Lesson Guide

In

Elementary Mathematics

Grade 6

Chapter II

Rational Numbers

Ratio and Proportion

Reformatted for distribution via DepEd LEARNING RESOURCE MANAGEMENT and DEVELOPMENT SYSTEM PORTAL

DEPARTMENT OF EDUCATION BUREAU OF ELEMENTARY EDUCATION

in coordination with ATENEO DE MANILA UNIVERSITY

2010

INSTRUCTIONAL MATERIALS COUNCIL SECRETARIAT, 2011

Lesson Guides in Elementary Mathematics Grade VI Copyright © 2003 All rights reserved. No part of these lesson guides shall be reproduced in any form without a written permission from the Bureau of Elementary Education, Department of Education.

The Mathematics Writing Committee

GRADE 6

Region 3

Dolores A. Umbina – Olongapo City Zenaida P. Gomez – Pampanga Teresita T. Tungol – Pampanga

Region 4-A Margarita Rosales – Lucena City Segundina B. Gualberto – Batangas Estelita Araullo – Rizal

Henry P. Contemplacion – San Pablo City National Capital Region (NCR)

Teresita L. Licardo – Quezon City Lilia T. Santos – Quezon City Eleanor Interia – Quezon City Elfrida V. Marquez – Manila Victoria C. Tafalla – Valenzuela

Bureau of Elementary Education (BEE)

Rogelio O. Doñes Robesa Hilario

Ateneo de Manila University

Girlie N. Salvador

Support Staff

Ferdinand S. Bergado Ma. Cristina C. Capellan Emilene Judith S. Sison Julius Peter M. Samulde Roy L. Concepcion Marcelino C. Bataller Myrna D. Latoza Eric S. de Guia – Illustrator

Consultants

Fr. Bienvenido F. Nebres, SJ – President, Ateneo de Manila University Carmela C. Oracion – Ateneo de Manila

University Pacita E. Hosaka – Ateneo de Manila

University

Project Management

Yolanda S. Quijano – Director IV Angelita M. Esdicul – Director III

Simeona T. Ebol– Chief, Curriculum Development Division Irene C. de Robles – OIC-Asst. Chief, Curriculum Development Division

Virginia T. Fernandez – Project Coordinator

EXECUTIVE COMMITTEE

Jesli A. Lapus – Secretary, Department of Education Jesus G. Galvan – Undersecretary for Finance and Administration

Vilma L. Labrador – OIC, Undersecretary for Programs and Projects Teresita G. Inciong – Assistant Secretary for Programs and Projects

Printed By: ISBN – 971-92775-5-6

iii

TABLE OF CONTENTS

Introduction .................................................................................................................................. iv Matrix ........................................................................................................................................ v

II. RATIONAL NUMBERS

J. Ratio and Proportion

Forming Ratio and Proportion ........................................................................................ 1 Reducing Ratios to Lowest Terms ................................................................................. 4 Finding the Missing Term in a Proportion ...................................................................... 9 Word Problems Involving

Direct Proportion ............................................................................................... 13

Partitive Proportion ............................................................................................ 16

Inverse Proportion ............................................................................................. 19

iv

I N T R O D U C T I O N

The Lesson Guides in Elementary Mathematics were developed by the

Department of Education through the Bureau of Elementary Education in

coordination with the Ateneo de Manila University. These resource materials

have been purposely prepared to help improve the mathematics instruction in

the elementary grades. These provide integration of values and life skills using

different teaching strategies for an interactive teaching/learning process.

Multiple intelligences techniques like games, puzzles, songs, etc. are also

integrated in each lesson; hence, learning Mathematics becomes fun and

enjoyable. Furthermore, Higher Order Thinking Skills (HOTS) activities are

incorporated in the lessons.

The skills are consistent with the Basic Education Curriculum

(BEC)/Philippine Elementary Learning Competencies (PELC). These should be

used by the teachers as a guide in their day-to-day teaching plans.

v

MATRIX IN ELEMENTARY MATHEMATICS

Grade VI

COMPETENCIES VALUES INTEGRATED STRATEGIES USED MULTIPLE INTELLIGENCES

TECHNIQUES With HOTS

J. Comprehension of Ratio

and Proportion

1. Forms ratio and proportion

for groups of objects/

Numbers

1.1 Use colon (:) and

fractions in writing

ratios and proportions

1.1.1 Reduces a ratio

to lowest terms

1.2 Find a missing term

in a proportion

2. Application of ratio and

proportion

2.1 Set up a proportion for

a given situation

2.2 Solve word problems

with:

2.2.1 Direct proportion Industry and diligence Write equation Group work

2.2.2 Partitive proportion Self-esteem Use of tables Puzzle, Group work, Reading

Friendliness, Sharing Draw pictures

2.2.3 Inverse proportion Kindness Write an equation Game, Group work

Concept development Role play

2.3 Describe answer in a

complete sentence with

proper labels/units

1

Forming Ratio and Proportion

I. Learning Objectives

Cognitive: Form ratio/proportion for groups of objects/numbers Use colon (:) and fractions in writing ratios and proportion

Psychomotor: Write ratio/proportion using colon or fraction form Affective: Show care in counting and handling things

II. Learning Content

Skill: 1. Forming ratio and proportion 2. Using colon (:) and fractions in writing ratios and proportions Reference: BEC PELC II K.1.1, 1.1.1 Materials: box with different objects Values: Carefulness

III. Learning Experiences

A. Preparatory Activities

1. Mental Computation/Drill

Solve for N.

a) 5 53

= N c) 51

6 = N e) 311 4 = N

b) 43

21

= N d) 8 54

= N

2. Review

Give the fractional part of the shaded portion.

a. b. c.

d. e.

3. Motivation Look around your room. What are the things you find inside?

2

B. Developmental Activities 1. Presentation

a. Activity 1 – Class Activity

1) Count the number of boys. (Let the pupils count.)

Count the number of girls. Ex. 25 boys and 28 girls 2) Guide the pupils to show the relationship of the number of boys to the number of

girls. How do you write the comparison of the number of boys to the number of girls using

fraction form? 2825

25 is the 1st term boys, 28 is the 2

nd term girls

Is there another way of writing it? How? 3) Let the pupils count other objects/things.

Let them give/write the ratio in 2 ways (colon and fraction forms). Ex. buttons to stones

shells to tansan

matchboxes to marbles

bottle caps to paper clips Did you count the objects/things correctly? Why? What will you do if you are given objects to use/manipulate? How will you handle them?

b. Activity 2 – Pair Activity

Present a Problem:

Ronald bought 3 pencils for 10 at Elen’s School Supply Store. Ruby bought 6

pencils for 20. Give the ratio of pencils to the amount of money of each child. Discussion: 1) What did Ronald and Ruby buy? How many pencils did each of them buy? How

much did each of them pay? 2) What are being compared in the problem? Write the ratio in 2 ways. 3) How many ratios did you write? 4) What can you say about the two ratios? Why? 5) How can we write the two ratios to show equality in two ways?

colon form - 3:10 = 6:20

fraction form 103

= 206

3 is called the first term 10 and 6 are called means 10 is called the second term 3 and 20 are called extremes 6 is called the 3

rd term

20 is called the 4th term

6) What do you call two equal ratios? 7) Provide other examples.

Ex. The ratio of chairs to tables is 8 to 2 or 4 to 1. Let the pupils write the ratio in 2 ways. Let them identify the terms, means, and extremes.

2. Fixing Skills/Exercises

1) Give the ratio of each of the following orally in two different ways.

a) squares to circles c) books to crayons

b) flowers to leaves d) basketballs to tennis balls

3

2) Using the diagram below, form ratios and proportions. Write them in two ways.

3. Generalization

1) What is a ratio? proportion? 2) What are the 2 ways of writing ratio/proportion?

C. Application

A. Find the ratio/proportion of the following. Use 2 different ways in writing them.

1) There are 10 buses at a station. If each bus has 6 wheels, what is the ratio of buses to wheels?

2) Every quarter each student submits 2 projects in EPP. Give the ratio of projects to quarters.

3) There are 3 caimito trees and 4 mango trees in Mang Tino’s orchard. While in Mr. Muñoz’ orchard, there are 6 caimito trees and 8 mango trees. Give the ratio of the mango to caimito trees in each orchard then write a proportion.

B. Read and analyze the problem.

A bag of M & M sweets contain just yellow and orange sweets. For every 2 yellow sweets, there are 6 orange sweets. Complete this table

Yellow 4 6

Orange 6 12

Total Sweets 24 40

Answer these questions. a. What is the ratio of orange to yellow sweets? b. If you have 8 yellow sweets how many orange sweets will you have? c. There are 32 sweets in the medium sized bag. How many will be yellow? d. In the extra large bag there are 40 sweets. What proportion are orange? e. You look in the sweet bowl and count out 16 yellow sweets. How many sweets are in the

bowl?

4

IV. Evaluation Write a ratio or proportion for each of the following: 1)

2) Eight compared to 28. 3) There are 5 kites to seven boys. 4) In a t-shirt factory, each box contains 3 t-shirts. Give the ratio of boxes to t-shirts. 5) In a camping, each boy scout will be given 4 hotdogs. If there are 5 boy scouts, 20 hotdogs will

be cooked. Write the proportion. V. Assignment

Form ratio/proportion for the following using 2 different ways. 1) 7 to 8 2) 3 to 5 is equivalent to 6 to 15 3) two barangays to 13 348 people 4) one boat to 3 people is equal to 6 boats to 18 people 5) 45 members of Glee Club to 30 members of Dance Club

Reducing Ratios to Lowest Terms

I. Learning Objectives

Cognitive: Reduce ratios to lowest terms Psychomotor: Write ratios correctly Affective: Keep oneself healthy

II. Learning Content

Skills: Reducing ratios to lowest terms Reference: BEC PELC II.K.1.1-1.1.1 Materials: concrete objects, cutouts, flash cards Value: Keeping oneself healthy/proper health habits

III. Learning Experiences

A. Preparatory Activities

1. Mental Computation

Have a review on finding the greatest common factor (GCF) of 2 or 3 given numbers. What is the GCF of the following? 12 and 16 18 and 30 6, 12 and 15 15 and 9 10 and 14

2. Review a) Conduct a review on reducing fractions to lowest terms. Let the pupils do this mentally.

Reduce these fractions to lowest terms:

108

, 1512

, 3018

, 93

, 206

b) Check up of assignment.

5

3. Motivation

Before presenting the problem situation, ask the pupils about their favorite drink for

“merienda,” ex. calamansi juice, tea, etc. * Tell them that calamansi juice is good because of

its nutritious value.

B. Developmental Activities 1. Presentation

a. Activity 1

1) Present this problem situation.

Mother is preparing calamansi juice. a) For each glass of calamansi juice, 5 pieces of calamansi are needed. b) If she makes 2 glasses, how many pieces of calamansi are needed? c) If she makes 3 glasses, how many pieces of calamansi are needed?

2) Analyze the problem by asking the pupils to identify:

a) What is asked? b) What facts are given? c) What strategies may be used to answer the problem?

3) Illustrate the problem:

a) b) c)

4) Ask: How many pieces of kalamansi are there in a glass of water in a? in b? in c?

Write the ratios:

1:5 or 51

2:10 or 102

3:15 or 153

5) Ask: Which of these ratios is expressed in lowest terms? (1:5) Why?

6

What other groupings can you do with b? c? How many groups were you able to make for b? c?

6) Provide additional situations similar to this problem for pupils’ practice. (Make use of objects that the pupils see in the classroom.)

b. Activity 2 1) Present an illustration of puto in a bilao. Ask pupils what it is and who likes to eat

puto for snacks. (If they know what it is made of, the nutritional values of the ingredients can be discussed.)

2) Say that the puto is sliced into 6 equal parts and mother took 2 of the slices to serve for merienda (take away 2 slices to show a bilao background.)

3) Ask: What is the ratio of the remaining slices of puto to the whole? Why?

Expected answers: 4:6 or 64

2:3 or 32

Which ratio do you think is in lowest terms? Why?

c. Group Activity 1) Present these illustrations. Let the LB’s (Learning Barkada) work on them. Allow 5

to 10 minutes.

2) What is the ratio of the shaded part to the whole rectangle? Why? Write 2 ratios that compare the number of shaded parts to the whole rectangle. _____; _____

(Expected Answer: 5:10; 1:2; 105 ,

21 )

3) What is the ratio of the shaded part to the whole circle? Why? Write the ratios. _____; _____

4) What is the ratio of the shaded part to the whole triangle? Why? Write the ratios. _____; _____

5) Ask the LB’s to present their work on the board. Encourage the pupils to compare their answers.

6) Lead the pupils to identify which of the ratios that they wrote are expressed in lowest terms.

7) Lead them to see that ratios can also be reduced or expressed in lowest terms, just like fractions. (Show examples.)

7

Note: Provide additional examples for more pupils’ practice on reducing a ratio to lowest terms.

d. Activity 3 – Comparing Measurements

1) Ask: a) Why do you think our school canteen also serves soup for our snack (recess)? b) How many school days in a week are we served soup? c) What is the ratio of the number of school days in a week to a week?

2) Lead the pupils to write the ratio; 5 days to 1 week. 3) Explain that 5 days and 1 week are quantities of the same kind that are expressed in

different units. To compare them, change them to the same unit. 4) Express 1 week in days. Since 1 week is 7 days. 5 days to 1 week = 5:7. Is the ratio

5:7 expressed in lowest terms? Why? 5) What is the ratio of 5 days to 1 month? 6) Is this expressed in lowest terms? Why? 7) What will be the ratio expressed in lowest terms? 8) For more practice let the pupils answer the following:

Instruct them to change the ratio to similar units (e.g., months to days) before reducing to lowest terms. a) Find the ratio of 25 minutes to 1 hour. b) Find the ratio of 8 years to 1 decade. c) Find the ratio of 8 days to 2 weeks.

2. Fixing Skills

A. Express the given ratio in simplest or lowest terms. 1) 8 hours to 10 hours 2) 40 minutes to 1 hour 3) 25 centavos to a peso 4) 2 dozens to 18 things 5) 264 km in 3 hours 6) 18 boys to 16 girls 7) 25 atis to 30 mangoes 8) 25 cm to 1 metre 9) * 50 people in 5 minutes 10) * 180 km to 20 litres of gasoline

* Note: If two different units cannot be changed to the same unit, then it becomes rate. The numbers then are used as is, so the ratio for no. 9 is 50:5, in lowest terms, 10:1.

B. Reduce the following ratios in lowest term. Choose the letter that corresponds to the

ratio of lowest term.

E = 3:4 I = 1:2 R = 2:9 T = 15:4

G = 1:6 N = 5:6 S = 1:4

4:8 15:18 30:8 18:24 6:27 15:20 8:32 60:16 7:14 25:36 4:24

What is the hidden word? ________________________________ 3. Generalization

Can a ratio be expressed in lowest terms? How?

8

C. Application

Study the table and answer the questions after it.

Things Quantity Cost

Stamps 10 50.00

Patches 15 180.00

Bookmark 20 300.00

Diary 12 300.00

1) In lowest terms, express the ratio of: a) stamps to patches b) bookmark to patches c) bookmark to stamps d) diary to patches e) bookmark to stamps

2) What is the rate or unit price of each item? IV. Evaluation

1) Check the ratios that are expressed in lowest terms.

a) 5:15 b) 9:20 c) 11:22

d) 8:15 e) 6:3 f) 2:14

g) 30:5 h) 14:15 i) 21:25

2) Reduce these ratios to lowest terms. a) 10:12 b) 9:15 c) 18:24

d) 21:27 e) 40:50

3) Write the ratios in lowest terms. a) 15 minutes to 1 hour b) 8 months to 1 year c) 50 cm to 1 m d) 5 m to 100 cm e) 3 decades to 20 years

V. Assignment

1) Which ratios are expressed in lowest terms? Box them. a) 10:4 b) 5:11

c) 20:7 d) 13:39

e) 16:19 f) 4:15

2) Reduce the ratios to lowest terms. a) 18:20 b) 9:36

c) 12:3 d) 28:50

e) 11:44 f) 60:12

3) Write a ratio for each. Express the ratio in lowest terms. a) 30 cm to 7 dm b) 18 hours to 3 days c) 3 months to 2 years d) 5 centuries to 1 millennium

9

Finding the Missing Term in a Proportion

I. Learning Objectives

Cognitive: Find the missing term in a proportion Psychomotor: Write proportions correctly Affective: Help parents at home

II. Learning Content

Skills: Finding the missing term in a proportion Reference: BEC PELC II.K.1.2 Materials: flash cards, pictures Value: Helpfulness

III. Learning Experiences

A. Preparatory Activities

1. Review a) Checking of assignment. b) Have a review on equivalent fractions. What are equivalent fractions? Give examples of

equivalent fractions. Which of these fractions are equivalent?

86

and 43

41

and 164

2416

and 128

32

and 23

106

and 109

2. Drill

Conduct a drill on comparing fractions. How do we compare two given fractions? Use cross multiplication, then compare using >, <, or =.

164

41

15

210

108

43

65

32

43

86

14

312

3. Motivation

Before presenting the problem situation, ask the pupils if they help their parents at home especially during weekends. Elicit from the pupils the importance of helping one’s parents during weekends.

B. Developmental Activities

1. Presentation

a. Activity 1 – Whole Class Activity

1) Present this problem situation. During weekends, Zeny helps her mother sell buko juice. For every buko, Zeny

adds 4 litres of water. How many litres of water does she need for 3 bukos so that the taste will be the same?

10

2) After analyzing the problem, ask the pupils to illustrate it.

A B 3) Ask: What is the ratio in A? in B?

Is the second ratio equal to the first ratio? Why?

Therefore, 1:4 = 3:12 or 41

= 123

are equal ratios.

4) Introduce the term “proportion” based on the given example. Lead the pupils to see that a “proportion” is formed by 2 equal ratios.

5) Introduce the terms extremes and means. Discuss how to find the missing “extreme” or “means” using the given problem situation.

Buko buko

Litres litres or 1:4 = 3:N of water of water

Solution 1 x N = 3 x 4 or 1:4 = 3:N 1 N = 12 1 x N = 4 x 3

N = 12 1 1 N = 12

N = 12 1 N = 12

To check: x Or 1:4 = 3:12

x

12 = 12 12 = 12 Note: Provide similar examples for pupils’ practice or for fixing the

skill.

b. Activity 2 – Group Activity 1) Present some problems on the board and let the LBs answer them. Allow 5 to 10

minutes. Encourage the LBs to illustrate the problem before writing the proportion. Problem 1:

Zeny and her mother also sells hotcakes on weekends. Mother’s recipe needs for 3 eggs to make 5 hotcakes. Zeny wants to make 25 hotcakes. How many eggs will she need? Problem 2:

For 5 hotcakes, 2 tablespoons of sugar are needed. How many tablespoons does Zeny need to make 25 hotcakes?

2) Group reporting a) Let the LBs illustrate their solutions on the board. b) Check if the LBs wrote the correct proportions for the problems. c) Again, guide the pupils in finding the missing term or element. d) Ask questions to elicit the rule for finding the missing element in a proportion. Note: For pupils’ practice, additional examples or exercises may be given.

41

N3

123

41

=

=

11

c. Activity 3 1) Give activity cards to the learning barkada or team.

Sample contents of activity cards a) 9 compared to 6 is the same as 24 compared to 16. b) Some number n is to 8 as 9 is to 12.

c) Is 159 =

104 a true proportion. If not, make it a proportion.

d) 4 is to 12 as 5 is to 15. e) 24 is to n as 6 is to 11. f) Solve for n.

n8

= 2253

g) Which of the following are proportions. Explain why you say they are or are not proportions. If they are not a proportion, make them a proportion.

32

= 23

9015

= 122

8:20 = 30:100 5:4 = 15:12

2) Ask a representative of each team to make a report of their answers. 3) Check and analyze the answers.

2. Fixing Skills (Individual Work) 1) Tell whether the following given ratios are proportions. Write Yes if they are and No if

not, change one term to make them proportions.

a) 54

and 3528

b) 18:45 and 4:12

c) 4820

and 3615

d) 3:21 and 6:40

e) 9:12 and 27:32

2) Solve for the missing term and check.

a) 321 =

43

b) 5490 =

39

c) 9x =

6349

d) 52 =

30n

e) 17n =

519

f) 103 =

n24

g) 4n =

2432

h) n5 =

6923

i) 71 =

x28

j) 73 =

y21

3) Tell whether n = 22 is a reasonable answer to complete each proportion. Explain how you decided?

a.) =

b.) =

c.) =

d.) =

e.) =

3. Generalization

What is a proportion? What are the terms or elements in a proportion? How do we find the missing element in a proportion?

12

C. Application Solve for the missing term. 1) What number compared to 10 is the same as 28 compared to 5?

2) Lisa saves 60 in 4 weeks. At this rate, how long will it take her to save 300.00? 3) Six compared to 11 is the same as 84 compared to what number? 4) A motorist traveled 240 km in 3 hours. At the same rate, how long will it take to travel 400

km? 5) A scale 3.5 cm on a map represents an actual distance of 175 km. What actual distance

does a scale distance of 5.7 cm represent? 6) There are 24 students in a Grade VI class. During a tree planting day, they planted a total of

24 mahogany and narra seedlings. Each girl planted 3 mahogany seedlings and every three boys planted 1 narra seedling. Find how many mahogany and narra seedlings each are planted by the students? (Answer: 18 mahogany seedlings and 6 narra seedlings)

IV. Evaluation 1) Find the missing term:

a) 74 =

14x

b) 8n =

205

c) y3 =

129

d) 5

10 = t

12

e) 6n =

2420

2) Solve for the missing term in each proportion.

a) 6:n = 8:12 b) m:7 = 6:21 c) 20:24 = x:6

d) y:6 = 28:84 e) 14:21 = 2:n

3) Write a proportion then solve for the missing term. 14 girls to 5 boys; how many boys to 28 girls 3 batteries to 1 flashlight, how many batteries to 4 flashlights

5 mangoes for 18; how much for 15 mangoes

V. Assignment

1) Find the missing term to make the proportion true.

a) 9:N = 27:15 b) N:8 = 12:32 c) 5:3 = 25:N

2) Complete the sentences. a) Ten books is to 5 pupils as ______ books is to 15 pupils.

b) Three bananas for 4.00 as 12 bananas for _____. c) Seven boy scouts to a tent as 42 boy scouts to _____ tents.

3) Cross out the ODD-MAN. a. 3:4 6:8 10:12

b. 2:3 3:4 4:6

c. 5:15 1:3 5:10

d. 9:6 4:2 15:10

13

Word Problems involving Direct Proportion

I. Learning Objectives

Cognitive: Solve word problems involving direct proportion Psychomotor: Write proportions correctly Affective: Practice diligence and industry

II. Learning Content

Skills: Solving word problems involving direct proportion Reference: BEC PELC II.K.2.2.1 Materials: flash cards, problem-solving chart Value: Industry and diligence

III. Learning Experiences

A. Preparatory Activities 1. Review

a) Checking of assignment. b) Present simple and short problems involving ratio and proportion for the pupils to answer

orally. Ex. Three boiled camotes sell for 5. How much do 9 pieces cost?

Two boys can paint 5 desks in 1 day. How many desks can 10 boys paint?

Four medium-sized onions cost 15. At most, how many pieces can I buy

with 45? 2. Drill

Conduct a drill on finding the missing term in a proportion. Use flash cards and have pupils answer orally:

n3

= 159

3:x = 6:10

6n

= 46

3:4 = 27:x

115

= n35

x:9 = 12:18

3. Motivation

Before presenting the problem situation, ask the pupils what they do on weekends. Let them realize that they can earn extra money during weekends if they are industrious or hardworking.

B. Developmental Activities

1. Presentation a. Activity 1 – Use a Problem Opener

1) Present this problem.

Roy and Al sell newspapers on weekends to earn extra money. For every 3 newspapers that Roy sells, Al sells 5. If Roy sold 15 newspapers, how many did Al sell?

2) Analyze the problem. a) What are given?

14

b) What is being asked? c) Illustrate the problem.

Roy

Al Explain the illustration. Encourage the pupils to think of other strategies to

answer the problem. d) Set up a proportion.

Roy or Roy : Al = Roy : Al

Al 3:5 = 15:N

Explain that the proportion is called a direct proportion. The quantities change in the same direction. As the number of newspapers that Roy sells increases, the number of newspapers that Al sells also increases.

e) For pupils’ practice, give them another problem. Let them work in groups. The sign on the store window says “Magazine for sale, buy 3, take 2.” How

many magazines must I buy if I want to take 10 magazines for free? Have pupils show their solution on the board. Check if they were able to write the proportion correctly.

2. Fixing Skills

1) At the school canteen: a) 3 pieces of pad paper cost 45 cents.

21 pieces of pad paper cost _____.

b) 4 colored pencils cost 25. 12 colored pencils cost _____.

c) 2 boiled bananas cost 3.50. 10 boiled bananas cost _____.

2) Solve the problems. a) A motorist travels 275 km in 5 hours. How far can he travel in 9 hours at the same

speed? Proportion: __________ Answer: __________

b) Two buses can transport 130 people. How many buses are needed to transport 780 pupils?

Proportion: __________ Answer: __________

c) The scale on a road map is 1 cm to 50 km. How far apart are 2 towns represented

on the road map by 214 cm?

Proportion: __________ Answer: __________

3. Generalization

How do you solve problems involving direct proportion. What must you remember when setting a direct proportion?

C. Application

A. Read and solve.

1) The ratio of the areas of 2 squares is 1:4. The area of the smaller square is 36 cm2.

How long is each side of the bigger square?

53

N15

=

15

2) Triangle ABC is similar to XYZ. If AB = 6 cm, BC = 9 cm, AC = 12 cm, find YZ and XZ.

B. Read the problem and complete the table to answer the question.

In a school survey for Grade 6, it found that for every 3 girls there were 4 boys.

Girls 3

Boys 12

Total no. of Pupils 35

a) What is the ratio of girls to boys? b) What is the proportion of boys? c) In a class of 28 pupils, how many are girls? d) There are 56 pupils in Grade 6. How many are boys? e) How many pupils are there in Grade 6 class if there are 36 boys?

IV. Evaluation

Analyze each problem and write a proportion to solve it. Draw a diagram to help you when necessary.

1) A tree casts a shadow of 12 metres when a 5-metre pole casts a shadow of 4 metres. How tall is

the tree?

2) At the rate of 3 items per 100, how much will 12 items cost? 3) A car travels 72 km on 8 litres of gasoline. At the same rate, about how far can it travel on 11

litres of gasoline? 4) The ratio of duck eggs to chicken eggs in an egg store is 2:7. If there are 312 duck eggs in the

store, how many chicken eggs are there? 5) The ratio of men to women working for a construction company is 10:3. If there are 21 women in

the construction company, how many men are there? V. Assignment

1) Write a proportion for each problem, then find the missing term.

a) The ratio of 2 numbers is 3:5. The larger number is 30. What is the smaller number? b) There are 3 teachers to 125 pupils during the school program. How many teachers were

there if there were 2 500 pupils? c) The ratio of male teachers to female teachers in our school is 2:9. If there are 108 female

teachers, how many teachers are male?

2) Dolor cooks for her family. Her recipe for ube jam is 4 cups of boiled and mashed ube, 1 cup sugar, and 3 cups milk. If the recipe is good for 2 small bilao, how can Dolor modify it for 5 bilaos?

B

A C X

Y

Z

10

16

Word Problems involving Partitive Proportion

I. Learning Objectives

Cognitive: Solve word problems involving partitive proportion Psychomotor: Illustrate word problems through drawings or tables Affective: Accept things given with open heart

II. Learning Content

Skills: Solving word problems involving partitive proportion Reference: BEC PELC II.K.2.2.2 Materials: worksheets Value: Industry and diligence

III. Learning Experiences

A. Preparatory Activities

1. Drill Find the hidden message. What’s hello in the Hawaiian language? Give the missing

element to form a proportion. Write in the blanks the letters that correspond to the answer.

1 2 3 4 5 Code:

A = 12 E = 8 H = 35 L = 9 M = 5 O = 6

1) 83

= 32

2) 5

= 6335

3) 6040

= 9

4) 72

= 10

5) 20 = 3018

2. Review

What is a direct proportion? How would you set up the proportion?

3. Motivation (Group or Pair Activity)

Identify the missing information in the following problems. a) Joel bought a sandwich in the canteen. Chris also bought 3 sandwiches. How much did

Chris pay? b) One hundred sixty-five boys and two hundred eighty-four girls attended the parade. (Let

the pupils explain their answers.)

17

B. Developmental Activities

1. Presentation a. Activity 1 – Story Problem

Joy and Dale are twins. They always share their things equally. Even their mother

gives them the same amount of anything, whether money, toys, candies, and others. But one day, their father gave them 5 chocolates, 2 chocolates for Joy and 3 chocolates for Dale. 1) What do you think each of the girls felt? 2) Why did their mother give them things equally? 3) If you were one of the girls, what will you do? 4) Is it alright to have the same amount of things as your other siblings? Why?

Joy and Dale found out that there are things that can not be shared equally. So one

day, their mother gave them 150 so that the ratio is 2:3, 2 parts for Dale and 3 parts for Joy. How much did each girl receive? 1) What is asked in the problem? 2) What are the given facts? 3) How can we find the answer? Use a table to show the relationship.

Dale Joy Sum of Money

2 x 1 = 2 3 x 1 = 3 5

2 x 2 = 4 3 x 2 = 6 10

2 x 3 = 6 3 x 3 = 9 15

2 x ___ = ___ 3 x ___ = ___ 150

4) What amount will you multiply to the 2 parts of Dale and 3 parts of Joy so that the

sum will be 150?

2 x ___ = ___ 3 x ___ = ___ 5) What number did you get?

Dale → 2 x 30 = 60 Joy → 3 x 30 = 90

60 + 90 = 150 6) As an equation, we write it as:

2n + 3n = 150

5n = 150

n = 150 5

n = 30 7) To check:

(2 x 30) + (3 x 30) = 150

60 + 90 = 150

60 is the amount received by Dale.

90 is the amount received by Joy. b. Activity 2

ANNOUNCEMENT The Glee Club and the Dance Club are auditioning members for the forthcoming

stage presentation. All interested pupils must see Miss Ruby Hilario for the audition on Monday, 1:00 p.m. at Rm. 25.

18

After one week, 72 pupils were accepted. The Glee Club and the Dance Club agreed that the ratio of participants is 4:5, respectively. How many pupils were chosen for each club?

Discussion on the announcement: 1) What is the announcement about? 2) What are the important information to be written in the announcement? 3) Why do pupils joined clubs such as Dance Club and Glee Club? 4) Why is it good to join clubs such as Dance Club and Glee Club? Discussion on the mathematical sentence: 1) What is asked? 2) What are the given data? 3) What is the relationship of 72 pupils to the ratio 4:5? 4) How can you solve the problem?

Let the pupils give their guesses.

Let them also give ideas on what operation/s to be used. 5) Guide the pupils to illustrate the relationship.

Glee Club Dance Club Total

4 x 1 = 4 5 x 1 = 5 9

4 x 2 = 8 5 x 2 = 10 18

4 x 3 = 12 5 x 3 = 15 27

4 x ___ = ___ 5 x ___ = ___ 72 6) What operation/s are you going to use to find the answer? 7) What will you multiply to 4 and 5 to have a sum of 72? What is the equation?

Guide the pupils to give this equation:

4n + 5n = 72

Let them solve for the answer. 4n + 5n = 72 9n = 72

n = 72 9 n = 8 (4 x 8) + (5 x 8) = 72 32 + 40 = 72

2. Fixing Skills – Group/Pair Activity (Use activity cards)

Analyze and solve each problem. 1) Two numbers are in the ratio 5:3. If the sum is 88, find the 2 numbers. 2) The ratio of chairs to tables is 2:7. There are 180 chairs and tables in a party. How

many are there of each kind? 3) The sum of two numbers is 215. If the ratio is 2:3, find the larger number.

3. Generalization

How do you solve word problems involving partitive proportion? What are the processes involved?

19

C. Application

Solve the given problems.

1) The salary of two workers is in the ratio 3:4. They received 12,250.00. How much did each worker receive?

2) The ratio of men to women at a college is 7 to 5. How many women students are there if there are 350 men?

3) The ratio of Math books to other books in a class is 8 to 5. How many Math books are there if there are 247 books in all?

4) Three boys sold garlands in the ratio of 2:3:4. Together they sold 225 garlands. How many garlands did each boy sell?

5) Two numbers are in the ratio of 4:7. If the difference between the two numbers is 48, find the sum of the two numbers.

6) The teacher-pupil ratio in a preschool class is 1:20. The number of girls is 5

3 of the number of

boys. If there are 200 boys, how many teachers are there? Ans. 16 IV. Evaluation

Read and analyze, then solve the problems. 1) The ratio of cats to dogs is 6:5. There are 495 dogs and cats in a certain barangay.

a) How many cats are there? b) How many dogs are there?

2) Three numbers are in the ratio 2:5:7. If their sum is 504, what are the three numbers? a) first number b) second number c) third number

V. Assignment

Analyze and solve the problems carefully. 1) The ratio of doors to windows is 1:5. There are 186 doors and windows in a building. How many

doors are there? windows? 2) The ratio of the angles of a triangle is 3:4:5. Find the measure of each angle. 3) Three numbers are in the ratio 1:4:7. Find the second number if their sum is 276. 4) The difference between two numbers is 40. They are in the ratio 9:7. What are the numbers?

5) The ratio of a string divided in 3 parts is 211 :

212 :3. How long is each piece if their sum is 28?

Word Problems involving Inverse Proportion

I. Learning Objectives

Cognitive: Solve word problems involving inverse proportion Psychomotor: Write or set up an inverse proportion correctly Affective: Be generous enough to care for the less fortunate and the needy

II. Learning Content

Skills: Solving word problems involving inverse proportion Reference: BEC PELC II.K.2.2.3 Materials: problem solving chart, flash cards Value: Kindness

20

III. Learning Experiences A. Preparatory Activities

1. Drill

Have a drill on finding the missing term in a proportion. Let the pupils answer these orally:

8n

= 249

n6

= 2118

49

= 16n

8n

= 2415

35

= n25

2. Review Recall the steps in solving problems involving direct proportion. Find out if the pupils

have mastered the skill in setting up a direct proportion through a game. 1) Group pupils into 3. 2) Flash these sample problems in a card for them to answer. 3) The first group to give the correct answer is given a point. 4) The group with the most number of correct answers wins. Sample Problems: a) 10 pieces of polvoron sell for 3 pesos

40 pieces of polvoron sell for _____ b) During recess, the ratio of pupils to teachers eating in the canteen is 7:3. If 84 pupils eat

in the canteen, how many teachers eat in the canteen? c) Four out of 5 pupils buy buko juice every day. How many of the 350 pupils buy buko

juice? 3. Motivation

Ask the pupils if they have visited some of the places that care for the physically handicapped, aged, orphans, etc. Discuss the importance of these places, and the value of helping our less fortunate brothers.

B. Developmental Activities

1. Presentation

a. Activity 1 – Using a Problem Opener

Problem 1:

I have enough money for a vacation of 12 days if I spend 500 a day. For how

many days will my money last if I decide to spend only 400 a day. 1) Teacher writes the problem in an index card. 2) Teacher groups the pupils into 4. 3) Teacher gives each group an index card with the problem. 4) Each group solves the problem and presents the solution to the whole class. 5) Analyze the problem:

a) What is asked? b) What are given? c) How can we solve the problem? Elicit possible solutions.

Since the pupils are not yet familiar with this type of proportion, explain what an

inverse proportion is. Show how an inverse proportion is set up. Lead the pupils to see how an inverse proportion differs from a direct proportion. Solution:

Original Amount New Amount

New No. of Days Original no. of Days =

21

Therefore:

400N = 500 x 12 400 N = 6000

N = 6000 400

the answer is N = 15 6) Lead or guide the pupils to observe that in an inverse proportion, the quantities

change in opposite directions, that is, as one quantity increases (no. of days), the other decreases (amount of money). In a direct proportion, the quantities are either fixed or they change in the same direction.

Problem 2:

An orphanage has enough bread to feed 30 orphans for 12 days. If 10 more orphans are added, how many days will the same amount of bread last? 1) Guide the pupils in setting up the inverse proportion after analyzing the problem.

Solution: Therefore:

40 N = 30 x 12 40 N = 360 N =

The answer is N = 9 Note: If the pupils still find difficulty setting up this type of proportion, (inverse), give them more examples for practice or fixing of skills.

2. Fixing Skills

Pupils may solve these problems individually then they can discuss these with their partners. Analyze and solve the problems: 1) If 4 farmers can plow a 3-hectare land in 6 days, how long will 8 farmers do it? 2) Twelve painters can paint a building in 10 days. How many painters are needed to paint

it in 6 days? 3) A house contractor has enough money to pay 8 workers for 15 days. If he adds 4 more

workers, for how many days can he pay them at the same rate? 3. Generalization

What is an inverse proportion? How does it differ from a direct proportion? How do we solve for an inverse proportion?

C. Application Analyze then solve the problems.

500 400

N 12

=

Original No. New No.

New No. Original No. =

(Orphans) (Orphans)

(Days) (Days)

30 40

N 12 =

360 40

22

1) If 8 men can build a house in 90 days, in how many days can 20 men working under the same conditions as the 8 men build the house?

2) A carpenter working 8 hours a day could finish a piece of work in 6 days. How many days could he finish a similar piece of work by working 10 hours a day?

*3) A worker can finish the painting job in 15 days, another worker can finish only 75% of that job for the same time. At first the second worker painted for several days and then the first worker joined him and together they finished the rest of the work for 6 days. Find how many days each worker worked and what percent of the work has been done by each one of them? Solution: First we will find the daily production of each worker. If we take the whole work as one unit, the production of the first one is 1/15 and the production of the second is 75% of 1/15 ,i.e. 75/100 x 1/15 = 1/20 Let’s say that the second worker worked alone for x days. Then his finished job will be x/20. For 6 days the work that they had done is 6 x (1/15 +1/20) = 6 x 7/60 = 7/10 The sum of x/20 and 7/10 gives the whole work , which is 1. So we get the equation: x/20 +7/10 = 1, where x = 6. Therefore, second worker painted for 6 +6 = 12 days and the first one only 6 days. The painting job done by the second worker is 12 x 1/20 = 60/100 = 60%, and the first worker is 6 x 1/15 = 40/100 = 40%

IV. Evaluation

Set the following proportions and solve. 1) A stock of food is enough to feed 50 persons for 14 days. How many days will the food last if 20

more persons will be added? 2) Four equal pumps can fill a tank in 42 minutes. How long will 6 pumps of the same kind fill the

tank? 3) If 3 farmers can plow a field in 4 days, how long will 6 farmers do it? 4) Five sewers can finish 200 children’s dresses in 8 days. How many days will it take 10 sewers to

finish the same number of children’s dresses? 5) I have enough money to have a vacation of 12 days. If I send 500 a day for how many days

will my money last if I decide to spend only 400 a day?

V. Assignment

Solve these problems. 1) Four teachers can finish interviewing 100 applicants for the school entrance examination in 5

days. If the interview period is to be finished in 2 days only, how many teachers should there be? 2) Sixty boxes are needed to pack 720 brownies in batches of 12. How many boxes are needed if

the brownies are packed in batches of 18? 3) Mr. Datu has enough money to pay 8 workers for 15 days. If he adds 4 more workers, for how

long can he pay them at the same rate?