Welcome to week 3 of our Beyond Invert and Multiply book study! Last week, we looked at decomposing numbers to help with fraction addition and examined the 6 mistakes kids often make when subtracting fractions and mixed numbers. That post got lost in cyber space for a few days but is back in action now so if you missed it, be sure to check it out!

Here is the posting schedule.

July 15th Part 2: Addition and Subtraction with FractionsJuly 22nd Part 3: Multiplication and Division with Fractions

July 29th Part 4: Discourse with Fractions

### Chapter 5 Developing Awareness: Multiplication and Division Problem Types

Several years ago when reading the Common Core standards, I stumbled upon a few tables in the glossary that I found so interesting. One was the table about the different problem types for addition and subtraction word problems that we talked about 2 weeks ago and the other was the table for the problem types for multiplication and division that is featured in this week's reading. Here is the table they are referring to:

If you need to see a larger or easy to print version of this table, check it out here! |

I printed this table and tacked it to my planning bulletin board and began making over my math class. I started at the fact level before moving up to multi digit multiplication and division and finally fractions and decimals. Now my students get to experience all 9 problem types at each level. Some problem types are harder than others especially if your students have never been exposed to that type of problem. You can read more about how I address these problem types here. I encourage all teachers to take a good look at their curriculum and see what problem types are not being represented. Make sure your fraction multiplication and division units contain a diverse group of problems!

### Chapter 6 Making Sense: Multiplication with Fractions

This chapter got me excited for teaching fractions this fall! I know I will be working with an intervention group of sixth graders right off this fall and there were so many ideas in here that made me think about this group. I still have 31 days before I am due back at school but this book is really making me think about heading back!

I loved the term "constructive struggling" used at the beginning of the chapter. Apparently this term is used in the book Faster Isn't Smarter which looks like a book I will definitely have to check out! I have seen some of students' best ideas come from constructive struggling yet it can be so hard for teachers to let kids struggle. I used to be that teacher who couldn't stand to see kids struggle and would jump in way to soon. This is something I worked on with peer conferencing a few years back and I still like to check in with colleagues when we are co-teaching about when we should jump in and when we should leave kids with a little disequilibrium.

The research around how kids learn fraction multiplication always interests me and it seems each time I read a new fraction book, I get a new take on how to teach it better. I really liked the ideas presented about teaching fraction multiplication (and multiplication in general) from a measurement standpoint. I always struggle to help kids understand how finding a fraction of a fraction translates to multiplication because I have never been able to connect it to whole number multiplication. On page 9, this paragraph made me stop in my tracks: "Given the problem 4 X 3, if one considers a multi-unit length of 4, iterated three times or a multi-unit length of 3 iterated four times, one arrives at 12. This could be thought of as "four iterations of length 3" or "three iterations of length 4" or the shorter "four of 3" or "three of 4."" There is my example of how the word of can be used in whole number multiplication. Right there! I can't wait to see how this will help me this year!

Next week we will be wrapping up this book study and I will be getting busy preparing for back to school! Look for a lot more blog posts again in August as I tackle the beginning of the year things and get back to teaching!

I love reading your posts about this book. Even though I teach second grade, I find the content of the book exciting. I will definitely pass your book study on to one of my former fifth grade teaching friends. I have just gotten in to reading your posts for this study--am going to also check out Faster Isn't Smarter. Thanks so much!

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