Fractions, Common Core & Calculators

I see so many examples of “common core math problems” that people are disparaging, whether from misunderstanding, a poorly asked question, poorly trained teacher, politics, mislabeling….the list is endless.  I thought it might be interesting to take a problem that is designed to support one of the CC standards, and show the many ways it could be approached to, in fact, show the spirit of the Common Core. In other words, show how students could analyze and try to solve it from different perspectives and multiple ways, which is really what the Common Core epitomizes – perseverance, multiple pathways, and justification of your thinking to deepen understanding and make connections.

Here’s the standard I’ve chosen. (Fractions always seems to be the bane of teachers and students, so it seemed a good one!):

Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)

This is a grade 5 standard. To put in context, students were introduced to fractions in 3rd grade, and have worked with adding, subtracting, multiplying, dividing, unit fractions, and  equivalency through 3rd and 4th grades. What this standard also asks is for students to create a story context for the equation – relate the mathematical skills to the context of a real-world situation to help them make sense of the problem. You can help this along by providing a real-world context.  A simple way to do this is to use recipes, and since we are in the holiday spirit, my context is going to be a chocolate chip recipe!

Recipe – Coconut Chocolate Chip Cookies (makes 1 dozen)

• 1-1/2 cups graham cracker crumbs
• 1/2 cup all purpose flour
• 2 tsp. baking powder
• 1-2/3 cups sweetened condensed milk
• 1/2 cup unsalted butter, softened
• 1-1/3 cups flaked coconut
• 3/4 lb. semisweet chocolate chips
• 1 cup chopped nuts

Preheat oven to 375°F. In a small bowl, mix graham cracker crumbs, flour and baking powder. In large mixer bowl, beat sweetened condensed milk and butter until smooth. Add graham cracker crumb mixture; mix well. Stir in coconut, chocolate chips and nuts. Drop by rounded tablespoonfuls onto ungreased cookie sheets. Bake 9-10 minutes, or until lightly browned.

Clearly, 1 dozen cookies is not enough! It’s fun to ask students how many they think would be appropriate for the class – how many cookies should each student get? How many should the teacher get? Based on these discussions, how many cookies are needed?  Which then leads to how many “recipes” do we need to make?

Let’s say there are 30 students, plus the 1 teacher, and it’s determined everyone gets 3 cookies. So – 31 x 3 = 93 cookies. How do we determine how many times we have to increase the recipe?  (Look at all the great math discussion we are having, in a real-world context, and just having some group discussion!)  Hopefully students then realize we need to divide the 93 by 12. (Oh no…doesn’t come out to a nice number.  Now what?!! Rounding? 7.75 recipes – what kind of math is that?) Ok – so, we have figured out that we should make the recipe 8 times – which will give us approximately 96 cookies. And now you actually get into the standard and have students multiplying the ingredients by 8 (but wait – you could have also done 7.75, changed it to its equivalent fraction form of 7  3/4 and used this too – maybe differentiate your students, and give some groups that to work with and others use the 8?)

If you’ve noticed, before we even start the actual math specified in the standard we chose, we have had in-context conversations, analyzed the problem, made mathematical decisions, rounded….so much math and Common Core related practices happening already (see Mathematical Practices). When we do focus on the standard, i.e. multiplication of the fractions by a whole number or a mixed number (depending on whether you chose 8 or 7.75), there is context and purpose. Set those students loose! Here’s where multiple pathways kicks in – let them decide how they are going to determine the amount of each ingredient needed. Give them the tools they might use – i.e. paper, pencil, manipulatives, dynamic software, calculator, etc.,  and let them make the decision on how they will approach solving the situation. You will get some visual models, some equations, a mixture of both maybe….that’s the idea behind a truly Common Core problem – students have the opportunity to approach the solution in the way that best fits THEIR understanding of the problem and their mathematical application. As long as they can justify their approach to the solution, it’s all perfectly okay. And if you have students share their approaches, it allows for rich conversation and the making of connections.

Possible examples (using 1-2/3 cups condensed milk ingredient):

  1. Equation: 1-2/3 x 8 = 5/3 x 8 = 5/3 x 8/1 = 40/3 = 13-1/3 cups
  2. Visual (drawing)IMG_2270
  3. Visual (dynamic math software for example)    2015-12-17_14-23-42
  4. Calculator (in this case, the fx-55Plus) I have included a quick video of how to use the calculator, which natural fraction number capability, to perform this calculation. Technology is another method students should be able to utilize when approaching problems. The important component here is to have students explain why they entered the calculation they did, what it means in context, and what the solution represents.

Hopefully you get the idea of how a simple problem is Common Core, not because of the problem itself, but due to the questions you ask, the discussions you encourage, the opportunities you provide for students to approach solutions in multiple ways, and the resources you provide to support their efforts. You need to be willing and able to accept multiple approaches and not expect every student to understand and solve in the same way. (THIS is where Common Core gets a bad rap – when ONE way is expected (because it’s in the textbook or because it’s easier to grade, for example), which is actually NOT Common Core! ) Do we eventually want students to be able to do mathematics in the most efficient way? Maybe – what I think is more important is that students have the chance to understand and make connections so that they can use the mathematics in different situations. So what if the way they understand it takes longer – math is NOT about speed, it’s about applying and using it to help solve a problem.

Anyway, hopefully you have some ideas. And – you now have a recipe too, just in time for the holidays! Happy baking!