Beyond Invert and Multiply Book Study Part 4

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We have made it to our fourth and final week of this book study!  It has been a great summer learning about fractions and I am excited to be back in the classroom putting what I have learned to good use.  If you have missed previous posts, you can check them out by clicking on the links below

July 8th Part 1: What is a Fraction?

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 8 Developing Awareness: Six Strategies for Fostering Student Talk About Fractions

This section of the book steps away from the ideas about developing fraction reasoning and instead focuses on how to structure your class so that you can provide opportunities for kids to talk about fractions.  A lot of the strategies presented in this section are good teaching strategies that are also very beneficial classroom practices in general.  

Strategy 1: Using Strategic Tasks

If you want your students to build conceptual knowledge about fractions rather than just learning the steps of a formalized procedure, you need to make sure you are giving them tasks that have something worth talking about.  They might be open ended, able to be solved in multiple ways and will make your students think.  Selecting a task that they can use some prior knowledge to build up from is also a good idea.  I think the ability to pick strategic tasks comes with time and practice.  If the task you pick isn't working the way you want it to, try something else!

Strategy 2: Creating Records of Thinking

Having students record their thinking or ideas in some way is a great way to share their thinking with classmates and to help organize their thoughts.  I used to have kids copy down their thinking on an overhead transparency or on the board when it was time to share ideas.  Now that I have a document camera in every room, this has gotten so much easier to share what kids are thinking.  Some kids might need help recording their thinking when they are first starting out but it is a great strategy for building student talk.

Strategy 3: Building Visual Models

Get great math tools into your students hands!  Fraction manipulatives such as fraction strips, or Cuisenaire rods as well as number lines are a great way to build conceptual understanding and engage kids in a hands on way.  

Strategy 4: Reasoning with Benchmarks and the Number Line

I love how using benchmarks helps kids think about their calculations and the reasonableness of their answers.  Thinking about how close a number is to 1/2 or 1 can really help kids see if their answers make sense.  This is estimation at its finest.  

Strategy 5: Using Talk Moves

Out of all the talk moves described in this chapter, wait time was the one I needed to work on the most.  I was one of those teachers who would ask a question and expect an immediate response.  Since purposefully working on my wait time, I have noticed a huge difference in the number of kids who are ready to participate.  For a lot more about talk moves, check out this book.  

Strategy 6: Asking Students to Turn and Talk

This is one of my favorite strategies as a teacher and as a learner.  When I have a chance to talk to someone about my ideas, it helps me organize them, develop them and be better able to articulate my thinking.  I find the same is true for my students and I use turn and talk many times each day.  

How do you incorporate student talk in your fraction lessons?

Monday Made It: Teacher Essentials Kit

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Back to school fever seems to have gripped the nation!  I know some schools go back next week but I have 3 more blissful weeks left!  However, I am already headed into back to school mode and spent this morning shopping and creating.  I have many more made its to accomplish before back to school but today I want to share with you one I managed to get done today.  When you are done head over to 4th Grade Frolics for a lot of back to school inspiration.  

I was cruising around the dollar store this morning getting my back to school organizational essentials.  I was thinking about all the things I needed to get done at home and in my classroom before I have even less time.  One thing I had been thinking about was creating an emergency box of essentials to keep in my classroom.  I have had various supplies stuffed into my filing cabinet for years but I really wanted to do this in a more organized fashion.  Because I have already dropped a lot of my own cash on classroom essentials, I wanted to keep this low budget.  I picked all of this stuff up at the Dollar Tree and the grand total was $10 and about 10 minutes of time.

Here is the list of essentials I included:

- Plastic box to hold it all

- Mini sewing kit

- Tylenol

- Tums

- Cough drops

- Travel toothbrush & toothpaste

- Mini hairbrush & hair ties

- Sugar free gum

- 2 kinds of hair clips

- Mini deodorant 

This covers the essentials for me!  I will be set when I leave the house without brushing my hair or when I finish my coffee on the way to work and really need to brush my teeth.  It will also be great to have a sewing kit because I have been known to rip off buttons or seams while at work.  I also but in the essentials for when I am not feeling well.  I am feeling much more prepared for back to school!  This would also make an excellent gift for a teacher friend.  I made one for a friend who is starting her first teaching job.  It includes the same things I put in mine, and I just tied a little bow around it for presentation.  

What are you working on this week?  

Beyond Invert and Multiply Book Study Part 3

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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 8th Part 1: What is a Fraction?

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 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! 

Beyond Invert and Multiply Book Study Part 2

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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 8th Part 1: What is a Fraction?

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!

Beyond Invert and Multiply Book Study Part 1

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Welcome to the first installment of our Beyond Invert and Multiply book study.  You can order this book at Math Solutions or on Amazon.  There is a great video introduction to this book here if you want to learn more about it.  

This book is written about fraction operations and is a follow up to Beyond Pizza and Pies.  I have received several emails and Facebook messages from folks asking which book they should get if they have to pick just one.  My answer to that is that if you teach grades 2-3, go with Beyond Pizzas and Pies.  If you teach grades 4 and up you REALLY should have both.  If your students lack the foundation knowledge covered in Beyond Pizzas and Pies, many of the activities suggested in Beyond Invert and Multiply will be a struggle for your students.  

Chapter 1: Fractions As Numbers

I love the way this chapter starts off with the classroom scenario of the human number line.  This is a great way to introduce kids to big ideas about number lines and doing it in a big way like this makes it very memorable.  I feel like kids really get the idea of equal spacing when doing a human number line and I think this is a great way to review what number lines are before moving into working with fractions on a number line.

The chapter lays out these 5 foundational understandings about fractions that kids need in order to compute with understanding.  These are the things that you will find your students tripping up on when learning fraction operations.  These are the things that are worth devoting some class time to before jumping into fraction operations.  They are all big conceptual ideas and together form the foundation for fraction understanding. 

• Fractions are numbers and follow the same rules and properties as other rational numbers. 
• The unit fraction is the building block of fractions.
• Fractions can be decomposed and recomposed in infinite ways.
• Equivalent fractions represent different ways of naming the same value.
• All rational numbers can be expressed as fractions in the form a/b, where b does not equal 0. 

The other thing that really struck me in this chapter was how important number lines are.  Over and over again you see how they can help kids see fractions as numbers.  The activities and DVD clips in this section do such a great job capturing the power of the number line.  It is hard to believe that less than 10 years ago, I had never used a number line for fractions in my own mathematics or with my students.  

Chapter 2: Addition and Subtraction with Fractions

This chapter dives headfirst into the research on problem types for addition and subtraction word problems.  These problem types are used with whole numbers and should also be used with fractions. If you use the Common Core for your standards, these problem types are detailed in the glossary.  I wish this table was right in the standards themselves because many people miss it.  I have included a snapshot below!  If it is out of focus or hard for you to read, you can click here to see a better copy.  

If your students have never experienced these problems with whole numbers, I would be hesitant to introduce them with fractions.  I think it is worth the time to go back and do some of these problem types with whole numbers before expecting kids to do them with fractions.  When my school first started paying attention to all the different problem types, we started with the primary students on single digits, moved up to double digit computation and then finally used the different problem types with fractions.

Which problem type is this?  Feel free to peek at the table above! Grab these cards here:) 

How would your students do on this one?  Would they use subtraction or a missing addend? Which problem type is this? 

I think there is a lot of value in sitting down and writing some of each of the problem types for your students to use.  It can be challenging at first, but gets much easier with time.  If you want to explore some problems that have already been written, you can check these out!
Addition and Subtraction Facts
Double Digit Addition and Subtraction
Fraction Addition and Subtraction

There was also a chapter on the different problem types in Children's Mathematics.  You can read my thoughts on that here!

Your turn!  Please tell us how you are doing with fractions in the comments below and what you thought about this week's reading.  Has your school addressed the different problem types for addition and subtraction?  How are your students doing at developing the 5 foundational understandings for fractions?  

Beyond Pizza and Pies Book Study Part 4

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You made it!  This is the fourth and final week of our Beyond Pizza and Pies book study.  Next Wednesday we will be jumping into the companion book, Beyond Invert and Multiply.  You can order this book at Math Solutions or on Amazon.  There is a great video introduction to this book here if you want to learn more about it.  

If you missed the previous posts, you can check them out by clicking on the links below! 

June 10th Chapters 1&2
June 17th Chapters 3&4
June 24th Chapters 5&6
July 1st Chapters 7&8

Chapter 7: The Multiple Meanings of Fractions

When I started teaching, I was really good at using the area model.  Everything we did related to fractions was all about the area model.  This chapter focuses on the many different ways we see fractions in life and on the three different models kids should be seeing for fractions in the classroom.  Here are the 3 kinds of models kids should be seeing in the classroom.

Area Model
This is the model that almost all programs include.  It is looking at partitioning and finding fractions of shapes like circles, rectangles and squares.  Each shape has its advantages and drawbacks but I find that my students get the most mileage out of the rectangle.  Circles can be very limiting because they are so hard to partition equally.  I also like rectangles because if you partition them vertically it leads nicely to transitioning to a number line.

Set Model
This is when you look at finding fractions of a group of things.  Find half of the kids in the class or find three-fourths of the package of markers.  Having experience with set models really helps kids when you get to questions involving fractions and whole number multiplication such as what is two- fifths of 25?

Number Line Model
Fractions are numbers and it just makes sense to think about them on a number line.  I love how the number line model leads to great questions about equality and the density of fractions (the idea that there are always more numbers between any two numbers).

The power in these models is really using them all and moving between them flexibly.  As McNamara states, "helping students understand the meaning of fractions in different contexts builds their understanding of the relevant features of different fraction representations and the relationships between them."

I love the activities presented in this chapter and always try to make sure I have some version of 7.1 included in each grade that has a fraction unit.  The idea of connecting the math they are learning to the real world is so important and these kinds of lessons really help.  Lesson 7.3, where students write fraction problems that can be solved with various models looks great!  I have never done a lesson like this during the fraction unit and I think it would make a great addition.  I love how this book is helping me refine my plans for next year.

Chapter 8 Comparing Fractions: Do You Always Need a Common Denominator? 

I think the only way I ever compared fractions in elementary school was with a common denominator.  Even when I began teaching, I really didn't have a good understanding of other ways kids (and adults!) might compare fractions.  As part of my journey to learn more about teaching math, I have read a lot about these ideas and now I have a good understanding about different strategies for comparing fractions.  I love presenting kids with various fraction activities and watching them develop and refine these strategies for themselves.  It is always so fun to watch!

I was surprised by the "What's the Research?" section of this chapter.  It is amazing how few kids could estimate the answer to a fraction addition problem.  This is a problem I would love to take back to school with me in the fall and give to the sixth graders.  I think it would be a good way to check in with how we are doing as a school on our quest to teach kids how to think about math rather than just doing procedures.  

I can't wait to read your thoughts on this week's book study!  Remember to come back next week for the second part of our summer focus on fractions.

If you are looking for more ideas about teaching fractions, check out this blog hop on squishing fraction misconceptions!  Just click on the links at the bottom of each post.