mathematics success – level d t609ntnmath.kemsmath.com/level d teacher notes/level d lesson...

Mathematics Success – Level D T609

LESSON 21: Fraction and Decimal Equivalence

[OBJECTIVE]

The student will be able to express a fraction with a denominator of 10 as an equivalent fraction with a denominator of 100 and use this technique to add two fractions with respective denominators of 10 and 100. The student will be able to use decimal notation for fractions with denominators of 10 and 100.

[PREREQUISITE SKILLS]

basic understanding of a fraction, understanding of place value

[MATERIALS]

Student pages S198–S209Transparencies T624, T626, T628, T630, T632, T634, T636, T638, and T640Centimeter cubes (50 per student pair)Colored pencils

[ESSENTIAL QUESTIONS]

1. What are equivalent fractions?2. How do you change a fraction to a decimal when the denominator is 10 or 100?3. How do you change a decimal in the tenths or hundredths place to a fraction?

[WORDS FOR WORD WALL]

equivalent fractions, numerator, denominator, decimal, tenths, hundredths

[GROUPING]

Cooperative Pairs (CP), Whole Group (WG), Individual (I)*For Cooperative Pairs (CP) activities, assign the roles of Partner A and Partner B to students. This allows each student to be responsible for designated tasks within the lesson.

[LEVELS OF TEACHER SUPPORT]

Modeling (M), Guided Practice (GP), Independent Practice (IP)

[MULTIPLE REPRESENTATIONS]

SOLVE, Verbal Description, Graphic Organizer, Pictorial Representation, Concrete Representation

Mathematics Success – Level DT610

[WARM-UP] (5 minutes – IP, WG, I) S198 (Answers on T623.)

• Have students turn to S198 in their books to begin the Warm-Up. Students will compare fractions with numerators of 1 and different denominators. Monitor students to see if any of them need help during the Warm-Up. Encourage students to use fraction strips or draw pictures of the fractions to assist with finding the correct answer. Give students 3 minutes to complete the problems and then spend 2 minutes reviewing the answers as a class. {Verbal Description, Pictorial Representation (if necessary), Concrete Representation (if necessary)}

[HOMEWORK] (5 minutes)

Take time to go over the homework from the previous night.

[LESSON] (Day 1: 60 minutes – M, GP, IP, WG, CP), (Day 2: 60 minutes – M, GP, IP, WG, CP)

---------------- Day 1 Equivalent Fractions and Adding Fractions ---------------

SOLVE Problem (3 minutes – WG, GP) T624, S199 (Answers on T625.)

Have students turn to S199 in their books, and place T624 on the overhead. The first problem is a SOLVE problem. You are only going to complete the S step with students at this point. Tell students that during the lesson they will learn how to add fractions with denominators of 10 and 100 and change them back and forth from fractions to decimals. They will use this knowledge to complete this SOLVE problem at the end of the lesson. {SOLVE, Graphic Organizer}

Equivalent Fractions – Concrete Representation (15 minutes – M, WG, GP, IP, CP) T624, T626, S199, S200 (Answers on T625, T627.)

10 minutes – M, GP, GP, WG: Have students turn to S199, and place T624 on the overhead. Students will work in partners with the concept of equivalent fractions before adding or changing them to decimals. Use the modeling box below to guide students to converting decimals from tenths to hundredths. Assign the roles of Partner A and Partner B. {Concrete Representation, Graphic Organizer, Verbal Description}



MODELING

Equivalent Fractions – Concrete Representation

Step 1: Have students look at the grids below the SOLVE problem. Both of these pictures represent one whole. However, they are each broken into different sections.

• Partner A, identify how many sections are in the grid on the left. (10)

• Discuss with students that the picture on the left represents tenths.

• Partner B, identify how many sections are in the grid on the right. (100)

• The picture on the right represents hundredths.

Step 2: Use the graphic organizer on S200 and the grids on S199 to complete the following activity.

• Students will take the centimeter cubes and represent the fraction 310

using the grid on the left. Students should make 3 groups of 10 cubes and place them on the grid.

Step 3: Have students answer Questions 3 and 4 in the graphic organizer at the top of S200.• Partner A, identify how many of the vertical strips are covered. (3)

Record.

• Partner B, explain what fraction represents the area of the chart

covered by centimeter cubes. ( 310) Record.




Step 4: Have Partner A slide the three strips of ten over to the hundredths chart.

• Partner B, determine if the number of centimeter cubes changed when we slid them to the next chart. (No, we never changed the number of cubes.)

• Have students answer Questions 6 – 8.

• Partner A, explain how many squares are covered in the hundredths chart. (30) Record.

• Partner B, identify the fraction we write to represent the hundredths

chart. ( 30100) Record.

• What can we conclude about the two fractions we created? (They are equal or equivalent fractions.)

• What are the equivalent fractions. ( 310 = 30

100) Record.

3 minutes – IP, CP: Students will work on page S199 to complete the models for 210 = 20

100 and 510 = 50100. Students should then

complete Questions 9 and 10 on the graphic organizer at the bottom of S200. {Concrete Representation, Verbal Description, Graphic Organizer}

2 minutes – WG: Go over the answers for the graphic organizer and make sure that students see the equivalency between tenths and hundredths. {Concrete Representation, Verbal Description, Graphic Organizer}

Equivalent Fractions – Pictorial Representation (15 minutes – M, GP, CP, WG, IP) T628, S201 (Answers on T629.)

8 minutes – M, GP, CP, WG: Have students turn to S201, and place T628 on the overhead. Students will now apply their understanding of the concrete examples of equivalent fractions to work with the pictorial representation. {Pictorial Representation, Verbal Description, Graphic Organizer}


MODELING

Equivalent Fractions – Pictorial Representation

Step 1: Have students look at Problem 1 in the graphic organizer that is provided.

• Partner A, identify the fraction we are working with. ( 310)

• Partner B, determine how many bars we shade to represent 310 in the

tenths grid. (3)

Have students shade the 3 bars of the tenths grid.

• How many squares do we shade in the hundredths grid? (30)

• Why do we shade 30 in the hundredths grid? (30 in the hundredths grid covers the same amount as 3 in the tenths grid.)

• Have Partner A explain to Partner B what connection they see with the tenths and hundredths. ( 3

10 = 30100) Record in the fourth column.

Fraction TENTHS HUNDREDTHS EQUIVALENT FRACTIONS

1. =

Step 2: As a whole group, complete Question 2. Use the same steps used for Question 1. Reinforce the use of the wording “seven tenths equals seventy hundredths”, rather than “seven over ten equals seventy over one hundred.” We want students to recognize that they are parts over the whole when they see the numerator over the denominator.


310

310

30100


5 minutes – IP, CP: Students will work on page S201 to complete themodels for Problems 3 and 4. {Pictorial Representation, Verbal Description, Graphic Organizer}

2 minutes – WG: Go over Problems 3 and 4 as a whole group. Make sure students are using the correct terminology for tenths and hundredths. {Pictorial Representation, Verbal Description, Graphic Organizer}

Adding Fractions with Denominators of 10 and 100 (27 minutes – M, GP, CP, WG, IP) T630, T632, S202, S203 (Answers on T631, T633.)

15 minutes – M, GP, CP, WG: Have students turn to S202, and place T630 on the overhead. Make sure each student has 2 different colored pencils. Students will begin exploring addition of fractions with denominators of 10 and 100. {Pictorial Representation, Verbal Description, Graphic Organizer}

*Teacher Note: The goal of this lesson is not for students to use a mathematical algorithm for adding fractions. We want to focus on pictorially representing the two fractions and adding them based on the picture.

MODELING

Adding Fractions with Denominators of 10 and 100

Step 1: Have students look at Problem 1.

• Partner A, identify the fi rst fraction we want to model. ( 410)

• Partner B, explain how 410 will be shaded. (by coloring 4 of the vertical

bars or 4 out of the 10) Have students shade 410.

• Partner A, identify the second fraction we want to model. ( 32100)

• Partner B, explain how 32100 will be shaded. (by coloring 32 of the 100

squares) Have students shade 32100.

Explain to students that in order to add these two fractions, the denominators must be the same. Have students look back on S201 at Problem 4.

• How did this example use a pictorial model to show equivalent fractions? (by shading the same area in the tenths grid and in the hundredths grid)



Step 2: In Row 2, shade the model for the equivalent fraction of 410 by shading

40 of the 100 squares.

• Partner A, identify the fraction that should be written below the hundredths grid? ( 40

100) Four tenths are the same as how many hundredths when we shade the last picture? ( 40

100) Have students shade the 40 hundredths in Column 3.

• Partner B, do we need to change the 32 hundredths to an equivalent fraction? Why or why not? (No, because the denominator is already hundredths.)

Have students shade the 32 hundredths in Column 3 in a different color.

Step 3: Partner A, identify the fraction that can be used to represent the Picture of the Sum in Column 3? ( 72

100) Record the sum under the picture model in Column 3.

Step 4: Complete Problem 1 by using the original problem in the column with the equivalent fractions and answer.

410 + 32

100 = 40100 + 32

100 = 72100

Step 5: Have students look at Problem 2 on S202.

• Partner B, identify the fi rst fraction we want to model. ( 510)

• Partner A, explain how 510 will be shaded. (by coloring 5 of the vertical

bars or 5 out of the 10)• Partner B, identify the second fraction we want to model. ( 11

100)• Partner A, explain how 11

100 will be shaded. (by coloring 11 of the 100 squares)

Explain to students that to add these two fractions the denominators must be the same.

• How did students use a pictorial model to show equivalent fractions? (by shading the same area in the tenths grid and in the hundredths grid)



Step 6: In Row 4, shade the model for the equivalent fraction of 510 by shading 50

of the 100 squares.

• Partner B, identify the fraction that should be written below the hundredths grid. ( 50

100) Five tenths are the same as how many hundredths when we shade the last picture? ( 50

100) Have students shade the 50 hundredths in Column 3.

• Partner A, do we need to change the 11 hundredths to an equivalent fraction? Why or why not? (No, because the denominator is already hundredths.)

Have students shade the 11 hundredths in Column 3 in a different color.

Step 7: Partner B, identify the fraction that can be used to represent the Picture of the Sum in Column 3? ( 61

100) Record the sum under the picture model in Column 3.

Step 8: Complete Problem 2 by writing the original problem in the column with the equivalent fractions and answer.

8 minutes – IP, CP: Students will work on page S203 to complete Questions 3 through 6. Partner A will complete Questions 3 and 4. Partner B will complete Questions 5 and 6. Students will share their solutions and discuss answers when all problems are complete. {Pictorial Representation, Verbal Description, Graphic Organizer}

4 minutes – WG: The last four minutes will be used to discuss the answers to Questions 3 through 6. Use this time to discuss vocabulary and the concepts that were studied today. {Pictorial Representation, Verbal Description, Graphic Organizer}



-------------------- Day 2 – Fraction and Decimal Equivalence -------------------

Tenths as Fractions and Decimals (15 minutes – M, GP, CP, WG) T634, S204 (Answers on T635.)

15 minutes – M, GP, CP, WG: Day 2 will begin with students learning how to transform a fraction with a denominator of ten to a decimal. Students will be working in cooperative pairs. Assign roles of Partner A and Partner B to students. {Pictorial Representation, Verbal Description, Graphic Organizer}

MODELING

Tenths as Fractions and Decimals

Step 1: Have students open their books to page S204, and place T634 on the overhead. Explain to students that another way to represent a fraction is as a decimal.

• Where have you seen decimals before? (Examples: money, time in a race, etc.)

• What place values do you see to the right of the decimal point in our Place Value Chart? (tenths, hundredths)

Ten

s

On

es

Ten

ths

Hu

nd

red

ths

Remind students that in Part 1 of the lesson they used tenths and hundredths to add fractions.

Step 2: Have students look at the graphic organizer at the bottom of page S204.

• Partner A, explain Step 1. (Shade the fraction in the second column.) Have students shade 3

10.

• Partner B, explain how to write the fraction in words. (three tenths) Record in both graphic organizers.



Step 3: Partner A, identify how many tenths we have. (3) Record in the chart in Column 3.

Step 4: Explain to students that in Column 4 we will write the value in the Place Value Chart. The Place Value Chart shows the value of each digit in a number. Each place value column can only have one digit in it. We use a zero for a place holder in the ones place for decimals less than 1.

How will we write the decimal of three tenths? [We have no ones so we use a zero as a place holder. Our Written Words column tells us we have 3 tenths. (0.3)] Fill in the chart at the top and the graphic organizer at the bottom of the page.

• What would the decimal look like if the fraction was four tenths? (0.4) Six tenths? (0.6)

Fraction Tenths Written Words Place Value Chart

Three tenths

Tens

Ones

Tenth

s

Hundre

dth

s0 3

Hundredths as Fractions and Decimals (17 minutes – M, GP, CP, WG) T636, S205 (Answers on T637.)

17 minutes – M, GP, CP, WG: Have students turn to S205, and place T636 on the overhead. Students will see how to write a fraction with a denominator of 100 as a decimal. Students will have the opportunity to apply what they have learned with tenths and the pictorial representation of hundredths. {Pictorial Representation, Verbal Description, Graphic Organizer)


310



MODELING

Hundredths as Fractions and Decimals

Step 1: Use the graphic organizer at the bottom of the page to guide the steps for this activity.

• Partner A, identify the fraction we want to represent. ( 32100)

• Partner B, explain how we can represent the fraction. (by shading in 32 of the sections on the grid)

Have students complete Step 1 by shading 32 of the boxes.

Step 2: Complete Step 2 in the chart by having the students write the value in words in both charts. (thirty-two hundredths)

Step 3: Partner A, how many hundredths do we have? (32) Record.

Fraction Hundredths Written Words Place Value Chart

32100

thirty-two hundredths

Tens

Ones

Tenth

s

Hundre

dth

s

0 3 2

Step 4: In Step 4, start with a shaded picture of 32 hundredths and break it down into tenths and hundredths. Students should come up with 3 tenths and 2 hundredths. Using these two shaded pictures, students can then place the appropriate number of tenths (3) and the appropriate number of hundredths (2) in the boxes of the columns labeled “Tenths” and “Hundredths”.

+=



Step 5: Have students write the value of the tenths and hundredths in the place value chart in both graphic organizers. Remind students to use a zero as a place holder in the ones place whenever the decimal is less than 1.

Tens

Ones

Tenth

s

Hundre

dth

s

0 3 2

Step 6: Complete the graphic organizer by writing the decimal in Step 6 (0.32)

Practice with Fraction and Decimal Equivalence (21 minutes – M, WG, GP, CP, IP) T638, S206 (Answers on T639.)

8 minutes – M, WG, CP, GP: Have students turn to S206, and place T638 on the overhead. Problems 1 – 4 offer students the opportunity to practice the skill of fraction and decimal equivalence. {Pictorial Representation, Verbal Description, Graphic Organizer}


MODELING

Practice with Fraction and Decimal Equivalence

For this activity, students are given one representation of the fraction, and they must complete the other boxes from that one piece.

Step 1: Direct students’ attention to Problem 1.

• Partner A, identify what information is given. (the fraction)

• Partner B, explain how we can represent the pictorial representation. (by shading 8 of the vertical columns) Record.

• Partner A, describe how we can write the fraction in words. (eight tenths) Record.

• Partner B, identify how eight tenths is written as a decimal. (0.8) Record.

Fraction Tenths or Hundredths Gird

Written Words Place Value Chart

810

eight tenths

Ones

Tenth

s

Hundre

dth

s0 8

10 minutes – IP, CP: Have students complete Problems 2–4 by fi lling in the missing representations for each number. {Pictorial Representation, Verbal Description, Graphic Organizer}

3 minutes – WG: Go over the answers to Problems 2–4 as a whole group to make sure that the students have the correct answers. {Pictorial Representation, Verbal Description, Graphic Organizer}



SOLVE Problem (5 minutes – GP, WG) T640, S207 (Answers on T641.)

Remind students that the SOLVE problem is the same one from the beginning of the lesson. Complete the SOLVE problem with your students. Ask them for possible connections from the SOLVE problem to the lesson. (Help students recognize that the pictorial representation of the two fractions can be applied to this word problem. They may not understand the algorithm of adding fractions with common denominators quite yet, but students can use the ideas with which they are familiar.) {SOLVE, Pictorial Representation, Verbal Description, Graphic Organizer}

If time permits… (10 minutes – IP, CP) S208 (Answers on T642.)

Have students complete Problems 1 – 4 in the chart on S208.

[CLOSURE] (2 minutes)

To wrap up the lesson, go back to the essential questions and discuss them with students.

• What are equivalent fractions? (Fractions that are equal; they have different names, but the same value.)

• How do you change a fraction to a decimal when the denominator is 10 or 100? (If the denominator is 10, place the numerator of the fraction directly to the right of the decimal point. Follow the same steps when the denominator is 100, unless the numerator is less than 10, then you must place a 0 in the tenths places and the numerator in the hundredths place.)

• How do you change a decimal in the tenths or hundredths place to a fraction? (If the digit is only in the tenths place, make that digit the numerator and make the denominator 10. If numerals are both in the tenths and hundredths place, take those digits and place them in the numerator over a denominator of 100.)

[HOMEWORK] Assign S209 for homework. (Answers on T643.)

[QUIZ ANSWERS] T644–T645

.

The quiz can be used at any time as extra homework or to assess how students progress on understanding equivalent fractions and decimals.


mathematics success – level d t609ntnmath.kemsmath.com/level d teacher notes/level d lesson...

Documents