Rational Numbers

•  Fractions •  Decimals •  Percents •  It is important for students to know how

these 3 concepts relate to each other and how they can be interchanged.

Fraction Vocabulary

•  It is important that vocabulary terms are taught to students. –  Fraction - part of a whole - all parts are equal. –  Numerator - top number (number of parts considered). –  Denominator - bottom number (number of equal parts). –  Greatest Common Factor (GCF) - largest common

factor for both the numerator and denominator that is larger than 1.

–  Least Common Multiple (LCM) - the smallest common multiple of two or more denominators.

–  Improper Fraction - fraction that is greater than 1. –  Mixed Number - whole number and a fraction.

Fraction Models

•  Region or Area – Surface is divided into equal parts – Rectangular or circular

•  Set – Whole is a set of objects and subsets of

the whole make up the fractional parts. – Example: You have 12 pieces of candy.

You give your friend 5 pieces. They have 5/12 of the candy.

Fraction Models

•  Length or Measurement – Length is compared instead of area. – Number lines – Line segments

Reading and Writing Fractions •  Students must know the following terms to be able to

use fractions both orally and in written form. –  Halves, thirds, fourths, fifths, sixths, sevenths, etc.

•  Students will need to read and write fractions with guidance from teachers and quick reinforcement if they are reading and/or writing fractions incorrectly.

•  Writing fractions –  Show students fractions with either concrete objects or

pictures. Instruct them that the denominator is the number of parts and the numerator is the number of shaded parts.

Beginning Fraction Concepts

•  When students are beginning to learn about fractions they should be working with concrete materials. –  Fraction tiles –  Fraction circles

•  Fractions should also be related to real life experiences. Example: Give half of your candy to your brother.

Beginning Fraction Concepts

•  Identifying/Counting Fractional Parts –  Identify the number of equal parts. – Count the number of the parts being

named. – This will assist students in understanding

that 1/8 is smaller than 1/4, which is a concept that is difficult for students to understand at first.

Beginning Fraction Concepts

•  Relationship to 1 –  Students must quickly identify if a fraction is less

than 1 (1/2), equal to 1 (5/5), or greater than 1 (7/5). –  Best demonstrated by using concrete fraction pieces

that can be manipulated. –  Denominator larger than the numerator - less than 1

(proper fraction). –  Numerator and denominator are the same number -

equal to 1. –  Numerator larger than denominator - greater than 1

(improper fraction).

Beginning Fraction Concepts

•  Mixed Numbers - 7 1/8 – Students must understand that it is the

same as 7 + 1/8. – Reading mixed numbers - 7 and 1/8. – Writing mixed numbers - 7 1/8.

Beginning Fraction Skills

•  Comparing Fractions –  Students must understand that the more parts in a

whole, the smaller the fraction. –  Compare using <,>,= signs.

•  1/2 > 1/4 4/8 < 2/5 4/8 = 1/2

–  Students must also realize that 7/8 is closer to 1 than 3/4.

–  Concrete fraction pieces will assist students, as they can put pieces on top of each other and compare the differences.

Error Analysis of Beginning Fraction Concepts

•  Students should complete pages 137-150 of the Error Analysis. –  Identify the error –  Complete two problems following the student’s

method –  Provide 2 strategies for teaching the student. –  Check your answers compared to the author’s

Adding & Subtracting Fractions

•  Like Denominators – This should be fairly simple if students

have a good understanding of the beginning fraction concepts.

– 3/4 - 1/4 = 2/4 7 4/9 + 1 1/9 = 8 5/9 –  If students are having difficulty, a review of

beginning concepts is necessary.

Adding & Subtracting Fractions

•  Unlike Denominators –  Students should first explore with fraction pieces.

(See handouts for examples) –  Students need to be taught the concept of

common multiples. Students must know their multiplication facts to be able to do this accurately and quickly.

•  For students who have difficulty, fraction bars may be of assistance. (See handouts for example).

–  Once students determine the least common denominator they write equivalent fractions and solve the problem.

Multiplying Fractions •  Begin with concrete objects. (see handouts for

examples). •  It is important to emphasize that the denominators

do not have to be the same in multiplication problems.

•  Algorithm - should only be taught after students have add ample opportunities to practice conceptually. –  Multiply the numerators - write the answer. –  Multiply the denominators - write the answer. –  Read the problem.

Dividing Fractions

•  Dividing fractions is the inverse of multiplying fractions. Most are taught to invert the divisor and multiply. While students can do this with little difficulty, most do not understand what they are doing or why they are doing it.

•  Providing students with story problems may assist them in understanding. Example: Maria has 1 1/2 hours to complete her test. If the test has 6 questions, how much time can Maria spend on each question?

Writing Fractions in Simplest Terms

•  Teach the concept of finding the greatest common factor (GCF). –  Determine factors of the numerator. –  Determine factors of the denominator. –  Determine the greatest common factor.

•  4 = 1,2,4 12 = 1,2,3,4,6,12 GCF = 4 4 x ? = 4 4 x 1 = 4 4 = 1 4 x ? = 12 4 x 3 = 12 12 3

Website

•  Students should explore the following website that shows how fractions can be taught.

•  http://nlvm.usu.edu/en/nav/category_g_2_t_1.html

•  There are several fraction activities to explore.

Error Analysis

•  Students should complete pages 151-166 AND 172-180 of the Error Analysis. –  Identify the error –  Complete two problems following the student’s

method –  Provide 2 strategies for teaching the student. –  Check your answers compared to the author’s

Decimals

•  Equal parts of a whole. •  The use of Base 10 Blocks will assist

students in conceptually understanding decimals. – Flat = Whole numbers – Rod = Tenths – Cube = Hundredths

Reading and Writing Decimals

•  As with fractions, it is important that students are able to understand decimals both in written form and verbally. –  Tenths, hundredths, thousandths, etc. –  .7 = seven tenths –  .77 = seventy-seven hundredths –  .501 = Five-hundred one thousandths –  4.53 = Four and fifty-three hundredths

Comparing Decimal Values

•  Students often think the more numbers to the right of the decimal, the larger the number.

•  Using manipulatives to show decimals will assist students with this (see handouts for examples).

•  .4 > .06 .754 < .75

Adding & Subtracting Decimals

•  Students should have mastered addition and subtraction up to 4-digit numbers (with and without regrouping) prior to working with decimals.

•  When beginning the problems should have the same place value.

•  When adding and subtracting students should be taught to first bring down the decimal point and then compute the problem.

Multiplying Decimals •  Proficiency in multiplying whole multi-digit numbers is

essential before beginning to work with decimals. •  The use of Base 10 Blocks will assist students in

understanding. •  Algorithm

–  Multiply problem. –  Starting from the right, count the number of decimal points

and place the decimal point in the problem. –  Important to provide examples with zero in both the factors

and the product.

Dividing Decimals

•  Proficiency in dividing whole numbers is necessary prior to working with decimals.

•  Algorithm – Bring the decimal point directly up. – Solve as a regular division problem.

Percents

•  Can be introduced once students have a strong sense of fractions and decimals.

•  What part of 100% belong to a certain group. (see handouts for examples)

•  Common uses of percents: –  Sales tax of 6.25% –  Worth 25% of your grade –  Put 50% of your money in a savings account –  We have raised 84% of our goal.

Relating Fractions, Decimals, Percents

•  It is important to be able to switch between fractions, decimals, and percents and know which one is easiest to use in different situations.

Relating Fractions, Decimals, Percents

•  A shirt cost $35.00. The sign says it is 33% off. Will you multiply 35 x. 33 or 35 x 1/3?

•  If you have 60 baseball cards and you have to share half with your brother, will you multiply 60 x .50 or 60 x 1/2?

Common Fraction, Decimal, Percent Equivalents

•  3/4 .75 75% •  1/2 .50 50% •  1/3 .33 33% •  1/4 .25 25% •  1/5 .20 20% •  1/10 .10 10%

Error Analysis in Decimals

•  Students should complete pages 167-171 AND 181-184 of the Error Analysis. –  Identify the error –  Complete two problems following the student’s

method –  Provide 2 strategies for teaching the student. –  Check your answers compared to the author’s

References •  Ashlock, R.B. (2005). Error patterns in computation. Upper Saddle, NJ:

Prentice Hall.

•  Hudson, P., & Miller, S.P. (2006). Designing and implementing mathematics instruction for students with diverse learning needs. Boston: Allyn & Bacon.

•  Kennedy, L.M., Tipps, S., & Johnson, A. (2004). Guiding children’s learning of mathematics (10th ed.). Belmont, CA: Thomson- Wadsworth

•  Sovchik, R.J. (1996). Teaching mathematics to children. New York: Longman. •  Van De Walle, J.A. (2004). Elementary and middle school

mathematics: Teaching developmentally (5th ed.). Boston: Allyn & Bacon.