Math Chapter 1 - Integers

Mathematics for Junior High School - Year 7PB Student’s Book Chapter 1 - Integer 1

In this chapter, you will learn how to use negative numbers, draw integers in a number line, compare integers, and put integers in a sequence. The key terms that you need to understand are integers, positive integers, and number line. Integers consist of natural numbers (the counting numbers 1, 2, 3, 4, ...), a negative of a natural numbers (-1, -2, -3, -4, ...), and zero. A large proportion of mathematics has been devoted to integers because of their immediate application to real situations.

You may learn the integers by first understanding the descriptions andexamples in each chapter; trying to solve each problem grading from the simplest to the most complex, and applying the use of integers in the real life. You may learn integers by uncovering their components and relationship among them. Section 1.2 discusses operation of integer. Section 1.3 discusses problem solving. The following is the map of integers and their components:

Chapter 1Integers

Integers

Positive IntegersNegative Integers

Addition Subtraction Multiplication Division

Zero

Operation

Mathematics for Junior High School - Year 72 Student’s Book Chapter 1 - Integer 3Mathematics for Junior High School - Year 72 Student’s Book Chapter 1 - Integer 3

Learning ObjectivesTo use negative •numbersTo draw integers in a •number lineTo compare integers•To put integers in a •sequence

Key Terms:integers•positive integers •negative integers•number line•

We sometimes need to predict when the rain will come and what the maximum or minimum temperature will be like. You also sometimes need to determine the temperature of water. To indicate correctly the temperature of water we can use a thermometer. Do you have any experience on how to read a thermometer? From the history, you learn that Celsiusdefined“upside-down”scaletosettheboilingpoint of pure water at 0 °C (212 °F) and the freezing point at 100 °C (32 °F); while Jean Pierre Cristin inverted the scale with the freezing point at 0 °C (32 °F) and the boiling point at 100 °C (212 °F).

The above problems of indicating the temperature are some of the examples of the application of integers. An integer can represent the result of calculation i.e. addition, subtraction, multiplication or division. An integercanbedistinguishedas“negative”,“zero”,or“positive”.Variable-lengthrepresentationsofintegers,can store any integer that fits in the computer’smemory.

Itisdifficulttoimaginedoingmanythingswithout using integers. People wake up in the morning and then read the integers on the clock. You might use the integers when you call someone and write his or her phone number. Your teacher gives you homework and you may check your homework on a certain page of your text book. You can easily find the integers by

using a calculator.Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.

Section 1.1Integers and Their Symbols


A. Positive and Negative NumbersIn a weather forecast, it is predicted that it will

rain in city P with a minimum temperature of 1ºC below zero and a maximum temperature of 6ºC. It will be sunny city Q with a temperature of 6ºC and cloudy in city R with a temperature of 10ºC. Can whole numbers 0, 1, 2, 3, ... represent the situation above?

Look at the picture of a thermometer on the left. What numbers are on the picture?

You may write the temperature of 5 degrees above zero as 5ºC, and write the temperature of 5 degrees below zero as -5 ºC. The number 5 is read as positive 5 while the number –5 is read as negative 5. The numbers 5 and -5 can be drawn on a vertical or a horizontal line as follows.

-5 0 +5

Horizontal number line

A group of integers; ..., -3, -2, -1, 0, 1, 2, 3, ... , can be symbolized as I. The three dots (...) on the right and the left mean that the numbers continue without ending to both directions.

The complete number line of integers can be drawn as follows.

0 is neither positive nor negative integers

Positive integers

0 is neither positive nor negative

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Negative integers

-6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6

+5

0

-5

Remember

= is read “is equal to”< is read “is less than”> is read “is greater than”

3 < 5 is read “three is less than five”5 > 3 is read “five is greater than three”


Example 1 a. Determine integers between -5 and 4. Solution:

-4, -3, -2, -1, 0, 1, 2, 3

b. Write all even integers between -6 and 11. Solution: The even integers between -6 and 11 are -4, -2, 0,

2, 4, 6, 8, and 10

Use the number line above to, answer the following questions

What is the number on the left of 0 with the same •distance as the distance from 0 to 2? The number is called the opposite of 2.What is the number on the right of 0 with the same •distance as the distance from 0 to -4? The number is called the opposite of-4. What is the sum of -4 and its opposite? •What is the oppositeof 6?•What is the opposite of -5?•Withoutusinganumberline,findtheoppositeof12.•Withoutusinganumberline,findtheoppositeof-15.•Does every integer have an opposite?•What is the sum of an integer and its opposite?•

Suppose 5 and -5 are on a number line. How many units are there from 0 to 5? How many units are there from 0 to -5? Two numbers are said to be the inverse each other if they have the same distance from 0 on the number line, but have a different direction. What are

other numbers that are the inverse of each other?

B. Comparing and Ordering IntegersLet 5 and 3 be on a number line. Which number has

a longer distance from 0? Which number has a shorter distance from 0? What other numbers have a shorter distance than 5? What other numbers have a longer distance from 0 than 3?


Problem 1

Problem 2

On a horizontal number line, the numbers located on the left side of another number are always less than those on the right side. For instance, since 3 is on the left side of 5, 3 is less than 5, written as 3 < 5. Since 5 is on right side of 3, 5 is greater than 3 or 5 > 3.

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

decrease increase

What happened if we use a vertical number line?

Example 2 Substitute the sign on -4 -7, to <, >, or =

Solution: -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

The number -4 is located on the right side of -7. Therefore, -4 > -7.

Arrange the following numbers from the least to the greatest.a. 5, -3, 6, -6, 2, 4, -1b. 9, -5, 6, -12, 17, 8, -14

In a mathematics test, the score for the right answer is 4, for the wrong answer is -1, and the for failing answer is 0. Complete the table below (use a spreadsheet, if possible) and then sort the students out based on their scores from the highest to the lowest.

Names

The Number

of Correct

Answers

The Number of Wrong Answers

The Number of Unanswered Questions

Total Score

Abdullah 5 3 2 5 x 4+3 x (-1)+2.0 = 17Aminah 6 4 0Galuh 5 2 3Zainul 8 2 0Nabila 8 1 1Zaty 8 0 2Hamidah 7 1 2Yusuf 7 3 0


Task 1.1 1. Draw a number line. Put each of the following numbers on your number line.

a. –1 b. 4 c. –7

d. –9 e. 2 f. 8

2. Write an integer expressing a temperature of 14 degrees Celsius below zero.

3. Write the inverses of the following integers.

a. 13 b. -8 c. 150 d. -212

4. Write three opposing pairs of situations. For example, stepping up two steps and stepping down two steps on a staircase.

5 Substitute the sign to <, >, or =.

a. 0 -8 f. 76 -239

b. 1 -7 g. -999 -99

c. -12 -5 h. -999 -99

d. -3 -7 i. -45 -45

e. -66 5

6. Write the following integers from the least to the greatest.

a. -2, 3, 4, -1

b. 3, -2, 0, -7

c. 4, -5, -2, 3, -1

d. -12, 0, -3, 9, 98, -10, 54

e. -1, 0, -11, -101, -111, 101,

7. Write the following integers from the least to the greatest.

a. -10, 8, 0,-6, 5 c. 0, -12, -3, -5-64

b. 56, -56, -40 d. 75, -3, -4, 12, 0, 9, -10

8. Write an integer which is between the integers below.

a. -7 and 3 c. -5 and -13

b. 0 and -6

9. Why is any negative integer less than any positive integer? Explain.

10. Write down the steps to determine whether an integer is greater or less than another integer.

11. Which of the following sentences is true?

a. -4 > -3 c. -3 > -4

b. 4 < -3 d. -4 > 3


Learning Objectives:To use negative •numbersTo draw integers in a •number lineTo compare integers•To put integers in a •sequence

Key Terms:integers•positive integers •negative integers•number line•

A. Mental Mathematics Mental mathematics or mental calculation is often

used in daily life. In a store, for instance, sometimes we do mental calculation to check whether if our money is enough to buy things. Have you ever done any mental calculation?

There are several techniques used for doing metal calculations. We will discuss four of them: properties, compatible numbers, compensation, and left-to-right methods.

1. PropertiesProperties Commutative, Associative, and

Distributive play an important role in simplifying calculations so that they can be performed mentally.

Example 1 Calculate the following mentally.1. 25 + (38 + 35)2. 32 x 19 – 14 x 323. (4 x 9) x 254. 97 + 86 5. 78 + 39Solution:1. 25 + (38 + 35) = (38 + 35) + 25 = 38 + (35 + 25) = 38 + 60 = 98.

(Use commutative and associative properties) 2. 32 x 19 – 14 x 32 = 32 x 19 – 32 x14 = 32 x (19 – 14) = 32 x 5 = 160.

Use commutative and distributive properties)3. (4 x 9) 25 = (9 x 4) x 25 = 9 x (4x 25) = 9 x 100 = 900. (Use commutative and associative properties)

Section 1.2Operations of Integers


Problem 1

Problem 2

4. 97 + 86 = 97 + (3 + 83) = (97 + 3) + 83 = 100 + 83 = 183. (Use associative property) 5. 78 + 39 = 70 + 30 + 8 + 9 = 100 + 17 = 117.

(Use commutative and associative properties)

Calculate mentally using properties.a. (64 + 35) + 26b. (8 x 17)

Example 2

2. Compatible NumbersCompatible numbers are numbers whose sums,

differences, products, or quotients are easy to calculate.

77 and 23 are compatible numbers under addition because it is easy to get their sum.4 and 25 are compatible numbers under multiplication because it is easy to multiply them.800 and 40 are compatible numbers under division because it is easy to divide 800 by 40.

1. Calculate the following mentally using properties and/or compatible numbers.

a. (8 x 17 ) x 25 b. 76 x 25 c. 2320 : 802. Calculate mentally using compatible number

estimation. a. 73 x 87 b. 56 x 284 c. 5304 : 26 d. 1606 : 73

3. CompensationThe process of reformulating a sum, difference,

product, or quotient to one that is more easily obtained mentally is compensation.


Example 3

Example 4

1) Calculate 26 + (33 + 18). Reformulate 33 + 18 mentally as 34 + 17, to obtain 26 + (33 + 18) = (26 + 34) + 17 = 60 + 17 = 77. 2) Additive compensation is used when 97 + 89 = 96 + 90 = 100 + 863) 67 – 28 can be thought of as 69 – 30.

The use of compensation in subtraction is called the equal addition method. The same number is added to both 67 and 28.

4) Multiplicative compensation 78 x 5 can be rewritten as 39 x 10.

1) Calculate 456 + 332. First add the hundreds (400 + 300 = 700);

then add the tens (50 + 30 = 80); and add the ones (6 + 2= 8),

to obtain 700+ 80 + 8 = 788.2) Calculate 257 +169. Youcanfirstadd200+100=300,300+50+60=410, 410+7+9, to obtain 426. Alternatively, 257 + 169, youcanfirstadd257+100=357, 357 + 60 = 417, 417 + 9 = 426 3) The multiplication 7 256 can be thought of mentally

as 7 200 + 7 50 + 7 6 using the distributive law, or 7 250 + 7 6 = 1792.

1. Calculate mentally using the indicated method.a. 468 – 329 using the equal addition methodb. 543 + 378 using additive compensationc. 198 + 676 using the left-to-right methodd. 88 x 125 using multiplicative compensatione. 9 x 236 using left-to-right method

2. Calculate mentally left to right.a. 234 + 655 b. 566 + 343 c. 678 – 457

4. Left-to-Right MethodLeft-to-Right method is one of the methods to do

calculation. You may understand this method by learning the following example.


b. One-Column/Two-Column Front-EndThe one-column front-end method always

provides low estimates in addition problems as well as in multiplication problems

Example 6Estimate 397 + 153 ? To estimate the result of addition 397 + 153, you may think that 300 + 100 = 400.

Example 5

B. Estimation with IntegersMany daily calculations are simply solved

by estimating the value. There are two types of computational estimation: Front-End Estimation and Rounding. The symbol “≈” is used to show that theresult is only approximation. It is read as approximately equal to.

1. Front-End Estimationa. Range Estimation

The following example shows how ranges can be obtained in addition and multiplication.

Find a range for the answer to these computations by using only the front-end digits.1) 156 + 378 2) 393 × 54 Solution:1) Sum 156 + 378

Low estimate 100 + 300 = 400High estimate 200 + 400 = 600Thus, the range for the answer is from 400 to 600.

2) Product 393 × 54 . Low estimate : 300 × 50 = 15,000 High estimate: 400 × 60 = 24,000


Example 7

Example 8

1) Estimate 264 + 51 + 326. Using one-column front-end method since there are no hundreds in 51 that yield 200 + 300 = 500.

2) Estimate 264 + 51 + 326.Using the two-column front-end method that yields260 + 50 + 320 = 630, which is closer to the exact answer 641 than the 500 obtained using the one-column front-end method.

Notice:The two-column front-end method also provides a low estimate for sums and products. However, this estimate is closer to the exact answer than one obtained from using only one column.

Tofind397+142,think 300 + 100 = 400 and, 97 + 42 is about 140. Thus the estimate is 400 + 140 = 540.

c. Front-End with Adjustment This method enhances the one-column front-end method. Unlike one-column or two-column front-end estimates, this technique may produce either a low esti-mate or a high estimate.

Solve the following problems:

1. Estimate each of the following using the four front-end methods (i) one-column, (ii) range, (iii) two-column, and (iv) with adjustment.

a. 5674 + 4325

b. 3457 + 5764 + 85 + 567

c. 34786 + 787 + 4534 + 341

2. Find a range estimate for these products.

a. 2937 × 259

b. 459 × 89,943

c. 2347 × 55


Example 9

Example 10

a. Round Up (Down) Rounding is the process for approximating a number

to the nearest value, which may be less than or greater than the original number.

1) The number 364 which is rounded up to the nearest tens place is 370 since 364 is between 360 and 370 and 370 is above 364.

2) The number 364 which is rounded down to the nearest tens place is 360.

3) The number 2367 which is truncated to the hundreds place is 2300.

b. Round a 5 UpThe most common rounding technique which is used

in schools is the round a 5 up method.

1) Round 255 to the nearest ten;The round a 5 up method always rounds such number up, so 255 rounds to 260.

3. Estimate using the method indicated.

a. 402 × 621 using one-column front-end

b. 1300 × 45 using range estimation

c. 5257 – 1768 using two-column front-end

d. 4167 + 984 using front-end adjustment

2. Rounding Rounding is perhaps the best known computation-al estimation technique. The purpose of rounding is to replace complicated numbers with the simpler ones. Consider the four types of rounding estimation:

1. round up (down)

2. round a 5 up

3. round to the nearest even

4. round to compatible numbers.

Notice:• 251 to 254 are

rounded to 250.• 256 to 259 are

rounded to 260.


Example 11

Example 12

Example 14

Example 13

c. Round to the Nearest EvenRounding to the nearest even can be used to avoid

errors of accumulation in rounding.

If 255 + 385 (= 640) is estimated by rounding up to the tens place or rounding a 5 up, the answer is 260 + 390 = 650.By rounding down, the estimate is 250 + 380 = 630. Since 255 is between 260 and 250 and the 6 in the tens place is even, while 385 is between 390 and 380 and the 8 is even, rounding to the nearest even method yields 260 + 380 = 640.

d. Round to Compatible Numbers

1) A reasonable estimate of 66x97 is 66x100 (= 6600). Notice:Here, only 97 needed to be rounded to obtain an estimate mentally.

2) The division problem 2326 : 45 can be estimated mentally by considering 2400 : 40 (= 60). Notice:Here 2326 was rounded up to 2400, and 45 was rounded down to 40 because 2400 : 40 easily leads to a quotient since 2400 and 40 are compatible numbers with respect to division.

A reasonable estimate of 27 x 39 is 25 x 40 = 1000. The numbers 25 and 40 were selected since they are estimates of 27 and 39 respectively, and are compatible with respect to multiplication. Notice:The rounding up technique would have yielded the considerably higher estimate of 30 40 = 1200, whereas the exact answer is 1053.

Estimate by rounding to compatible numbers in two different ways.(a) 52 x 31 (b) 267 : 34Solution:(a) 52 x 31 = 50 x 31 = 1550 52 x 31 = 52 x 30 = 1560 (The exact answer is 1612.)(b) 267 : 34 = 240 : 30 = 8 267 : 34 = 280 : 40 = 7 (The exact answer is 7 with remainder 29)


Communication Activity

1.Estimateusingtheindicatedmethod.(Thesymbol“≈”means“isapproximately.”)

1. Estimate 1261 + 5457 by rounding down to the nearest hundreds place.

2. Estimate 4450 – 2750 by rounding to the nearest even hundreds place.

3.Estimate 655 – 296 by rounding a 5 up to the nearest tens place.

2.Roundasspecifieda.368 down to the tens placeb.854 to the nearest even hundreds placec.692 to the nearest tens placed.7548 to the nearest hundreds placee.5637 to the nearest thousands place

Read and think each of the following situation. Then

explain what you think about each situation.Read and think about each of the situations. Then •explain what you think about each situation.Your car usually gets about 12 km per litre on the •highway. Your gas tank is about one-half full and it is 40 km to the next gas station. What do you think about this situation?Your plane is scheduled to leave at 06.00 and it usually •takes about one hour to get to the airport. Should you overestimate or underestimate the time needed to get to the airport when deciding what time to leave?Make up an estimation problem where you would want •to (a) overestimate (b) underestimate.

C. Adding IntegersAt the elementary school, you have learned about

addition, subtraction, multiplication and division of integers. Here are some situations to represent the operation.

Suppose your football team played in a tournament. Unfortunately,yourteamlost5goals inthefirstgame


and lost another 3 goals in the second game. We can use a model to draw the number of goals that your team lost.

The dark model represents the number of goals that your opponent scored.

1. What is the third number?

2.Useamodeltofindthefollowingaddition.

a. -4 + (-6)

b. -1 + (-8)

c. -5 + (-2)

3. What is the sign for an addition of two negative numbers?

The white model shows the number of goals that your team scored.

4 a. Write the mathematical sentence for the model.

b. What is the sign for the result of the addition of two positive numbers?

In the next games, however, your team scored 3 goals but your opponent scored 5 goals. The model for the situation is:

−5 +3 ...

1

-5

−5 −3 ....

-3

−5 −3 ....

....

−5 −3 ....

-1

+4 +3 ....

+4 +3 ....

+4 +3 ....


5. a. Determine the third number.

b. Did your team score more? Why?

You can also use the number line to model an addition of integers. For instance, use the number linetofindthesumof-5and+3.

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

-5

+3

Step 1 : Start at 0. Move 5 units to the left to obtain -5.

Step 2 : To show the sum, that is positive 3, move three steps to the right to get -2. Thus, -5 + 3 = -2.

A. For an addition with two positive numbers, such as 5 + 3, we can do the following:

1. Add one number to another by using algebraic tiles.

2. If you use a number line, start from zero and move 5 units to the right until you reach the position of number 5. Then move 3 units to the right to get 8. Thus, 5 + 3 = 8.

B. For an addition involving two negative numbers, such as -5 and -3, we can do the following.

1. Add each value without regarding the negative sign, that is 5 + 3 = 8, and then give a negative sign to the result, that is -8. Therefore, (-5) + (-3) = -8.

Take five positive tiles.

Take another three positive tiles so that we have 8 positive tiles.


2. If you use a number line, start from zero, and movefiveunits to the left toget–5.Then,movethree units to the left to get -8. Therefore, (-5) + (-3) = -8.

C. For an addition involving one negative number and one positive number, such as -7 + 2, we can do the following.

1. Take the following procedure.

2. Use a number line. Start from zero, and move 7 units to the left to get –7. Then, move 2 units to the right to get –5. Therefore, –7 + 2 = -5. Calculate

a. 12 + 9

b. -23 + 14

c. 36 + (-49)

d. -89 + (-25)

e. 124 + 0

D. Subtracting IntegersIn a restaurant there are two refrigerators. The

firstrefrigeratorhasatemperatureof-5o C, while the second one -3o C.Tofind thedifferencebetween thetemperatures of the two refrigerators, you should do a subtraction.

Add two positive tiles to remove two negative tiles. (Remember one positive tile and one negative tile make zero)

Draw seven negative tiles.

How many negative tiles are left?

Draw seven negative tiles.

Add two positive tiles to remove negative tiles. (Remember one positive tile and one negative tile make zero).

How many negative tiles are left?


1. Which expression shows the subtraction of -2 from -5; that is -2 - (-5) or -5 – (-2)? Why? The number line can be used in subtraction.

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Start from zero. Move two steps to the left. Start from −2. Move five steps to the right (opposite of -5) to get 3. Then, −2 – (−5) = 3

Example 15

Start from zero. Move two steps to the left.Startfrom-2.Movefivesteps to the right (opposite of -5) to get 3.Then, -2 – (-5) = 3

E. Multiplying and Dividing IntegersWe come back to the diver mentioned at the

beginning of this chapter. The diver, for instance, has a constant rate of 4 metres per second for 3 seconds. You can draw the diving using a vertical number line.

1. a. How far is the diver from the sea susface after 3 seconds?

b. Which integer shows the position of the diver?

Basically, every subtraction can be changed into addition. 1. 7 –(-8) = 7 + 8 = 15 (see addition at A)2. –18 – 5 = -18 +(-5) = 23 (see addition at B)3. 15 – 7 = 15 + (-7) = 8 (see addition at C)Calculate.a. 34 -13 b. -76 - 45c. 34 - (-59)d. -148 + (-101) e. -36 + 32 f. -18 - (-57)


You can use repeated addition or multiplication to show the movement of the diver.

Repeated Addition Multiplication

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

2. Find the result of each multiplication using repeated addition.

a. 2(-5)

b. 4(-2)

3. Using commutative principles, you can write 3 (-4) as -4 x 3. What is the result of -4 x 3, -5 x 2, and -2 x 4?

Thepatterncanbeused tofind theproductof twonegative numbers.

Work in Groups

Find the product of :

a. 4 x 3 3 x 3 2 x 3 1 x 3 0 x 3 -1 x 3-2 x 3 -3 x 3

b. 3 x-3 2 x -3 1 x -3 0 x -3 -1 x -3 -2 x -3 -3 x -3 -3 x -3

c. Based on your work above, solve the following problems.

a. -12 x -3 d. -9 x -3

b. -8 x -2 e. -6 x -2

c. -4 x -1 f. -3 x -1

4. Look at your work in number 1 and 2 and then answer the following questions using your prediction. What is the product multiplied?

a. a positive number multiplied by a positive number is a . . . . number . . . .

b. a positive number multiplied by a negative number is a . . . . number . . . .


Problem 8

c. a negative number multiplied by a negative number is a . . . . number . . . .

d. zero multiplied by any number is . . . .

Product of Integers

The product of two integers having the same signs is •a positive integer. The product of two integers having different signs is a •negative integer.The product of any integer with zero is zero•

Calculate :a. 13 × 4 d. -25 ×(-14) b. 24 × (-12) e. -15 × 0 c.-8 × 24

Multiplication and division are opposite to each other. You can use the relation between multiplication and division to set rules in division of integers.

5. Use what you know about the opposite operation to findtheresultofthefollowingdivisions.

3×4 = 12 12 ÷ 4 = 12 ÷ 3 = 3×(-4) = -12 -12 ÷ (-4) = -12 ÷ 3 = (-3)×(-4) = 12 12 ÷ (-4) = 12 ÷ (-3) =

6.Ingroupsoftwoorfourstudents,findthesignofthedivision of the following integers.

1 a. 16 ÷ 4 2 a. 12 ÷ 3 b. 12 ÷ 4 b. 9 ÷ 3 c. 8 ÷ 4 c. 6 ÷ 3 d. 4 ÷ 4 d. 3 ÷ 3 e. 0 ÷ 4 e. 0 ÷ 3 f. -4 ÷ 4 f. -3 ÷ 3 g. -8 ÷ 4 g. -6 ÷ 3 h. -12 ÷ 4 h. -9 ÷ 3


3. Based on your work in number 1 and 2, solve the following problems.

a. -12 ÷ -3 d. -9 ÷-3 b. - 8 ÷ -2 e. -6 ÷ -2 c. - 4 ÷ -1 f. -3 ÷ -1

4. Look at your work in number 1, 2 and 3 above and then answer the following questions according to your prediction.

a. The result of dividing a positive number by another positive number is a . . . number.

b. The result of dividing a positive number by a negative number is a . . . number.

c. The result of dividing a negative number by a positive number is a . . . number.

d. The result of dividing a negative number by another negative number is a . . . number.

E. Dividing IntegersDividing integers involves rules similar to those

for multiplication. To divide two numbers with the same sign, divide their absolute values and give the resulting quotient a positive sign.

Calculate:

a. 144 ÷ 3 b. -246 ÷ 6 c. 248 ÷ (-8) d. -120 ÷(-10)

Calculate:

a. (82 × 4) ÷ 2 b. (-23 + 36) × 5 c. 23 × ( 34 – 21)

Here are some properties of integers:

1. Commutative property for addition 4 + 5 = 5 + 4

2. Commutative property for multiplication 4×5 =5×4

The commutative property does not apply to the subtraction of integers.


Example 16

Problem 11 Copy and complete the following table using the properties: “commutative, associative, or closure”.

Operation Property that applies

Property that does not apply

AdditionSubtractionMultiplicationDivision

Explain each of your answers. Give examples of the properties that do not apply to support your answer.

Relation with the Real WorldLook at the chessboard picture on the left side. The board has black and white squares. How many squares are there?Howdoyoufindit?Do you remember the formula of the area of a square and the volume of a cube?

a. Copy and complete the table below.

No Side of the square (cm)

Area of the square (cm2)

1 42 53 8

Are integers closed over multiplication? •Examine whether the product of two integers is also an •integer. Explain your answer. Examine whether the division of integers applies the •closure property. Why? Clarify your answer.

3. Associative property for addition 4+(5+6) = (4+5) +6

4. Associative property for multiplication 4 x (5 x 6) = (4 x 5) x 6

5. Closure property for addition

The operation of addition is closed with respect to the integers. Why?


b.Howdoyoufindtheareaofeachsquareabove?c. Find the results of 42, 52, 82 d. Copy and complete the table below.

No No Side of the square (cm)

Area of the square (cm2)

1 . . . 162 253 64

e.Howdoyoufindthelengthofasquare’ssidewithagiven area?• To find a positive number the square ofwhich is

16 meanstofindthesquarerootof 16 written as. Therefore, 16 = 4.

• To find a positive number the square ofwhich is25 means to find the square root of 25 written

as 25 . Therefore, 25 = 5.f. What is the square root of 64?

Is there any number other than 4 the square of which is • 16 ? Certainly, that is -4. But you should remember that 16 is only 4. Why?

If • 0a ≥ ,then a is the number which is not negative and the square of which is a .

Find out the value of

a. 49

b. 100

c. 225

Problem 12

Problem 13 Find the the length of a square’s side if the area isa. 81 cm2

b. 100 m2

c. 289 m2


Activitiesa. Complete the table below

No Cube Edge (cm) Cube Volume (cm3)

1 2 . . .

2 3 . . .

3 5 . . .

b.Howdoyoufindthevolumeofthecube?c. Find the results of 23, 33 , 53

d. Copy and complete the table below.

No Cube Edge (cm) Cube Volume (cm3)

1 . . . 8

2 . . . 27

3 . . . 125

e.Howdoyoufindthe lengthofacube’ssidewithagiven volume?• Tofindthenumberthecubicrootofwhichis8

meanstofindthecubicrootof8,writtenas 3 8 . Therefore, 3 8 = 2.

• Tofindthenumberofthecubicrootofwhichis27meanstofindthecubicrootof27,writtenas 3 27. Therefore, 3 27 = 3.

• 3 8− = -2, because (-2)3= -8.What is the cubic root of 125?

Problem 13 Find the results of :a. 3

62 b. 10003 c. −273 d. −1253

Problem 14 Find out the side of the cube with a given volume of : a. 64 cm2

b. 216 m2


Look at the following multiplication.

a. 33 = 3 × 3 × 3 and 32 = 3× 3 33 × 32 = ( 3 × 3 × 3 ) × ( 3× 3 ) = 3 × 3 × 3 × 3 × 3 = 35 = 33+2

So, 33 × 32 = 33+2

b. 26 = 2 × 2× 2 × 2 × 2 × 2 and 24 = 2 × 2 × 2 × 2

26 ÷ 24 = 22

6

4

= 2 2 2 2 2 2

2 2 2 2× × × × ×

× × ×

= 2×2 = 22 = 26-4. So, 26 ÷ 24 = 26-4

c. (23)2 = (2×2×2)2 = (2×2×2) × (2×2×2) = 2×2×2×2×2×2 = 26 = 23x2. So, (23)2 = 23x2.

Distributive Property of Multiplication over Addition and Subtraction 1. If a, b and c are integers, then they satisfy: a × (b + c) = (a × b) + (a × c). Such a property is called the distributive property of

multiplication over addition.For example: 12 (6 + 13) = 12 × 19 = 228or (12 × 6) + (12 × 13) = 72 +156 = 228Therefore, 12 ×(6 + 13) = (12 × 6)+(12 × 13)

The Exponent of Integers

If a, m, and n are integers, then•am × an = am+n

•am ÷ an = am-n

•(am)n = amxn


1. Cluster estimation is used to estimate sums and products when there are several numbers that cluster near a single number. The addend in 681 + 708 + 697 cluster around 700. Thus 3 700 = 2100 is a good estimate of the sum.

Estimate the following using cluster method.a. 356 + 361 + 342 + 353b. 896 + 907 + 888 + 904c. 49 × 53 × 47 d. 32 × 28 × 29 × 31

2. What is interesting about the quotient obtained 2,108,658,976 by 8? Do this mentally.

3. Critical thinking. One student calculated 96 – 49 as 96 – 50 = 46 and 46 + 1 = 47; thus 96 – 49 = 47. Another student calculated 96 – 49 as 96 – 50 = 46 and 46 – 1 = 45. Determine which of these two methods is valid. Explain your answer.

Task 1.2

2. If a, b and c are integers, then they satisfy: (a × b) – (a × c) = a × (b – c) and a × (b + c) = (a × b) + (a × c).

Such a property is called the distributive property of multiplication over subtraction.For example: 22 × (16-3) = 22 × 13 = 286or (22 × 16) - (22 × 3) = 352 - 66 = 256Therefore, 22 × (16-3) = (22 × 16) - (22 × 3)For example:51×49 = (50+1) × 49 = (50+ 1) × (50 -1) = 25000 -50 + 50 -1

= 2499Does the property above also apply to a division? Explain.


4. Calculate.a. 24×(56 -23) b. (21 - 46) × 14 = c. (127 - 43) ÷ 2 d. 44 × (125 ÷5) =

5. Open Question. Write down a mathematical statement involving a positive integer and a negative integer so that the sum of the two numbers is ...

a. negative integers. b. zero. c. positive integers.

6. Examine the result of 242 4 = 60.5. It shows that the division of integers does not hold the property of ...

7. A staircase consists of 12 footsteps. Ani stands up on the fourth footstep, and then she moves three steps upward.

a. On what number of footstep will Ani stand up after stepping up for the second time?

b. If the distance between footsteps is 48 cm, how highisAni’spositionabovethefloorlevel?Explainyour answer.

c. If the distance between footsteps is 48 cm, how high isAni’spositionabovethefloorlevelaftersteppingup for the second time? Clarify your answer.

8. Copy the magic square in the left side. Put the integers –4, –3, –2, –1, 0, 1, 2, 3, 4 so that the vertical sum, the horizontal sum, and the

diagonal sum are zero.

9. Find the two numbers having a sum of –5, while their product is 4.

10. The temperature drops on at an average of 3 degrees per hour. If the temperature is 35 degrees at 12.00, what will the temperature be at 3.00 pm?

11. Is the statement below true or false? “The sum of a positive integer and a negative integer

isanegativeinteger”. Give an example to clarify and support your answer.


12. Compute the squares of non-negative integers from 11 to 15.

13. Find the square roots of 4, 9, 16, 49, 81, and 100.

14. Calculate the cubic power of the three non-negative integers from 4 to 8.

15. Compute the cubic root of the numbers 64, 216, 729, and 1000.

16. There is a square tile. Its side length is 30 cm. If the area of a square = side × side, what is the area of the tile?

17. Pradnya takes $58 with her on a shopping trip to the mall. She spends $18 on new shoes and another $6 on lunch. How much money does she have after spending these expenses?

a. $34 c. $52

b. $40 d. $24


Problem-solving guidelines:

Understanding the •problemDeveloping a plan and •selecting strategiesCarrying out the plan•Checking the answer•

Problem-Solving Strategies:

Guessing and checking•Drawing a table/•diagramWriting an equation•Simplifying the problem•Looking for patterns•Using logical reasoning•

Example 1

Draw a DiagramTo repair the well pump, Pak Jarsih went down the well as far as 7 metres. Draw a number line to show his position from the ground level and write down the number.

Solution:1. Understanding the problem a. What is the unknown?

• PakJarsih’spositioninthewell • Thenumbertoshowhispositionb. What are the data?

Pak Jarsih went down the well as far as 7 metres. c. Is there any other information?

The position must be represented by a number line.

2. Developing a plan and strategy Because of the expression ‘go down’, a vertical number

line is used. The strategy is drawing a diagram. The calculation starts from 0, moving down for 7 units. The number shows the position.

3. Carrying out the plan Pak Jarsih’s position is 7 metres from the ground level.

Because it is under the ground level or in the well, the number is –7.

Zero (0) shows the position on the ground level. 4. Checking the answer Because the number sign is negative and is relevant

to the distance, –7 is acceptable.

0 -1

-2

-7

Section 1.3Problem Solving


Solving easier problems and investigating patternsWhat is the value of 22009 in regular notation?Solution:1. Understanding the problem a. What is the unknown? The value of 22009. b. What are the data? The exponential number 22009

2. Developing a plan and strategy The exponential number 22009 means the multiplication of 2 as many as 2009 times. Because 2009 is a large number, the result of the exponentiation will be very large. Even a calculator cannot show the expected result. Therefore, the problem will be solved by using exponents 1,2,3 and so on. By using small exponents, we can obtain the results easily. Then the results are compared to see whether there are patterns. The patterns are determined by units numbers of 22009. Thus the strategies are solving easier problems and identifying patterns.

3. Carrying out the plan 21 = 2, the units number is 222 = 4, the units number is 4 23 = 8, the units number is 824 = 16, the units number is 625 = 32, the units number is 226 = 64, the units number is 427 = 128, the units number is 828 = 256, the units number is 629 = 512, the units number is 2From the obtained results, the units numbers are, respectively, 2, 4, 8, 6, 2, 4, 8, 6, 2……Therefore, the pattern is the repetition of the unit numbers after the exponent 4 and its multiplication. It means that each number which is a multiple of 4 from 1 to 2009, the units number is 6. For example, the exponent 2008, which is exactly divisible by 4, has the units number 6. Because 2009/4 = 502 with the remainder 1, the units number of 2009 is not 6, but the firstnumberafter6,thatis2.

Example 2


Exercise 1 Solve the following problems using one or more strategies.1. Look at the photograph below. Can you estimate how

many chocolate chips are shown?2. The average attendance at a football game last

year was 1755 people. The table below shows the attendance at each game this year comparing to the last year’s average. What was the total attendance for all 6 games above or below last year?

Game 1

Game 2

Game 3

Game 4

Game 5

Game 6

+357 -144 -250 +347 +420 -1883.Aspiderisrunningdownthestairsfromthefirstfloor

of an old lady’s house to the basement below. It stops every5stepstocatchafly.Ifthereare26stepsabovethegroundand14below, howmany fliesdoes thespider catch?

4. The lowest point in Asia is the Dead Sea, 400 metres below the sea level. The lowest point id the US is the Death Valley, 86 metres below sea level. How much higher is the Death Valley than the Dead Sea?

5. A tetramino is constructed of four congruent squares that have common sides.

a. How many tetraminos can be constructed? b. Which tetramino has the shortest perimeter?

4. Checking the answer 24 = 1628 = 256212 = …6,Because 2009 is not exactly divisible by 4 (with the remainder 1), the number after 6 is 2. So the answer is acceptable.

Mathematics for Junior High School - Year 732 Student’s Book Chapter 1 - Integer PB

Reflection 1. Indicate the type of integer which is the most

difficulttounderstand.2. Indicate the type of integer which is the easiest to

understand. 3. Are the descriptions and examples of the integers

in this chapter easy to learn? If not, explain.4. Do the descriptions and examples of the integers

in this chapter make you more active in learning mathematics?

SummaryIn this chapter, you have learned:

1. The relationship among positive integers, zero and negative integers.

2. The use of negative numbers in daily life. 3. The use of zero in daily life. 4. The use of positive numbers in everyday life. 5. The characteristics of the addition operation

of integers. 6. The characteristics of the subtraction operation

of integers. 7. The characteristics of the multiplication operation

of integers. 8. The characteristics of the division operation

of integers.

Education

Math Chapter 1 - Integers