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One of the biggest issues that I have discovered is that first of all we are not teaching the student the correct vocabulary and then they are applying the opposite to many different situations where they do not need it. Another issue that I came across was that it does not really teach them with the conceptual understanding piece. It does not help them with division aspect. They just learn the trick and then later their will be confusion because math is constantly building off of itself so it is important that you develop that deeper meaning when first introducing this concept to students.

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By Tina Cardone, Posted March 2, 2015 -
Few phrases make me cringe the way I do when I hear, "Ours is not to reason why; just invert and multiply." A student's job in math class is to reason, and a teacher's job is to help the students see that math makes sense. Understanding division of fractions is complicated, bringing together many ideas that build up to a method that makes sense, especially in the tricky case of a fraction divided by a fraction. Liping Ma writes about research asking teachers to explain what it means to divide by a fraction. Most U.S. teachers were startled to find how difficult this was for them.

What if we start with the simpler case of a whole number divided by a fraction?

Christopher Danielson posed the question below (#tmwyk is his project Talking Math With Your Kids) and asked how we would think it through. I was surprised by how I arrived at my answer.

I was able to use my understanding of fractions to go from unwieldy part (3/4) to unit fraction (1/4) to whole. I realized later that I was solving for the total rather than answering the question, "How many more?" There are many different strategies for this question, some of which include finding how many more directly without calculating the total. You can check out the conversation to see other approaches. Max Ray recently made a presentation on a similar prompt (7 cups of dog food, divided into 2/3 cup servings). Check out his presentation that includes several examples of student methods for solving the problem: Ignite Talk from AMTNJ.

I was surprised how automatic this process was and that it didn't require "the" standard algorithm. I completed all the steps in the standard approach to dividing by a fraction but I (1) didn't need to recognize the problem as requiring division by a fraction, and (2) knew why I was completing each step.

How many students think this way? I have to admit I don't think this way when I see a problem without context. If I saw

I would multiply 30 by the reciprocal of 3/4. But students could see why we do that if they were encouraged to take some more time to explore what it means to divide by a fraction.

Division and multiplication are inverse operations. We can write

for any fact family. This is how students approach integer division; there's no reason not to approach fraction division the same way:
http://mathmistakes.org/?p=328
What steps did we take? First, we divided by the numerator to get a unit fraction. Then we multiplied by the denominator to get the whole. When students repeat this process, first in context, then in more general cases, they will recognize the pattern. When someone recognizes the pattern, celebrate! And share that this pattern has a name. Dividing by the numerator and multiplying by the denominator can be completed in one step (multiply by original denominator over original numerator); this new thing is called the reciprocal. Taking two steps to complete this process is still efficient, but the idea of the reciprocal becomes important, so students should be introduced to the term.

The phrase "multiply by the reciprocal" is preferable to "same, change, flip," or any other mnemonic. Reciprocal is a precise term that reminds students why we are switching the operation. I see many students who use language like "same, change, flip" without understanding where it comes from. This leads to mistakes like this one:

This student doesn't appear to know the difference between "flipping" a fraction and "flipping" the sign of a number. The overuse of the word opposite can further compound errors because the vocabulary is open to interpretation.

Once students have an understanding of dividing a whole number by a fraction, it's time to tackle dividing two fractions. The procedure is the same, but there are a few ways to build intuition. Again, use the phrase "multiply by the reciprocal," but only after students understand where this algorithm comes from.

If the last problem looked like the previous examples, it would be easier. So let's rewrite with common denominators:

If students are asked to solve enough problems in this manner, they will want to find a shortcut and will look for a pattern. Show them (or ask them to prove!) why multiplying by the reciprocal works. One way to show this is in the following way:

In this case, students discover that multiplying by the reciprocal is the equivalent of getting the common denominator and dividing the numerators. This is not an obvious fact. Students will only reach this realization with repeated practice, but practice getting common denominators is a great thing for them to be doing! More important, the student who forgets this generalization can fall back on an understanding of common denominators, whereas the student who learned a trick after completing this exercise once (or not at all) will guess at the rule rather than attempting to reason through the problem.

Tina Cardone, @crstn85, is a high school teacher at Salem High School in Salem, Massachusetts. She is the author ofNix the Tricks and blogs about her teaching at DrawingOnMath.blogspot.com.

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By Tina Cardone, Posted March 2, 2015 - Few phrases make me cringe the way I do when I hear, "Ours is not to reason why; just invert and multiply." A student's job in math class is to reason, and a teacher's job is to help the students see that math makes sense.

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One of the best ways to get students using number sense is to have them apply it to real life situations. This will get them thinking about what they are doing and help them transfer that knowledge to other situations. One great activity that I found dealt with cooking. In the activity students practice three ways of doubling fractions using representations, addition, and multiplication. They also convert between improper fractions and mixed numbers. The activity is leveled and help them build upon their knowledge so that they can have a better understanding. Their is also a video of someone making 24 cup cakes and I like it because it shows the process that they go through when working with fractions to solve their dilemma.

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While doubling a cupcake recipe, students practice three ways of doubling fractions using representations, addition, and multiplication. They also convert between improper fractions and mixed numbers.

4-7

1 hour

Use these resources to create a simple assessment or video-based assignment with the Lesson Builder tool on PBS LearningMedia.

1. Have the students work in pairs for this activity.

2. Read the following to your students: "In a Cyberchase For Real episode, Bianca is helping a friend bake cupcakes for a party. Bianca's friend is supposed to bake 24 cupcakes, but her recipe only has the ingredients listed for 12, so they have to double the recipe. When you have to adjust the amounts of a recipe, there may be several strategies you can use. In this activity, you will try out ways of adjusting amounts in these recipes and develop your strategies."

3. Distribute Handout 1: Grandma Gittel's Favorite Cupcake Recipe .

4. Ask the students to complete the ingredients list for 24 cupcakes, and fill in the table in Handout 1.

5. Tell the students that they will watch a video clip in which Bianca and her friend try to figure out the amount of ingredients to use to make 24 cupcakes.

6. Play the The Right Amount of Cupcake Mix Video and ask students to write down Bianca's ingredient list as they watch.

7. Ask the students to compare their results in Handout 1 to Bianca's ingredient list in the video clip. Did they notice that they had to double the ingredients before viewing the clip?

8. Tell the students that now that they have doubled the ingredients in the recipe, they will try out three different strategies for doubling.

9. Distribute Handout 2: Three Ways to Double Fractions .

10. Ask the students to complete Handout 2. Encourage them to compare the three methods for doubling as they complete the handout.

11. Tell the students that as they examine recipes, they will find that some ingredients are listed in fractional form. Some of these will be improper fractions, and some may be mixed fractions. Because all these forms are common, they need to learn how to convert them.

12. Distribute Handout 3: Converting Mixed Numbers and Improper Fractions .

13. Ask the students to read and discuss Handout 3.

14. Discuss converting mixed numbers and improper fractions with the students. Make certain that the students understand the distinctions between mixed numbers and improper fractions.

15. Tell the students that doubling mixed numbers is probably easiest if they convert the mixed number to an improper fraction, and then double the improper fraction.

16. Distribute Handout 4: Doubling a Mixed Number .

17. Ask the students to read Handout 4, and complete the problems.

18. Discuss with the students any problems they had in doubling the mixed numbers, or converting mixed numbers to improper fractions, and vice versa.

Assessment: Level A (proficiency): Students are asked to adjust ingredients for 24 and 48 cupcakes from the basic recipe for 12 cupcakes.

Assessment: Level B (above proficiency): Students are asked to double a variety of fractions, and represent them in both numerical form and rectangular graphical representations.

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Using Recipes to Add Fractions and Convert Improper Fractions to Proper Fractions or Mixed Numbers

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While doubling a cupcake recipe, students practice three ways of doubling fractions using representations, addition, and multiplication. They also convert between improper fractions and mixed numbers. 4-7 1 hour Use these resources to create a simple assessment or video-based assignment with the Lesson Builder tool on PBS LearningMedia. 1.

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Estimating is an important concept for students to be able to understand because of how often that we use it. Estimating in a way allows students to be able to determine if their answer is reasonable. This makes them apply what they know about numbers and apply it to the question that they are answering. This directly links to what students should know when they are working with fractions. In the activity that I have provided student will be working with number line to help them estimate their fraction Then it builds off of itself and had the students just estimate it by looking at the fraction that they are given and explain why they picked it.By preforming this activity it will help students to estimate an answer so that they can determine if their answer is reasonable.

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Kaiulani Ivory Akpan Eliot Hine Middle School Washington, DC

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Activity 1 - DON’T SKIP THE MANIPULATIVES

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Activity 1 - DON’T SKIP THE MANIPULATIVES

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Using manipulatives is key in getting students to understand how fractions work. One concept that I find is especially key is teaching the students about equivalent fractions. For this activity students are presented with a card that has a fraction on it and they will have to build it with the tiles. From there they are asked to build equivalent fractions with the other tiles by building and proving if they are or if they are not equivalent fractions. This allows students to see a visual and also allows them to build those connections and build that number sense. This lays the groundwork for a lot of activities that we do with fractions.

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Making Sense of Invert and Multiply

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As elementary teachers, we rarely have the opportunity to explore division of a fraction by a fraction. When we do, it's normally accompanied with Keep-Change-Flip or the saying "Yours is not the reason why, just invert and multiply."

Both are conceptual cripplers.

I've been drafting the 4th installment of the Making Sense Series involving fractions and I'm sharing this post as more of a personal reference should K-C-F make its way round these parts again...and I'm sure it will.
Side note: A while back Fawn and Christopher each shared a post about division of fractions using common denominators. Both posts left lots of math residue and are well worth your time.
Let's start with a model for 3/5 ÷ 1/4.

At some point along the way it becomes inefficient for students to draw models once the conceptual understanding is established. As students represent measurement division of fractions they should be formally recording their thinking.

From here students generate their own algorithm (shortcut). They begin to recognize that they will always get a denominator of "1 whole" so they begin to purposefully leave it out. In doing so, they become more efficient in the procedure of dividing fractions.

Some students begin to eliminate the green and red steps from the above equation because they're seen as repetitive. We've even had one student that "invented" and generalized cross multiplication for division of fractions as they searched for ways to record fewer numbers and symbols.

It looked something like this...

I keep reminding myself that if fractions are the gatekeeper to algebraic reasoning then I need to slow the process down and conceptually understand what's happening. This includes K-C-F.

As students understand the power of creating a whole number denominator they begin to search for more efficient ways to get 1 whole. It's here that they being to explore with equivalent fraction and the idea of using the reciprocal to create a whole.

What I've realized is that aside from complex fractions, the underpinnings of this equation are developed in the elementary grades.

5.NF.3 Interpret a fraction as division of the numerator by the denominator (a÷b = a/b).

4.NF.1-Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b).

3.NF.3.c- Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.

Just like before, students will look to generalize the equation above and find shortcuts. They'll do so by eliminating repetitive steps which would leave them with...

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As elementary teachers, we rarely have the opportunity to explore division of a fraction by a fraction. When we do, it's normally accompanied with Keep-Change-Flip or the saying "Yours is not the reason why, just invert and multiply." Both are conceptual cripplers.

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Our IS to reason why, NOT just
invert and multiply!

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Our IS to reason why, NOT just
invert and multiply!