First of all, one risk of all algorithms, including “invert and multiply”, is that the students just memorize a procedure without fully understanding the concepts behind it.  When facing a real-life situation, students will not be given a written problem; rather, they will need to figure out what the problem is asking before actually solving it.  Knowing to flip the second fraction will not be very useful when students must infer the fractions and decide which one is the dividend and which is the divisor.  

Speaking of authentic contexts, another drawback of the “invert and multiply” approach is a loss of focus on units.  When working with fractions in settings such as medicine or baking, it is imperative that the unit is known.  By executing an algorithm, students can easily forget that 2/3 is referring to ounces of an amoxicillin dose or that 1 3/4 corresponds to the number of cups of sugar needed for a partial recipe.  

The last main complaint that I have about the “invert and multiply” mindset is that it discourages higher-level critical thinking.  If I asked a student to explain how he or she got his or her answer, a regurgitation of steps would demonstrate true learning.  By encouraging modeling, invented strategies, and other means, students are much more engaged in the content.  For instance, let’s say there is a problem asking how many ½ cup servings of ice cream are in a quart, or four cups, requiring a student to calculate 4 ÷ ½.  A student using the algorithm may very well give an answer of eight, but the unit could be wrong or missing since they don’t fully understand the relationship between the numbers.  Someone taught with alternate methods would reason that there are two half cup servings in one cup, and since the container has four cups, there are 4×2, or eight, half cup servings.  In a way, the second student would discover the algorithm through problem solving, giving him or her a much more thorough understanding of it than someone whose teacher spoon-fed it to the class.

 

            One potential activity to reinforce this paradigm shift for students is to make a big number line between zero and one as a class on the wall.  Each student would be given a sticky note with a fraction (greater than zero and less than one) written on it.  Students would take turns placing their sticky notes where they think they belong in between zero and one.  As the number line fills up, I would give students the opportunity to change the locations of their fractions as needed.  For example, a student might put three eighths at a certain point on the number line only to later realize that it belongs halfway between one fourth and one half.  This activity will be valuable to students when checking their answers to several types of problems, such as division.  If two thirds is divided by three fifths, students will know that the quotient should be greater than one since the dividend is greater than the divisor.

            I have an idea for an activity involving manipulatives that can address a misconception about adding and subtracting.  In the past, a handful of my students had difficulty understanding that only the numerators of fractions get added or subtracted.  To demonstrate this concept while having some fun in the process, I could give each student a roll of Smarties, which has 15 pieces of candy in it.  Each piece would represent 1/15, meaning that the whole roll would be 15/15, or one.  Students would know that they only have 15 pieces, so it would not make sense to end up with a denominator greater than that.  In addition to showing the students the importance of only adding or subtracting numerators, this activity could be used to practice adding and subtracting fractions.  I could make a worksheet that asks students to list their quantities of each color and add or subtract different ones.  Each student’s situation would be specific to him or her, creating a very personalized learning experience.  On top of that, the students would get to eat the Smarties when they’re done!

            Since I had already discussed addition, subtraction, and division of fractions on my wall, I sought a drawing activity for multiplication.  Among all the ones that I encountered, my favorite was the area model.  I feel that it illustrates the standard algorithm for fraction multiplication extremely well.  For instance, let’s say that 2/3 and 4/5 were being multiplied.  A student would start by separating a square into three vertical, congruent rectangular columns, forming three thirds.  Two of the thirds could then be shaded in with a pencil to show 2/3.  To demonstrate fifths, the student would then divide the square horizontally into five congruent rectangular rows.  Four rows within the original shaded region would represent the product of 2/3 and 4/5, so one row would need to be erased, yielding eight shaded boxes, or 8/15.  This activity is quite versatile.  It can give meaning to the standard algorithm, serve as an alternative to it, or even be used to check an answer.  Just like many other activities, its specific use will depend on students’ needs.