Elementary Math Elementary Math Support: Support: Computation with Computation with FractionsFractionsSession 8Session 8

April 4, 2013April 4, 2013

Thinking About FractionsThinking About Fractions

Premature attention to rules for fraction Premature attention to rules for fraction computation has a number of serious computation has a number of serious drawbacks. drawbacks. None of the rules help students think about None of the rules help students think about

the operations and what they mean.the operations and what they mean. Armed only with rules, student have no Armed only with rules, student have no

means of assessing their results to see if means of assessing their results to see if they make sense. they make sense.

What does NCTM say? What does NCTM say?

““The main focus on fractions and decimals in The main focus on fractions and decimals in grades 3-5 should be on the development of grades 3-5 should be on the development of number sense and informal approaches to number sense and informal approaches to addition and subtraction.”addition and subtraction.”

““In grades 6-8, students should expand their In grades 6-8, students should expand their skills to include all operations with fractions, skills to include all operations with fractions, decimals, and percents.” decimals, and percents.”

Developing Fraction Developing Fraction ConceptsConcepts

Begin with simple, contextual tasks.Begin with simple, contextual tasks. Connect the meaning of fraction Connect the meaning of fraction

computation with whole number computation with whole number computationcomputation

Let’s discuss: Let’s discuss: What does 2 ½ x ¾ mean?What does 2 ½ x ¾ mean? What does 2 x 3 mean? What does 2 x 3 mean?

Developing Fraction Developing Fraction ConceptsConcepts

Let estimation and invented methods Let estimation and invented methods play a key role in developing strategies.play a key role in developing strategies.

Should 2 ½ x ¾ be more than 1? Should 2 ½ x ¾ be more than 1? Should it be more or less than 3? Should it be more or less than 3?

Developing Fraction Developing Fraction ConceptsConcepts

Explore each of the operations using Explore each of the operations using models.models. Use a variety of modelsUse a variety of models Have students defend their solutions using Have students defend their solutions using

modelsmodels

Informal ExplorationInformal Exploration

Using nothing other than simple drawings, how Using nothing other than simple drawings, how would you solve this problem without setting it would you solve this problem without setting it up in the usual manner with common up in the usual manner with common denominators?denominators?

Paul and his brother were each eating the Paul and his brother were each eating the same kind of candy bar. Paul had ¾ of his same kind of candy bar. Paul had ¾ of his candy bar. His brother still had 7/8 of his candy bar. His brother still had 7/8 of his candy bar. How much candy did the two boys candy bar. How much candy did the two boys have altogether? have altogether?

Adding and Subtracting Adding and Subtracting FractionsFractions

The myth of common denominatorsThe myth of common denominators How could you solve the following How could you solve the following

problems w/o first finding common problems w/o first finding common denominators? What models might you denominators? What models might you use? use?

¾ + 1/8 ¾ + 1/8 2/3 + 1/22/3 + 1/2 ½ - 1/8 ½ - 1/8 1 ½ - ¾ 1 ½ - ¾

Next Step: Developing Next Step: Developing the Algorithm the Algorithm

Like Denominators: Like Denominators: Help children see that the top number counts, and the bottom Help children see that the top number counts, and the bottom

number is what is being counted. number is what is being counted.

Unlike Denominators: Unlike Denominators: Consider 5/8 + 2/4. Consider 5/8 + 2/4. Let students use pie pieces to get the result 1 1/8 using any Let students use pie pieces to get the result 1 1/8 using any

approach. approach. Note that the model shows two fractions that make one whole, Note that the model shows two fractions that make one whole,

with 1/8 remaining. with 1/8 remaining. Key Question: How can we change this problem into one of Key Question: How can we change this problem into one of

the easy ones where parts are the same? the easy ones where parts are the same? 5/8 + 2/4 is the same as 5/8 + 4/8. 5/8 + 2/4 is the same as 5/8 + 4/8.

Adding & Subtracting Adding & Subtracting Continued…Continued…

Next try some fractions where both Next try some fractions where both denominators need to be changeddenominators need to be changed 2/3 + ¼ 2/3 + ¼ Focus attention on rewriting the problem in a form Focus attention on rewriting the problem in a form

that is like adding “apples to apples”, where parts of that is like adding “apples to apples”, where parts of the fractions are the same.the fractions are the same.

Be sure students understand the new problem is the Be sure students understand the new problem is the same as the original problem. same as the original problem.

Demonstrate this point with models. (pg 268 for Demonstrate this point with models. (pg 268 for examples) examples)

Mixed NumbersMixed Numbers

Avoid layers fractions with yet another Avoid layers fractions with yet another rule.rule.

Include mixed numbers with all of your Include mixed numbers with all of your activities with addition and subtraction, activities with addition and subtraction, and let students solve them in ways that and let students solve them in ways that make sense to them. make sense to them.

It works best to work with whole numbers It works best to work with whole numbers first, then deal with the fraction parts. first, then deal with the fraction parts.

Multiplying Fractions Multiplying Fractions

When working with whole numbers, we would When working with whole numbers, we would say 3 x 5 means three sets of five. This is a say 3 x 5 means three sets of five. This is a good place to start. good place to start.

Simple story problems are a good way to Simple story problems are a good way to develop this concept. develop this concept. There are 15 cars in Michael’s toy collection. 2/3 of There are 15 cars in Michael’s toy collection. 2/3 of

the cars are red. How many red cars does Michael the cars are red. How many red cars does Michael have? have?

Wayne filled 5 glasses with 2/3 liter of soda in each Wayne filled 5 glasses with 2/3 liter of soda in each glass. How much soda did Wayne use? glass. How much soda did Wayne use?

Unit Parts without Unit Parts without SubdivisionsSubdivisions

You have ¾ of a pizza left. If you give 1/3 of You have ¾ of a pizza left. If you give 1/3 of the leftover pizza to your brother, how much of the leftover pizza to your brother, how much of a whole pizza will your brother get? a whole pizza will your brother get?

Gloria uses 2 ½ tubes of blue paint to paint the Gloria uses 2 ½ tubes of blue paint to paint the sky in her picture. Each tube holds 4/5 ounce sky in her picture. Each tube holds 4/5 ounce of paint. How many ounces did Gloria use? of paint. How many ounces did Gloria use?

(Example Models, pg 271)(Example Models, pg 271)

Subdividing Unit PartsSubdividing Unit Parts

When the pieces have to be subdivided When the pieces have to be subdivided into smaller parts, the problems become into smaller parts, the problems become more challenging: more challenging:

Zack had 2/3 of the lawn left to cut. After Zack had 2/3 of the lawn left to cut. After lunch, he cut ¾ of the grass he had left. lunch, he cut ¾ of the grass he had left. How much of the whole lawn did Zack cut How much of the whole lawn did Zack cut after lunch? after lunch?

Developing the AlgorithmDeveloping the Algorithm

Shift from contextual problems to computation Shift from contextual problems to computation after students have had ample time to explore after students have had ample time to explore fraction multiplication, and modeling. fraction multiplication, and modeling. Let’s use a model for 3/5 of a set of ¾ Let’s use a model for 3/5 of a set of ¾ Draw a square representing ¾ (lines in one Draw a square representing ¾ (lines in one

direction)direction) Using the same square (lines in a different direction, Using the same square (lines in a different direction,

show 3/5. show 3/5. What do you notice? What do you notice? What’s shaded, partly shaded, not shaded?What’s shaded, partly shaded, not shaded? What do you notice about rows and columns? What do you notice about rows and columns?

Developing the AlgorithmDeveloping the Algorithm

There are three rows and three columns in the There are three rows and three columns in the “product” (double shaded area), or 3 x 3 parts.“product” (double shaded area), or 3 x 3 parts.

The whole is now five rows, and 4 columns, so The whole is now five rows, and 4 columns, so there are 5 x 4 parts in the whole.there are 5 x 4 parts in the whole.

Product = 3/5 x 3/4 = Number of parts in product = 3 x 3 = 9Product = 3/5 x 3/4 = Number of parts in product = 3 x 3 = 9 Kind of parts Kind of parts 5 x 4 = 20 5 x 4 = 20

Dividing FractionsDividing Fractions

““Invert the divisor, and multiply is Invert the divisor, and multiply is probably one of the most mysterious probably one of the most mysterious rules in elementary math. We want to rules in elementary math. We want to avoid this mystery at all costs! avoid this mystery at all costs!

Start with the division of whole numbers.Start with the division of whole numbers. There are two meanings of division: There are two meanings of division:

partition and measurement. partition and measurement.

Informal Exploration: Informal Exploration: Partition ConceptPartition Concept

Partition problems can be both sharing Partition problems can be both sharing problems or rate problems. problems or rate problems. 24 apples to be shared among 4 friends24 apples to be shared among 4 friends If you walk 12 miles in 3 hours, how many If you walk 12 miles in 3 hours, how many

miles do you walk per hour? miles do you walk per hour?

All partition problems askAll partition problems ask How much is one? How much is one?

Examples:Examples:

Note: As you work through these problems, Note: As you work through these problems, think about, “How much is a whole?” and “How think about, “How much is a whole?” and “How much for one?”much for one?”

Cassie has 5 ¼ yards of ribbon to make three bows Cassie has 5 ¼ yards of ribbon to make three bows for birthday packages. How much ribbon should for birthday packages. How much ribbon should she use for each bow if she want to use the same she use for each bow if she want to use the same length of ribbon for each one? length of ribbon for each one?

Mark has 1 ¼ hours to finish his three household Mark has 1 ¼ hours to finish his three household chores. If he divides his time evenly, how many chores. If he divides his time evenly, how many hours can he give to each? hours can he give to each?

Fractional DivisorsFractional Divisors

Note: It is still enormously helpful to ask “How Note: It is still enormously helpful to ask “How much is one?”much is one?” (pg 275) (pg 275)

Elizabeth bought 3 1/3 pounds of tomatoes for Elizabeth bought 3 1/3 pounds of tomatoes for $2.50. How much did she pay per pound? $2.50. How much did she pay per pound?

Aidan found out that if she walks really fast during Aidan found out that if she walks really fast during morning exercises, she can cover 2 ½ miles in ¾ of morning exercises, she can cover 2 ½ miles in ¾ of an hour. She wonders how fast she is walking in an hour. She wonders how fast she is walking in miles per hour. miles per hour.

Informal Exploration: Informal Exploration: Measurement ConceptMeasurement Concept

13 divided by 3 means “How many sets of 3 13 divided by 3 means “How many sets of 3 are in 13?”are in 13?”

If you have 13 quarts of lemonade, how many If you have 13 quarts of lemonade, how many canteens holding 3 quarts each can you fill?canteens holding 3 quarts each can you fill?

You are going to a birthday party. From Ben and You are going to a birthday party. From Ben and

Jerry’s ice cream factory, you order 6 pints of ice Jerry’s ice cream factory, you order 6 pints of ice cream. If you serve ¾ of a pint of ice cream to each cream. If you serve ¾ of a pint of ice cream to each guest, how many guests can be served? guest, how many guests can be served?

More Examples:More Examples:

Farmer Brown found that he had 2 ¼ Farmer Brown found that he had 2 ¼ gallons of liquid fertilizer concentrate. It gallons of liquid fertilizer concentrate. It takes ¾ gallon to make a tank of mixed takes ¾ gallon to make a tank of mixed fertilizer. How many tanks can he mix?fertilizer. How many tanks can he mix?

(Hint: Think about how many sets of ¾ (Hint: Think about how many sets of ¾

are in a set of 9/4.) are in a set of 9/4.)

Developing AlgorithmsDeveloping Algorithms

There are actually two different There are actually two different algorithms for the division of fractions.algorithms for the division of fractions.

1) To divide fractions, first get common 1) To divide fractions, first get common

denominators, and then divide numerators.denominators, and then divide numerators.

(pg 277 for picture model) (pg 277 for picture model)

Algorithms ContinuedAlgorithms Continued

The second algorithm for division of fractions is The second algorithm for division of fractions is the “Invert and Multiply” algorithm.the “Invert and Multiply” algorithm. A small pail can be filled to 7/8 full using 2/3 of a A small pail can be filled to 7/8 full using 2/3 of a

gallon of water. How much will the pail hold if filled gallon of water. How much will the pail hold if filled completely?completely? Ignore for a moment that the amount of water is 2/3 gallon. Ignore for a moment that the amount of water is 2/3 gallon. Draw a picture of the bucket that is 7/8 full. Label the Draw a picture of the bucket that is 7/8 full. Label the

water as 2/3. water as 2/3. Remember, our task is to find out how much is in the whole Remember, our task is to find out how much is in the whole

bucket, or 8/8.bucket, or 8/8. What do we need to do? What do we need to do?

Inverse Bucket Analogy!Inverse Bucket Analogy!

Because the water is seven of the eight parts full, Because the water is seven of the eight parts full, dividing the water by 7 and multiplying that dividing the water by 7 and multiplying that amount by 8 solves the problem. amount by 8 solves the problem.

Therefore, take 2/3 divided by 7, and multiply by Therefore, take 2/3 divided by 7, and multiply by 8. 8.

This is the same as 2/3 x 8/7 (Viola! Multiplying This is the same as 2/3 x 8/7 (Viola! Multiplying by the inverse by the inverse ) )