Skill. 19 Simplify Fractions. Using Skill 19 COMMON ERRORS. 85 Holt Matemáticas. OBJECTIVE Write a fraction in simplest form

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1 Skill Simplify Fractions Using Skill OBJECTIVE Write a fraction in simplest form Write on the board. Ask: What are some factors of the numerator? ( and ) What are some factors of the denominator? ( and ) How many factors do and have in common? (one) What is it? () Direct the students attention to the definition of simplest form. Ask: Why can you say is in simplest form? (because the only common factor of the numerator and denominator is ) Review the definition of simplest form, explaining that is the greatest common factor of both and. For Step, have students tell you the factors of and of. Then for Step, write the pairs of common factors on the board as students name them. (, ;, ;, ;, ; and, ) Ask: What is the greatest common factor? () For Step, help students realize that is equal to. So, when a fraction is divided by, the value of the fraction does not change. Then ask: How can you tell that is in simplest form? (because the only common factor of the numerator and denominator is ) TRY THESE Exercises model the type of exercises students will find on the Practice on Your Own page. Exercises and common factors Exercise more than common factors PRACTICE ON YOUR OWN Review the example for simplifying a fraction greater than at the top of the page. Ask: How can you tell that is greater than? (The numerator is greater than the denominator.) CHECK Determine if students know how to find factors and the greatest common factor of the numerator and denominator and use the GCF to find the simplest form. Success is indicated by out of correct responses. Students who successfully complete the Practice on Your Own and Check are ready to move to the next skill. COMMON ERRORS Students may divide the numerator by one factor and the denominator by a different factor. Students may use a common factor but not the greatest common factor. Students who made more than errors in the Practice on Your Own, or who were not successful in the Check section, may benefit from the Alternative Teaching Strategy on the next page. Minutes Holt Matemáticas

2 Optional Alternative Teaching Strategy Modeling Simplest Forms of Fractions OBJECTIVE Use models to find the simplest form for a fraction MATERIALS fraction circles for halves, fourths, eighths, and sixteenths Minutes An alternate method for finding the simplest form is to use,, or as a trial divisor and keep reducing the fraction until it is in simplest form. Have students use fraction circles. For example, write on the board. Have students find the fraction circle for eighths and shade of the parts. Continue: Suppose we again divide the numerator and denominator by. What is the result? ( ) Have students find the fraction circle for halves and shade of the parts. Say: Suppose we divide the numerator and denominator by. What is the result? ( ) Have students find the fraction circle for fourths and shade of them. The circle models help students to visualize that any further division is not possible. At the same time they see that there can be more than one equivalent fraction for many fractions. Repeat the activity to find the simplest form for. Holt Matemáticas

3 Nombre Simplificar fracciones Una fracción está en su mínima expresión cuando el máximo común divisor, o MCD, del numerador y del denominador es. Escribe en su mínima expresión. Paso Escribe los factores del numerador y del denominador. Paso Busca los factores comunes. Encierra en un círculo el máximo común divisor numerador,,,,, (MCD). denominador,,,,, numerador,,,,, denominador,,,,, Paso Divide el numerador y el denominador entre el MCD. Inténtalo La mínima expresión de es. Escribe cada fracción en su mínima expresión. Halla el MCD y enciérralo en un círculo. Halla el MCD y enciérralo en un círculo. Halla el MCD y enciérralo en un círculo.,,,,,,,, Divide el numerador y el denominador entre el MCD.,,,,,,, Divide el numerador y el denominador entre el MCD.,,,,,, Divide el numerador y el denominador entre el MCD. Ir a siguiente página Holt Matemáticas

4 Nombre Practica por tu cuenta Escribe en su mínima expresión. Paso Haz una lista con todos los factores del numerador y del denominador. Encierra en un círculo el MCD. Paso Divide el numerador y el denominador entre el MCD.,,,,,,,,,, ó Escribe cada fracción en su mínima expresión. Encierra en un círculo el MCD.,,,,,,, Divide el numerador y el denominador entre el MCD. Encierra en un círculo el MCD.,,,,,, Divide el numerador y el denominador entre el MCD. Encierra en un círculo el MCD.,,,,,, Divide el numerador y el denominador entre el MCD. Escribe los factores. Halla el MCD. Luego, escribe la fracción en su mínima expresión. MCD MCD MCD mínima expresión mínima expresión mínima expresión Escribe la fracción en su mínima expresión. Comprueba Escribe la fracción en su mínima expresión. Holt Matemáticas

5 Skill Write Equivalent Fractions Using Skill OBJECTIVE Use different methods to write equivalent fractions MATERIALS fraction models Minutes Discuss with your students the meaning of equivalent. (equal in value, same amount) Direct students attention to Model A. Ask: To find equivalent fractions, do the fraction models have to be the same length? (yes) How many parts is the first fraction model divided into? () How many parts is the second fraction model divided into? () How many parts in the second model are equal to part in the first model? () Direct students attention to Model B. Ask: What is the factor that both the numerator and the denominator are multiplied by? () Emphasize that both the numerator and denominator are multiplied by the same number. Ask: What is the value of? () Then recall that any number multiplied by is that number. So, when the fraction is multiplied by, the result is an equivalent fraction,. Continue to ask similar questions as you work through Model C. TRY THESE Exercises model the type of exercises students will find in the Practice on Your Own page. Exercises Equivalent fractions using models Exercises Equivalent fractions using multiplication and division; mathematical cues provided PRACTICE ON YOUR OWN Review the example at the top of the page. Ask students to explain how they know that the models are equivalent and why the numerators and denominators are multiplied or divided by the same number. CHECK Make sure that the students multiply or divide the numerator and denominator by the same number. Success is indicated by out of correct responses. Students who successfully complete the Practice on Your Own and Check are ready to move to the next skill. COMMON ERRORS Students may forget to multiply or divide both the numerator and the denominator by the same number. Students who made more than errors in the Practice on Your Own, or who were not successful in the Check section, may benefit from the Alternative Teaching Strategy on the next page. Holt Matemáticas

6 Optional Alternate Teaching Strategy Modeling Equivalent Fractions OBJECTIVE Use fraction models to model equivalent fractions MATERIALS models in halves, thirds, fourths, fifths, sixths, eighths, and tenths Minutes You may wish to have the students work in pairs. Have partners take turns. One student models the fraction with the fraction models while the other student records the results on paper. Write? on the board. Distribute the fraction models. Direct students to select the fraction model that shows. Ask: How many equal parts is the fraction model divided into? () How many parts are shaded? () Explain that this fraction model is called the halves fraction model. Now point to the fraction?. Ask: Which fraction model should you choose next? (One that is divided into equal parts.) Which fraction model showing equal parts is equivalent to? (The one with of the equal parts shaded.) Explain that this fraction model is called the fourths fraction model because it represents four equal parts. Have students compare the shaded parts of the two fraction models. Ask: How many shaded parts in the fourths fraction model are equal to the shaded part in the halves fraction model? () Have students record the answer on their papers. Then demonstrate how they can find the same answer by multiplying both the numerator and denominator by. Repeat this activity using fraction models with fractions such as and ; and. For each model, demonstrate how to use multiplication to find the equivalent fraction. When the students show that they understand how to find equivalent fractions, remove the fraction models and have them use multiplication. Holt Matemáticas

7 Nombre Escribir fracciones equivalentes Las fracciones equivalentes representan la misma cantidad. Puedes usar diferentes métodos para hallar fracciones equivalentes. Modelo A Haz un modelo para mostrar tercios. Sombrea. Haz otro modelo para mostrar sextos. Modelo B Multiplica el numerador y el denominador por el mismo número para hallar una fracción equivalente. Modelo C Divide el numerador y el denominador entre el mismo número para hallar una fracción equivalente. y representan partes del mismo tamaño. Por lo tanto,. Por lo tanto,. Por lo tanto,. Inténtalo Escribe la fracción equivalente. Ir a siguiente página Holt Matemáticas

8 Nombre Practica por tu cuenta Razona: Multiplica o divide para hallar una fracción equivalente. Multiplica Divide Escribe la fracción equivalente.???????????? Comprueba Escribe la fracción equivalente. Holt Matemáticas

9 Skill Compare Fractions Using Skill OBJECTIVE Compare fractions Minutes Begin by having students compare whole numbers and then like fractions. Present these examples: Ask: Is less than, equal to, or greater than? (less than) Display these symbols: < >. Ask: Which of these symbols do you use to show that is less than? ( ) Display. Point out that the fractions and have like denominators. Ask: When you compare two fractions, and the denominators of the fractions are the same, what part of each fraction do you look at? (the numerator) Why? (The numerator tells how many equal parts are being considered. When the denominators are equal, then the size of the equal parts are the same. So, compare the number of parts considered in each fraction.) Continue: Is less than, equal to, or greater than? (less than) How do you know? ( is less than ) Which symbol do you write to show this? ( ) Display. For the first example, recall how to write equivalent fractions. For Comparing Mixed Numbers, ask: What if you are comparing and, which mixed number do you think is greater? ( ) How can you tell? (Because you can compare the whole numbers: and ; since is greater than, then.) For Comparing Fractions Greater than, ask: How can you tell that a fraction is greater than? (The numerator is greater than the denominator.) TRY THESE Exercises model the type of exercises students will find on the Practice on Your Own page. Exercise Fractions less than Exercise Mixed numbers Exercise A fraction less than and a fraction greater than Exercise Fractions greater than PRACTICE ON YOUR OWN Review the example at the top of the page. Have students explain why they should change to a whole number. CHECK Determine if students can compare fractions less than or greater than one. Success is indicated by out of correct responses. Students who successfully complete the Practice on Your Own and Check are ready to move to the next skill. COMMON ERRORS Students may compare numerators without first checking to make sure the denominators are the same. Students may compare the fraction parts of two mixed numbers when it is only necessary to compare the whole-number parts. Students may not know how to find equivalent fractions. Students who made more than errors in the Practice on Your Own, or who were not successful in the Check section, may benefit from the Alternative Teaching Strategy on the next page. Holt Matemáticas

10 Optional Alternative Teaching Strategy Compare Fractions on a Ruler OBJECTIVE Use a ruler to compare fractions MATERIALS inch ruler showing halves, fourths, eighths, and sixteenths Minutes Provide students with rulers or number lines labeled in intervals of halves, fourths, eighths, and sixteenths. If you wish, students can use customary rulers to make their own number lines. Present like fractions to compare:. Students find the point for. Then they find the point for. Remind the students that as they move from left to right on the number line, the numbers are greater. Ask: Which fraction is greater? ( ) Next, show a pair of unlike fractions less than one:. Some students may believe that is greater than, because the numeral is greater than. Use the ruler as a model to help students recognize that the intervals for fourths are greater than the intervals for eighths. Ask: Are these fractions, and, equivalent fractions? (no) How do you know? (They have the same numerator, but different denominators.) First have students locate and on the ruler. Guide students as they determine that is less than. Ask: How do you compare these fractions when you do not have a ruler? (Find equivalent fractions with a common denominator) Then have students find how many eighths are in by counting the eighth-inch marks on the ruler. Display and. Continue in a similar manner to compare other fractions. Point out that sometimes students only have to compare the whole number parts of mixed numbers to determine the greater number. () () inches () () = () () As students demonstrate facility with comparing fractions on the ruler, provide some examples to be completed without the ruler. Holt Matemáticas

11 Nombre Comparar fracciones Compara fracciones Compara y. Como los denominadores son diferentes, escribe fracciones equivalentes con el mismo denominador. Compara los numeradores. Como, entonces Razona: es múltiplo de, entonces vuelve a escribir como una fracción equivalente con denominador.. Por lo tanto,. Compara números mixtos Compara y. Como <, entonces Por lo tanto,. Compara fracciones mayores que Compara y. Como, entonces. Si, entonces.. Compara y Por lo tanto, Por lo tanto,.. Inténtalo Compara. Escribe, ó. Mismo denominador? Mismos números cabales? Mismo denominador? Menor que o mayor que? Compara los números cabales primero. Son iguales.. Como los denominadores son diferentes, vuelve a escribir cada fracción como una fracción equivalente con el mismo denominador. Compara los numeradores. Cada fracción es mayor que. Divide el numerador entre el denominador. Compara los números cabales. Como, entonces. Ir a siguiente página Holt Matemáticas

12 Nombre Practica por tu cuenta Compara Mismo denominador? y. La fracción es mayor que. Divide el numerador entre el denominador. Compara los números cabales. Como, entonces Por lo tanto,. Compara. Escribe, ó.? Mismos números cabales? Mismo denominador?. Mismo denominador? Menor que o mayor que? Mismos números cabales? Mismo denominador? Menor que o mayor que? Comprueba Compara. Escribe, ó. Holt Matemáticas

13 Skill Write Fractions as Decimals Using Skill OBJECTIVE Write a decimal for a fraction Minutes You may wish to begin by reviewing the different ways to show division. Explain to students that to write a fraction as a decimal, divide. Direct students attention to Step. Ask: Which number is the divisor in the fraction? () Which number is the dividend? () Which number is the divisor in the fraction? () Which number is the dividend? () Direct students attention to Step. Show how to place the decimal point and a zero in the dividend. Ask: Where do you place the decimal point in the quotient? (above the decimal point in the dividend) Direct students attention to Step. Point out that they divide decimals as they would whole numbers. Then work through the division steps with students. Emphasize how to write zeros in the dividend until the remainder equals zero or until they have the number of decimal places in the quotient that they need. TRY THESE In Exercises students divide to write a fraction as a decimal. Exercise Write as.. Exercise Write as.. Exercise Write as.. PRACTICE ON YOUR OWN Review the example at the top of the page. Ask: Which number in the fraction is the divisor? () Which number is the dividend? () Discuss how to place the decimal point in the quotient and then divide as if dividing whole numbers. CHECK Determine if students know how to divide the numerator of a fraction by the denominator to find an equivalent decimal. Success is indicated by out of correct responses. Students who successfully complete the Practice on Your Own and Check are ready to move to the next skill. COMMON ERRORS Students may misplace the decimal point or not write it at all in the quotient. Students may make errors in dividing because they have not yet mastered their division facts. Students may forget to write a zero in the quotient and may write the quotient. as.. Students who made more than error in the Practice on Your Own, or who were not successful in the Check section, may benefit from the Alternative Teaching Strategy on the next page. Holt Matemáticas

14 Optional Alternative Teaching Strategy Use Questioning Strategies to Understand Equivalent Fractions and Decimals Minutes OBJECTIVE Find equivalent fractions with denominators of,, or, to write equivalent decimals Recall with students that a fraction can be written as an equivalent decimal. Remind students that all decimals can be written as fractions with denominators of,,,, and so on. Guide students to understand that if they write a fraction as an equivalent fraction with a denominator of,, or,, then they can write the equivalent decimal. Display?. Mention that one way to write equivalent fractions is to multiply the numerator and denominator by the same number. Ask: What number times equals? () Say: If we multiply the denominator by, what must we do to the numerator? (Multiply it by.) Display. Ask: Is a multiple of? (yes) How many tenths equal? () How do you write tenths as a decimal? (.) Point out that the fraction is equivalent to the decimal.. Next, display?. Ask: What number times equals? () If we multiply the denominator by, what must we do to the numerator? (Multiply it by.) Display:. Ask: How do you write hundredths as a decimal? (.) Point out that the fraction is equivalent to the decimal... Display?. Use the same questioning as in the previous two examples. Point out that the fraction is equivalent to the decimal... Repeat the activity using the fractions (,.), (,.), (,.),.), and (,,.).. (, Holt Matemáticas

15 Nombre Paso Divide como lo harías con números cabales, hasta que el residuo sea o hasta que tengas la cantidad de posiciones decimales que necesitas..... Escribir fracciones como decimales Paso Divide el numerador entre el denominador. Recuerda que puedes pensar en una fracción a como a b. b Paso Escribe un punto decimal seguido de un en el dividendo. Luego coloca un punto decimal en el cociente. Ir a siguiente página.... Escribe cada fracción como decimal. Divide para escribir una fracción como decimal. Escribe y como decimales. Por lo tanto,. y Inténtalo Holt Matemáticas

16 Nombre Practica por tu cuenta Razona: Primero, escribe la fracción como un problema de división. Luego escribe un punto decimal y un cero en el dividendo. Coloca un punto decimal en el cociente. Si es necesario, escribe más ceros en el dividendo... Escribe cada fracción como decimal Comprueba Escribe cada fracción como decimal. Holt Matemáticas

17 Skill Decimal Operations Using Skill OBJECTIVE Perform operations with decimals Minutes Have students look at the four different examples. Ask: What is important to remember when adding or subtracting decimals? (Align the decimal points.) Ask: When multiplying decimals, how do you determine where the decimal point goes in the product? (Count the total number of decimal places in the factors and then move the decimal point that many positions to the left in the product.) What do you have to remember to do when dividing a decimal by a decimal? (Change the divisor to a whole number by moving the decimal point in the dividend and divisor.) TRY THESE In Exercises students add, subtract, multiply and divide using decimals. Exercise. Exercise. Exercise. Exercise. PRACTICE ON YOUR OWN Review the examples at the top of the page. CHECK Determine if students know how to perform operations with decimals. Success is indicated by out of correct responses. Students who successfully complete the Practice on Your Own and Check are ready to move to the next skill. COMMON ERRORS Misalignment of decimal points. Encourage students to use lined paper or graph paper to keep decimal aligned. Mistaking place values. When adding or subtracting numbers with a different number of place values (..), encourage students to add zeros or use a place value chart to align the places correctly. Misplaced decimal point in the product. Have students count the number of decimal points in each factor and then place the decimal point in the product. Missing the decimal point in the quotient. Have students immediately place the decimal point in the quotient. Students who made more than errors in the Practice on Your Own, or who were not successful in the Check section, may benefit from the Alternative Teaching Strategy on the next page. Holt Matemáticas

18 Optional Alternative Teaching Strategy Perform Operations with Decimals OBJECTIVE Perform operations with decimals MATERIALS index cards with problems written on them Minutes If students are having difficulty with adding, subtracting, multiplying, or dividing decimals, you may need to provide four separate instructional periods to review each skill. Students might recognize that the processes for adding and subtracting decimals are similar but that different procedures are needed to multiply and divide decimals. To help develop number sense, prepare a set of cards with exercises similar to these: A... B... C... D... Without computing, have students identify either orally or in writing, what is necessary to carry out each operation. For A, students should realize the need to rewrite the problem and align the decimal points. They should recognize the value of adding a zero to. so that both numbers have the same number of decimal places. For B, students should realize the need to rewrite the problem and align the decimal points. Students should recognize the value of adding a zero to. so that each number has the same number of decimal places. You might ask students if regrouping will be necessary when they subtract. For C, students should recognize the need to rewrite the problem vertically. Students should recognize that multiplying decimals is very similar to multiplying whole numbers. Ask students how many decimal places the answer will have (). For D, students should recognize the need to rewrite the problem. They should know it is necessary to move the decimal point two places in both the dividend and divisor. Holt Matemáticas

19 Nombre Operaciones con decimales Sumar decimales Halla... Restar decimales Halla... Multiplicar decimales Halla... Dividir decimales Halla... Escribe el problema de manera que los puntos decimales estén alineados... suma. Escribe el problema de manera que los puntos decimales estén alineados.... Si es necesario, agrega. ceros como marcadores. de posición y reagrupa al restar. Escribe el problema de nuevo. Determina el número de posiciones decimales en el producto.. posiciones decimales. posición decimal. posiciones decimales Escribe el problema de nuevo... Convierte el divisor en un número cabal y mueve el punto decimal en el dividendo.... Divide como lo. harías con números cabales. Inténtalo Resuelve..... Alinea los decimales posiciones decimales. Alinea los decimales.. posición decimal posiciones decimales.... Mueve el punto decimal. Ir a siguiente página Holt Matemáticas

20 Nombre Practica por tu cuenta Razona: Para sumar y restar decimales, alinea los puntos decimales. Razona: Cuando multiplicas decimales, determina el número de valores posicionales que tendrá el producto. Razona: Cuando divides decimales, recuerda mover el punto decimal para dividir entre un número cabal. Resuelve.... alinea.... posiciones decimales. posición decimal posiciones decimales.... mueve el decimal Resuelve Comprueba Resuelve Holt Matemáticas