II. Answering and explaining.

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Quiz I

1. Which of the following numbers has the greatest value?

A. –8.3

B. |–7.7|

C. 2

D. |4.5|

E. 6.8

Quiz II

1. If m and n are negative integers, which of the following must be true?

I. m + n < 0

II. mn > 0

III. mn > n

A. I only

B. II only

C. I and II

D. I and III

E. I, II and III

2. If v and w are both odd integers, which of the following could be an even integer?

A. vw

B. v + w + 1

C. v/w

D. 2(v + w)

E. 2v +w

Answers and explanations

Quiz I

1. B is correct. The question asks for the greatest value, so you want the number farthest to the right on a number line. The two absolute values convert to 7.7 and 4.5. Of all five numbers, 7.7 is the greatest, so B is the best answer.

Quiz II

1. E is correct. Adding two negative numbers will just result in an even smaller (further to the left) negative number. So m + n < 0 and I is true. You know that a negative multiplied by a negative results in a positive number. So mn > 0 and II is true. This also means that mn is greater than any negative number, including n. Therefore mn > n and III is true. So the answer is E.

2. D is correct. You know that an odd integer multiplied by an odd integer results in an odd integer. So vw is odd and you can eliminate A. An odd integer plus an odd integer results in an even integer. So v + w is even, but v + w + 1 is odd and you eliminate B. Dividing an odd integer by an odd integer gives you either an odd integer, such as 15/5=3, or a non-integer, such as 17/5=3 2/5. So eliminate C. An even number times an odd is even, so 2v is even. Adding an even to an odd is odd, so 2v + w is odd and you should cross off E. As we saw earlier, v +w is even and multiplying by 2 keeps it even. (Even if v + w were odd, multiplying by 2 would make it even.) So D is the best answer.

III. Playing a trick with numbers.

Write down your house number or your telephone number or the total amount of the coins in your pocket.

Multiply this by 2.

To this add 5.

Multiply that by 50.

Then add your age.

Now add 365 (days in year).

Then subtract 615.

The last two numbers will be your age, and the other numbers will be your house address, total amount of coins in your pocket or whatever you decided upon.

IV. The ‘Terribly Stressed’ game

Count from 11 to 31 – but only say every second number (for example 1, 3, 5).

1. Count backwards from 29 to 17.

2. Count backwards from 113 to 91

Practice set 8

Four Basic Operations in Mathematics

3 + 2 = 5 ADDITION (to add) 1) three plus two is five 2) the sum of three and two equals five 3) three increased by two is equal to five 4) two added to three equals five

6 – 2 = 4 SUBTRACTION (to subtract from) 1) six minus two is four 2) six decreased by two equals four 3) the difference of six and two equals four 4) two subtracted from six equals four

3 × 2 = 6 MULTIPLICATION (to multiply by) 1) three multiplied by two is six 2) three times two equals six 3). the product of three and two equals six 4) three times as large as y means multiply three 3 times y

8 ÷ 2 =4 DIVISION (to divide by) 1) eight divided by two equals four 2) the quotient of eight and two 3) two divided into eight

Inverse operations

addends minuend

3 + 2 = 5 the sum 6 – 2 = 4 the difference

subtrahend

multiplicand dividend divisor

3 × 2 = 6 the product 7 ÷ 2 = 3 the quotient and the part

Multiplier which is left is a remainder 1

I. Use this mind-map ‘Four basic operations in Mathematics’ as a topic activator to speak about the basic operations in Arithmetic.

II. Complete the crossword with words referring to the basic operations of Arithmetic. The first letter of words is given for you to help.

₁M

₂R

₃D

₄ Q

₅A

₆M

₇A

₈D

₉D

₁₀S

₁₁P

₁₂M

₁₃D

₁₄F

₁₅M

₁₆M

₁₇S

1. The number by which we multiply

2. The result which we obtain in the operation of subtraction

3. The number to be divided

4. The result of division

5. The elementary branch of mathematics

6. The inverse operation of division

7. The number to be added

8. The process of finding how many parts of a number is contained in another one

9. Ten _____ by three is equal to seven

10. The number to be increased

11. The result of multiplication

12. The number from which we subtract

13. The _____ of nine and six is three

14. Numbers to be multiplied

15. The sign of subtraction

16. The number which is multiplied in the operation of multiplication

17 The process of finding the difference of two numbers

III. Read the following numerical expression to your partner. S/he will fill in the blank with the missing operation symbol and says the operation which the given numbers are subjected to. The first example is done for you.

Expression		Operation
ten multiplied by six point four	10 _____ 6.4	Multiplication
thirty times seventeen	30 _____ 17
fifteen point three decreased by point three three	15.3 _____ 0.33
Eighty-two increased by eleven	82 _____ 11
the sum of nineteen and twelve	19 _____ 12
the difference of sixty-five and fifty	65 _____ 50
twenty-two subtracted from two hundred and two	202 ____ 22
the quotient of one-tenth and six-fourths	1/10 ____6/4
sixty-two divided by four	62 _____ 4
One added to seventeen	1 _____ 17
The product of twelve and eight	12 _____ 8
two divided into one thousand	1000 _____ 2

IV. Read the following numerical expressions to your partner. S/he will write down what you say. Check the answers. Then ask to repeat the given expressions. Listen to see if s/he repeats correctly. Correct your partner’s incorrect answers.

1. three times twenty – five plus four

2. eighty-one divided by nine

3. thirty decreased by seventeen

4. two times the sum of fifteen and eight

5. the quotient of negative one hundred and fifty-two

1. nine-fifths minus two-thirds

2. one-sixth subtracted from three-halves

3. the product of four-thirds and eleven-sixths

4. the quotient of one-half and two thirds

5. eleven divided by fifteen-sixteenths

1. point two five times one point six

2. point six plus six point one

3. subtract one point double zeros from six point one

4. the quantity, point five minus point one, times three

5. twice the quotient of one point nine and one point two

V. Look at each numerical expression written in symbols and signs. Then say it in words. Your partner will listen to see if you repeat correctly and correct your incorrect answers.

.01 + .9 – 1.7	.067 + 3.004
2(15 + 8) – 13	2(61 – 24)
3(20 + 3) + 41	38.4 ÷ 6.01
59.9 ÷ 3.8 – .21	2/3 – 1/10
(4/3) × (11/6) × (15/16)	(15/32 ) × (6/7)
√19 – 6	√75–25
√15 + 50	(93 – 43) ÷ 2
(50 +86) 10	√90 + √144

Practice set 9

ALGEBRA came from the Arab word “aljabr”

Letters of the Latin alphabet are used to represent numbers

An algebraic expression is an expression in which several numbers represented by letters or by letters and figures are connected by means of signs

A. deals with the operations of rational/irrational numbers, equations, logarithms, functions, graphs and complex numbers

The turning point in the history of algebra was the 16^th century

The French mathematicians Viet and Descartes introduced the systematic use of letters of the Latin alphabet

The most important new demands in algebra come from topology, analysis and algebraic geometry

These signs indicate the order of the operations which the numbers must be subjected to

A. is one of the most rapidly changing areas of Mathematics, because it is sensitive to all the trends, which originate in other areas of Mathematics

I. Use this mind-map ‘Algebra’ as a topic activator to speak about Algebra (its origin and some facts from its history).

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