I had some great discussion last week around the two ideas below. I will tackle them one at a time so as not to overload our working memories.
The Improper Fraction Dilemma
When adding the fractions 4 1/2 - 2 1/9, is it better to show Grade 8 students the "fail-safe" method of using improper fractions
4 1/2 - 2 1/9 = 9/2 - 19/9
= 81/18 - 38/18
= (81 - 38)/18
= 43/18
= 2 7/18
or to show them that we can subtract the whole parts and then the fractional parts, but noting that sometimes a "borrow" is necessary?
4 1/2 - 2 1/9 = 4 - 2 + 1/2 - 1/9
= 2 + 9/18 - 2/18
= 2 + (9 - 2)/18
= 2 7/18
Note that if we were to switch the place of the 1/2 and 1/9 then a borrow from the whole part (indicated with **) is necessary since we obtain a (-7) as the numerator of our fractional part:
4 1/9 - 2 1/2 = 4 - 2 + 1/9 - 1/2
= 2 + 2/18 - 9/18
= 2 + (2 - 9)/18
= 2 + (-7)/18
= 1 + 18/18 + (-7)/18 **
= 1 + (18 - 7)/18
= 1 11/18
This is an interesting question, since the biggest concern I was hearing from the Grade 8 teacher I was working with was that the fractional skills of the students were not solid enough to handle teaching them the borrowing technique. So the teacher opted to show them only the improper fraction technique method, which works in all instances and students do not have to deal with borrowing if a negative numerator occurs.
One of the biggest roadblocks I find at the university level when I am teaching fractions, is my student's reluctance to accept that improper fractions are not necessary. I often get solutions that look like
23 25/113 - 20 12/113 = 2624/113 - 2272/113
= (2624 - 2272)/113
= 352/113
= 3 13/113
rather than
23 25/113 - 20 12/113 = 23 - 20 + 25/113 - 12/113
= 3 (25 - 12)/113
= 3 13/113.
We can see that the second approach is favourable since there are no unusually large numbers in the numerators, and there is also no borrowing. Most students don't get to a correct final answer if they use the first approach due to the heavy computations involved.
So to me, the real trouble here is that students are likely never shown the borrowing technique at all. So students are not really thinking about the best or most efficient way of answering a mixed number subtraction question. I do agree with the teacher I was working with: it is important for students to see one, and only one, method for addition/subtraction of fractions when it is first introduced. After students have mastered that method, then we can branch out and explore other, more efficient methods. While the improper fraction method is not necessarily the most efficient, it does reinforce ideas like mixed number conversions and simplification of fractions. Time is also another factor to consider: is it better to only show the improper fraction approach and have the students very good with one method (saves time), than to show both approaches and have students weak with two different methods (takes more time)?
I thought a bit about this question and came up with the following idea. Why not introduce basic fraction arithmetic earlier in the curriculum? Why does it happen so late (in Grade 7 and Grade 8)? If we introduced it around Grade 5, then students will have had a few years to grasp the abstractness of fractional addition and subtraction. Then by Grade 7 or 8, we can have them decide which strategy to use when solving a mixed number question like this.
The Cancelling Common Factors Dilemma
When multiplying fractions, we can either decide to multiply the numerators together, then multiply the denominators together
8/5 x 5/2 = (8 x 5)/(5 x 2)
= 40/10
= 4
or we can decide to cancel any common factors, then multiply
8/5 x 5/2 = 4/1 x 1/1
= (4 x 1)/(1 x 1)
= 4.
You can clearly see that if we decide to cancel off common factors first, then the multiplications we do in the second step are much simpler. I was a bit shocked to see the first method being used over the second one. Although, I was ensured that the second method is where the students would get to by the end of the chapter. The Grade 8 teacher told me that the multiplication skills of the students were very weak, so this was a way for the students to review their multiplication facts.
This brought up two thoughts in me. The first is why we care about the idea of cancelling off common factors. When working with rational expressions, we are often concerned with making our expressions simpler by cancelling off "prime" linear or quadratic expressions. Students who are not shown how to do this with rational numbers tend to not grasp the idea when it becomes more abstract. So I will always encourage teachers to eventually have students master cancelling off common factors when multiplying or dividing rational numbers - this way the leap to multiplying and dividing rational expressions is not so large.
The second thought was regarding the statement "the multiplication skills of my students are very weak". This suggests that curriculum is not sufficiently doing its job. According to the curriculum "recall of the multiplication ... facts up to 5 x 5 is expected by the end of Grade 4" and "recall of multiplication facts to 81 ... is expected by the end of Grade 5". However, students are entering Grade 7 and 8 without the basic facts committed to long term memory. Without this, how can we hope for them to be able to work with integers, or with fractions?
Why is it that students do not have a good grasp of the basic facts? One major reason may be due to the fact that these outcomes occur so late in the curriculum. The earlier we can introduce the basic facts and allow students to play with them, the more our students are going to encounter them. The more their working memories encounter the basic facts, the more likely it is that the basic facts will be stored long term. To be having our students only have the facts up to 9 x 9 memorized by the end of Grade 5 is really tragic. Since there is no mention of multiplication facts before the Grade 4 curriculum, our Grade 2 or 3 teachers looking at the curriculum document outcomes may not think about introducing multiplication facts at a more age-appropriate time.
Sorry for a long post, but I just had to get some of these opinions off my mind. Have any of you had similar experiences with fractions? I would love to hear your stories.
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