Engaging students: Adding and subtracting fractions with unequal denominators

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Kristin Ambrose. Her topic, from Pre-Algebra: adding and subtracting fractions with unequal denominators.

What interesting (i.e., uncontrived) word problems using this topic can your students do now?

Cooking is a great example of where you frequently add and subtract fractions with unequal denominators. For example, here is a real-world word problem I came up with for adding and subtracting fractions in cooking:

You are making dinner tonight and you’re having Lemon Chicken with Scalloped Potatoes. The recipes for these can be found below (and yes they are real recipes that you can use in real life).

 Scalloped Potatoes4 med. potatoes ¼ cup flour 4 tbsp. butter 2 cups milk 1 cup grated cheese Dash of garlic powder and white pepper Salt and pepper to taste Instructions: Preheat oven to 350°. Peel and boil potatoes, then set aside to cool. Make 2 cups of cream sauce by melting the butter and blending in the flour. Stir constantly, slowly adding the milk. Stir until the sauce thickens. Add grated cheese and spices. Slice potatoes and arrange in casserole dish. Pour sauce over potatoes. Sprinkle with paprika and bake for 10 minutes at 350°.

Lemon Chicken:

½ lb. boneless chicken breasts

1/8 cup flour

¼ tsp. salt

1 tbsp. butter

½ tsp. lemon pepper seasoning

½ cup of asparagus

1 lemon

Instructions:

1. Cover the chicken breasts with plastic wrap and pound until each pieces is about a ¾ of an inch thick. Place the flour and salt in a shallow dish and gently toss each chicken breast in the dish to coat. Melt the butter in a large skillet over medium high heat; add the chicken and sauté for 3-5 minutes on each side, until golden brown, sprinkling each side with the lemon pepper directly in the pan.
2. When the chicken is cooked through, transfer to a plate. Add the lemon slices and chopped asparagus to the pan. Make sure the lemon slices are on the bottom so that they caramelize and pick up the browned bits left in the pan from the chicken and butter.
3. When the asparagus is done and the lemons are golden brown, add the chicken back to the pan and rearrange everything (lemons on top) so it looks nice for serving.

You only have a half a cup of flour left in your pantry. Looking at the recipes above, do you have enough flour to make dinner? Or do you need to go to the grocery store to buy more flour?

In order to solve this problem students would first have to add the different amounts of flour for each recipe (1/4 + 1/8 = 3/8). Then students would have to subtract this amount from the amount of flour they had to see if they would have enough (1/2 – 3/8 = 1/8). Since 1/8 cup of flour would be left, they have enough flour to make dinner.

How can this topic be used in your students’ future courses in mathematics or science?

It would be difficult to do mathematics without knowing how to add and subtract fractions with unequal denominators. In mathematics when adding or subtracting fractions, it doesn’t always work out nicely where the denominators are the same, so it’s important to be able to solve problems even when the denominators are different. One example of this is summations. Take $\sum_{n=1}^4 \frac{1}{2n}$; what this equation really means is to add 1/2+1/4+1/6+1/8=25/24 or 1 1/24. Therefore adding fractions with unequal denominators could arise in summations. Also, in Algebra students will study quadratic functions and the factors of quadratic functions often take a form similar to something like (x+a)(x-b), with a and b being numbers. Students will have to know how to multiply these factors out and simplify the expressions. For example, a set of factors could be $(x+\frac{1}{2})(x-\frac{2}{3})$. When multiplied out students will have $x^2 + \frac{1}{2}x-\frac{2}{3}x - \frac{1}{3}$. Students will have to know how to subtract 2/3 from 1/2 in order to simplify the expression.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

YouTube can be a good source for finding videos to engage students in a topic. In particular, I found a short, funny video that reminds students of the significance of fractions. Here is the link to the video: https://www.youtube.com/watch?v=CBy8QbZyzy4. It makes a difference when a superhero only saves half of your stuff and not all of it. Just like you wouldn’t want only half your things saved, you wouldn’t want to add 2/3 of a cup of flour to a recipe that only calls for 1/4 a cup, or you wouldn’t want to fill up 2/3 of your tank of gas if it was already 1/2 of a tank full. Understanding fractions and how to add and subtract them is an important part of daily life.

I also found another video that demonstrates where fractions can come into play in science. Here is the link to the video: https://www.youtube.com/watch?v=hLGDJFGAmic. The YouTube channel ‘Numberphile’ in particular has many interesting videos involving numbers and mathematics, and would be a great resource for finding interesting videos to engage students.