## Rounding Numbers

Purpose

In this unit students will discuss why it is important to round numbers sensibly. They will practice choosing and rounding to a sensible level of accuracy for different contexts.

Achievement Objectives
NA5-6: Know and apply standard form, significant figures, rounding, and decimal place value.
Specific Learning Outcomes
• Round whole numbers sensibly in context.
• Round decimal numbers sensibly.
Activity

#### Session 1

1. Rounding numbers for newspaper headlines. The government of Outer Australis reports spending \$33,883,641.31 in the 2002-2003 financial year. Discuss how to put this number into a newspaper headline. A sensible answer here is "Government spends \$34 million last year".
While many students will round the 8 hundred thousand ‘up’ because it is over ‘5’ it is desirable to present this as a number line and see which end the number is closer to. Discuss why this line has 33,500,000 in the middle and why 33,883,641.31 is roughly where the 1. Student Exercises. Round these numbers suitably for use in newspaper headlines
• Quality Stores make a profit of \$3,493,631.
• Prime Minister paid \$251,419.91 last year.
• Scientist estimates there are 56,409,100 possums in New Zealand.
• Cost of producing cheese drops to 101.8 cents per kilogram due to improved efficiency at the cheese factory.
• A milk factory reports it bought 27,309,604 litres of milk from farmers last year.
1. 7 plates are to be sold for \$75. The price of 1 plate is 75 ÷ 7 = \$10.71428571. Discuss how to round to get these different answers:
• the supermarket charges to the nearest cent so one plate is priced at \$10.71.
• the petrol station does not use cents only multiples of 5 cents. So it charges \$10.70
Number lines are very useful.

Supermarket: Petrol Station: 1. Student Exercise. Complete the table.
 Cost per Item Items Total Cost Number Calculator Supermarket Petrol Station Plates \$75.00 7 10.71428571 \$10.71 \$10.70 Tins of Soup \$49.40 13 Cola cans \$104.10 97 Frozen Peas \$71.11 21 Fly Spray \$811.00 206 Chocolate bars \$111.00 73

#### Session 2

1. Jerry measures the width of the school football field and finds it is 70.4m wide. Discuss why this means the true width is between 70.35m and 70.45m. Melissa measures the length of the field and finds it is 101.8 m long. Discuss why it is between 101.75m and 101.85m long. Jerry and Melissa now collaborate to find the area by using the calculator. Unfortunately 70.4 x 101.8 = 7166.75 is not a sensible answer. Discuss why this is overly accurate given the possible errors in the data.
Discuss why 70.35 x 101.75 < True Area < 70.45 x 101.85 that is
7158.1125 m2< True Area < 7175.3325m2. Rounding the lower and upper areas to 3 significant figures is a sensible answer but rounding to 2 decimal places (7166.75) is not. In practice an answer like 7170m2 is good enough for practical problems like how much grass seed is needed resow the field.
2. Student Exercise. Complete the table for area of rectangles.
 Length Width Area Lower Measured Upper Lower Measured Upper Lowest Rounded Highest 101.75 101.8 101.85 70.35 70.4 70.45 7158.1125 7170 7175.3325 111.7 61.3 88.7 9.81 202 11.62 181.7 161.7 9.31 6.34 1.61 0.86
1. Discuss why, when multiplying numbers from measurements the answer is never more accurate that the lowest number of significant figures in the factors.
2. The area of a rectangle is 100.4m and has one side of 20.4m. So the other side is
100.4 ÷ 20.4 = 4.92156862m.
Discuss why the exact answer is between 100.35 ÷ 20.45 and 100.45 ÷ 20.35 that is 4.907m and 4.936m. Discuss why 4.9m is a sensible answer.

#### Session 3

How many significant figures are there in a number?

1. At the Olympics the running track is 400m long. If this number has 1 significant figure the length of the track is between 350 and 450 which is ridiculous.
Discuss and fill in the table
 Length Number of Sig. Figs. Least possible length Largest possible length Sensible 400m 1 350 450 No 400m 2 395 405 No 400m 3 399.5 400m 4 399.95 400m 5

So a 400m running track is laid out to an accuracy of 5 significant figures or perhaps 6.

1. Student Exercises.
• Julie says she is about 50 metres from home. How many significant figures are in this number?
• A building plan shows the length of a room is 6700mm long. How many significant figures?
• The Lotto people predict a first division prize of \$1,400,000. How many significant figures?
• The census shows New Zealand population is 4 270 000. How many significant figures?
1. A supermarket works out its prices to be to the nearest cent. Suppose a kilogram of chicken costs \$14.91. Discuss why it is more sensible to say this number has been rounded to 2 decimal places rather than 4 significant figures.
2. If trigonometry has been studied, discuss calculated lengths need to be rounded sensibly. For example; Discuss why a sensible answer for x is between 2.6 and 2.8. So 2.7 is reasonable.

1. Student Exercises.   #### Session 4

Rounding decimals by using number lines.

1. To round 16.469 to 1 decimal place discuss why the choices are either 16.4 or 16.5 and so a suitable number line is Discuss why the middle is 16.45 and fill in ten divisions and add the arrow for 16.469. Discuss why 16.469 is rounded to 16.5 and not 16.4.

1. Student Exercises. Students who have a limited understanding of decimal place value often find dividing a scale into 10 and labelling each division difficult. It is worth doing nevertheless to assist their understanding. Draw number lines divided into ten labelled parts to solve these problems.
 13.89143 (1 decimal place) 707.6341 (2 decimal places) 3.04192 (3 sig. fig.) 80915.81 (nearest 10) 0.0473816 (3 decimal places) 4.004916 (4 sig. fig)
1. This a true story. Mr. Brown rounded 14.486 to the nearest whole number by rounding 14.486 to 14.49 by the "over 5" rule. Then he rounded 14.49 to 14.5 by the same rule. Then he rounded 14.5 to 15 by the rule. Unfortunately this is wrong. Discuss why. This highlights the danger of the rule versus realising that the number is rounded to the closer of the left hand end or right hand end on a number line.