Journey through the CCSSM

Recall From Yesterday… 3.NF.1 Understand a fraction as the quantity formed by 1 part when a whole is partitioned into b equal parts. Understand as the quantity formed by parts of size .

Recall Also… 3.NF.3d--Compare two fractions with the same numerator or the same denominator by reasoning about their size. … Inspired by Greg Tang, NCTM Conference, 2012

Compare… What is the size of each piece? and What is the size of each piece? What about the number of pieces? So which fraction is larger? Inspired by Greg Tang, NCTM Conference, 2012

Journey through the CCSSM 3.NF.2 - Understand a fraction as a number on a number line; represent fractions on a number line diagram.         …..shift to LINEAR Why? Inspired by Greg Tang, NCTM Conference, 2012

2 Dimensions How can we compare the figures below? These shapes have the same area. But that’s very hard to “see” in this model. Equivalence in area is a pretty deep concept and can be difficult to perceive at any level. So area models bring a level of complexity to a task.. .may be move confusing than helpful for some kids. Inspired by Greg Tang, NCTM Conference, 2012

2 Dimensions How can we compare the figures below? These shapes have the same area. But that’s very hard to “see” in this model. Equivalence in area is a pretty deep concept and can be difficult to perceive at any level. So area models bring a level of complexity to a task.. .may be move confusing than helpful for some kids. Inspired by Greg Tang, NCTM Conference, 2012

To use the Area Model of Fractions You Must: Know what area is, Identify the area of the part, Identify the area of the whole, Compare the two areas by direct or indirect measurement Peter Gould Powerpoint 2012 - http://www.curriculumsupport.education.nsw.gov.au/primary/mathematics/elni/num_forum/fraction_cont.pdf

Traditional Fraction aCTIVITIES Ask… how are kids likely thinking about this visual?

Traditional Fraction Activities How about this? Area or Counting?

1 Dimension How can we compare these figures? It’s much easier to make 1 dimensional comparisons! Kids often have much more experience with length. It’s not that we avoid the area model, but we should respect that fraction concepts will often be easier to develop in a linear setting. It is MUCH Easier to make the comparison in 1 Dimension!

Base 10 connections Connections… do kids really see the 1000 piece as made up of 1000. Wouldn’t linear be more convincing.

Journey through the CCSSM 3.NF.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram. a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

1 2 Bring out here that kids need to see a point on the numberline as representing a number whose value is it’s distance from zero. In particular, 2/3 can be though of as 2 iterations of a length that is 1/3 long. So it is both a single point and a distance. (i.e. a “dot” at 2/3 and the length represented by the yellow.

IES Practice Guide IES guide for fractions is available online. (Link will be posted on Marti site). We will probably do more with it in the summer PD. But just wanted to let them know the research basis for the standards and for our work in Marti.

IES Practice Guide Recommendation 2 Help students recognize that fractions are numbers and that they expand the number system beyond whole numbers. Use number lines as a central representational tool in teaching this and other fraction concepts from the early grades onward.

Focus on Fractions Book Consider Big Idea “Modeling is a means to the mathematics, not the end” (page 1) …..What does it mean to you? Why should students interact with a variety of models?

Focus on Fractions Book Reflect on your own instruction using table 1.2 on page 25

CCSSM – what about 4th grade? 4.NF.2 – Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. ….

Compare… What can we say about the size of the pieces? and What can we say about the size of the pieces? What about the number of pieces? So which fraction is larger? Which missing pieces is smaller (1/7)… so 6/7 is closer to one… therefore larger. Although this comparison isn’t explicitly required in 3rd grade, it follows naturally from the understandings expected in 3rd grade. IN this case, students should see that 3/5 is less than one and 8/7 is more than one. Inspired by Greg Tang, NCTM Conference, 2012

Compare… and What benchmark could help us compare? Benchmark to 1 Inspired by Greg Tang, NCTM Conference, 2012

Compare… and What benchmark could help us compare? Benchmark to ½ Inspired by Greg Tang, NCTM Conference, 2012

Compare… and Find common numerator .... 6..... So 6/68 is larger because the size of the pieces would be larger!) and Inspired by Greg Tang, NCTM Conference, 2012