2E Adding and subtracting decimals����������������������������������������������������������������������������������

Learning intentions

Inter-year links ✔ I can add and subtract positive and negative decimals. Years 5/6 Adding and subtracting decimals Year 7 4C Adding and subtracting decimals

Adding and subtracting decimals • Decimals can be added or subtracted in the same way as whole numbers using the vertical 3.02 algorithm. Add or subtract digits in each column from right to left. + 4.15 _ ➝ Make sure that the decimal points are aligned and the digits are in the correct place values. 7.17 _ • When dealing with negative decimals: ➝ consider the direction of both numbers on the number line, • if they have the same direction, find the sum • if they have different directions, find the difference ➝ consider if the result will be positive or negative. • the sum will have the same sign as the calculation • the difference will be positive if the positive number is larger • the difference will be negative if the negative number is larger • Simplify the calculation by considering the operation and the sign applied to each decimal.

Question Diagram Conclusion Calculation Result

3.12 + 4.2 +3.12 +4.2 –1 0 1 2 3 4 5 6 7 8 sum Sum in the positive direction 3.12 +4.20 _ 7.32 Positive result of 7.32 –3.12 – 4.2 –4.2 –3.12 –8 –7 –6 –5 –4 –3 –2 –1 0 1 sum Sum in the negative direction −( 3.12 + 4.20 _ 7.32 _) Negative result of −7.32 –3.12 + 4.2 +4.2 –3.12 –3 –2 –1 0 1 difference Difference in the positive direction

1 10 4.2 0 − 3.12 _ 1.08 _ Positive result of 1.08 3.12 − 4.2 –4.2 +3.12 –2 –1 10 2 3 4

Difference in the negative direction

1 10 4.2 − 3.12 _ 1.08 _

Negative result − −( _ of −1.08

0 _)

+ ( + ) = + − ( + ) = − + ( − ) = − − ( − ) = + DRAFT ONLY - NOT FOR SALE

difference

Example 2E.1 Adding and subtracting decimals

Use vertical addition and vertical subtraction to calculate each result. a 14.507 + 8.63 b 45.93 − 30.498

THINK

1 Estimate the result by rounding to the nearest whole number. 2 Write the numbers underneath each other.

Ensure each decimal point and digits are lined up. 3 Replace blank spaces with additional zeroes so that each number contains the same number of decimal places. 4 Add or subtract the digits in each column as you would for whole numbers. Ensure the decimal point in the answer is lined up with the decimal points in the calculation. 5 Check that your estimate is a close approximation of the result and that the decimal point is in the correct position.

WRITE

a 14.507 + 8.63 ≈ 15 + 9 = 24 14.507 + 08.630 __ __ 1 41 .507 + 08.630 __ 23.137 __ 14.507 + 8.63 = 23.137 24 ≈ 23.137 b 45.93 − 30.498 ≈ 46 − 30 =16 45.930 − 30.498 _ _ 2 10 45.93 0 − 30.428 _ 15.502 _ 45.93 − 30.498 = 15.502 16 ≈ 15.502 Example 2E.2 Adding decimals with negative signs Calculate each result. a −3.256 + 5.89 b −1.68 + (−3.45)

THINK

a 1 Estimate the result by rounding to the nearest whole number. 2 Determine the direction of each number on the number line. –3.256 moves to the left on the number line and 5.89 moves to the right on the number line. 3 Determine if your calculations require you to find the sum or the difference of the decimals. The directions are different, so find the difference. 4 Subtract the decimals using the vertical algorithm. 5 Determine the sign of the result. Since the value of the positive decimal is greater, the result will be positive. Check your answer against your estimate.

WRITE a −3.256 + 5.89 ≈ −3 + 6 = 3 +5.89–3.256 0 1 2 3–1–2–3–4–5 5.89 8 0 10 1 DRAFT ONLY - NOT FOR SALE −3.256 _ 2.634 _ −3.256 + 5.89 = 2.634 3 ≈ 2.634

b 1 Estimate the result by rounding to the nearest whole number. Simplify the signs. 2 Determine the direction of each number on the number line. Both –1.68 and –3.45 move to the left on the number line. 3 Determine if your calculations require you to find the sum or the difference of the decimals. The directions are the same, so find the sum. 4 Add the decimals using the vertical algorithm. 5 Determine the sign of the result. Since both values are moving in the negative direction, the result will be negative. Check your answer against your estimate.

b −1.68 + (−3.45) = −1.68 − 3.45 ≈ −2 − 3 = −5 –3.45 –1.68 0 1–1–2–3–4–5 1 1.68 +3.45 _ 5.13 _ −1.68 + (−3.45) = −5.13 −5 ≈ −5.13 Example 2E.3 Subtracting decimals with positive and negative signs Calculate each result. a 7.58 − 8.6 b 4.206 − (−13.797)

THINK

a 1 Estimate the result by rounding to the nearest whole number. 2 Determine the direction of each number on the number line. 7.58 moves to the right on the number line and –8.6 moves to the left on the number line. 3 Determine if your calculations require you to find the sum or the difference of the decimals. The directions are different, so find the difference. 4 Subtract the decimals using the vertical algorithm. 5 Determine the sign of the result. Since the value of the negative decimal is greater, the result will be negative. Check your answer against your estimate. b 1 Estimate the result by rounding to the nearest whole number. Simplify the signs. 2 Determine the direction of each number on the number line. Both 4.206 and 13.797 move to the right on the number line. 3 Determine if your calculations require you to find the sum or the difference of the decimals. The directions are the same, so find the sum. 4 Add the decimals using the vertical algorithm. 5 Determine the sign of the result. Since both values are moving in the positive direction, the result will be positive. Check your answer against your estimate.

WRITE a 7.58 − 8.6 ≈ 8 − 9 = −1 –8.6 +7.58 2 4 6 8 10 120–2–4–6 8.6 5 0 10 − 7.58 1.02 7.58 − 8.6 = −1.02 −1 ≈ −1.02 b 4.206 − (−13.797) = 4.206 + 13.797 ≈ 4 + 14 = 18 +4.206 +13.797 3 6 9 12 18 21150–3–6 13 1 .7 1 1 97 + 4.206 _ 1 DRAFT ONLY - NOT FOR SALE 18.003 _ 4.206 − (−13.797) = 18.003 18 ≈ 18.003

ANS p490 ✔ If the signs are the same, find the sum! ✔ If the signs are different, find the difference! ✔ Make sure that the decimal point in each number is lined up under the other when adding or subtracting decimals.

For example,

✔ When adding trailing zeroes after the decimal, you are not changing the number. You are just writing a zero that was already implicitly there.

Exercise 2E Adding and subtracting decimals 1 Adding and subtracting decimals can be shown on the number line. Instead of starting at the first number of the calculation, we can start at zero and treat the first number as a positive or negative jump along the number line.

For example, 1.2 − 0.4 is shown on the number line below, starting at zero. 0 0.1 0.2

1, 2, 3(a, c, e), 4, 5–8(1st column), 9–12, 13(a, b), 15, 16(a) +1.2

–0.4 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 For each number line below, write the calculation shown, starting from zero.

a

–0.5 0 0.5 1 1.5 2 2.5 3

b

–2–1.5–1–0.5 0 0.5 1 1.5 2 2.5 3

c

–0.6–0.8–1–1.2–1.4–1.6–1.8–2 –0.4–0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

2, 3(d–f), 4, 5–8(2nd column), 9(c, d), 12, 13(a, c), 14, 16, 17(a, b)

3(d–f), 5–8(2nd column), 9(d), 13(c, d), 14, 17(c, d), 18, 19 100 000.3 12.000 003–DRAFT ONLY - NOT FOR SALE

d

–0.6–0.8–1–1.2–1.4–1.6–1.8–2 –0.4–0.2 0 0.2 0.4 0.6

a

5.27 +4.83 _

b

18.9 + 7.35 _ 3 Use vertical addition to calculate each result.

c

0.085 1.32 +0.647 _

d

17.378 6.003 + 5.999 _

a b c d e 23.82 + 17.13 + 124.06 + 82.95 5.613 + 9.27 + 11.3856 + 7.5 324.7 + 76.29 + 4.93 + 285.064 74.0106 + 2.93 + 28.738 + 0.9167 0.0034 + 0.21 + 0.024 + 0.0402 UNDERSTANDING AND FLUENCY f 99.999 + 14.24 + 4.748 + 0.0284 4 Use vertical subtraction to calculate each result. a 158.29 − 75.46 b 2051.384 − 1265.73 __ c 0.35 − 0.2156 _

2E.2 2E.3 5 6 7 8 9 10 d 70.308 − 44.057 _ Use vertical subtraction to calculate each result. a 9.104 − 6.3728 b 265.7 − 88.504 c 5061.03 −1266.735 d 73412 − 8743.56 e 9000 − 0.555 f 6.5677 − 4.5678 Calculate each result. a 3.2 + (−5.8) b −3.2 + (−5.8) c −3.2 + 5.8 d 51.294 + (−3.028) e −4.1928 + 0.285 f −0.048 + (−495.19) Calculate each result. a 4.78 − 8.48 b −4.78 − 8.48 c −4.78 − (−8.48) d −3.541 − 58.242 e −274.28 − (−0.4827) f 9.402 − 10.611 Evaluate the following. a 5.8 − 9.9 + 3.2 b 19.53 + 3.78 − 32.05 c 100 − 210.15 + 27.3 d 0.0032 + 0.1 − 0.352 Evaluate the following. a (+3.7) − (+5.2) + (−6.1) b (+4.8) + (−9.7) + (+10.11) c (−17.3) − (+12.4) − (−30.8) d (−85.23) + (−0.0573) − (+2.90204) The circuit of a fun run has five different stages. The lengths of the stages are: 1.435 km, 2.790 km, 1.902 km, 2.285 km, 3.613 km. First estimate the distance around the full circuit by rounding each length to the nearest whole number and then calculate the exact _DRAFT ONLY - NOT FOR SALE distance.

PROBLEM SOLVING AND REASONING 11 Lee Redmond once held the record for the longest fingernails on both hands (see table), after not cutting them for 30 years.

Fingernail lengths (cm) Right hand Left hand

Thumb 76.4 80.0

Index finger 72.3 76.4

Middle finger 74.1 76.7

Ring finger 73.6 76.2

Little finger 71.6 73.6 a What is the total length of the fingernails on: i her right hand ii her left hand? b Which hand has the longer fingernails? Suggest a reason for this. c What is the total length of the fingernails on both hands? 12 Consider the calculation 7.2 − 9.4 + 2.3.

a Which of the following is not equivalent to the original calculation? A (7.2 − 9.4) + 2.3 B (7.2 + 2.3) − 9.4 C 7.2 − (9.4 − 2.3) D 7.2 − (9.4 + 2.3) E 2.3 − (9.4 − 7.2) b Explain why the answer to part a is not equivalent to 7.2 − 9.4 + 2.3 and write a similar expression that would be equivalent. c For any calculation involving only addition and subtraction, if all subtractions are performed first, will the answer always be correct? Explain why or why not. 13 Calculate the following sums and differences by using addition and subtraction of the equivalent fractions.

For example, 12.3 + 4.21 = 123 _ 10 + 421 _ 100 = 1230 _ 100 + 421 _ 100 =

1651 _ 100 = 16.51.

14 a −17.32 + 4.0032 b 9.999 + (−50.91) c −0.003 − (−0.0502) d −90.06 − 73.0012 Jimmy currently has $42.50 saved up. He wants to buy a new game that costs $69.99. Jimmy’s parents agree to pay the difference if Jimmy pays them back. Jimmy earns $4.50 a week in pocket money but can increase it by completing various chores. Each chore has a limit of how often Jimmy can complete it for pocket money. The chores and how much Jimmy earns from them are listed below. • unload and hang the washing up to dry, $1.10 twice a week • fold the laundry and put it away, $0.90 twice a week • take out the rubbish bin, $0.45 daily • dust the shelves and bookcases, $1.70 weekly • help prepare dinner, $2.10 daily a Can Jimmy pay his parents back in two weeks? State how much Jimmy still owes or how much extra he earns. b If Jimmy wants to save up for another $69.99 game after he has paid his parents back, what is the minimum number of weeks it will take? DRAFT ONLY - NOT FOR SALE 15 Look back over your calculations for adding decimals together. What do you notice about the number of decimal places in the calculation and the number of decimal places in the result? Determine the number of decimal places in the result of a sum where a number with x decimal places is added to a number with y decimal places.

b Use the fact that −(4.2 − 3.12) = −4.2 − (−3.12) by the distributive law to explain why we can subtract 3.12 from 4.2 to calculate 3.12 − 4.2. 17 Subtraction can be rewritten as adding a negative number. As addition is associative and commutative, this allows us to rearrange the order of calculations. We can then add all the positive numbers together and all the negative numbers together. The result can be determined by finding the difference between the two numbers.

For example, −3.1 + (−2.4) + 4.9 − (−9.3) − 6.2 = (−3.1) + (−2.4) + (+4.9) + (+9.3) + (−6.2) = [(+4.9) + (+9.3)] + [(−3.1) + (−2.4) + (−6.2)] = 14.2 + (−11.7) = 2.5

Write the following as the difference of the total positive and negative sums by making use of the associative and commutative laws, then evaluate. a 9.2 − 7.2 + (−4.1) − (−4.8) + 7 − 2.6 b −28.19 − (−49.18) + 99.24 + (−72.09) − (−13.25) c 0.028 − 0.18 − (−0.291) + (−2.59) − 12.01 + 9 d −70.18 − 81.8 + 29.09 − (−45.29) − 29.90 + 17.7 18 You begin by facing North. You turn 197.35° clockwise, then 87.194° anti-clockwise, then 3.813° clockwise, then 99.24° clockwise, then 100.01° anti-clockwise, and finally 32.021° anti-clockwise.

a In what direction (clockwise or anti-clockwise) and by how many degrees are you now facing relative to North? b How many degrees did you turn in total, regardless of direction? 19 Fill in the blank boxes with the correct number. Every number is formed by adding the number left below and right below in the box. Your algebra skills may be useful for solving this problem.

CHALLENGE PROBLEM SOLVING AND REASONING 4139 434.64 130.42191.4 100.6 104.2 DRAFT ONLY - NOT FOR SALE

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Interactive skillsheet Adding and subtracting decimals Investigation How large is your hand? Topic quiz 2E