If one was to categorize every SAT question from the very first SAT ever published, I believe we would find the following type of algebraic ratio problem one of the most common type. Even with all the exposure students now have to SAT problems, my direct experience is that many students still struggle with these types of questions.
WHY?
More importantly, are these types of problems important enough in the CCSSM to justify the time investment to introduce them in middle school and reinforce in secondary algebra classes? IMO, ABSOLUTELY!
If h hens consume a total of p pounds of feed per day, then, at this rate, how many pounds of feed would c hens consume in x days?
Not only was a similar question the recent SAT Question of the Day on the College Board web site, the statistics were also published:
35620 responded (up to the time I checked)
31% correct
So, about 7 out of 10 students attempting this question online got it wrong.
Note: The actual question was followed by 5 choices, allowing students to plug in numbers and test each choice, but I chose to focus on the question here rather than test-taking strategies.
IMO, the College Board hires highly competent math people who write succinct, accurate and helpful online solutions but this only scratches the surface. It only suggests one particular approach and has little to do with Instructional Strategies and the various ways children develop these important ideas.
REFLECTIONS...
1. Where are ratio concepts introduced for the first time in the CCSSM? K? 1st? 4th 5th?
2. By your own estimate, how many of these kinds of questions appear as sample problems or homework exercises in your elementary/prealgebra/algebra texts?
3. Do you believe ALL your students receive adequate exposure to and review of these?
4. Would you be willing to share some of your favorite methods of laying the groundwork for and developing the skills and concepts needed for your students to be successful with ratio problems and ultimately algebraic types? If I take a risk, would you?
Putting myself out there...
The simplest and most instinctive approach usually makes the most sense, doesn't it? We know how we learn best and the same is true of all students. Do you accept the following as a truism, an essential tenet of teaching and learning mathematics?
EVERYONE LEARNS BETTER WHEN PRESENTED WITH CONCRETE NUMERICAL RELATIONSHIPS BEFORE TACKLING ABSTRACTIONS. FURTHER, THE COMPLEXITY OF LANGUAGE SHOULD BE GRADUALLY INCREASED, STARTING WITH THE MOST ACCESSIBLE INFORMAL PHRASES.
For example,
If 6 hens eat a total of 12 pounds of feed each day, how many pounds of feed would one hen eat in one day?
When first introduced, should our focus be on which operation to perform? In my view, our goal should be to develop number sense, in this case, ratio sense.
We all know that a powerful construct for developing ratio/proportion sense is the idea of first reducing the information to a UNIT.
Many of us were taught this way and most children tend to think like this at first.
Scaffolding...
If 6 hens eat a total of 12 pounds of feed each day, how many pounds of feed would nine hens eat in one day?
Working from one hen consumes 2 pounds per day, the child can usually move on to 9 hens eat 9x2 or 18 pounds per day.
Two points here...
First, I believe it is important to routinely use a variety of equivalent phrases:
"in one day" vs. "each day" vs. "per day."
Secondly, I would encourage students who can reason proportionally to share this with the group:
"Well, if 6 hens eat 12 pounds, then 3 hens will eat half as much or 6 pounds, so 9 hens will eat 12+6 or 18 pounds."
Teaching conceptually means NOT SETTING UP A PROPORTION initially. Procedures and algorithms turn off the child's sense-making and stifle intuition and number sense. You can fight me on this all you want, folks, but you will not win here on my blog!
So when do we introduce proportion problems involving variables and what are some good ways to solve the original problem?? I'll allow my readers to figure that out for themselves...
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