BASIC EDUCATION ASSISTANCE FOR MINDANAO LEARNING GUIDE

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FIRST YEAR - MATHEMATICS

REAL NUMBER SYSTEM

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Basic Education Assistance for Mindanao (BEAM) project. Prior approval must be

given by the author(s) or the BEAM Project Management Unit and the source must

be clearly acknowledged.

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Mind Map

The Mind Map displays the organization and relationship between the concepts and activities in this Learning Guide in a visual form. It is included to provide visual clues on the structure of the guide and to provide an opportunity for you, the teacher, to reorganize the guide to suit your particular context.

Stages of Learning

The following stages have been identified as optimal in this unit. It should be noted that the stages do not represent individual lessons. Rather, they are a series of stages over one or more lessons and indicate the suggested steps in the development of the targeted competencies and in the achievement of the stated objectives.

Assessment

All six Stages of Learning in this Learning Guide may include some advice on possible formative assessment ideas to assist you in determining the effectiveness of that stage on student learning. It can also provide information about whether the learning goals set for that stage have been achieved. Where possible, and if needed, teachers can use the formative assessment tasks for summative assessment purposes i.e as measures of student performance. It is important that your students know what they will be assessed on.

1. Activating Prior Learning

This stage aims to engage or focus the learners by asking them to call to mind what they know about the topic and connect it with their past learning. Activities could involve making personal connections.

Background or purpose

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Strategy

DECODING. A strategy used to translate data or a message from a code into the original language or form. In the context of this activity, students will simplify certain expressions. After which, students will look for the corresponding answers on the choices to decode the words that will satisfy the given challenge.

Material

 activity sheets (refer to Student Activity 1.1 to 1.3 on pages 16-17)

Activity 1: “Linking Words”

Instructions:

1. Organize the class into 6 groups. 2. Provide each group an activity sheet.

 Group 1 and 4 – Student Activity 1.1  Group 2 and 5 – Student Activity 1.2  Group 3 and 6 – Student Activity 1.3

3. Set a time allotment for them to perform the task.

4. Let them present their outputs to the class for comparison.

Formative Assessment

Roam around. Make sure that everybody participates actively in performing the task. Check the output of each group. Refer to the answer key on page 19.

Roundup

The students would have performed the fundamental operations involving whole numbers.

2. Setting the Context

This stage introduces the students to what will happen in the lessons. The teacher sets the objectives/expectations for the learning experience and an overview how the learning experience will fit into the larger scheme.

Background or purpose

In this stage, the students will tell whether they agree, disagree or have no idea about the given statements involving operations on integers.

Strategy

AGREE/DISAGREE RESTRUCTURED. A strategy that will help students organize data to support a position for or against an idea. It promotes students' thinking about the content. A “Don't Know” column is added to find out the concept which the students do not have prior knowledge.

Materials

 enlarged agree/disagree chart

(refer to Teacher Resource Sheet 1 on page 20)

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Activity 2: “Be Counted”

Instructions:

1. Prior to the activity, post the chart on the board.

2. Ask the students to answer each of the given statements in the 1st_{column of the} chart. Tell them whether they agree, disagree or don't know on it.

3. Instruct them to raise their hands accordingly as each item in the 1st_{column is} read. Tally their responses on the appropriate boxes of the BEFORE columns. 4. Note the misconceptions and address them in the succeeding activities. 5. Keep the chart for this will be revisited in stage 6.

Formative Assessment

Ensure that all students are actively participating in the task.

Roundup

The students would have told whether they agree, disagree or have no idea about the given statements involving operations on integers.

3. Learning Activity Sequence

This stage provides the information about the topic and the activities for the students. Students should be encouraged to discover their own information.

Background or purpose

In this stage, the students will be able to:

 define integers;

 describe opposite quantities in real life;  determine the absolute value of a number;

 solve simple absolute value equations using the number line;  perform fundamental operations on integers; and

 solve problems involving integers.

Strategies

 INTERACTIVE LECTURE. This strategy provides students with a general outline to give

them a framework for thinking about a subject and to structure their notetaking. This type of lecture involves students by focusing their attention on key concepts. It emphasizes information transfer at the knowledge, recall, and comprehension levels of learning.

 GAME. This strategy enables the students to learn and develop their mathematical

skills in a fun and interesting way. It caters students whose comprehensions are activated using visuals. This enhances their logical/mathematical, kinesthetic and spatial intelligences.

Materials

Activity 3 - Bingo!

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Activity 4 – Perfect Combination

 cards (refer to Teacher Resource Sheet 2 on pages 24-25)  activity sheet (refer to Student Activity 4 on page 26)  manila paper

 pentel pen

Teacher's Input 1

Begin this stage by illustrating the number line to the class, and then ask the following questions.

Figure 1 Questions:

1. What do you see in the figure?

2. How many types of numbers are illustrated on it? What are they? 3. Are -2 and 2, -5 and 5, and -4 and 4 equal? Why?

4. What are the natural numbers in the figure? How do you call those numbers that are not classified as natural numbers?

5. How do you name the given figure?

6. What do you think is the collective name of the set of numbers on the number line?

THE NUMBER LINE

Number line is the line representation of numbers.

All numbers can be pictured as points on it. The reference point (origin) on the number line is 0. The numbers to the right of 0 are positive numbers, and the numbers to the left of 0 are negative numbers. The coordinate is the number paired with a point on the number line.

Two integers that are same distance from zero in opposite directions are called opposites. The integers, -1 and 1, -2 and 2, -3 and 3, and so on, are opposite to each other. Thus, opposites as used in Mathematics are denoted by signed numbers.

In terms of direction, going south is the opposite of going north; in terms of length, short is the opposite of long.

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Exercises 1: Use the number line to identify the letters that correspond to the coordinates described below.

1. The reference point on the number line. 2. The natural numbers.

3. The positive numbers. 4. The negative numbers.

5. The point halfway between D and H.

6. The point that is same distance from the origin as H.

7. The point when moving 4 units to the left of the reference point. Possible answers:

1. E

2. F, G, H and I 3. F, G, H and I 4. A, B, C and D

5. F 6. B 7. A

ABSOLUTE VALUE OF AN INTEGER

Refocus the students' attention on the number line, then ask the following questions.

Questions:

1. How many units are there from 0 to +_5? 2. How many units are there from 0 to -6? 3. How many units are there from 0 to -2?

The number of units from 0 to any given integer represents the absolute of the integer.

Thus, the absolute value of -5 is 5. This is indicated as |-5| = 5, and read as “the absolute value of -5 is 5.”

Other examples:

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The absolute value of -17 or |-17| is 17. The value of |+_{8| + |-13| is 8 + 13 is 21.}

SOLVING SIMPLE ABSOLUTE VALUE EQUATIONS USING THE NUMBER LINE

Expressions with the absolute value symbol can be simplified.

Illustrative Examples:

1. Solve and illustrate |x| = 3 using the number line. The distance from 0 to x is

3 units.

Therefore, x = -3 or x = 3.

2. Solve and illustrate |x - 1| = 3 using the number line.

To find the solution to this sentence, x must be a number whose distance from 1 is 3. Thus, think of starting at 1 and moving 3 units in both directions on the number line. You are arrive at -2 and 4 as the solutions.

Therefore, x is equal to -2 or 4.

The diagram shows that |x - 1| = 3 is equivalent to x – 1 = -3

x = -3 + 1 x = -2

x - 1 = 3 x = 3 + 1 x = 4

Exercises 2:

A. Give the absolute value of each of the following. 1. |+_10|

2. |-24|

3. |-13.7| + |+_7.4| 4. |-763| - |+_348| B. Solve and illustrate using the number line.

5. |x| = 5 6. |m| = 8

7. |x - 4| = 7 8. |x + 9| = 12 Possible answers:

1. 10 2. 24 3. 21.1 4. 415

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Activity 3: “Bingo!”

Instructions:

1. Reorganize the class into groups of 5 or as desired.

2. Distribute to each group the activity sheets on pages 21-22. 3. Set a time allotment for them to complete the activity. 4. Let them post their outputs for comparison.

5. Check their outputs using the answer key on page 23.

FUNDAMENTAL OPERATIONS ON INTEGERS A. Addition of Integers

The sum of integers can be obtained by using either the number line or rules.

1. Find the sum of +_{6 and -2 by using the number line.}

To add the integers, start moving 6 units to the right of zero. This movement represents +_{6. From this position, move 2 units to the left to represent -2. Since the} final position is 4 units to the right of zero, therefore the answer is 4.

2. Add -5 and -3.

To add -5 and -3, start moving 5 units to the left of zero. From this position, move further 3 units to the left. Thus, -5 + (-3) = -8

From the given examples, we can say that:

B. Subtraction of Integers

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1. +_{8 - (}+_{4) = ?}

What number must be added to +_{4 that will result to}+_8? In symbols, +_{4 + __ =}+₈

The subtraction of signed numbers and the idea of looking for the missing addend can be explained using the number line.

+_{4 +}+₄₌+_{8. Therefore,}+_{8 - (}+_{4) =}+_4.

2. +_{7 - (-2) = ?}

What number must be added to -2 that will result to +_7? In symbols, -2 + __ = +₇

-2 + +₉₌+_{7. Therefore,}+_{7 - (-2) =}+_9.

The given examples above show that subtraction is the inverse of addition. Thus,

C. Multiplication of Integers

Multiplication is a short cut of repeated addition. The multiplication of signed numbers can be explained through the number line.

1. The product of -4 and 3.

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2. Find the product of +_{2 and}+_3.

(+_{2) x (}+_{3) = (}+_{2) + (}+_{2) + (}+_{2) =}+_{6. Therefore, (}+_{2) x (}+_{3) =}+_6.

The multiplication of two negative numbers can be seen through this pattern: -3 x (+3) = -9 The first column is always -3.

-3 x (+2) = -6

-3 x (+1) = -3 The second column decreases by 1 in each item. -3 x 0 = 0

-3 x (-1) = +3 The result increases by +3. -3 x (-2) = +6

-3 x (-3) = +9 -3 x (-4) = +12

From the pattern, it can be seen that as the multiplier decreases by one, the product increases by three. When the point where multiplication of two negative numbers is reached, the product becomes positive. This shows that when two negative numbers are multiplied, the product is positive.

D. Division of Integers

Division is the inverse operation of multiplication. Hence, division of integers can be explained using the rules in multiplication of integers.

1. +_{30 ÷ (}+_{5) = ?}

What number must be multiplied by +_{5 that will result to}+_30?

+_{5 x ___ =}+_{30. In this example, the answer is}+_{6. Thus,}+_{30 ÷ (}+_{5) =}+_6. 2. +_{30 ÷ (-5) = ?}

What number must be multiplied by -5 that will result to +_30?

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What number must be multiplied by +_{5 that will result to -30?}

+_{5 x ___ = -30. In this example, the answer is -6. Thus, -30 ÷ (}+_{5) = -6.} 4. -30 ÷ (-5) = ?

What number must be multiplied by -5 that will result to -30? -5 x ___ = -30. In this example, the answer is +_{6. Thus, -30 ÷ (-5) = 6.}

The examples show that division as the inverse operation of multiplication can be written as a multiplication problem. The rules for multiplication of integers can also be applied to division of integers.

Activity 4: “Perfect Combination”

Instructions:

1. Reorganize the class into 8 groups.

2. Distribute to each group the activity sheet on page 26 and the other materials. 3. Set a time allotment for them to complete the activity.

4. Let them post their outputs for comparison and discussion.

Formative Assessment

Ensure the active involvement of the students in the different activities. Check their outputs.

Roundup

The students would have defined integers, described opposite quantities in real life, determined the absolute value of a number, solved simple absolute value equations using the number line, performed fundamental operations on integers, and solved problems involving integers.

4. Check for Understanding of the Topic or Skill

This stage is for teachers to find out how much students have understood before they apply it to other learning experiences.

Background or purpose

In this stage, the students will demonstrate knowledge and skills related to integers.

Strategy

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Materials

 cut-outs of the following: ✔ Learning Stations 1–5 ✔ tasks involving integers on

Teacher Resource Sheet 3, pages 27-28

 cartolina/manila paper  pentel pen

 masking tape

Activity 5: “Turn Around”

Before conducting the activity, prepare the following: 5 learning stations inside the classroom

cut-outs of problems to be pasted on each station

Instructions:

1. Organize the class into 5 groups.

2. Assign one group in each station to solve the task/s with time limit.

3. Let them proceed to the next station in a clockwise direction after giving the signal.

4. End up the activity when they have completed all the tasks.

5. Finally, ask them to consolidate their outputs on a manila paper and present it for comparison and discussion.

Formative Assessment

Check each group's outputs.

Roundup

The students would have demonstrated knowledge and skills related to integers.

5. Practice and Application

In this stage, students consolidate their learning through independent or guided practice and transfer their learning to new or different situations.

Background or purpose

In this stage, the students will explore and solve some real-life problems involving integers.

Strategy

PROBLEM SOLVING. Teaching students how to effectively solve problems will provide them with useful lifelong skills. Problem solving models, such as working mathematically model, break problem solving into a step by step process:

CLARIFY What is the problem asking you to do or find out? What is given in the problem?

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USE Use the tools to gain an answer to the problem.

INTERPRET Is this answer reasonable? Can you check it using another method?

Materials

 Teacher Resource Sheet 3 on page 29  23 blocks or dice

 manila paper

 pentel pen  masking tape

Activity 6: “Act It Out”

Instructions:

1. Organize the class into eight (8) groups.

2. Assign in every group a leader to facilitate the given tasks.

3. Prepare two sets of each problem under Teacher Resource Sheet 3 on page 29 in order to have two groups solving a common task.

4. Distribute the materials and let them answer the task at a given time.

5. Instruct the two groups with the same task to compare their answers and reach a consensus.

6. Then, let them finalize their answers on a manila paper for presentation and discussion.

Formative Assessment

Check the outputs of the students. Refer to page 30 for the answer key.

Roundup

The students would have explored and solved some real-life problems involving integers.

6. Closure

This stage brings the series of lessons to a formal conclusion. Teachers may refocus the objectives and summarize the learning gained. Teachers can also foreshadow the next set of learning experiences and make the relevant links.

Background or purpose

In this stage, the students will consolidate their learning on operations on integers by revisiting the chart used in stage 2.

Strategy

AGREE/DISAGREE RESTRUCTURED. A strategy that will help students organize data to support a position for or against an idea. It promotes students' thinking about the content. A “Don't Know” column is added to find out the concept which the students do not have prior knowledge.

Materials

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Activity 7: “Be Counted: A Revisit!”

Instructions:

1. Prior to the activity, post again the chart on the board. 2. Ask the students to complete the AFTER columns of the chart.

3. Tell them to raise their hands accordingly as each item in the 1st_{column is read.} Tally their responses on the appropriate boxes of the AFTER columns.

4. Address the misconceptions if there are any.

Formative Assessment

Ensure the active participation of each student in performing the activity.

Roundup

The students would have consolidated their learning on operations on integers.

Teacher Evaluation

(To be completed by the teacher using this Teacher’s Guide) The ways I will evaluate the success of my teaching this unit are: 1.

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STUDENT ACTIVITY 1.1

Linking Words

Objective: Perform the fundamental operations involving whole numbers.

The numbers to the left of zero on the number line are called __________ numbers.

For you to determine the answer, do the following steps.

1. Perform the fundamental operations on whole numbers in the boxes.

2. Match your answers to these letters.

3. Fill in each box with the appropriate letter that corresponds to your answer.

STUDENT ACTIVITY 1.2

Linking Words

Two integers that are the same distance from zero in opposite directions are called ______________.

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3. Fill in each box with the appropriate letter that corresponds to your answer.

STUDENT ACTIVITY 1.3

Linking Words

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TEACHER RESOURCE SHEET 1

Agree/Disagree Chart

Directions: Enlarge this chart on a manila paper.

STATEMENT

BEFORE

AFTER

Agree Disagree Don't Know

1. An integer is a set of

numbers that consists of whole numbers together with

negative numbers that are the opposite of the nonzero numbers.

2. Zero is not an integer.

3. The absolute value of -3 is 3 and 5 is -5.

4. The sum of two negative numbers is a negative number.

5. The difference of two negative numbers is a negative number.

6. When x – 1= 2, then the value of x is either 1 or 3.

7. The product of two negative numbers is a negative number.

8. The quotient of a positive and a negative number is a negative number.

9. The sum of -7, 12 and -15 is 10.

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STUDENT ACTIVITY 3

Bingo!

Worksheet 1

Objectives:

 Determine the absolute value of the given integers.

 Solve simple absolute value equations using the number line.

Directions:

1. Solve the following problems below.

2. Find the answers in the Bingo board and color them.

3. Shout “Bingo!” if the colored boxes will form either horizontal, vertical or diagonal line.

A. Give the absolute value of each of the following.

1. |+_9| 2. |-3| 3. |+_53|

4. |+_{471| + |-298|} 5. |-871| - |+_710| 6. |+_{78.91| + |-92.09|}

B. Solve and illustrate using the number line.

7. |x| = 12 8. |b| = 451 9. |x – 2| = 6

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STUDENT ACTIVITY 3

Bingo!

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STUDENT ACTIVITY 3

Answer Key for Bingo!

A. 1. 9 2. 3 3. 53 4. 769 5. 161 6. 171

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TEACHER RESOURCE SHEET 2

Perfect Combination

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STUDENT ACTIVITY 4

Perfect Combination

Objective

: Solve problems involving fundamental operations on integers.

1. Use all the cards to obtain the possible numbers when these are all

combined.

2. All numbers and operations must be used once, and the answers must

be integers.

3. Consolidate your answers on a manila paper.

POSSIBLE COMBINATIONS

ANSWERS

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TEACHER RESOURCE SHEET 3

Turn Around

Directions:

1. Prepare the following cut-outs using cartolina/manila paper.

2. Write the following problems for every learning station on a manila paper.

STATION 1

A magic square is a square in which all of the numbers in each row, column, and diagonal add up to the same number. The integers in the magic square below when added

horizontally, vertically or diagonally give the sum of -15.

Problem: In the magic square below, each row, column, and diagonal add up to 3. Find the missing numbers.

STATION 2

Find the product of the integers below. Then, arrange the products from least to greatest and explain the pattern.

a. -10 x 0 b. -8 x 2 c. -4 x -2

d. -8 x -2 e. -3 x -8 f. +_{8 x -1}

5

0 -1

2 -4 -9 -2

-3 -5 -7

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STATION 3

Complete this subtraction table. Subtract each integer in the first row by each integer in the first column.

STATION 4

Use the number line to identify letters that correspond to the coordinates described below.

1. The reference point on the number line. 2. The point halfway between B and H.

3. The point that is same distance from the origin as A.

4. The point when moving 3 units to the right of the reference point. 5. The point when moving 4 units to the left of the reference point.

STATION 5

A student said that if he divides two integers, their quotient is equal to the quotient of their opposites. For example: 45 ÷ 5 = 9, and -45 ÷ -5 = 9.

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TEACHER RESOURCE SHEET 3

Act It Out

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TEACHER RESOURCE SHEET 4

Act It Out – Answer Key

The elevator went up 6 floors, down 5 floors, then down 10 floors to get to floor 1. Let x = floor level where Joy lives.

Equation: x + 6 – 5 – 10 = 1 x – 9 = 1 x = 1 + 9

x = 10. So Joy lives on the 10th_floor. Therefore, Jay lives on the 16th_floor.

Use integers +_{6, -5, -10 and 1.}

Combine all the integers: +_{6 + (-5) + (-10) + 1 = 10. Therefore, Jay lives on the 10}th_floor.

Possible combinations at the lowest cost: 2 kg + 5 kg + 5 kg + 5 kg = 17 kg

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For the Teacher:

Translate the information in this Learning Guide into the following matrix to help you prepare your lesson plans.

Stage

_1.

Activating Prior Learning

2.

Setting the

Context

3.

Learning

Activity Sequence

4.

Check for

Understanding

5.

Practice and

Application

6.

Closure

Strategies

Activities from the Learning Guide

Extra activities you may wish to include

Materials and planning needed

Estimated time for this Stage