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Lesson Guide
In
Elementary Mathematics
Grade 5
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
Chapter II
Rational Numbers
Ratio and Proportion
INSTRUCTIONAL MATERIALS COUNCIL SECRETARIAT, 2011
Lesson Guides in Elementary Mathematics Grade 5 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 5
Region 3
Nelia D. Bamba – Pampanga Magdalena P. Garcia – Angeles City
Region 4-A (CALABARZON)
Gundalina C. Gonzales – Batangas Gerlie Ilagan – Quezon Consuelo Caringal – Batangas
National Capital Region (NCR)
Teresita P. Tagulao – Pasig City Alyn G. Mendoza – Mandaluyong City Emma S. Makasiray – Pasig/San Juan Ester A. Santiago – Valenzuela Milagros Juakay – Pasig/San Juan Lucia Almazan – Manila Julie La Guardia – Valenzuela
Bureau of Elementary Education (BEE)
Federico L. Reyno Noemi B. Aguilar
Ateneo de Manila University
Grace Uy 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 – Principal, 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 – OIC, Undersecretary for Finance and Administration
Vilma L. Labrador – Undersecretary for Programs and Projects Teresita G. Inciong – Assistant Secretary for Programs and Projects
Printed By: ISBN – 971-92775-4-8
iii
TABLE OF CONTENTS
Introduction .................................................................................................................................. iv Matrix ........................................................................................................................................ v II. RATIONAL NUMBERS
Ratio and Proportion
Comparing and Writing Ratios .............................................................................................. 1 Reducing Ratios in Lowest Terms ........................................................................................ 5 Equal Ratios ......................................................................................................................... 9
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 V
COMPETENCIES VALUES INTEGRATED STRATEGIES USED MULTIPLE INTELLIGENCES
TECHNIQUES With HOTS
II. RATIONAL NUMBERS
Comprehension of Ratio and Proportion
Reducing ratios to lowest terms Love for Mother Earth
Modeling, Use of tables
Global themes, Charts √
1. Express the ratio of two numbers
1.1 Visualize the ratio of two given sets of objects
Appreciation, Cooperation
Modeling, Use of tables, Finding a
pattern
Tables, Self-awareness, Cooperative groups
√
1.2 Express the ratio of two numbers by using either colon (:) or a fraction
Appreciation Use of tables, Concept
development
Cooperative groups, Hands-on activities
2. Forms Proportion for Numbers Cooperation Cooperative learning, Finding a pattern, Concept
development
Cooperative groups √
1
Comparing and Writing Ratios
I. Learning Objectives
Cognitive: Compare the two quantities using ratio Psychomotor: Write ratios in 2 ways Affective: Appreciate use of ratio in real-life situations
II. Learning Content
Skill: Naming and writing ratios in two ways Reference: BEC-PELC II.E.1.1-1.2 Materials: flash cards, cutouts, real objects Value: Appreciation for use of ratio
III. Learning Experiences
A. Preparatory Activities
1. Mental Computation (Drill)
Reducing fractions to lowest terms as review of previous lesson
Use flash cards (pen-and-paper drill)
B. Developmental Activities
1. Presentation Strategy 1: Using Actual Pupils in Naming Ratio (Use of tables)
Mechanics: a. Let the pupils count the number of girls and boys in their respective rows. b. Let the pupils write their answers on the board. c. Tabulate the data on the board as follows:
Row number Number of Boys Number of Girls Number of Pupils in a Row
d. Ask the following questions:
How many pupils are there in each row?
How do you compare the number of boys to the number of girls in Row 1? Row 2?
e. Introduce the concept of ratio as the comparison of two quantities. f. More examples:
Compare the number of boys in Row 1 and 2.
Compare the number of girls in Rows 3 and 4.
Compare the number of pupils in Rows 2 and 4. g. Suggested answers to be written on the board:
The ratio of girls to boys in Row 2 is 3 to 8.
The ratio of the pupils in Row 1 to the pupils in Row 2 is 10 to 11. h. Lead pupils to state that ratios can also be written in other ways:
2
i. Going back to the examples used in class activity, ask:
If I ask for the ratio of boys to girls in Row 1 and Row 2, can I interchange the numbers in the ratio?
Why or why not? j. Lead pupils to conclusions that the terms of a ratio are not commutative. k. Give more examples or have pupils form other ratios by using objects in the
classroom.
Strategy 2: Use of Concrete Objects or Cutouts (Modeling)
Materials: concrete objects or cutouts a. Teacher places different objects on the table like notebooks, pencils, crayons, and
sheets of paper. b. Ask pupils to count the number of objects in each set and write their answers on the
board. Sample answers:
Set 1 – 2 notebooks, 3 pencils Set 2 – 5 crayons, 10 sheets of paper
c. Ask: How will you compare the number of notebooks with the number of pencils? (There are 2 notebooks for every 3 pencils.)
d. The teacher introduces the term “ratio”. e. The ratio of notebooks to pencils is 2 to 3. f. Define ratio as the comparison of two quantities. g. Can we interchange the terms in a ratio? Why or why not? h. Write the ratios in 2 ways (Refer to letter h of Strategy 1) i. Give more examples or have pupils form ratios using actual objects.
Example: objects in the classroom
2. Fixing Skills
A. Write a ratio for each of the following; first as fraction in lowest terms and then with the colon (:).
1) 5 out of 12, , 5:12
2) 12 chairs to 3 tables ______, _______
3) 1 day to a week ______, _______
4) 10 boys to 20 girls ______, _______
5) 2 pencils for 5 pesos ______, _______
B. Write a ratio in lowest for each of the following.
1) 30 cm to 8 dm ______
2) 18 in to 3 ft ______
Colon Form Fraction Form
The ratio 3 to 8 can also be written as 3:8
8
3
The ratio 10 to 11 can also be written as 10:11
11
10
3
3) 15 days to 1 month ______
4) 18 hours to one day ______
5) 455 m to 1 km ______
6) 8 years to 3 decades ______
7) 2 T-shirts for P190 ______
8) 5 yoyos for 5 boys ______
C. In all-girl class of 45 pupils, 12 wore black shoes, 17 wore brown shoes and the rest wore white rubber shoes.
1) What is the ratio of girls wearing black shoes to those wearing brown shoes? 2) What is the ratio of those wearing brown shoes to those wearing white rubber
shoes? 3) What is the ratio of white rubber shoes to those wearing black and brown shoes?
3. Generalization
What is ratio? Ratio is the comparison of two quantities or given sets of objects. It is also a pair of
numbers that compare two quantities in the same unit by division. What are the 2 ways of writing ratio?
We write ratio using colon or in fraction form.
As an extended lesson: Ratios can also be defined as a comparison of “2 or more” quantities. The teacher
may cite these quantities being compared such as “2 ball pens to 3 notebooks to 5 pencils”, which can be written as 2:3:5 in colon form but seldom used or written in fraction
as 5
3
2
.
C. Application
1. Write the ratio using colon or fraction form.
a) 5 , 6 The ratio of balls to balloons =
b) 2 , 8 The ratio of triangles to circles =
2. Visualize the ratio of these sets of objects by using illustrations or drawings. a) 15 apples to 3 plastic bags b) 9 boxes to 45 candies c) 1 ball pen to 3 notebooks d) 3 blouses to 2 skirts e) 15 blue marbles to 10 red marbles
4
3. During the open tournament in basketball, a team played 45 games. It won 35 games and lost 10.
a) Compare the number of games won with the number of games lost. b) Compare the number of games lost with the number of games won. c) Compare the number of games won/lost with the total number of games.
IV. Evaluation
A. Write the ratio of the following in 2 ways. (colon form and fraction form) 1) 3 books, 5 bags – ratio of books to bags 2) 10 candies, 2 chocolate bars – ratio of chocolate bars to candies 3) 7 boys, 9 girls – ratio of girls to boys 4) 3 dogs, 8 cats – ratio of dogs to cats 5) 6 forks, 12 spoons – ratio of spoons to forks
B. Read the word problem and answer the questions that follow.
In a field trip by the Grade V class, 10 children took the caterpillar ride, 15 took the merry-go-round, 3 took the Condor ride, 5 took the roller coaster and the rest just walked around. If there are 35 pupils in the Grade V class, express the following ratios:
1) The number of pupils who took the caterpillar ride to the number of pupils who took the
roller coaster. 2) The number of pupils who did not take any ride to the total number of pupils in class. 3) The number of pupils who took the Condor ride to the number of pupils who took the
merry-go-round. 4) The total number of pupils to the number of pupils who took the caterpillar. 5) The number of pupils who took the merry-go-round to the number of pupils who did not
take any ride.
C. Write the ratio in 2 ways. 1) ratio of unshaded to shaded parts 2) ratio of all the parts to unshaded parts
3) ratio of all the parts to unshaded parts
4) ratio of apples to mangoes 5) ratio of all the fruits to apples 6) ratio of vowels to consonants in the English alphabet
5
V. Assignment
Write the following ratios in 2 ways.
1. number of days in a week to the number of months in a year. 2. number of hours in a day to the number of hours in a week 3. 30 cm to 2 m 4. number of eggs in a dozen to the number of eggs in 3 dozens 5. 700 grams to 2 kilograms
Reducing Ratios to Lowest Terms
I. Learning Objectives
Cognitive: 1. Reduce ratios to lowest terms 2. Solve word problems using ratios Psychomotor: Write ratios in lowest terms Affective: Demonstrate love for Mother Earth by recycling
II. Learning Content
Skill: Writing ratios in lowest terms, solving word problems Reference: BEC-PELC II.E.1.3 Materials: flash cards, cutouts, real objects Value: Love for Mother Earth by recycling
III. Learning Experiences
A. Preparatory Activities
1. Drill
Mental computation on reducing fractions to lowest terms
Strategy 1: Traveling Game
Materials: flash cards Mechanics: a. Pupil no. 1 challenges the person seating directly beside or behind him. b. Teacher flashes a card. The pupil who gives out the correct answer first moves on to
challenge the next pupil. Losing pupil sits on the vacant chair. c. Winner continues to move until he loses. In this case, he sits on the chair of the new
winner. d. Pupils who has traveled the farthest from his original seat may be crowned “King or
Queen” of the Day.
2. Review
Review definition of ratio and the two ways of writing ratio (colon and fraction form)
3. Motivation
a. Use patterns to help you complete the table below.
6
Recycle Campaign
Number of Bottles
(kg) 1 2 7 10 15
points 5 10 15 60
b. Describe a pattern you found from the table. c. Write the ratio for the lowest term for 15 points; 10 kg; 15; 60 points d. What benefits can be derived from the recycle campaign activity?
B. Development Activities
1. Presentation Strategy 1: Use Concrete Objects or Cutouts to Answer Problem Opener
(Modeling)
Materials: several bottles of soft drinks or cutouts
Joel and Josie went to the park to help clean up by picking discarded bottles of softdrinks to be recycled. Joel picked 24 bottles while Josie got 12 bottles. What is the ratio of the number of bottles gathered by Joel with the number of bottles gathered by Josie in lowest term?
a. Teacher shows to class the bottles. Ask what are given? What is being asked? b. Ask the following:
What are given?
What is being asked?
Would you have done the same thing as what Joel and Josie did?
In what ways can you help Mother Earth? Cite some ways of recycling.
Why do we recycle? c. Have a pupil show the number of bottles Joel and Josie each got. Lead pupils to
give the ratio. d. Ask: How else can we represent the ratio of the number of bottles of Joel to the
number of bottles of Josie? e. Lead pupils to seeing that 4:2 is the same as 2:1 or for every bottle Josie gets, Joel
gets 2. f. Tell pupils that like fractions, ratios may be reduced to lowest terms by dividing the
numerator and denominator by a common factor. Take note, though, that 2:1 or 1
2
cannot be written as simply 2 because ratio is a comparison of a two quantities.
Mention also that we do not write ratios as mixed numbers (e.g. 8
11 ).
g. Solve the problem by making a table or an organized list. Write the table on the board. Have pupils use the bottles to find the data for the table.
h. Solve the problem. i. Provide more exercises in reducing ratios to lowest terms and problem solving.
Softdrink bottles
Joel 4
Josie 2
Total 6
7
Strategy 2: Using Pupils to Solve Problem Opener
In the launching of the Class Recycling Day, there were 40 girls and 25 boys participated in the activity. What is the ratio of the girls to the boys in lowest term?
a. Ask the following:
What are given?
What is being asked?
Why is there a need to recycle?
What things can we recycle? b. Call on 5 boys and 8 girls to come up front. c. Ask the pupils to name the ratio of boys to girls. What is another way of naming the
ratio? d. Proceed as in Strategy 1. Ask leading questions which were used in letters e and f. e. Solve the problem using a table or an organized list. f. Provide more exercises.
2. Fixing Skills
A. Write a ratio for each of the following. Give each lowest terms.
1) 15 boys to 30 girls _______ 2) 5 boxes to 60 oranges _______ 3) 2 jeep for 44 passengers _______ 4) 7 days to 3 weeks _______ 5) 6 decades to 1 century _______
B. Find the ratios of the time spent for the different activities.
AJ’s Daily Activities No. of Hours
Sleep 7
School 6
Exercise 1
Studying/Doing Assignments 2
Recreation 1
Meals and Personal Hygiene 2
Other Activities 3
1) spent sleeping to a whole day _______ 2) spent in studying to the time in recreation _______ 3) spent in school to a whole day _______ 4) spent in others to meal and person and al hygiene _______ 5) spent in exercise to sleeping _______
C. Solve the following:
1) Two numbers are in the ratio 3:5. If the smaller number is 12, what is the bigger
number? 2) The ratio of the age of CA to that of RJ is 3:5. If CA is 10 years old, how old is
RJ? 3) A piece of rope is 36 m long. If you cut it into 2 pieces so that the pieces are in
the ratio 4:5, how long will each piece be?
8
3. Generalization
What did you learn today? How do we reduce ratios to lowest terms?
We divide the numerator and denominator by a common factor until the two numbers have the number 1 as the only common factor.
C. Application
A. Express the ratio of the first quantity to the second quantity and reduce to simplest form.
1) 2 teachers to 46 pupils 2) 4 books to 10 students 3) 12 flowers to 4 vases 4) 21 garbage cans to 14 classroom 5) 36 glasses of juice to 30 sandwiches
B. In a class, the ratio of boys to girls is 4:5. If there are 16 boys, how many girls are there?
C. Complete the table using the cubes.
No. Ratio of : Edges Areas of 1
Face Total Surface
Areas Volumes
1 A to B 1:2 1:4
2 C to A
3 D to E
4 B to D
5 A to E
IV. Evaluation
A. Reduce these ratios to lowest terms.
1) 10: 5 4) 30
15 =
2) 3: 12
3) 20: 4 5) 8
6 =
C
A
D
1 dm 2 dm 3 dm 4 dm 5 dm
E
9
B. Write the following ratios to lowest terms: 1) 25 wins in 30 games 2) 8 red balls to 12 blue balls 3) 25 dm to 2 m 4) What is the ratio of the letter O to all the letters in the word COLOR? 5) What is the ratio of Saturday and Sunday to the days in a week?
V. Assignment
A. Express these ratios in lowest term.
1) 10
100 2) 2
120
3) 500
50 4) 4000
4
5) 2
200
B. Write these ratios in lowest term.
1) 14: 10 2) 24: 16 3) 6: 9 4) 20: 20 5) 12: 18
Equal Ratios
I. Learning Objectives
Cognitive: 1. Identify equal ratios
2. Find the missing term in equal ratios Psychomotor: Write equal ratios in two ways Affective: Appreciate the value of good nutrition to one’s health
II. Learning Content
Skill: Identifying equal ratios Reference: BEC-PELC II.E.1.4 Materials: pictures Value: Appreciating the value of proper nutrition
III. Learning Experiences
A. Preparatory Activities
1. Drill
Naming Ratios
Strategy: “ Buhay Reporter”
10
Materials: pictures for each of the following ratios will be posted in different parts of the room a. (5) red cars to (6) white cars b. (3) handbags to (4) hats c. (6) mayas to (4) eagles d. (5) kites to (7) tops Mechanics: a. The teacher will tell the pupils that they will be acting as reporters. They will go
around the room and look for something “to report on”. b. They will specifically report on the “news items” related with ratios. c. Remind them that they will name the ratios in 3 ways. d. At the teacher’s signal, the pupils will go around. After 2 minutes they will be
asked to report individually.
2. Review
Reducing ratios to lowest terms
Let the pupils recall the ratios used in the drill. Teacher may ask, “Which ratio is not in the lowest terms?” Other ratios aside from 6 is to 4 will be provided for the pupils to identify.
3. Motivation
Teacher asks the following questions: a. Do you know how to cook? What recipes can you cook?
Let them express that it is important to maintain the ratio of the amount of each
ingredient to preserve the good taste of the food and its nutritional value.
B. Development Activities
1. Presentation
Strategy 1: Use of Tables (Looking for Patterns)
a. Present the following: Two eggs are needed to make 7 pancakes. How many eggs will be consumed for 28 pancakes?
b. Make a table like this:
Eggs 2 4 6 8
Pancakes 7 14 21 28
c. Remind the pupils that the table can be completed by finding equivalent fractions.
What is the missing number in 7
2 =
28
8
d. What other fractions are equivalent to 28
8 ?
e. Lead the pupils to discover the pattern on the table. f. Give more tables to complete.
Boys _____ 8 _____ _____
Girls 3 6 9 12
11
Strategy 2: Using of Price List (Listing) a. Present the following price list:
Figure A Figure B
b. Ask: What is the ratio of the pencils to the price in Figure A? In Figure B?
Expected answers: A. 2 : 5 or A. 5
2 B. 10
4
B. 4 :10 c. Tell the pupils to write the ratios as fractions.
Ask: What can you say about the ratios 5
2 and
10
4?
How did you know they are equal?
5
2 =
10
4
2 4 = 20 5 10 = 20
2. Fixing Skills
A. Fill in each blank to complete the sentence based on the given proportion,
16:28 = 4:7
1) The first term is _____.
2) The second term is _____.
3) The third term is _____.
4) The last term is _____.
5) Why are 16:28 and 4:7 equal ratios? _____
B. Solve for the missing term in each proportion.
1) n
8 =
45
14
12
2) 7
5 =
35
x
3) y
36 =
6
8
4) 12
18 =
m
12
5) 9
2 =
18
10
C. Tell whether x = 25 is a reasonable answer to complete each proportion. Explain
how you decided.
1) x
45 =
50
90 3)
99
x =
108
60
2) 120
150 =
20
x
4) 48
33 =
80
x
3. Generalization
Lead the pupils to answer these questions. When are two ratios equal? How can we build a set of equal ratios?
The process of multiplying the means and extremes is called getting the cross products. We can use cross products to find out if two ratios are equal.
C. Application
A. Complete the table to build a set of equal ratios.
Petals 5 20
Leaves 12
B. Solve:
1) If 12 pencils are bought for P60, how much will you pay for 25 pencils at the
same rate? 2) The ratio of two numbers is 3:5. If the smaller number is 45, what is the larger
number? 3) Two numbers are in the ratio 4:5. If the sum is 130, find the two numbers?
4) A car travels 120 km in 2 h and a train travels 40 km in 30 min. Find the ratio of
the speed of the car to that of the train.
13
IV. Evaluation
A. Identify which of the following show equal ratio. Write Yes or No on the blank.
1) 4
3 = 16
12 ______
2) 2
3 =
4
6 ______
3) 5
4 = 15
13 ______
4) 36
6 =
6
1 ______
5) 2
9 =
16
72 ______
B. Give 3 ratios that are equal of the following.
1) 2 : 5 4) What is the next ratio 2) 4 : 1
3) 2 : 14 8
3 =
16
6 =
5) = 12
10 =
15
10, What is the first ratio?
C. Complete the equal ratios.
1) 3
2 = 6
2) 2 : = 9 : 5
3) 3 = 3
2 4) : 21 = 8 : 3
5) 12
24 = 12
6) Two numbers are in the ratio 2:3. If the bigger number is 6, what is the smaller number? 7) What number compared with 8 is the same as 6 compared with 24? 8) There are 18 roses for every 6 roses. How many roses are there if there are 3 roses? 9) There are 7 children for every 2 adults in a plaza. How many adults are there if there are
21 children? 10) Two numbers are in the ratio 3:5. If the difference is 12, what are the numbers?
14
V. Assignment
A. Find the unit rate.
1) 180 kilometers in 3 hours
2) 75 stools in 3 weeks
3) 300 words in 5 minutes
B. Form ratios equal to the given ratios.
2:7 5 to 3