Chapter 12 – Fractions and Decimals: Meanings and Operations

1. Chapter 12 – Fractions and Decimals: Meanings and Operations Chelsea Marshall, Michelle Roberts, and Tori Flint

2. Introduction • Fractions and decimals have been troublesome for many students over the years. • One reason could be that curriculum may focus too much on symbolization and operations. • There are two ideas associated with fractions and decimals: partitioning and equivalence. • Partitioning refers to sharing equally. • Equivalence refers to different representations of the same amount.

3. Introduction • The development of computational fluency should include fluency with common fractions. • It is not necessary to complete the study of fractions before moving onto decimals. • Teachers may introduce decimals after a foundation has been built with fractions.

4. Conceptual Development of Fractions • Helping students develop the necessary conceptual understanding is better done with a curriculum that emphasizes many representations- physical, pictorial, verbal, real-world, and symbolic. • Materials and Literature can help teach fractions- fraction bars, pattern blocks, and books, like The Hershey’s Milk Chocolate Fraction Book or The Doorbell Rang.

5. Three Meanings of Fractions • Three distinct meanings of fractions: • Part-whole- A whole has been partitioned into equal parts and a certain number are being considered. • Quotient- Fraction considered as a division problem. • Ratio- Fraction represents a ratio. Ex. 3/5 could mean 3 boys for every 5 girls in a class. • Focus is usually on part-whole, but ignoring the others might be one source of students difficulty.

6. Models of the Part-Whole Meaning • Region • Length 1 2 3 • Set • Area-pg.292 In the Classroom 12-1

7. Making Sense of Fractions • Partitioning- Idea of making equal shares by separating a whole into equal parts. • Let the children do the partitioning. • Introduce the words halves, thirds, fourths, and so on. • Once children are familiar with the words for fractional parts, begin counting the parts.

8. Making Sense of Fractions • After children can name, count, and compare using oral language, then it is appropriate to teach written words and symbols. • Draw a model. • Students should be able to go from a part to a whole. • Ex. This is 3/5 of a cake, draw a picture of the whole cake. • Must have a solid understanding of equivalence.

9. Ordering Fractions and Equivalent Fractions • Concrete Models- Fraction bars • Pictorial Models • Symbolic Representation- It is easier to compare two fractions that are represented with similar denominators. Ex. 3/7 and 2/5 or 3/5 and 2/5

10. Mixed Numbers and Improper Fractions • Through models, you can lead children naturally into mixed numbers and improper fractions, even as they are learning the initial concepts of fractions.

11. Operations with Fractions • The key to helping children understand operations with fractions is to make sure they understand fractions, especially the idea of equivalent fractions. • Children gain a better understanding of operations with fractions if they learn to estimate answers. • Using whole numbers and benchmarks ( 0, ½, 1) makes it easier to decide if an answer is reasonable.

12. Addition and Subtraction • Begin with problems that involve joining and separating instead of symbolic sentences like ½ + ¼=. • These problems, with pictorial models, help students • See that adding and subtracting fractions can solve problems similar to problems involving whole numbers • Develop an idea of a reasonable answer • See why a common denominator is necessary when adding or subtracting

13. Multiplication • One of the simplest algorithms: multiplying the numerators gives the numerator, and multiplying the denominators gives the denominator. • However, learning this algorithm does not give students insight into why it works or when to use it. • Types of Multiplication Problems: • Whole number times a fraction • Fraction times a whole number • Fraction times a fraction

14. Division • Let students draw pictures. • Use problems instead of simple equations to solve. • Look at rhymes on page 305 In the Classroom 12-7.

15. Conceptual Development of Decimals – Quick Facts • Relate decimals to what is already known – FRACTIONS! • Also, relate to everyday experiences – relevancy drives remembering! • An understanding of decimals comes from an understanding fractions.

16. Relationship to Common Fractions – Tenths • Before introducing the names of decimal places, review that a tenth is a portion of something broke into ten equal parts. • Link students prior knowledge of place value to learning the place values of decimals.

17. Relationship to Common Fractions – Tenths • The language of decimals should be consistent. 0.3 and 3/10 are both said as “three-tenths” • Also, students need to learn that and is used to show the decimal point. For example, 2.3 or 2 3/10 would be said as “two and three-tenths” • Saying decimals and fractions like this helps students connect them to each other.

18. Let’s play a game!! Directions Reminder Heads = Decimal Tails = Fraction EXTENDED LEARNING Use sixteen-tenths, or Five and eleven-tenths • Teacher selects numbers and says them aloud. • Students get to chose how to write them – fraction or decimal, and after the teacher flips a coin and that’s the deciding factor for who gets the point.

19. Relationship to Common Fractions – Hundredths • Use a hundreds chart to show students that the whole square is one unit. Have students color 23 little squares and ask what portion is colored. Allow answers like 23/100 or 2/10 + 3/100 to and connect the place value interpretation to the decimal notation.

20. Relationship to Common Fractions – Thousandths and beyond! • “Older students often do not understand small decimals such as 0.234 or 0.0003, partly because there is less emphasis on these decimals but also because teachers often expect children to generalize from hundredths to all other places.” • Never expect what you don’t teach!!

21. Relationship to Common Fractions – Writing F. as D. • With careful explanation, students should be able to switch back and forth between fractions and decimals fairly easily. • Students need to learn, either through the use of a calculator or by learning equivalent fractions, that, for example, 4/5 is equivalent to 0.8. • Likewise, it should be discussed that 1/3 can be represented as 0.33 or 0.333, and so on, but actually the 3 keeps repeating, so we sometimes have to round up.

22. Relationship to Common Fractions – Place Value • We must be consistent while teaching place value and show our students that it is just like on the other side of the decimal, 10 tenths bumps the number up to the ones place. • This is a great little box that I saw in the book:

23. Operations with Decimals – Addition and Subtraction • Much simpler than with fractions • Always remember to line up your decimals – if helpful use grid paper like previously shown • Difficulty arises while students are doing word problems or when problems are given in a horizontal format (EX: 5.69 + 7.24)

24. Operations with Decimals – Multiplication and Division • Multiplying and dividing decimals by a whole number is much easier than having two decimals • Students can of course begin to think of it as repeated addition, but we all hope they move away from that and find other means. • Let’s look at the two other ways to multiply decimals the book discusses on page 310!

25. Operations with Decimals – Multiplication and Division • To divide a decimal by a whole number you complete regular long division and bring the decimals up. • However, instead of having a remainder, you keep adding a zero at the end of the number inside the box until the bottom difference is also zero. • In Middle School students begin to understand multiplying and dividing more and they will begin putting more decimal decimal problems together.

26. Cultural Connections • In 3500 B.C., the Egyptians were one of the first cultures to make use of fractions. • Today, in Japan, they do little work with fractions until the 5th grade, where they are expected to add and subtract fractions with like denominators. • Some countries teach decimals before fractions, and vice versa. • Many countries use a comma instead of a period for the decimal point.

27. A Glance At Where We’ve Been • Teachers should make sure that children can figure out a whole from a fractional part and that they understand the idea of equivalent fractions. • The use of concrete models, pictorial models, and symbolic representations to build children's ability to order fractions and find equivalent fractions.

28. A Glance At Where We’ve Been • To teach addition and subtraction of fractions, begin with problems and pictorial models that help children: 1. See how adding and subtracting fractions can solve problems similar to problems with whole numbers 2. Develop an idea of a reasonable answer 3. See why a common denominator is necessary.

29. A Glance At Where We’ve Been • To teach multiplication of fractions children need to: 1. Understand the meaning of multiplication 2. Estimate the size of an answer