approach like this one for learning the times
tables, feel free to share it.