work out what number needs to be added to each of these to give the answer 10.
3 and 2/5
4 and 5/12
7 and 7/24
Answers
Answer:
10.
Example: (2 + 3) 2
In this expression the brackets around the addition of 2 + 3 indicate that it is 2 + 3 that is raised to the power of 2, NOT just 3. We must work out the brackets first and then square the answer. 5 2 = 10
Example:
Actual rule: Fractions should be treated as if the numerator is in brackets and the denominator is in brackets and the fraction bar (the "vinculum") is division.
Example:
We work out the square root first to get 3 and then do the division and multiplication working from left to right. 12 ÷ 3 x 2 is equal to 8.
We cannot do anything with until it has been simplified. Also, you will notice once again that if we do the multiplication before the division then we will get a different answer.
What to remember:
Work on one level at a time, starting at the top and going down.
Within each level, work from left to right.
Brackets
powers, roots and fractions
multiplication and division
addition and subtraction
Activities
Four fours
Using four fours and any mathematical operations and signs you wish, can you make every number from 1 to 20. Can you make every number up to 100?
For example, (4 +4) x 4 - 4 = 28 and 4 + (4 x 4) - 4 = 16.
This is an excellent activity for a class to do over a week. Make a large chart with a space for one or more expressions for each number. Students can enter their expressions on the class chart after they have been checked. The teacher can decide what signs are allowed.
Manipulating expressions
6 + 17 - 15 x 4 ÷ 3
By inserting brackets into this expression (as many as you like, wherever you like) make expressions with as many answers as you can.
The correct answer when there are no brackets is
6 + 17 - 15 x 4 ÷ 3 = 3.
This set of inserted brackets changes the answer to 8.66:
6 + (17 - 15) x 4 ÷ 3 = 6 + 2 x 4 ÷ 3 = 6 + 8 ÷ 3 = 6 + 2.66 = 8.66
Using calculators
Not all calculators have correct order of operations built in. More sophisticated calculators have programmed logic which enables them to use the standard mathematical conventions. Others just process the information/keystrokes exactly as they are entered.
Example: If you need to calculate 1 + 5 x 7 and enter these 6 key presses:
1
+
5
x
7
=
some calculators give 42 (1 + 5 gives 6, multiply by 7 gives 42) and others give 36 (multiply first so 5 x 7 = 35, add 1 + 35 giving 36).
The second is the correct answer for the expression.
Find out how your calculator works and check to see if it has brackets to help be precise. Learn how to use the memory to keep intermediate answers.
Quick quiz
1. Using the example 10 - 1 - 2 , show why you need to follow the correct order of operations.
2. Calculate the following expressions:
(a) 11 x (3 + 2) x 4 ÷ 2
(b) 7 - 18 ÷ 2 x 3 + 5
(c) 42 ÷ 3 x 7
3. Calculate
9 + 4 ÷ 2 x 7 - 6 ÷ 3 - 4 x 2 + 8 ÷ 2 + 3 x 3
4. Using the expression in question 3, make 3 alternative expressions and answers by inserting brackets.
5.
Find the answer to,
showing the method you have used to ensure you follow the correct order of operations, e.g. checklist, colour scheme, arrows etc.
6. Calculate the following expressions:
(a) 32 ÷ 42 x (3 - 8)
(b) 81 ÷ (4 - 7)3
(c)
(d)
(e)
7. Find out how to use your calculator to evaluate the expressions in question 6.
8. Bernie is in the process of landscaping the gardens of two new townhouses. If he buys 30 bundles of 12 wooden planks for the fence for each house and 15 bundles of 10 hardwood planks for the decking for each house, write an expression for the total number of planks bought and then work it out. If Bernie then returned 2 bundles of the wooden fence planks but bought 5 extra bundles of the hardwood planks, write a new expression and then work out the answer.
9.
Two thirds of all Year 8 students, one quarter of all Year 9 students, only 30 Year 10 students and two fifths of Year 11 and 12 students combined ride their bike to school. If there are 99 Year 8 students, 124 Year 9 students, 111 Year 10 students, 65 Year 11 students and 50 Year 12 students attending the school, how many students ride their bike to school.
Write a mathematical expression for the number of students who ride to school and then find the answer.
10.
(300 ÷ (10 x 2)) x 4. Create an appropriate worded problem from this mathematical expression.
To view the quiz answers, click here.
Monster multiple brackets example!
3 + ((4÷2)x7) - (6÷3) - ((4x2) + ((8÷2) + (3x3)))
=
3 + ((4÷2)x7) - (6÷3) - ((4x2) + ((8÷2) + (3x3)))
=
3 + ((2)x7)) - (2) - ((8) + ((4) + (9))
=
3 + (14) - (2) - (8 + (13))
=
3 + (14) - (2) - (21)
=
3 + 14 - 2 - 21
=
17 - 2 - 21
=
15 - 21
=
-6
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