This is a continuation from the previous lesson.
Equations with Fractions. I start out by writing an equation on the board that has fractions in it:
A mood of gloom will usually settle upon the room. “This isn’t as hard as it seems,” I’ve been known to say which gets mixed responses.
I’ll ask for the common denominator of the fractions and then have them convert the fractions resulting in:
“What should I do next if I want the variable to have a positive coefficient?” They will want to subtract (8/20)x from both sides:
Completing the process we get:
I have them solve it and walk around the room inspecting notebooks. This problem has a negative answer which I assure them is fine, and utter other pithy phrases such as “go with it” or, “sometimes life gives you negative numbers.” (Students sometimes tell me “You sound like my dad.”)
Solving it they should get -80.
The key is to find the common denominator and convert all fractions. In that way it is easier to decide how to proceed if you want to have a positive coefficient.
I try some more examples:
“Getting a fraction for an answer, like getting a negative number, is fine; it doesn’t mean it’s wrong,” I tell them. “Fractions are numbers too; treat them with kindness,” I’ve been known to say. Many students resist leaving an answer as a fraction. They feel it must be changed into something else, like a decimal. I have heard from other teachers who, like me, try to frame it in such a way that it will feel like they are being allowed to break the rules about fractions. That is, it’s OK to leave a fraction as is, even if it’s in improper form; it isn’t necessary to change it to a mixed number.
Some students want to change them to decimals in the erroneous belief that it is more accurate to do so. I tell them that for some numbers for which the decimal form is repeating that they might introduce error. My final appeal is that this is preparation for algebra where fractions are much more useful in working with expressions and equations. The bottom line is that students greatly dislike fractions. One teacher told me: “It’s been a struggle but I simply refuse to allow mixed numbers or decimals when solving linear equations. I tell them ‘Fractions are like castor oil; nasty, but it will do you right!’ ”
The key is not to overload them; so to show them it’s not all bad, I include examples that have whole number solutions (albeit negative like this one):
“Find the common denominator first.” I circulate around the room to check what they’re doing.
They should get:
Multiplying All Terms by Common Denominator. In a previous lesson on solving two-step equations that contain fractions, I had said that these problems can also be solved by multiplying each term by the lowest common denominator to eliminate all fractions and that they would be addressed in a later chapter. This is that chapter and now is the time!
I caution that in seventh grade, some students will like using this procedure; others will find it confusing. I explain that it is a method for them to use if they choose to do so. They will come across it again when they take algebra and solve equations with rational expressions (i.e., fractions).
I will tell them that there is a shorter way to do these problems, by getting rid of all fractions in the equation.
The common denominator is 15. So we need will multiply both sides of the equation by 12:
Students may want to write 15 as 15/1 to help them cross-cancel and multiply.
Example:
I have them do this independently.
Problems with Decimals. In the section on word problems, there was a problem for which the equation was:
I will ask what can we multiply by to remove the decimals. “What if we multiply by ten? Will that eliminate the decimals?”
If there is no consensus I’ll multiply and they quickly see that the multiplier should be 100:
I make it clear that it’s really what makes it easy for them. Some students don’t like working with large numbers, and working with decimals might be easier in such cases.
The next example I let them choose how they want to solve it, and check their notebooks as I circulate around the class:
Answer using elimination of decimals:
Homework. Textbooks vary and I usually pick problems from a variety of them including “Basic Algebra” (Dolciani) and other elementary algebra books. I try to keep the problems simple and limit them to 15 problems of which two or three are word problems. The rest should have some fractions, and some decimals.