Directed numbers
AS WE continue our review of selected topics from the syllabus with Directed Numbers, I will share with you the answers to the problems given for homework last week.
Evaluate the following:
(i) –3 x – 5 = 15 ii) – 48 ÷ 8 = – 6 (iii) – 4 – 1 + 3 = – 2 (iv) 3a x –5b = – 15ab (v) 5/12 + 7/6 – 4/3
The solution to question (v) is based on the conversion of the three fractions to the same denominator. This denominator is 12, the LCM of the existing denominators 12, 6 and 3. (Please note that the LCM of 12, 6 and 3 is 12.)
5/12 + 7/6 – 4/3
= (5 x 1) + (7 x 2) – (4 x 4) 12
= 5 + 14 – 16 = 3 = 1 12 12 12 12 4
If the above posed no difficulty, then you are ready to consider exam-type questions.
Prior to doing so, I do recommend that you pursue a comprehensive review of the number system. You should, no doubt, be familiar with odd and even numbers, but what about natural numbers (1, 2, 3 …), whole numbers (0, 1, 2 ….), integers ( ..– 2, – 1, 0, 1, 2 …), rational or irrational numbers, fractions and decimals? In reviewing these, please pay attention to: Proper fractions (5/6 ) Improper fractions ( 17/12 ) Mixed numbers ( 3 7/12 )
APPLICATION OF THE FOUR ARITHMETIC OPERATIONS TO VULGAR FRACTIONS
Having reviewed the application of the four arithmetic operations to integers, select fractions and decimals, we may now consider exam-type questions. In applying multiple operations to vulgar fractions, students are required to observe the correct law with respect to applying the order of the operations as follows:
1) B: Brackets
2) O: Of (Multiply) 3) M: Multiply 4) D: Divide 5) A: Add
6) S: Subtract
BOMDAS identifies the order in which the operations should be carried out and must always be obeyed. Where an expression has multiple operations, then the operations within the brackets are evaluated first, if they exist. Multiplication, or ‘Of’, is done before division, while division is done before addition, and so on.
Let’s now practise the use of BOMDAS.
1) Calculate the value of:
2 1 + 9 x 2 ÷ 1 1
3 2
Convert the mixed numbers to vulgar fractions
= 7 + 9 x 2 ÷ 3
3 2
In this case, three operations are involved.
Using BOMDAS, we do the multiplication first: That is 9 x 2 = 18 = 7 + 18 ÷ 3
3 2 We then do the division: ( ) = 7 + 18 ÷ 3 = 7 + 18 x 2
3 2 3 3 = 7 + 12
3
And finally the addition: = 7 + 12 = 2 1/3 + 12
3
= 14 1/3
( )
2) 2 x 21 +13
4 5
BOMDAS directs that we evaluate the brackets first (despite the fact that we are required to add):
■(21 )=
+ 13 9 + 8 .
4 5 4 5 Using the LCM of 4 and 5, that is 20, we get
= ( 9 x 5) + (8 x 4)
20
= 45 + 32 = 77
20 20
To complete the problem, we now multiply:
2 x 77 = 77 = 7.7 20 10
3) Calculate the value of: 6 1 – 1 5
3 6
1 1 x 2 2 2 3
This is a typical exam-type question, so please note it well.
Using BOMDAS, we first note that the line represents brackets and so the numerator may be evaluated first. (It is also appropriate to evaluate the denominator first.)
6 1 – 1 5 = 19 – 11
3 6 3 6 Initially convert mixed numbers to a vulgar fraction.
The LCM of 3 and 6 is 6 = ( 2 x 19) – (1 x 11)
6 = 38 – 11 = 27 6 6
POINTS TO NOTE
Evaluating the denominator: 1 1 x 2 2
2 3
Convert to vulgar fractions and cancel 3 x 8 = 24 = 4
2 3 6
Dividing: = 27 ÷ 4 = 27 x 1 = 9 6 6 4 8 The above assumes that you are able to manipulate fractions. If you are not able to, including cancelling, you need to get help in this specific area.
■ In solving a problem such as question 3, you may first evaluate either the numerator or the denominator. You may verify this by finding the solution beginning with the denominator.
■ Finding the LCM CORRECTLY is a very important step in the solution. If you have difficulty with this step, you should resolve these at this time.
■ As question 3 requires the exact value, you are not allowed to express the fraction in decimal form. If this is done, then your answer would be different from 9 and you may be penalised.
8
■ Your working must always be clearly shown in logical sequence as presented above.
Let us now work the following together:
EXAMPLES
1) Using a calculator, or otherwise, determine the exact value of :
(5.1)2 – (8.25 + 5.31).
SOLUTION
Given (5.1)2 – (8.25 + 5.31) Using the recommended approach, we first evaluate the brackets using the calculator.
(5.1)2 = 26.01 and (8.25 + 5.31) = 13.56
As there are brackets, you may do the second bracket first, as long as the substitution is done appropriately.
= 26.01 – 13.56 = 12.45
Ans = 12.45
2) Determine the values of: a) 6.35 ÷ 1.024 b) 2.35 x 4.12 – 3.22 c) √ 8.6091 d) (0.3)3 + (0.5)2 e) 5.62 x 3.7462
Solutions
Using the calculator: a) 6.35 ÷ 1.024 = 6.201 b) 2.35 x 4.12 – 3.22 = 6.462 c) √ 8.6091 = 2.934 d) (0.3)3 + (0.4)2 = 0. 187 e) 5.62 x 3.7462 = 5.62 x 14.033 = 78.865
Please be reminded that it is important to get this, the first question on the exam paper, correct. It naturally builds your confidence. Always remember to apply BOMDAS. Even if the individual operations are done correctly, the appropriate order is required to get the correct answer.
I close this week with the following:
1. 7 x 3 + 12 ÷ 3 – 5 =
2. Calculate the value of x( )
6 7 + 1
3 4
technique like emotive language, descriptive language, figurative language, narrative voice, diction or slang. After identifying the technique, there should be a discussion on how well it helped to achieve the purpose of the piece. For non-print stimuli, like pictures, cartoons or photographs, note body language and the pictorial features. Comment on the effect of your chosen stimulus, highlight its emotional appeal or discuss its relevance to the issue/topic/theme chosen.
Again, you can comment on the use of language in each piece in a separate paragraph.
REFLECTION 3
This reflection should be done once you have finished most of your tasks. It seeks to capture how the process of completing the SBA has contributed to your personal development. Have your collaborative skills been enhanced? Have you become better at conducting research? Are you better at presenting in public? Do you now have better time-management skills? Identify three ways in which you have grown as a person/student and discuss each way in a separate paragraph.
REMINDER: Share your reflections with members in your group. This will help you to improve your reflections and is evidence of group activity. You should also submit them to your teacher for feedback.
I will stop here for now. Next week, I will share samples of reflective paragraphs; and I will also focus on the indicators of group activity, the written report and the oral presentation.
See you next week!