DR. BETH BOS TEXAS STATE UNIVERSITY

CAMTJULY 2010

Fractions for ELL Learners

Learning the Language of Math Providing Linguistic Support to All Learners

Learning the Language of Math

• Pedagogy

1/3 x 15 =__________

2 Challenges

In our classrooms, we have students with:• Language Diversity

• Diversity in Mathematical Understanding

BICS (Basic Interpersonal Communication Skills)

CALP (Cognitive Academic Language Proficiency)

Part of the reason English learners struggle in mathematics is that rather than being language free, mathematics uses language that is a highly compressed form of communication where each word or symbol often represents an entire concept or idea. In a literature text, readers can comprehend a passage if they are familiar with 85%-90% of the words. The other words and their meanings can often be gleaned through context. Mathematics problems, on the other hand, generally require the student to understand nearly every word as there is seldom enough context provided with the problem to assist with unfamiliar words or concepts. Another problem that English learners encounter is that sometimes they recognize a word, but the meaning they know for the word is different from the intended meaning and therefore does not help them understand the problem. (14)

• Language minority students often appear to be English proficient yet perform poorly in content areas because they lack the proficiency in content specific vocabulary.

• This inhibits the development of academic skills.

• Students who lack English skills often find themselves farther & farther behind.

Acquiring the First Language

• We learn the names of common objects• The items are repeatedly introduced visually

& physically.• We learned through direct experiences with

concrete examples and were provided a context embedded environment.

Our second language is acquired in this way.

The language of math is acquired in this way.

4 Stages of Language Use in the Mathematics Classroom

10 Strategies for Helping English Language

Learners

• Ask questions and use prompts• Practice wait time• Modify teacher talk• Recast (interchange) mathematical ideas and terms• Pose problems that have familiar contexts• Connect symbols with words• Reduce the stress level in the room• Use “Think-Pair-Shares”• Use “English Experts”• Encourage students to “retell”

Discourse Structures

Emphasizing receptive and expressive roles for students as well as teachers.

• Pay deliberate attention to having students read, write, and say mathematical words & symbols

• Point out and discuss meaning of new words/ symbols and write them on the board.

• In addition to modeling the correct use of mathematical language, the teacher must also support the correct use by students.

• For some students, especially ELL, aural word recognition may not be accurate or quick enough for the students to be able to follow the discussion.

• If the talk is unclear or chaotic, the benefits of discourse are greatly reduced.

Teachers should use:Voice inflectionModified speechReduced pacePausesBody language

Irregularities

Syntax: Homonyms (some/sum; whole/hole) Synonyms (add, plus, combine, join, increased by) Semantic differences between and ordinary language (difference, odd)

Oral names for symbols and their written form:

Why doesn’t it make sense to write 50032?

Why don’t you write 3/5 as 3.5?

Why do you say x over 3 for x/3 and not x thirds?

Precision in our language and explicitness in our teaching is the key!

Terms that are used loosely or incorrectly are often the source

of student errors and misunderstandings.

What operation do you need to solve this?

Jason ate 3 times as many cookies as Mike. Jason ate 6 cookies. How many cookies did Mike eat?

6 ÷ 3 = x

How do you read this?15-12=

What is the difference between 15 and 12?

12 and how many more make 15? Students who can say the problem to themselves in flexible ways can choose a more efficient method.

VocabularyIntroduce vocabulary only after developing understanding of the related mathematical ideas

• In order for a vocabulary word to become part of one’s personal repertoire, it must be used in meaningful ways close to 30 times.

p. 351

Task: Come up with as many different ways to model 2/3.

Open Number Line

Creating a Fraction Kit

Task:You will create your own fraction kits. You will be creating one whole, halves, thirds, fourths, fifths, sixths, eighths, tenths, and twelfths.As you are completing this task, think about what children need to know and understand to be able to make their own kits.

half = mediofourth = cuartothird = tercerafifth = quintosixth = sextoseventh = séptimoeighth = octavoninth = novenotenth = décimoeleventh = undécimotwelfths = duodécimo

Comparison of Fractions

Consider ways to reason when comparing these fractions.

5/7 or 3/7 3/8 or 3/4

5/4 or 8/9

15/16 or 9/10 1 1/3 or 6/3

Conceptual Thought Patterns

More of the same-size parts.More of the same-size parts.

Same number of parts but different Same number of parts but different sizes.sizes.

More or less than one-half or one More or less than one-half or one whole.whole.

Distance from one-half or one whole Distance from one-half or one whole (residual strategy–What’s missing?)(residual strategy–What’s missing?)

½, 1, 0,

3/83/8 3/103/10 6/56/5

7/477/47 7/1007/100 25/2625/26

7/157/15 13/2413/24 14/30 14/30

16/1716/17 11/911/9 5/35/3

8/38/3 17/1217/12

RecordingRecording Sheet for ComparingSheet for Comparing

CompareCompare

1

2

3

4

1 3

2 4andCompareCompare

1

2

3

4

6644

<

3 1 2

4 2 8

1

2

3

4

64

<

By how much?By how much?

We can see that the size of this difference is 2 pieces of the newly doubled cut cake. Now all we need to do is to decide how to record this. Each cake is now shared with a total 8 shares. Using “share with” and “take” we have.

3 1 2

4 2 8

3 1

4 2

1

2

3

4

64

<

We can see a total of 10 pieces of he newly double cut cake. We have 1 full cake and 2 pieces left over . To record this using the faction symbol we can record “share with” and “take” as follows:

3 1 10

4 2 8

How much do we have in all?How much do we have in all?

Practice

Divide your paper into four sections.a. Comparison

b. Addition c. Subtraction ( subtract the smaller from

larger number)

d. Word Problem

2 1 1 2 2 11. 2. 3.

3 2 4 3 5 21 3 3 2

4. 5.3 4 5 3

MultiplicationMultiplication

Area Model: Let’s look at the example of 2 x 3 = 6. The sketch of this would be a rectangle 2 units in one direction and 3 in the other giving rise to an area of 6 square units. This would look like::

2

33 2 x 3 = 62 x 3 = 6

3 x 2 = 63 x 2 = 6

6/3 = 26/3 = 2

6/2 =36/2 =3This shows how this single sketch can be used to represent the relationships between and among four related facts (fact families).

222

The sketch of 2x3 showed a rectangle 2 units in one direction and 3 in the other. In a like fashion should show us a rectangle in one direction and n the other. First look at the ½ ..

1

2

1

2 1

2Now draw 3/4

1

2

3

4

3

4

x

1

2 3

41

21

2

Now draw thesharedportion

From the picture there are 3 small pieces which are found in the rectangle that is 1/2 in one direction and 3/4 in the other. We can also see that there are now a total of 8 pieces making up the full cake. So the total area we are describing in this multiplication problem is

. The number of small pieces we have, i.e. the numerator of our answer can be described by the rectangle that is 1 x 3. The total size of the cake i.e. the denominator of our answer, is described by the rectangle that is 2 x 4. This is identical to the procedure of multiplying numerator x numerator and denominator x denominator.

PracticePractice

2 3 21. 3 2.

3 4 51 2

3. 22 3

1

2

3

4

64

<

How many ¾ are in ½. We know that ¾ is larger than ½ so it will go in less than one.

How much in each group?How much in each group?

½ ÷ ¾

1

2

3

4

64

How many ¾ are in ½. We know that ¾ is larger than ½ so it will go in less than one. Therefore we have 4/6.

How much in each group?How much in each group?

½ ÷ ¾

1 2 3 4

5 6

Practice

2 1 1 21. 2. 2

4 4 3 3

3 1 3 23. 4. 2

4 8 4 3

What is Relational Thinking?

Children using relational thinking draw upon their knowledge of fundamental properties of number, operations and equations to analyze a problem in the context of a goal structure and then to simplify their progress towards this goal.

¾ + ½THINK

¾ + ¼ + ¼ = 1 ¼

For example

7/5 + 4/5 =

7 x 1/5 + 4 x 1/5 = (7 + 4) x 1/5

11/5

Where 7a + 4a = (7 + 4)a = 11a

Three one-thirds make a whole candy bar and that one candy bar divided among

three people yields one-third of a candy bar to each.

2 + 3 = (1 + 1) + 31/3 + 1/3 = 2/3

Two candy bars are shared among three

children..

Holly: four halves

Jeremy is making cupcakes. He wants to put ½ a cup of frosting on each cupcake. If he makes 4 cupcakes for

his birthday party, how much frosting will he use to frost all of

the cupcakes?

John represented each cup of frog food with a rectangle, then divided each rectangle in half and notated “1/2” on each half to show how

much food Mr. W’s frogs could eat in a day. He then counted these to arrive at 20 days.

Mr. W has 10 cups of frog food. His frogs eat ½ a cup of frog food a day. How long can he feed his frogs before his food runs out?

The choices were ½, ¼, ¾, and 3/8.

It takes ____ of a cup of sugar to make a batch of cookies. I have 5 ½ cups of sugar. How many batches

of cookies can I make?

If 8 x 3/8 = 3

1/3 x (8x 3/8) = 1/3 x 3

(1/3 x 8) 3/8 = 1

8/3 x 3/8 = 1

3cups + 1 ½ cups + 1 cup

(8groups + 4groups + 8/3groups) 3/8

14 2/3 x 3/8

8 groups of 3/8 is 3 cups4 groups of 3/8 is 1 ½ cups

1 more cup needed

Two thirds of a bag of coffee weighs 2.7 pounds. How much would a whole bag of coffee weigh?