Percent Word Problems Revisited -- Making It Easier?

Since there has been some discussion about the complications in the previous post on percents, how do we as educators deal with adversity and transform it into a teachable moment. We're explaining a difficult problem that students are struggling with, so we try to explain it again, but to no avail. What do experienced educators do?
(A) Abandon the problem - it was simply too hard or they're not ready for it yet. Perhaps assign it for extra credit?
(B) Make the question simpler by removing some of the complexity. Consider a scaffolding approach?
(C) Re-think what prerequisite skills were needed?

In this case, let's re-work the problem as follows:

In the senior class, there are 20% more girls than boys. If there are 180 girls, how many more girls than boys are there among the seniors?

What do you think the results will be now that we've concretized the problem?
Do you think some students will make the classic error of taking 20% of 180 and subtract to obtain 144 boys? You betcha!! There's no getting around the issue of recognizing the correct BASE for the 20% in my opinion. Whether you consider the boys to be 100% and the girls to be 120% or you let x = number of boys and 1.2x = number of girls, the central issue is recognizing that you cannot take 20% of the girls! Sometimes students need to just use algebra as a tool - it's a great one! By the way, one student used algebra with ratios to solve this:
Boys = x
Girls = 1.2x
Difference = 0.2x
Therefore, the difference/girls = 0.2x/1.2x = 1/6 and 180 divided by 6 equals 30!

How would you have reworded the question to make it more accessible?

Dad's Math Book Makes the Grade

Thanks to Joanne Jacobs for this interesting article about how an engineer, a native of Ghana and now living in Detroit, decided to give up his full-time job to dedicate himself to writing a series of math texts to help his own children (who were struggling) and now others. After reading the full article I must say I want to see more details and samples of what he's written. According to the article, his materials adhere to his state's standards and apparently blend the traditional algorithms and skill development with conceptual understanding. Gee, does this sound familiar! The fact that other parents and school administrators are now taking notice and want copies is fascinating. Textbook publishers pour tons of money into developing a new math series and along comes one individual who decided these materials were simply not working for his children. Is there a message here?

There are 20% more girls than boys in the senior class. What % of the seniors are girls? The confounding semantics of percents...

PLS READ THE COMMENTS. VERY ASTUTE OBSERVATIONS FROM OUR REGULAR CONTRIBUTORS TO WHICH I RESPONDED IN GREAT DETAIL. THIS MAY CONTINUE...

For now, I'll just leave the title as the problem to be discussed. Please consider how middle and high school students would approach this. How many might incorrectly guess that 60% of the seniors are girls and 40% are boys. If the problem were rephrased as a multiple-choice question, this type of error would occur frequently from my experience. What causes the confusion? Is it just the wording of the question or is there also an underlying issue regarding conceptual understanding of percents? Could it be that the question is asking for a percent and hasn't provided any actual numbers of students? What are the most effective instructional methods and strategies to help students overcome these issues? Certainly, algebraic methods would be a direct approach, but what foundation skills and concepts should middle schoolers develop even before setting up algebraic expressions?

Making Connections: An Algebra Problem -- Or Is it Geometry? Or Both?

Lines L and M are perpendicular. Line L contains (0,0). Lines L and M intersect at A(5,2). Find x.

Comments:
(1) Would students in geometry be more likely to consider some or all of the following: similar triangles, altitudes on hypotenuse theorems, areas, Pythagorean approaches? If this were presented as a coordinate problem in Algebra 2, what would be the most likely approach? Would some students use the distance formula?
(2) As always, it is our role as educators to present these kinds of challenges and to encourage students to think more deeply. Making connections between algebra and geometry happens naturally for some, but certainly not for all! We must enable this dialogue via the classroomm environment we establish and the kinds of questions we ask. It is important to first decide the goal. What concepts are we trying to develop? Slope? Ratios of corresponding parts? How many students never make the connection between the two!
(3) The answer is 29/5 or 5.8. Think of at least two methods!

Understanding Algebra Conceptually?

What do you think would be the results of giving the following Algebra 1 problems to your students before, during, and after the course?
Do you believe that either or both of these could be or have been SAT questions?
Do students normally have exposure to these kinds of problems in their regular assignments?
Do these kinds of questions require a deeper conceptual understanding of algebra?

1. Given: x² - 9 = 0
Which of the following must be true?

I. x = 3
II. x = -3
III. x² = 9



(A) I only (B) I, II only (C) I, III only (D) I, II, III

(E) none of the preceding answers is correct


2. Given: (a - b) (a² - b²) = 0
Which of the following must be true?

I. a = b
II. a = -b
III. a² = b²


(A) I only (B) I, II only (C) I, III only (D) III only (E) I, II, III

Are all triangles in the universe similar to 3-4-5?

(a) AB = 8, BC = 6 in rectangle ABCD. Find the lengths of all segments shown in the diagram above.
Specifically: BD, CF,DE, BE, FE, CE

Comments: This is another in a series of rectangle investigations. To deepen student understanding of triangle relationships and to provide considerable practice with these ideas, the question asks for more than just one result. Students should be encouraged to first draw ALL of the triangles in the diagram separately and recognize why they are all similar! Using ratios, students should be able to find all the segments efficiently. One could also demonstrate the altitude on hypotenuse theorems as well!

(b) In case, students need a bit more of a challenge, have them derive expressions for all of the above segments given that AB = b and BC = a. To make life easier, assume b > a. This should keep your stronger students rather busy! This algebraic connection is powerful stuff. We want our students to appreciate that algebra is the language of generalization.

How many even 3-digit positive integers do not contain the digits 2,4, or 6?

More practice for students...
As indicated many many times on this blog, there is no substitute for experience!
The keys to success here are:
(1) Careful reading (do students often miss the key word!)
(2) Knowledge of facts (why is zero the most important number in life!)
(3) Knowledge of strategies, methods (multiplication principle, organized lists, counting by groups, etc)

If these problems are helpful for students, let me know...

Surviving the Math Wars - Both Ends Against the Middle

Identity Crisis...

I imagined the following dialogue taking place between a traditionalist (T) and a reformist (R) after both had finished reading the posts I've written on this blog for the past 6-7 months:

T: He is definitely a radical reformer. He uses phrases like investigations, explorations, discovery-learning, problem-solving, working in groups and encourages the use of the calculator for some activities. He is more concerned with conceptual understanding than with content and skills that all students must know. He actually believes that young children can think profoundly about concepts before they have completely mastered their skills. He talks 'less is more.' He has no documented research base for any of his wild ideas, and pretends that decades of classroom instruction are just as legitimate as a carefully developed research study. In what accredited journal of education research has he published?

R: Nonsense. He's one of your kind. He preaches strong foundation in basic skills, automaticity of basic facts, facility with percents, decimals and fractions and generally does not promote the use of the calculator in the lower grades. Other than some esoteric mortgage activity he wrote, most of his math challenges have little to do with the real world, focusing instead on number theory and combinatorial math, topics which are above the heads of most middle schoolers. His geometry problems are very traditional. He is often critical of calculator use. He focuses on standards and curriculum, even suggesting we need a nationalized math curriculum (and we know who will determine that!). He doesn't even applaud the efforts of his own national math teacher organization. He covers up his 'back to basics' approach by pretending he is a centrist. We all know one cannot be a centrist here. Either you're pregnant or you're not! He has no documented research base for any of his wild ideas, and pretends that decades of classroom instruction are just as legitimate as a carefully developed research study. In what accredited journal of education research has he published?

I know the regular readers of this blog know who I am and I know where their heart lies, but what can be done to move math education into a zone of reality that is sorely needed for our children. All these wonderful ideas but there is still so much confusion out there. How far have we come in the past dozen years or so to change the reality of a curriculum that is 'one inch deep and one mile wide'? Your thoughts are welcome...