## Wednesday, July 15, 2015

### Beyond Invert and Multiply Book Study Part 2

Welcome to week 2 of our Beyond Invert and Multiply book study!  Last week, we looked at some research around fractions and dove into the 12 different problem types for addition and subtraction word problems.  This week we will be taking a closer look at fraction addition and subtraction.
Here is the posting schedule.
July 15th Part 2: Addition and Subtraction with Fractions
July 22nd Part 3: Multiplication and Division with Fractions
July 29th Part 4: Discourse with Fractions

### Chapter 3: Making Sense: Addition with Fractions

Fraction addition used to be my enemy.  I remember struggling with fraction addition more than I struggled with any other topic in elementary school math.  I learned it in such a rote and procedural way and it didn't make a lot of sense to me.  Now when I think back to that experience and I read some of the research on fraction addition, I see myself in the classic cases of how not to teach fractions.  I was that kid who really didn't know what estimating was or how to do it and I had no sense of the magnitude of fractions.  In elementary school I would have really struggled with the problem they presented on page 42; "Estimate the answer to 12/13 + 7/8.  You will not have time to solve the problem using pencil and paper."   This is presented in a multiple choice format with the choices being 1, 2, 19, 21 and I don't know.  I would have been in the 76% of kids who could not answer this question correctly.

So how do I keep my students from following in my footsteps?  First I make sure they have a strong foundation in part to whole reasoning, equivalence and magnitude.  If they are not there yet with these foundational ideas about fractions, I give pull them for booster groups, do additional whole class lessons or meet with them during Guided Math time.  If you are looking for ideas for helping kids develop these foundational understandings, check out Beyond Pizzas and Pies or A Focus on Fractions.

It struck me when reading this chapter how similar fraction addition is to whole number addition.  We often treat it as a brand new topic but I think we miss out on connecting it to what kids already know about whole number addition.  The properties all still hold true and some of the strategies kids developed when learning basic addition can also help them when learning fraction addition.  I feel like the idea of decomposing numbers to make friendly tens and hundreds is something we are doing very well at my school.  However, I don't think that we are using this skill quite as well with fractions.  The video clips from lesson 3.4 really opened my mind to some new ideas to try this year.

My other big take away from this chapter was activity 3.6.  As someone who was terrible at estimating and didn't really get it, this activity would have really helped me out.  I also like that it could be used for any operation with any number.  This is definitely a routine that is getting added to my math classes.  This is the activity where you give students a problem and have them tell you all they can about the answer.  I particularly liked the sentence starters for those students who are stuck:

• The answer will be more than ______ because _________
• The answer will be less than ______  because _________
• The answer will be between _______ and ________ because ________

### Chapter 4 Making Sense: Subtraction with Fractions

This chapter really got me thinking about the mistakes kids make when subtracting fractions.  It seems like it as the inverse of addition, subtraction should be very similar but it always seems to trip up a lot more kids.  Here are examples of some of the most common mistakes for fraction subtraction.

• Seeing the numerators and denominators as separate whole numbers and subtracting across both of them.

• Finding a common denominator but not making the corresponding change to the numerator.

• When presented with mixed number subtraction, ignoring the fractional part and just subtracting the whole numbers or vice versa.

• For a problem where the minuend is a whole number and the subtrahend is a fraction, thinking the whole number has the same denominator as the fraction.

• The borrowing issues! On the left the student does 4 - 2 but then ignores the fact that is should be 2/4 - 3/4 and switches them around.  On the right, the student borrows a whole and turns it into 10/4.
• Context problems!  When writing word problems, there is a thin line between a fraction subtraction a fraction multiplication problem.  I like to give my students this little printable pictured below, have them cut the stories apart on the thin black lines and have them sort them according to which operation they could use to solve them.  Many students think they are the same at first glance.  Look closely!  A few words changed make for a big difference in the operation and approach for solving the problem.  If you want to try these with your students, you can grab them from Google Drive.   Grab this freebie here!
What mistakes do you see students making with fraction subtraction?  Leave your response in the comments section below or head over to my Facebook page and let us know what you think!