a Calculations You will learn how to: * Understand that brackets, indices (square and cube roots) and operations follow a particular order. * Use knowledge of the laws of arithmetic and order of operations (including brackets) to simplify \g decimals or fractions. calculations contai Starting point * how to apply order of operations to calculations that include brackets, positive indices and the four operations? For example, find (2 +3?) x5 = how to use laws of arithmetic and order of operations to simplify calculations (without brackets) containing decimals or fractions? For example, simplify and calculate: 2.2 + 3.9-0.2,25x17«4 Thi SO = you use order of operations with negative indices and with more complicated calculations = you use order of operations, inverse operations and equivalence to simplify calculations containing both decimals and fractions = you use order of operations to simplify calculations containing fractions and decimals. Find (3 + 2) and 3° + 2°. Are the answers the same or different? Explain this using order of operations Explain how the diagrams below represent (3 + 2 and 3? + 22. Find (4-2) and 4? - 2", Are the answers the same or different? Explain this using order of operations. Explain how the diagrams below represent (4~ 2)’ and 4? ~ 2".Use a calculator to do the calculations below. Make sure that in /9+ 16, the whole addition is underneath the square root sign, or else use brackets like this; (9+ 16) . Do the same for 25-9. In each pair, are the answers the same or different? Suggest why. v9=16 and ¥9 + Vi6 y25=9 and 25 - V9 Draw a diagram to represent each calculation. 1 Seago Did you know? | These expressions all have the same meaning: (9+ 16) VG +16) \9+16 | They tell you to do the addition first and then find the square root of the result. The answeris 5. | These expressions have the same meaning but it isa different meaning to those above: v9+16 VarI6@Oeina: a) 10-54 b) (10-5) x4 ) 102243 d) 102 (2+3) e) 345% V6 f) V4 x (2-5) g) 6x iB +2 hy 2844 Okina: a) 36+3°-2° by 12-12-79" oe xe d) \10?- 87 «5 2) 4x 9-G+4)) f) 10+ 920-12 g) 4x57+(3+2) hy) ¥o-2%7 i) (43+ 3x2) © ‘Technology question Use a calculator to find: ~ VE) x 18-6 +3 ~5Px aaa a7 )x3 a tae 9 G-5) x27 a) M12—4 +4 oy 2220576 ® SE g) 2x 5 410° hy) sae Dominic wants to find /9+ 16 He says, ‘In the correct order of operations, square roots come before addition.” His working is below. V9+16 = V9 + 16 344 =7 ‘State whether or not his calculation is correct. Explain your answer.Or the numbers 9, 11, 20 and 45 once each to make this calculation correct. (@ +@)+(@-@)=6 Find the missing numbers. a) 3+ (@-3x5)= 100 b) (29- Ylix 2)+3=7 Oirer brackets to make the statements correct. Tip a) 48412¢4e142=50 — b) 4841244 142-49 Wunernel panties 9 48412¢4%142=57 ‘one pair of brackets. Yr square root signs to make the statements correct. a) 94 16x 422=35 b) 94 16x 442-25 0 9416x422=10 d) 9416K422=17 ) 94 16x442=13 £)9416x442=7 ‘Think about | You can write one square root inside another. Calculate the innermost root first. For example, 22+ V742 = 2243 = V25 =5. | By writing two square root signs in 9 + 16 x 4 + 2, with one root inside the other, make the answer 5, | Find a different answer using one square root sign inside another. Discuss | Brackets and the order of operations are very | important in real life. Suppose two classes of 18 and 13 children are going to visit the Blue Mountains in | Australia on a school trip. The teacher takes two | bottles of water for each child. The calculation is | 2.x (18 + 13) = 62 bottles. What mistakes could | the teacher make if there were no brackets? Think about other examples where brackets and order of | operations are important in real life. Thinking and working mathematically activity Use +, -, x and + to make the calculations correct, JOM +3=2 (2) 28) +3 =4 9M 25+2=3 (7x 2-203" = 64 (506?) = (2s) =2 3x 18/2 x 2? +8? = 107 Discuss your strategies for solving these problems. Write your own questions like these.Serre een rer nt| pes Nastosty Worked example 2 (@) Do these calculations as efficiently as you can a) 0.25% 1.52%4 eds (d2) 9 99x18 d) 0.4 x28 +0.6 x28 a) 0.25 x 1.52 x4=0.25 x4 x 1.52 =1%1,52 = 1.52 ‘Swap numbers so the calculation starts with 0.25 « 4. This gives you 1, Which is easy ‘to work with. With multiplication a xb =b xa. This is the commutative law. weg (a+a)-(sara)+5 io = Group the numbers in a different way, so that the first calculation is 63 +4 This gives you 7, which is easy to work with Use the associative law to make the addition easier without changing the answer. + + 9) 99x 1.8=(10-0.1) x 1.8 = 10x 18-0.1x18 = 18-018 = 17.82 Partition 9.9 into 10-0.1 Then use the distributive law to make the multiplication easier without changing the answer.pairs of calculations give the same answer? A (161x5) x2 161 x (5x2) BaBx tt 48x 2-48x © (17x 29)-(7 x29) 24x29 D025+4 440.25 E G1x5.4)x09 31x (54%0.9) F 025+037+0.75 0.75+0.25+0.37 | It is possible to answer this question without | doing the calculations. i © Use efficient methods to find the answers. Show your working. a) 87+ (13 +128) b) 1.54+(25+39) 0) 175461425 d) 7.6+389+2.4 e) (3.71.4 12.25) +0.75 f) 42467458473 © se efficient methods to find the answers to the calculations below. Show your working. If an answer is a fraction greater than 1, write it as @ mixed number. 5,4),3 2.3 1,7), 18 a (g+ br gtigts 9 (+a) duZethed 24(3 3's (i242) 44 a ii+zedeg o) 2+(3+17) 9 23+(142)+3 @ Find efficient ways of doing these calculations. Show your working. a) 101 x 23 b) 99x 0.6 ©) 1.1%18 d) 09x67 e) 024x141 f) 06x99 © Simplify the calculations where possible, and find the answers, Show your working, a) 4x (2572) b) (1.50.7) x6 9) 0.25 x (1.8.x4) ) 0.5 x (0.2 x 15) @) 1.2 (0.7-0.3) f) 083258xdx 2011 a) 24x38 by Fx11x9 ay a 3x1, @3+(3+3) e) 3xgt2 4 h) 25x72 “oe Find the answer using an easier method. 49x7_ 19x? | Think about Write two methods for finding 53 «3 and two methods for finding 4.1. 2.1. For each calculation, decide whether you find one of the methods easier | than the other. 1 a 49x 7-197 Thinking and working mathematically activity Y Investigate which types of calculation obey the associ three numbers and two operations. Test different combinations of operations, which could be either the same, for example: Does (a = b) + cequal a + (b+)? law. Explore calculations with or different, for example: Does (ax b) + cequal ax (b + ¢)? ‘Summarise your findings. Try to write a general rule and explait Consolidation exercise “Amy works out 24 + (2 + 4) = 18. She does the following calculations: 2422=12 2424 1246=18 Explain why Amy is wrong, “Gabrielle's garden is a square shape with sides of length 9.8 m. The garden is to be sown with grass seeds. The grass seeds cost $8 per m: plus $10 for delivery. The gardener estimates that he will need (10 + 9.8") x 8 ~ $880, Gabrielle estimates the cost to be 9.8 x 8 + 10 = $90. Do you agree with the estimates? Explain your answer. @© Explain the difference between /25—9 and 25 ~9Okina: a) 2 x (16-1)? » 25 2-18 d) 3x98 e) 100-(10-@+1)P f) Veh = YB g) 50x 4-7 10? b) 4x 9- (17-8) @ insert a cube root sign to make each calculation correct, a) 125-27+8 14 b) 128 -8'-27=97 @ insert one set of brackets to make each calculation correct. a) 22545 %6= 140 b) 2 x545%6=240 ) 25456 = 150 @ Decide whether each statement is correct or incorrect. You can decide without actually performing the calculations. a) (45 x6) x82 =4.5 x (6x 82) b) 5x35 235248 x35 ©) (7.242)204=7.22(2+04) d) (9.54 3.7)-28=954 (37-28) 14(242)a(1 34(414,3)23,141,53 ov ge(§+g)= (543 9 §*(5+ io) a3 *3 "Ho © A éclass of children make paper chain decorations. They make the following lengths: * 15 chains are 2 of a metre long * 15 chains are } a metre long Write two methods for finding the total length of the paper chains. Find the total length and write it as a mixed number. © A cuboid is 5m long, 0.24m wide and 0.8m high. Find its volume, @® Sieg buys four items for $14.99 each, Use an efficient method to find the total cost. End of chapter reflection RO RGR is COC e koe STs In the order of operations, Use the correct order of square and cube roots have the operations with calculations ae same priority as indices. including square and cube roots. *) 2% (© ~ os) Treat calculations under a by SF a8 square or cube root sign like calculations in brackets: do them before other calculations. Some calculations can be Use the laws of arithmetic and Simplify and calculate: simplified by finding an efficient order of operations to simplify. order to do the calculations. calculations with fractions and a) 6.88 + (1.12 +3.97) decimals, including calculations 6) (2+ 12) +2 with brackets. 3 4 O15 xB 48x T d) 3.65 x 12- 1.65 x 12