Now the students are ready to start solving simple comparison problems with variables.
First the students are taught how to translate a phrase like "T is less than H" or "R is more than W" onto a number family.
The lesson might be taught like this:
Sometimes we refer to a number without telling which number it is. We can call that number J or B or any other letter. Here is a sentence that tells about two numbers: J is less than M.
We don't know which numbers J and M are, but we can put those numbers in a number family. J is less than M. So J is the small number. M is the bug number.
The big number goes at the end of the arrow. The small number goes close to the big number. Here's how you write it.
Then an example using "more," like "R is more than W," is taught.
After the students are firm on this skill, they are ready to tackle a translation like "J is 18 more/larger than K."
This must be a difficult skill because in CMC they scaffold the instruction by circling the number which tells how much more. Like this:
Eventually the scaffolding is faded.
To translate the problem, the students are instructed to ignore the circled number. This makes the problem identical to one they know how to translate -- "J is larger than K." They know that this can be translated into:
Then they are taught to place the circled number, 18, into the only available spot in the number family. Like this:
Once the students are firm on this skill, they can be given a problem that they know how to solve like "F is 12 more than 56." Now they should be able to translate this to a number family and solve for F.
Then the problem can be made more difficult by specifying two variables and the value of one of the variables, such as "R is 250 more than P. R is 881. What number is P?"
Finally, the students are ready to start solving real word problems like "Fran was 14 years older than Ann. Ann was 13 years old. How many years old was Fran?"
Here's how they are taught how to solve these kinds of problems:
This is a good stopping point. This represents a month worth of instructional time for the lessons and the practice. That's for lower performers, higher performers can probably learn this in about a week. Bear in mind that the students are learning and practicing about 10 other strands of material while all this is going on.
The value of this problem solving technique (like Singapore Math's bar graphs) is twofold. First, it reduces solving math problems to a systematic process; this will clarify the student's thought process. Second, the use of the written number families frees up the novice student's working memory which is taxed heavily in solving word problems. Given enough practice, these skills will become automatic for the student and lodged in long term memory. When this occurs, the burden on the student's working memory is becomes much less and the need for the number family prompt is diminished.
Teaser for next lesson: The students learn how to solve word problems like "Jerry weighed 72 pounds. Terry weighed 94 pounds. How much heavier is Terry than Jerry?" To solve problems like this, the students are taught the concept of moving forward and backward along the number family line.
I am actually disappointed with this approach. It seems to spend a lot of time emphasizing the technique of the line. I am also finding the reliance on a line with an arrow confusing. This looks a lot like a number line, yet the numbers above the line are placed in arbitrary positions and will need to be heavily compartmentalized (i.e. ignored) when actual number lines are introduced and the numbers above a line with an arrow on the end take on a different meaning.
ReplyDeleteMy step daughter is in seventh grade and her text book teaches nothing (and I mean nothing) but techniques. You multiply fractions by flipping the second one. That's it--no explanation for why this works. This technique seems to me to be the same.
It's not clear to me that Singapore actually teaches this particular problem via bar diagrams.
ReplyDeleteI really think it's these simple exercises such as these that appear in the student text that do the trick...
"What is 1 more than 56?"
"What is 1 less than 56?"
"What is 10 more than 56?"
"What is 10 less than 56?"
and then my kids have "magically" been able to handle problems such as
"____ is 12 more than 56" without relying on either the bar diagram, arrow diagram, or a step-by-step solution guide.
The arrow diagram is confusing to me as well. I originally thought they were trying to place the known and unknown quantities onto a number line. Why is F at the end of the arrow rather than on top of it? Why have an arrow head at all? Why must the addition problem be stacked vertically at the end of the arrow diagram? Are they trying to teach algebraic concepts to children who haven't even mastered basic mental arithmetic yet or does this simply a visual way of showing step three: "Figure out the number for the other letter."
Is this arrow diagram applicable to other kinds of problems later on in the program such as, "Fran is 4 years less than thrice the age of Ann?" or would a new method need to be taught for that?