need help teaching a pre-algera concept

[kfeusse]

my daughter has always had trouble reading and understanding word problems.  (I did too as a kid)...so now that we are into pre algebra and she is being asked to write equations from a word problem and she is nearly throwing the book in frustration.  Please give me something that might help her.  Here is an example of what she has to do:

 

The age of Sarah's mother is 3 years less than 3 times Sarah's age (s).  If Sarah's mother is 42, how old is Sarah?

 

then she needs to come up with: 3s-3=42  (which looks simple enough...but all of those words in the problem completely confuse my daughter.)

 

OR

 

an atom of zinc has 30 electrons.  An atom of oxygen has 2 more than 1/5 as many electrons as an aton of zinc.  How many electrons does an atom of oxygen have?

 

she needs to come up with 30/5 +2 =e.

 

Once she has the equatiion she has no problem solving it...it's coming up with the equation that she can't get.

 

PLEASE HELP.  I know it's only going to get harder from here.

 

 

[justasque]

Encourage her to write what she knows, define her variables, and break it down into small pieces.
Example:

The age of Sarah's mother is 3 years less than 3 times Sarah's age (s).  If Sarah's mother is 42, how old is Sarah?

Sarah's mother's age = 42
Sarah's age = s

3 times Sarah's age = 3s

Sarah's mother's age = (3 times Sarah's age) - 3

Sarah's mother's age = 3s - 3

42 = 3s - 3

 

Another approach is to use the sentence:

The age of Sarah's mother is 3 years less than 3 times Sarah's age (s).  If Sarah's mother is 42, how old is Sarah?

 

(The age of Sarah's mother) = 3 years less than 3 times Sarah's age (s).  If Sarah's mother is 42, how old is Sarah?

 

(The age of Sarah's mother) = 3 years less than (3 times Sarah's age (s)).  If Sarah's mother is 42, how old is Sarah?

 

(The age of Sarah's mother) = (3 times Sarah's age (s)) - 3 years.  If Sarah's mother is 42, how old is Sarah?

 

s = Sarah's age

 

(The age of Sarah's mother) = (3s) - 3 years.  If Sarah's mother is 42, how old is Sarah?

 

42 = 3s - 3

It helps to explain that this is NOT the kind of problem that she is expected to just look at and know how to proceed step by step to the answer.  This kind of problem has to be played with and teased apart, and there may be dead ends, and that is both OK and perfectly normal for higher level math problems.  The goal is to use various tools to explore the problem.  Tools include:
Draw a picture, make a table, write an equation, solve a simpler problem, act it out, use manipulatives, etc.

 

It also helps to give her some ideas of what she can do in the beginning - the basic approach even *before* she knows where she's going to go with it.  

--Read the problem to get the overall gist of it.

--Read it again to get a deeper sense.

--Read it again, and see if you can break it down into understandable parts.

--Write down what you know (Sarah's mom's age = 42)

--Define your variables (Sarah's age = s)

--Draw a diagram or picture if the problem lends itself to that.

Once she's worked through these steps, she'll be in a better place to begin to tackle solving the problem.

Let her know that even college-age math majors use this kind of approach to solve problems.  It's not knowing instantly how to solve a problem that makes one good at math.  Complex problems just can't be solved that way.  It's having these skills/tools and routines/habits, and working through them, that allow math majors to solve difficult problems.

[AK_Mom4]

DD13 struggled with word problems for a while as well.  We made up a "cheat sheet" of words to look for when building equations:

 

"IS" means equals

"Less Than" means subtraction

"More Than" means addition

"of" means multiplication when it's next to a number

"Per" means division when next to a number (Per Cent just means divided by 100)

 

Having a secret decoder ring she could look at when working the problems seemed to help her get started.

 

[letsplaymath]

When students have trouble with word problems, it is almost always in the translation stage. And once they start thinking "word problems are hard", their brain freezes, which makes it worse.

 

BUT your daughter has been talking and reading and making sense of words for most of her life, right? So that means she CAN DO THIS!

 

She just has to fight away the brain freeze (acknowledge the panic, shut her eyes, take a deep breath, relax) and think about what the words mean. Not what they mean in math, but what they mean as words. Just like reading a story or following the plot of a movie.

 

For example...

The age of Sarah's mother is 3 years less than 3 times Sarah's age (s).  If Sarah's mother is 42, how old is Sarah?

 

Part of the trouble with language is that it takes so many words to describe a single thing. And we like variety in our language, so writers tend to change the words they use. They don't repeat exactly the same words each time they mention the same thing. In the first sentence, the writer is talking about ages, but then in the second sentence, he switches to using the person's name when he still really means the age.

 

So, as a person who knows how to read, your daughter can look over a paragraph and understand it. Just read it as a story, not as math.

 

In five words or less, what is the writer talking about?

People's ages.

How many people?

Two: Sarah and her mom.

 

Okay, so let's simplify the problem by editing the writer's wordiness and making it more consistent. Focus on what our story is all about, the people and their ages. Ignore all the minor details, like the numbers, for now:

 

[Mom's age] is 3 years less than 3 times [sarah's age].  [Mom's age] is 42, how old is [sarah's age]?

 

Now, every declarative sentence, or every phrase that could stand as a sentence, is a statement about what is true in our story.

 

As a person who knows how to read, your daughter can identify sentences. How many declarative sentences, or phrases that could stand alone, do we have?

 

[Mom's age] is 3 years less than 3 times [sarah's age]. 

[Mom's age] is 42.

 

And your daughter can draw conclusions. Anyone who reads a book needs to sometimes read between the lines to understand what the writer means. Or what about movies: the camera zooms in on a shadowy hand with a knife, and then zooms out to show a pedestrian walking toward a dark alley. Your mind jumps from that pattern to the obvious conclusion, right?

 

Well, there is a pattern in two of our sentences above that should jump out just as obviously, now that we've edited away the excess words:

[Mom's age] is 3 years less than 3 times [sarah's age]. 

[Mom's age] is 42.

 

Conclusion:

42 is 3 years less than 3 times [sarah's age].

 

And because ALL of the things we are talking about are age years, we don't really need to keep the word "years." We can edit it out:

42 is 3 less than 3 times [sarah's age].

 

At this point, you should feel encouraged. This looks so much simpler than what we started with, doesn't it? We have mostly numbers and words that we know relate to things we can do with math. The editing is done, and now it's time to translate.

 

Let's start by translating the easiest part, the verb:

42 = 3 less than 3 times [sarah's age].

 

The next easiest to translate is the word "times" because when something is three times as big, we know that means it is multiplied by three.

42 = 3 less than 3 X [sarah's age].

 

And we can put in the "s" for Sarah's age, leaving out the times symbol, as is traditional in algebra:

42 = 3 less than 3s

 

Now the absolute hardest part: the phrase "less than." Translating subtraction is always tricky, because direction makes a difference.

 

This is where someone who is blindly using key words will get confused, but a reader who is looking at the words the way they would read a book---that is, looking at the meaning of the words as language, not just as "math"---can get it right.

 

What does it mean for one number to be three less than another number? Which of these subtractions matches the meaning of the words in our story:

42 = 3 - 3s

or

42 = 3s - 3

Our story says 42 is three less than something. So if we have that something and take away three from it, we should get 42, right?

 

Final translated equation: 42 = 3s - 3.

 

If you can read a book or follow the plot of a movie, you can also read and follow the plot of a math word problem.

 

It takes time at first, but like any task, it will gradually come easier:

• Read the words like a real story, as if they have meaning. Don't think of them as math.
• Start by getting the big picture. Say what the story is about in five words or less.
• Edit the sentences of the story to make them simpler.
• Look for patterns that let you draw conclusions.
• Keep everything in words, so you can use what you know about stories and plots to help you figure it out.
• Finally, when you've edited it down as simple as possible, start translating the words into symbols. Begin with whichever parts look easiest.
• But always remember to focus on meaning---don't get fooled by tricky words like "less than"!

[letsplaymath]

Let's do another example:

 

An atom of zinc has 30 electrons.  An atom of oxygen has 2 more than 1/5 as many electrons as an atom of zinc.  How many electrons does an atom of oxygen have?

 

What is the writer talking about?

Chemistry.

What atoms are made of.

Electrons in zinc and oxygen. <--(This is the most helpful summary.)

 

What declarative sentences (facts about our story) do we have?

An atom of zinc has 30 electrons. 

An atom of oxygen has 2 more than 1/5 as many electrons as an atom of zinc. 

 

Edit to make the sentences simpler.

Since we are talking about the number of electrons in zinc and oxygen, we will label them [zinc electrons] and [oxygen electrons].

 

"An atom of zinc has 30 electrons."

becomes:

[Zinc electrons] are 30.

 

"An atom of oxygen has 2 more than 1/5 as many electrons as an atom of zinc."

becomes:

[Oxygen electrons] are 2 more than 1/5 of [zinc electrons].
 

Look for patterns that let you draw conclusions.

[Zinc electrons] are 30.

[Oxygen electrons] are 2 more than 1/5 of [zinc electrons].

Conclusion:

[Oxygen electrons] are 2 more than 1/5 of 30.

 

Translate the words into symbols.

First, the verb, because it's easiest. And we'll put in "e" to stand for the number of oxygen electrons, since we're doing algebra:

e = 2 more than 1/5 of 30

 

"More than" is easy, because direction doesn't matter when you add:

e = 2 + 1/5 of 30

 

Now the hard part, because in math we always save the hardest part for last. What is "1/5 of 30"? Remember, you know how to talk and read. You know what words mean. What does it mean when someone says they have a fifth of something? Well, that means the something has been split into five equal parts, right? And what math operation splits something into equal parts?

 

e = 2 + 1/5 of 30

means:

e = 2 + 30 ÷ 5

But in algebra, we write division as fractions.

 

Final translation: e = 2 + 30/5

[kiwik]

The first one basically says 'if Sarah's mum was three years older she would be three times Sarah's age'. She is 42 so if she were three years older she would be 45 so Sarah is 15.

 

42+3 = 3S

[letsplaymath]

The first one basically says 'if Sarah's mum was three years older she would be three times Sarah's age'. She is 42 so if she were three years older she would be 45 so Sarah is 15.

 

42+3 = 3S

 

Yep. With beginner algebra problems, it can be much quicker to solve using common sense and a bit of arithmetic. Figuring out how to translate the relationship into algebra is the hard part.

[kiwik]

Yes I guess she does need to learn the algebra but applying a bit of logic does help you notice if your answer is stupid. I calculated the volume of a cylinder last week and I could see that my answer was out but a lot of people accept whatever answer their calculator spits out.

[justasque]

Yep. With beginner algebra problems, it can be much quicker to solve using common sense and a bit of arithmetic. Figuring out how to translate the relationship into algebra is the hard part.

 

Yes!  I usually explain to the savvy student who would rather do the problem the way they already know than do it the "new" way (in this case using algebra) that the textbook writer gave them a problem they already know how to do on purpose, so that they would be able to see how the new technique works on a problem they already understand.  I explain that if they just use the old technique, they may get the answer, but they are missing the opportunity to learn the new technique (which was the intent of doing this lesson in the first place), and that later there will be problems where the old technique won't work or (especially in the case where old technique = doing it in your head) will be too unwieldy to use.  If they don't use the new technique now, they will be stuck learning it later, when they won't have the advantage of being able to "check" the problem using the old technique.  (It helps to illustrate this by creating a big complicated-looking problem on the board with lots of x's and y's and parenthesis and square root symbols and exponents and other complicated stuff, and, with lots of flourish, solving it using the new technique.  And then explaining that the student will be able to do this too, easily, in a couple years, IF they learn the technique we are working on.)

 

So ideally they'll get both the non-algebra way to help them understand different ways to look at the problem, AND (since it's a pre-algebra class) the general approaches to breaking down a word problem.  More tools in the toolbox are always a good thing!

 

(And letsplaymath - I can't like your post(s) enough!  Thanks so much for writing all that out!!!)

[kfeusse]

Do any of you want to come and teach my daughter math?  :)

 

I can't thank you all enough for the time it took you to write out all that you did.  that was amazing!!!!  It was all very helpful...to me...at least.  But like someone said...she has brain freeze as soon as she sees any word problem on the page...much less one that requires the new skill of algebra. 

 

I will keep plugging away.

 

thanks again.