Elementary Math Support: Computation with Fractions Session 8 April 4, 2013

Thinking About Fractions  Premature attention to rules for fraction computation has a number of serious drawbacks.  None of the rules help students think about the operations and what they mean.  Armed only with rules, student have no means of assessing their results to see if they make sense.

What does NCTM say?  “The main focus on fractions and decimals in grades 3-5 should be on the development of number sense and informal approaches to addition and subtraction.”  “In grades 6-8, students should expand their skills to include all operations with fractions, decimals, and percents.”

Developing Fraction Concepts  Begin with simple, contextual tasks.  Connect the meaning of fraction computation with whole number computation  Let’s discuss:  What does 2 ½ x ¾ mean?  What does 2 x 3 mean?

Developing Fraction Concepts  Let estimation and invented methods play a key role in developing strategies.  Should 2 ½ x ¾ be more than 1?  Should it be more or less than 3?

Developing Fraction Concepts  Explore each of the operations using models.  Use a variety of models  Have students defend their solutions using models

Informal Exploration  Using nothing other than simple drawings, how would you solve this problem without setting it up in the usual manner with common denominators?  Paul and his brother were each eating the same kind of candy bar. Paul had ¾ of his candy bar. His brother still had 7/8 of his candy bar. How much candy did the two boys have altogether?

Adding and Subtracting Fractions  The myth of common denominators  How could you solve the following problems w/o first finding common denominators? What models might you use?  ¾ + 1/8 2/3 + 1/2  ½ - 1/8 1 ½ - ¾

Next Step: Developing the Algorithm  Like Denominators:  Help children see that the top number counts, and the bottom number is what is being counted.  Unlike Denominators:  Consider 5/8 + 2/4.  Let students use pie pieces to get the result 1 1/8 using any approach.  Note that the model shows two fractions that make one whole, with 1/8 remaining.  Key Question: How can we change this problem into one of the easy ones where parts are the same?  5/8 + 2/4 is the same as 5/8 + 4/8.

Adding & Subtracting Continued…  Next try some fractions where both denominators need to be changed  2/3 + ¼  Focus attention on rewriting the problem in a form that is like adding “apples to apples”, where parts of the fractions are the same.  Be sure students understand the new problem is the same as the original problem.  Demonstrate this point with models. (pg 268 for examples)

Mixed Numbers  Avoid layers fractions with yet another rule.  Include mixed numbers with all of your activities with addition and subtraction, and let students solve them in ways that make sense to them.  It works best to work with whole numbers first, then deal with the fraction parts.

Multiplying Fractions  When working with whole numbers, we would say 3 x 5 means three sets of five. This is a good place to start.  Simple story problems are a good way to develop this concept.  There are 15 cars in Michael’s toy collection. 2/3 of the cars are red. How many red cars does Michael have?  Wayne filled 5 glasses with 2/3 liter of soda in each glass. How much soda did Wayne use?

Unit Parts without Subdivisions  You have ¾ of a pizza left. If you give 1/3 of the leftover pizza to your brother, how much of a whole pizza will your brother get?  Gloria uses 2 ½ tubes of blue paint to paint the sky in her picture. Each tube holds 4/5 ounce of paint. How many ounces did Gloria use?  (Example Models, pg 271)

Subdividing Unit Parts  When the pieces have to be subdivided into smaller parts, the problems become more challenging:  Zack had 2/3 of the lawn left to cut. After lunch, he cut ¾ of the grass he had left. How much of the whole lawn did Zack cut after lunch?

Developing the Algorithm  Shift from contextual problems to computation after students have had ample time to explore fraction multiplication, and modeling.  Let’s use a model for 3/5 of a set of ¾  Draw a square representing ¾ (lines in one direction)  Using the same square (lines in a different direction, show 3/5.  What do you notice?  What’s shaded, partly shaded, not shaded?  What do you notice about rows and columns?

Developing the Algorithm  There are three rows and three columns in the “product” (double shaded area), or 3 x 3 parts.  The whole is now five rows, and 4 columns, so there are 5 x 4 parts in the whole.  Product = 3/5 x 3/4 = Number of parts in product = 3 x 3 = 9  Kind of parts 5 x 4 = 20

Dividing Fractions  “Invert the divisor, and multiply is probably one of the most mysterious rules in elementary math. We want to avoid this mystery at all costs!  Start with the division of whole numbers.  There are two meanings of division: partition and measurement.

Informal Exploration: Partition Concept  Partition problems can be both sharing problems or rate problems.  24 apples to be shared among 4 friends  If you walk 12 miles in 3 hours, how many miles do you walk per hour? All partition problems ask  How much is one?

Examples:  Note: As you work through these problems, think about, “How much is a whole?” and “How much for one?”  Cassie has 5 ¼ yards of ribbon to make three bows for birthday packages. How much ribbon should she use for each bow if she want to use the same length of ribbon for each one?  Mark has 1 ¼ hours to finish his three household chores. If he divides his time evenly, how many hours can he give to each?

Fractional Divisors  Note: It is still enormously helpful to ask “How much is one?” (pg 275)  Elizabeth bought 3 1/3 pounds of tomatoes for $2.50. How much did she pay per pound?  Aidan found out that if she walks really fast during morning exercises, she can cover 2 ½ miles in ¾ of an hour. She wonders how fast she is walking in miles per hour.

Informal Exploration: Measurement Concept  13 divided by 3 means “How many sets of 3 are in 13?”  If you have 13 quarts of lemonade, how many canteens holding 3 quarts each can you fill?  You are going to a birthday party. From Ben and Jerry’s ice cream factory, you order 6 pints of ice cream. If you serve ¾ of a pint of ice cream to each guest, how many guests can be served?

More Examples:  Farmer Brown found that he had 2 ¼ gallons of liquid fertilizer concentrate. It takes ¾ gallon to make a tank of mixed fertilizer. How many tanks can he mix?  (Hint: Think about how many sets of ¾ are in a set of 9/4.)

Developing Algorithms  There are actually two different algorithms for the division of fractions.   1) To divide fractions, first get common denominators, and then divide numerators.  (pg 277 for picture model)

Algorithms Continued  The second algorithm for division of fractions is the “Invert and Multiply” algorithm.  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 completely?  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 water as 2/3.  Remember, our task is to find out how much is in the whole bucket, or 8/8.  What do we need to do?

Inverse Bucket Analogy!  Because the water is seven of the eight parts full, dividing the water by 7 and multiplying that amount by 8 solves the problem.  Therefore, take 2/3 divided by 7, and multiply by 8.  This is the same as 2/3 x 8/7 (Viola! Multiplying by the inverse )