Kiss those Math Headaches GOODBYE!

Archive for September, 2010

Not all variables are created equal


Are all variables the same?

Does every variable serve the same purpose?

When you think about it, you’ll see that the answer is “no.” Variables serve different purposes. When we explain this to students, we help them understand how variables work. Explaining this helps students understand how algebra “works.” You’ll see what I mean in a moment.

Consider the famous slope-intercept equation:  y = mx + b

A student recently asked me:  Are the  x and y variables the same as the m and b variables? What a great opportunity to explain something important!

I explained that the x and y variables serve completely different purposes than  the m and b variables. Here’s how.

The variables m and b are what I call “identifier” variables. By which I mean that they help us identify a specific line. To explain that, I asked the student a set of questions about something everyone understands — home addresses.

What would happen, I asked, if someone wanted to know where I live, and I told him that I live at 942? The student replied that this would not be enough info.

Then I asked, what if I told this person only that I live on Vuelta del Sur (a street name where I live in Santa Fe, NM)? Again the student said that this would not be enough info.

But what if I told this person that I live at 942 Vuelta del Sur. This, the student realized, would be enough information to enable someone to find my house. (All they have to do is Google me, and they’ll have my house AND directions!)

I pointed out that a similar situation applies to lines.

If I have a specific line in mind, and I want someone else to know the line I’m thinking of, is it enough to give this person just the line’s slope? No, for it could be any line with this slope, of which there are infinitely many parallel lines. What if I don’t give the slope but I do give the line’s y-intercept? Still not enough, as there are infinitely many lines that run through this y-intercept. But what if I tell the person both the slope and the y-intercept. Aha! The student could see — through drawings I made of this situation on a coordinate plane — that when you provide both slope and the y-intercept, there is one and only one line that could be indicated.

 

Three lines — the red and blue lines have the ...

Red & blue lines have same slope, so slope alone does not indicate a specific line; Red and green lines have same y-intercept, so y-intercept alone does not identify a specific line.

 

I explained that variables like m and b, which help identify a specific line, are “identifier” variables; their job is to identify a specific line. If your students are more advanced, you can explain that there are other identifier variables in different kinds of equations. For example, in the equation of a parabola:   y – k  = a(x – h)^2, the identifier variables would be the variables a, h, and k.

But what about variables like x and y? What do they do? What is their purpose?

These variables, I explained, have a completely different purpose. I call variables like x and y “ordered-pair generators.”

To explain this, I show students a simple linear equation like  y = 2x, and demonstrate how, using a “T-table,” you can use this equation to generate as many ordered pairs as you’d like, ordered pairs like (0,0), (1,2), (2,4), (3,6), etc. Point out that you can keep going and going. And then explain that the purpose of the x and y variables is to generate the infinitely many points that make up the line.

So the m and b variables tell us where the line is, and the x and y variables allow us to find the infinitely many actual points on the line. The two sets of variables, while different in purpose, work together toward a common goal:  to give us the equation of a line.

There are other purposes that variables serve, of course. And I’ll probably describe some of the other purposes in future posts. But the main point is that it helps students to recognize that variables do serve different purposes. Armed with that understanding, they can make much more sense of algebra’s formulas and equations.

How to Multiply Even Numbers by 5 — FAST!


Time for a math trick …

Q:  How do you multiply an even number by 5 in lightning speed?

A:  Divide the number by 2, then tack on a “0.”

Example:   5 x 24

Divide 24 by 2 to get 12.

Tack a “0” onto 12 to get 120. Presto, nothing up your sleeve. It’s that easy.

Why does it work? Hint: Think about how we multiply by 10. Then think about how multiplying by 5 compares to multiplying by 10.

Rotated version of File:Symbol support2 vote.svg.

Image via Wikipedia


Try these for fun (answers at bottom of post):

a)  5  x  16

b)  5  x  8

c)  5  x  28

d)  5  x  64

e)  5  x  142

f)  5  x  2,468

g)  5  x  6,042

h)  5  x  86,432

j)  5  x  888,888


Answers:

a)  5  x  16 = 80

b)  5  x  8 = 40

c)  5  x  28 = 140

d)  5  x  64 = 320

e)  5  x  142 = 710

f)  5  x  2,468 = 12,340

g)  5  x  6,042 = 30,210

h)  5  x  86,432 = 432,160

j)  5 x 888,888 = 4,444,440

k)  5  x  2,486,248 = 12,431,240

Problem of the Week – Answer


Answer to the 9/24/2010 Problem of the Week

The problem —

At Gamesville High, students love their clubs. While 20% of the children in the Hex Club are also members of the Backgammon Club, 80% of children in the Backgammon Club are also members of the Hex Club. The Backgammon Club has 35 children. The question:  how many children are in the Hex Club?

Solution:  Here is the solution, provided by the only person who got it right (name below).

20%=0.2, 80%= 0.8.
Let X = the number of students in the Hex Club
Let Y = the number of students in the Backgammon Club
So:
0.2X=0.8 Y because the 20% and 80% are the same children, the same number of people. It just looks different because the percentages show a relationship between the total numbers in each club.

Speaking of total numbers, the problem tells us how many are in the Backgammon Club: 35. So:
Y=35

We now have two equations. We can substitute the value of Y from the second for the Y in the first and solve for X.
0.2X=0.8 (35)
0.2X=28
X=140
There are 140 students in the Hex Club

Sharron Herring

Sharron has answered my problems many times in the past. So thanks for sharing that solution, Sharon.

Next problem will be posted this Friday, Oct. 1.

Problem of the Week — 9/24/2010


Problem of the Week —  9/24/2010

At Gamesville High, students love their clubs. While 20% of the children in the Hex Club are also members of the Backgammon Club, 80% of children in the Backgammon Club are also members of the Hex Club. The Backgammon Club has 35 children. The question:  how many children are in the Hex Club?

To get the honor of being put into the Winner’s Circle, you need to get the correct answer and show how you arrived at it.

Please write your answers as comments on the blog post. Or alternatively, you can send it as an email to me:  josh@SingingTurtle.com

I will post the answer and name the five who make it to the Winner’s Circle on Monday.

P.S.:   Hex is a great board game, invented by two mathematicians. If you’d like to read a post about it, go here:

https://mathchat.wordpress.com/2010/01/27/play-a-game-meet-john-nash/

Screenshot from the program GNU Backgammon (Fr...

Backgammon, Image via Wikipedia

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.

Solution:

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
order.

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.

Finding domain and range — with color!


Have you ever noticed that a lot of students struggle with the idea of domain and range? This concept, taught mostly in Algebra 2,  often confuses students to the point where they cannot even identify the domain and range of a simple, continuous function.

I don’t really understand why students struggle with this concept, but I recently found a way of showing the idea that makes it considerably easier — using color to mark up a function.

Here’s an example of a problem where students need to figure out the domain and range by looking at a graph, like this:

What I have students do is use two colors to sort of “box in” the function. With one color, green in this case, students mark the left bound and right bound of the function by drawing vertical lines. And with another color, red, students mark the lower bound and upper bound by  drawing horizontal lines. I have students write in the phrases:  left bound, right bound, lower bound, and upper bound, like this:

Finally I ask students to figure out the domain and range by writing three-part inequalities for x and y, respectively, like this:


I’ve used this approach with a number of students, and so far no one has been unable to find the domain and range when using it. So it appears to be a winner. Try it yourself, either as you teach a concept, or as you re-teach it to those who are struggling.

Tuesday “Teasers”


A pair of Brain Teasers for Tuesday. If you’re a teacher, consider putting these up as a warm-up at the start of class.

Teaser # 1)  Poor Mr. and Mrs. Household. They have 10 boys, and each of their sons has one sister. How many children do Mr. and Mrs. Household have altogether?

Teaser #2)  While keeping track of their many children’s height, Mr. and Mrs. Household notice that one of their sons has an unusual characteristic — his height doubles every year until he reaches age 13, at which time he abruptly stops growing. How many years did it take this boy to reach one quarter of his maximum height?

Answer #1)   Eleven children, for only one girl is required in order for each boy to have one sister.

Answer #2)   The boy would reach 1/4 of his maximum height when he has his 11th birthday. Since his height doubles every year, he had to be half of his maximum height on his 12th birthday. And so he had to be half of that — or 1/4 of his final height — on his 11th birthday.

Rubik's Cube

Rubik's Cube

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Using Color to Show Perimeter


Many things look better in color, right?

So why should that be any different  in math?

I’ve found that taking a “colorful approach” to math not only makes mathematical objects look more interesting and pleasurable, it can also make mathematical concepts more clear.

Here’s an example from something I did today — I used color to show a shortcut for finding the perimeter of rectangular-ish objects.

I was tutoring a boy who had to find the perimeter of this figure:

Right object, find perimeter

This student did not see that there is a short-cut that could help him find the perimeter. I wanted to make this clear, so I reached for my color pencils and colorized both the left vertical segment and the two right vertical segments. My goal was to help the student see that the sum of the two right vertical segments equals the long left vertical segment.

The student realized this after I colorized it. Then I used a different color, red, to show that the sum of the two horizontal segments on top equals the longer horizontal bottom segment, like this:

At this point I felt that the student was ready to see the math that relates to the whole figure, so I wrote the math, using color to relate the numbers to the colors of the sides of the figure, like this:


At this point the student was able to see the shortcut in this kind of problem, which together we wrote as follows:


This is a fairly basic example of how color can, quickly and effectively, illustrate math concepts. Feel free to share examples of how you use color in your math lessons. I’m curious to learn (and share) a variety of ways, for I see that color has great potential.

Colouring pencils

Image via Wikipedia

Problem of the Week, 9/17/10: “Flying Flora”


Flying Flora, late as usual for her business meeting, speeds from Santa Fe to Las Cruces at 105 mph.

Off the Plaza in Santa Fe, New Mexico

Santa Fe Plaza

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.

Las Cruces, New Mexico

Organ Mountains, Las Cruces

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. Post your answers on Facebook, or send to:  josh@SingingTurtle.com All entries must be received by 10 p.m. this Wednesday, Sept. 22, to qualify for getting your name posted among the five first to get it right, on Friday.