Amy LeHew Elementary Math Facilitator Meeting October 2012

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Solutions?

Last time we looked closely at types of visual fraction models and considered the connection between the task and visual model used to solve it.

What visual fraction model is used by most students in our schools?

What visual fraction model is used the least?

Take a look at the 3 rd grade fraction standards   Highlight the phrase “visual fraction model” everywhere it appears.

Underline the phrase “number line” everywhere it appears.

Develop understanding of fractions as numbers.

1. Understand a fraction 1/ as the quantity formed by 1 part when equal parts; understand a fraction 2. Understand a fraction as a number on the number line; represent fractions on a number line diagram. Represent a fraction 1/ b b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b a / b a whole is partitioned into as the quantity formed by a parts of size 1/ equal parts. Recognize that each part has size 1/ b b . 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/ Recognize that the resulting interval has size a / b from 0. and that its endpoint locates the number a / b on the number line. 3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Turn and make a statement to your partner about something you learned while highlighting/underlining.

Support understanding of important properties of fractions ◦ ◦ ◦ ◦ Numerical unit Relationship between whole numbers and fractions Density of rational numbers (infinite # between any two) A number can be named infinitely many ways http://lmr.berkeley.edu/docs/Pt3-Ch13NCTM%20Yearbook07-4.pdf

 ◦ Most widely used fraction model? Area Partition pizzas, brownies, etc.  ◦ Limitations of area models Count pieces without attending to the whole (don’t distinguish between fractional part of a set from continuous quantity (area)).

How can number lines help students understand fraction concepts that are often obscured by area models?

Of the students who were successful, they counted “two-sixths” for the point depicted in the figure below: it is “six spaces and that’s two.”

   One-third is three pieces and one thing There isn’t another fraction name Don’t know

1 1 15 or 9 How do you know?

“The smaller the denominator, the bigger the fraction”

Read the classroom scenario on pages 20-22 ◦ When you finish, reflect silently on the following:   Do student in your building consider fractions as numbers? Do teachers?

What other generalizations do students and teachers make about fractions? Are some of these generalizations helpful?

Consider the following two statements. Declare Always True, Sometimes True If sometimes true, show an example and a counter example.

1. The larger the denominator, the smaller the fraction.

2. Fractions are always less than one.

3. Finding a common denominator is the only way to compare fractions with different denominators.

Which is larger 5 / 6 OR 7 / 8 ?

5 / 6 The smaller the number the bigger the pieces.

5 / 6 If the denominator is smaller, the piece is bigger 5 / 6 The piece is bigger than 1 / 1 8 / 6 7 / 8 Since 8 is bigger than 6 and 7 is bigger than 5 This one ( 7 / 8 because it has more pieces )

  How do we ensure students make meaning of the numerator and denominator?

How can we make sure students think of fractions as numbers

Using your blank number line sheet and Cuisenaire rods, label the first line in halves 1 2 nd 3 rd 4 th 5 th line in Thirds line in Fourths line in Sixths line in Twelfths 0

What number is halfway between zero and one-half?

What number is one-fourth more than one half?

What number is one-sixth less than one?

What number is one-third more than one?

What number is halfway between one-twelfth and three-twelfths?

What would you call a number that is halfway between zero and one-twelfth?

    How might this impact student’s understanding of fractions as numbers on a number line? (using the Cuisenaire rods to make a number) How does a number line diagram help students make meaning of the numerator and denominator?

This activity found on page 23-26 Look at page 27+ for using 2 wholes

Time has run out for the tortoise and the hare!

The Hare jumped five-eighths of the way The tortoise inched along to three-fourths of the way.

Who is winning?

How would constructing a number line diagram to solve this task help students make meaning of the numerator and denominator?

With fractions? Yea or Na

Use < = > to compare the following sum: (do not add) *You must represent your thinking on a number line diagram.

1 / 2 + 1 / 4 ______ 1 / 3 + 1 / 5

3.NF Closest to 1/2 3.NF Comparing sums of unit fractions 3.NF Find 1 3.NF Find 2/3 3.NF Locating Fractions Greater than One on the Number Line 3.NF Locating Fractions Less than One on the Number Line 3.NF Which is Closer to 1?

  Bring 3 rd grade unit 7: Finding Fair Shares Bring 4 th grade unit 6: Fraction Cards and Decimal Squares