Understanding Fraction Representations

Group 2 Session 2

Amy LeHewElementary Math Facilitator Meeting

February2013

Cut a sentence strip so that it is 5/6 units long.

Bring the ends of the sentence strip together to fold the strip of paper in half.

Label this point.

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.

Visual Fraction Models

Take a look at the 3rd grade fraction standards

Highlight the phrase “visual fraction model” everywhere it appears.

Underline the phrase “number line” everywhere it appears.

Numbers and Operations with Fractions

Develop understanding of fractions as numbers.1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal

parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. 2. Understand a fraction as a number on the number line; represent fractions on a number line diagram. 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.

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.

3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

b. 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.

c. 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.

d. 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.

Numbers and Operations with Fractions

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)).

Area Models

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

What visual fraction model is used the least?

In CMS

Most students were successful identifying whole numbers on a variety of number lines.

Of the students who were successful, they counted

Very few were successful in identifying fractions on a number line

“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

Most had difficulty generating equivalent names for the same point

Lay out 2-3 examples of number line work from your school. Label them 3rd, 4th, or 5th

Gallery Walk

What did this look like in CMS

How can number lines help students understand fraction concepts that are often

obscured by area models?

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 wayshttp://lmr.berkeley.edu/docs/Pt3-Ch13NCTM%20Yearbook07-4.pdf

Number Lines

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?

Classroom Scenario

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.

Always True? Sometimes True?

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 lines as indicated below

1st line in Halves2nd line in Fourths3rd line in Sixths5th line in Twelfths

Construct a Number Line

0 1

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?

Answer With A Partner

How might this impact student’s understanding of fractions as numbers on a number line? (using the Cuisenaire rods to identify fractions on a number line)

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

From Area to Linear

http://www.telegraph.co.uk/news/newsvideo/weirdnewsvideo/9246505/Americas-tallest-man-gets-measured-for-size-21-trainers.html

Building Up from a Unit Fraction

Inch Bricks

Amy plans to build an enclosure for her great danes that is 1/8 of a mile wide and 1/4 of a mile long. Her daughter argues, “that won’t be enough room! And proves her point by showing that…

Reading Group

Carefully cut out a strip of paper that has a length of 5/6

- Bring the ends of the strip together to fold the strip of paper in half. How long is half of the strip? Use your strip to mark this point on the number line.

- What two numbers can you multiply to find the length of half the strip? Write an equation to show this.

Unfold your paper strip so that you start with 5/6 again.

Now fold the strip of paper in half and then in half again.

i. How long is half of half of the strip? Use your strip to mark this point on the number line.

ii. What numbers can you multiply to find the length of half the strip? Write an equation to show this

http://s3.amazonaws.com/illustrativemathematics/illustration_pdfs/000/000/965/original/illustrative_mathematics_965.pdf?1343856895

How many 1" x 1" square tiles can you fit into a 12" by 8" gridHow many 1" x 1" square tiles can you fit into 3/4 of a 12" by 8" gridHow many 1/2" x 1/2" square tiles can you fit into a 7 1/2" x 9 1/2" gridHow many 1/4" x 1/2" rectangular tiles can you fit into a 10 1/2" x 8 1/2" grid

5.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)

Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.