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&2June 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.
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 ModelFractions 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.
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