Algebra/word problems

My sister gave me a tip yesterday that I have yet to try but it makes alot of sense.

Have your child replace the names of the people in the word problems with names of people that he knows. It aids in visualization.

Her son uses this technique and scores in the 99% in math.

I see this as the difficulty: “..he was having no problems until word problems were introduced. This unit…” It looks like whatever series he has has not been using word problems all along so students could build up incrementally strategies needed to solve them. Instead, they have lumped all the word problems into one unit and I bet when that is over they will seldom reappear. (In which case you can breathe a sigh of relief.)

For the immediate problem, try Linda F.’s strategy. But over the long haul kids need to know how to apply math and that is what the word problems are about. My ds also has capd (reasonably remediated) and word problems were very difficult when he was much younger. I worked extensively with him using Saxon math, which is excellent for this type of child. If I were you I would get a hold of the Saxon Algebra book and and do two or three of their word problems every day with him. We’re not up to that Saxon level yet, but if Algebra works the same as the other books in the series, the problems will start out as one step and gradually go to two steps and more. CAPD kids will have a real problem with three step word problems if they haven’t already had a lot of practice—to the point of overlearning—one and then two step word problems.

Thanks Linda and Mariedc, I will try these things. This is his first year in a non-ld math class and he has done well until this point. He says he did not have word problems in his LD math classes, so the terms are new for him. Thanks.

Since they haven’t had him doing word problems up to now (which I find shocking), I wouldn’t do the Saxon until after you’d gone through the It’s Elementary book pattim sugested on the other board. This really does have to be taught from the bottom up.

My son also has CAPD. I was wondering about the relationship between word problems and CAPD. Is it the direction following?

Beth

Not just that necessarily. Most CAPD kids have language difficulties that make word problems difficult. My ds needed help initially with some of the abstract words used in word problems, like “between.” Fortunately, the number of these is limited and can be taught using manipulatives. Then there is the syntax typically used in math problems, which can differ from straight prose, and difficulty visualizing the set up of the problem. Manipulatives or just drawing pictures on the paper helps here. Also sequencing of what happens first, second etc. (and what you need to do first and second, etc.) and ferreting out what information you need to answer the question and what is extraneous.

I thought Saxon was particularly good for teaching kids with language problems how to do word problems. In the lower grades, there is a lot of use of stories with manipulatives (the little bears are very cute) and the kids are prompted to set up a number sentences for each story (e.g. 4+5—but prior to using numbers they draw pictures). They have exercises for teaching concepts like between, and have an uncanny knack for doing so for the very words that tripped up my ds. There is a lot of overlearning of one step problems before introducing two step problems. Also, word problems are incorporated into every lesson. This is in huge contrast to the math texts my ds had in the lower grades, where there was lot of work on nonword problems for each unit with only two or three word problems stuck on the end, some of which had three steps. The sum total of word problems my ds had from school throughout fourth grade was not more than 20. In Saxon it would have been at least two every day.

Some of the notes in this thread mention the “between concept”. Could somebody give and example of a problem the uses the “between concept”?

My son’s fourth grade book is much the same. He always gets the word problems wrong but most of the time they are pretty hard. I think I need to do something systematically since he is clearly not learning how to do them at all at school.

Beth

The towns of Alpha and Beta are 30 miles apart. The town of Gamma is between Alpha and Beta. The distance from Gamma to Alpha is four times as far as the distance from Gamma to Beta. How far is Gamma from each other town?

Note: there are several other usages of “between” including sharing, number order, and others. The above is the most common I can think of.

I keep hearing this one over and over — and I keep hearing the same dead-end quick fix suggestions over and over too. So please excuse me if I sometimes sound frustrated!!

I am a math major, was in grad school math and part-time college math teaching until a thyroid problem floored me. So yes, I have done this before and this is from a long-term perspective, what actually succeeds to get you out of high school and through college.

Samantha is right that you need a strategy. Thanks.

One thing she did not mention is so-called “key words”. Some people will tell you that this is the solution. In general, they are *wrong*. This strategy will get you a bare pass, maybe, inthe first course, and leave you lost and floundering next year, with no foundation to fall back on — a bad dead end.

She is also right that so-called “word problems” are it. This is what math is all about.
I keep asking people two questions:
When was the last time you were in a business and people were running around frantically trying to fill in the answers on worksheets? Never, because worksheets aren’t what what you do in the real world; you solve real problems.
And how are problems in the real world going to be presented to you? In cartoons? No, in the real world people come up to you and explain problems in words. Or they send memos and explain problems in written words — like what you just did posting here, right?
So the phrase “word problems” is completely redundant. It’s like saying “water swimming”. What else are you going to do?

This isn’t to say that it is always easy —it isn’t. But it’s the core and purpose of the program, what you are there to learn, so it is worth taking a lot of time and effort over.

Samantha suggests the “four-step” problem-solving approach. That’s OK, but I find it sometimes not clear enough. Working with my tutoring students, I’ve come up with seven steps:

(1) *Summarize*
What facts are you given? List them in note form, in words.
What *exactly* are you asked to find? Note it down.
(2) *Visualize*
Draw a diagram
Draw a chart
Draw a graph
Make a table of values — experiment with different values and see what patterns you get
(3) *Relate*
Do you know any formulas, like d = rt? A(circle) = pi r squared? and so on?
Are you *given* any relationships, like John’s age = twice Sue’s age?
Can you *see* any relationships from your diagrams?
Did you *find* any relationships when you charted values? What system did you use to calculate values?
Do you *recognize* a familiar problem pattern that may be in an example in your text or notes?
(4) *Name*
Give SPECIFIC names to your variables. WRITE the expression “Let …”
Good: Let J = John’s age
NOT J = John (might be his weight, his height, his grade on SAT’s . . )
** Why this is a problem: the classic “mixture problem” where you mix say chlorine and water to purify, or peanuts and cashews to sell a mix; there is the *quantity* c = gallons of chlorine bleach, and then there is the *value* 0.25c = amount of pure chloride in the bleach; p = pounds or *quantity* of peanuts, but 3p = *value* of peanuts @ $3.00 a pound. You have to be absolutely clear what your variable means, or else you will have confusion coming back to bite you later.
(5) *Equation*
DO NOT write an equation without doing the first four steps. In a simple problem they may take one minute and three lines, but DO them. When you hit the hard stuff, you need the skills you were supposed to be developing while you did the easier stuff; if you skip the learning process on the practice problems, you will find out. The hard way.
With the relationships and word equations from (4) and the variables from (5), the equations should most of the time jump out at you. With a little practice, they seem fast and easy. The important point is to work through the *process* so that it *does* become easy.
(6) *Solve*
Use proper algebra techniques; don’t guess and don’t skip steps.
Sometimes in the value-charting method the correct answer falls in your lap; tha’s OK if it happens, but do the algebra anyway. The next time the answerr may not fall in your lap, and you’re here to learn the *method*. Also many problems have two or more answers, and you may get a surprise.
(7) *Check*
Any real algebra problem involves enough steps that anyone can slip up. *Everyone* makes mistakes; good mathematicians are the ones who have learned to fix them.
Check in your *first* equation, and go back and see if your answer *makes sense* in the real situation of the *original* problem. Reality check is fundamental.

At first this looks really long; it isn’t, I swear. Doing this, I can solve any high school/junior college problem as fast as I can write it. From Algebra 1 to Calculus 3, this method breaks any problem.
All the other people trying to “save time” are staring at a blank paper with blood coming out of their foreheads. Or scribbling and erasing and scribbling again. Or getting grades of F because their work was certainly fast, but it was wrong.
I strongly encourage students to *write*. For some reason, elementary teachers (most of whom became elementary teachers because they did *not* want to take math classes beyond the minimum) have the peculiar idea tha writing math down is bad and that the ideal would be to get the answer by divine inspiration and write it down in solitary splendour. This is the opposite of what real mathematicians do; real mathematicians (and scientists and engineers and technical workers) write everything down so they can keep track of all the details and check back for errors, same as any other responsible adult. Get your student to *write* the summary, *draw* the visualization, *write* the word equations for the relationships, *write* the variable naming, *write* the equations and the steps in the solution, and *write* the check (a few lines, but saves 10 to 20% on your grade).
I also have a campaign against pencils and erasing, This is a math class, not an erasing class. If you watch a weak math student, you will see that 3/4 or more of his time is spent erasing. If he just draws a line through the errors and moves on and does more *math*, instead of erasing, he will instantly be working three to four times as fast. And if, as often happens, he realizes that he was not wrong after all, he can go back to the part that was OK and continue on. Use a pen and do math, forget kindergarten training.

This is general advice; please get back to me if you have any specific questions.

Please note: the above is copyright; you are welcome to use it, but please do not re-publish wothout my permission. Thank you.

Victoria, In first grade my ds could not for the life of him even begin to know how to answer the question “Name a number between 20 and 30.” After a few days of doing the fork, knife, and spoon practices suggested in Saxon for the “between” concept he was able to answer the question effortlessly.

My sister gave me a tip yesterday that I have yet to try but it makes alot of sense.

Have your child replace the names of the people in the word problems with names of people that he knows. It aids in visualization.

Her son uses this technique and scores in the 99% in math.

I see this as the difficulty: “..he was having no problems until word problems were introduced. This unit…” It looks like whatever series he has has not been using word problems all along so students could build up incrementally strategies needed to solve them. Instead, they have lumped all the word problems into one unit and I bet when that is over they will seldom reappear. (In which case you can breathe a sigh of relief.)

For the immediate problem, try Linda F.’s strategy. But over the long haul kids need to know how to apply math and that is what the word problems are about. My ds also has capd (reasonably remediated) and word problems were very difficult when he was much younger. I worked extensively with him using Saxon math, which is excellent for this type of child. If I were you I would get a hold of the Saxon Algebra book and and do two or three of their word problems every day with him. We’re not up to that Saxon level yet, but if Algebra works the same as the other books in the series, the problems will start out as one step and gradually go to two steps and more. CAPD kids will have a real problem with three step word problems if they haven’t already had a lot of practice—to the point of overlearning—one and then two step word problems.

Thanks Linda and Mariedc, I will try these things. This is his first year in a non-ld math class and he has done well until this point. He says he did not have word problems in his LD math classes, so the terms are new for him. Thanks.

Since they haven’t had him doing word problems up to now (which I find shocking), I wouldn’t do the Saxon until after you’d gone through the It’s Elementary book pattim sugested on the other board. This really does have to be taught from the bottom up.

My son also has CAPD. I was wondering about the relationship between word problems and CAPD. Is it the direction following?

Beth

Not just that necessarily. Most CAPD kids have language difficulties that make word problems difficult. My ds needed help initially with some of the abstract words used in word problems, like “between.” Fortunately, the number of these is limited and can be taught using manipulatives. Then there is the syntax typically used in math problems, which can differ from straight prose, and difficulty visualizing the set up of the problem. Manipulatives or just drawing pictures on the paper helps here. Also sequencing of what happens first, second etc. (and what you need to do first and second, etc.) and ferreting out what information you need to answer the question and what is extraneous.

I thought Saxon was particularly good for teaching kids with language problems how to do word problems. In the lower grades, there is a lot of use of stories with manipulatives (the little bears are very cute) and the kids are prompted to set up a number sentences for each story (e.g. 4+5—but prior to using numbers they draw pictures). They have exercises for teaching concepts like between, and have an uncanny knack for doing so for the very words that tripped up my ds. There is a lot of overlearning of one step problems before introducing two step problems. Also, word problems are incorporated into every lesson. This is in huge contrast to the math texts my ds had in the lower grades, where there was lot of work on nonword problems for each unit with only two or three word problems stuck on the end, some of which had three steps. The sum total of word problems my ds had from school throughout fourth grade was not more than 20. In Saxon it would have been at least two every day.

Some of the notes in this thread mention the “between concept”. Could somebody give and example of a problem the uses the “between concept”?

My son’s fourth grade book is much the same. He always gets the word problems wrong but most of the time they are pretty hard. I think I need to do something systematically since he is clearly not learning how to do them at all at school.

Beth

The towns of Alpha and Beta are 30 miles apart. The town of Gamma is between Alpha and Beta. The distance from Gamma to Alpha is four times as far as the distance from Gamma to Beta. How far is Gamma from each other town?

Note: there are several other usages of “between” including sharing, number order, and others. The above is the most common I can think of.

I keep hearing this one over and over — and I keep hearing the same dead-end quick fix suggestions over and over too. So please excuse me if I sometimes sound frustrated!!

I am a math major, was in grad school math and part-time college math teaching until a thyroid problem floored me. So yes, I have done this before and this is from a long-term perspective, what actually succeeds to get you out of high school and through college.

Samantha is right that you need a strategy. Thanks.

One thing she did not mention is so-called “key words”. Some people will tell you that this is the solution. In general, they are *wrong*. This strategy will get you a bare pass, maybe, inthe first course, and leave you lost and floundering next year, with no foundation to fall back on — a bad dead end.

She is also right that so-called “word problems” are it. This is what math is all about.
I keep asking people two questions:
When was the last time you were in a business and people were running around frantically trying to fill in the answers on worksheets? Never, because worksheets aren’t what what you do in the real world; you solve real problems.
And how are problems in the real world going to be presented to you? In cartoons? No, in the real world people come up to you and explain problems in words. Or they send memos and explain problems in written words — like what you just did posting here, right?
So the phrase “word problems” is completely redundant. It’s like saying “water swimming”. What else are you going to do?

This isn’t to say that it is always easy —it isn’t. But it’s the core and purpose of the program, what you are there to learn, so it is worth taking a lot of time and effort over.

Samantha suggests the “four-step” problem-solving approach. That’s OK, but I find it sometimes not clear enough. Working with my tutoring students, I’ve come up with seven steps:

(1) *Summarize*
What facts are you given? List them in note form, in words.
What *exactly* are you asked to find? Note it down.
(2) *Visualize*
Draw a diagram
Draw a chart
Draw a graph
Make a table of values — experiment with different values and see what patterns you get
(3) *Relate*
Do you know any formulas, like d = rt? A(circle) = pi r squared? and so on?
Are you *given* any relationships, like John’s age = twice Sue’s age?
Can you *see* any relationships from your diagrams?
Did you *find* any relationships when you charted values? What system did you use to calculate values?
Do you *recognize* a familiar problem pattern that may be in an example in your text or notes?
(4) *Name*
Give SPECIFIC names to your variables. WRITE the expression “Let …”
Good: Let J = John’s age
NOT J = John (might be his weight, his height, his grade on SAT’s . . )
** Why this is a problem: the classic “mixture problem” where you mix say chlorine and water to purify, or peanuts and cashews to sell a mix; there is the *quantity* c = gallons of chlorine bleach, and then there is the *value* 0.25c = amount of pure chloride in the bleach; p = pounds or *quantity* of peanuts, but 3p = *value* of peanuts @ $3.00 a pound. You have to be absolutely clear what your variable means, or else you will have confusion coming back to bite you later.
(5) *Equation*
DO NOT write an equation without doing the first four steps. In a simple problem they may take one minute and three lines, but DO them. When you hit the hard stuff, you need the skills you were supposed to be developing while you did the easier stuff; if you skip the learning process on the practice problems, you will find out. The hard way.
With the relationships and word equations from (4) and the variables from (5), the equations should most of the time jump out at you. With a little practice, they seem fast and easy. The important point is to work through the *process* so that it *does* become easy.
(6) *Solve*
Use proper algebra techniques; don’t guess and don’t skip steps.
Sometimes in the value-charting method the correct answer falls in your lap; tha’s OK if it happens, but do the algebra anyway. The next time the answerr may not fall in your lap, and you’re here to learn the *method*. Also many problems have two or more answers, and you may get a surprise.
(7) *Check*
Any real algebra problem involves enough steps that anyone can slip up. *Everyone* makes mistakes; good mathematicians are the ones who have learned to fix them.
Check in your *first* equation, and go back and see if your answer *makes sense* in the real situation of the *original* problem. Reality check is fundamental.

At first this looks really long; it isn’t, I swear. Doing this, I can solve any high school/junior college problem as fast as I can write it. From Algebra 1 to Calculus 3, this method breaks any problem.
All the other people trying to “save time” are staring at a blank paper with blood coming out of their foreheads. Or scribbling and erasing and scribbling again. Or getting grades of F because their work was certainly fast, but it was wrong.
I strongly encourage students to *write*. For some reason, elementary teachers (most of whom became elementary teachers because they did *not* want to take math classes beyond the minimum) have the peculiar idea tha writing math down is bad and that the ideal would be to get the answer by divine inspiration and write it down in solitary splendour. This is the opposite of what real mathematicians do; real mathematicians (and scientists and engineers and technical workers) write everything down so they can keep track of all the details and check back for errors, same as any other responsible adult. Get your student to *write* the summary, *draw* the visualization, *write* the word equations for the relationships, *write* the variable naming, *write* the equations and the steps in the solution, and *write* the check (a few lines, but saves 10 to 20% on your grade).
I also have a campaign against pencils and erasing, This is a math class, not an erasing class. If you watch a weak math student, you will see that 3/4 or more of his time is spent erasing. If he just draws a line through the errors and moves on and does more *math*, instead of erasing, he will instantly be working three to four times as fast. And if, as often happens, he realizes that he was not wrong after all, he can go back to the part that was OK and continue on. Use a pen and do math, forget kindergarten training.

This is general advice; please get back to me if you have any specific questions.

Please note: the above is copyright; you are welcome to use it, but please do not re-publish wothout my permission. Thank you.

Victoria, In first grade my ds could not for the life of him even begin to know how to answer the question “Name a number between 20 and 30.” After a few days of doing the fork, knife, and spoon practices suggested in Saxon for the “between” concept he was able to answer the question effortlessly.