1A_Ch5(1)

5.2Estimation Methods and Strategies Relating to Numbers A Estimation Methods B Estimation Strategies Index 1A_Ch5(2)

Index A Benchmark Strategies Decomposition-recomposition Strategies B Measuring Groups of Objects C Using Formulas D 5.3Estimation Strategies in Measurement 1A_Ch5(3) Using Graph Papers E

The Significance of Estimation 1.Meaning of Estimation Index 5.1The Significance of Estimation 1A_Ch5(4) i.Estimation is the way of finding an approximate value of a number. In the case of a numerical expression, the value of the entire expression is estimated by taking an approximate value of each number in the expression for computation. ii.The result of estimating a numerical expression is called an estimated value or an estimate.

The Significance of Estimation 2.Reasons for Estimation Index 5.1The Significance of Estimation 1A_Ch5(5) There are many reasons for making estimation. Listed below are some of the main ones. i.To simplify calculations. ii.Limitations in measurement. iii.Value unknown. iv.Value varies. v.Estimation helps to understand the numbers better. vi.Checking reasonableness of results. + ExampleExample

In each of the following, which option A, B, C or D do you think will give the best estimated value for the given exact value (underlined)? Index + Key Concept 5.1.1Key Concept The Significance of Estimation 1A_Ch5(6) 1.The length of the bridge is km. A B. 18 C. 17 D The population of a city is A B C D The travelling time from Central to North Point is min. A. 12 B C D. 11

Estimation Methods Index 5.2Estimation Methods and Strategies Relating to Numbers 1A_Ch5(7) E.g.i.573  570(correct to the nearest ten) ii.573  600(correct to the nearest hundred) E.g.i.34.5  30ii.185  100 iii.1.24  1iv.0.89  0 1.Rounding off + ExampleExample 2.Front-end + ExampleExample A)

Estimation Methods Index 5.2Estimation Methods and Strategies Relating to Numbers 1A_Ch5(8) 3.Special Cases When we need to apply estimation to real-life problems, we have to adjust the degree of accuracy of our estimation according to the situation. There are two ways we often use to make the adjustment. A)

Estimation Methods Index 5.2Estimation Methods and Strategies Relating to Numbers 1A_Ch5(9) i.Rounding Up Sometimes we have to make an estimated value that is slightly larger than the exact value. We always advance 1 to the place which is to be corrected to, regardless of the digit that follows. ii.Rounding Down Sometimes we have to make an estimated value that is slightly smaller than the exact value. All the digits after the place which is to be corrected to are replaced by 0. 3.Special Cases + Example Example A) + Index 5.2Index 5.2

Index 5.2Estimation Methods and Strategies Relating to Numbers 1A_Ch5(10) Estimate the following numbers by rounding off. (a)4 620 (correct to the nearest hundred) (b)599 (correct to the nearest ten) (c)0.237 (correct to the nearest 0.01) (d) (correct to the nearest 0.01) (e)0.712 (correct to the nearest 0.1) Key Concept 5.2.1Key Concept 5.2.1

Index + Key Concept 5.2.1Key Concept Estimation Methods and Strategies Relating to Numbers 1A_Ch5(11) Estimate the following numbers by the Front-end method. (a)1 108 (b)4 253 (c)321 (d)12.8 (e)

Index 5.2Estimation Methods and Strategies Relating to Numbers 1A_Ch5(12) In each of the following, i.determine whether the underlined value in the expression should be estimated by ‘rounding up’ or ‘rounding down’; ii.write down the estimated value of the expression. (a)The volume of a box of apple juice is 254 mL. How many boxes of apple juice are needed to fill up a 1000 mL plastic box? Expression : 1000 ÷ 254 (b)Ann baked 164 cakes. She put 5 cakes in a box, how many boxes she needed to pack up all the cakes? Expression : 164 ÷ 5

Index 5.2Estimation Methods and Strategies Relating to Numbers 1A_Ch5(13) (a)i.Rounding down ii.Estimated value : 1000 ÷ 250 = 4 ∴ 4 boxes of apple juice are needed to fill up a 1000 mL plastic box. (b)i.Rounding up ii.Estimated value : 170 ÷ 5 = 34 ∴ She needed 34 boxes to pack up all the cakes. + Back to QuestionBack to Question + Key Concept 5.2.2Key Concept 5.2.2

Estimation Strategies Index 5.2Estimation Methods and Strategies Relating to Numbers 1A_Ch5(14) ‧ Estimation strategies help us to decide which estimation method we can use to make a better estimate. B) E.g.Estimate 11 – 2.3 – 2.1 – 2.05 – 1.91 – Take 2 as the cluster and the expression can be rewritten as 11 – 2 – 2 – 2 – 2 – 2 or 11 – (2 × 5). 1.Clustering + Example Example

Estimation Strategies Index 5.2Estimation Methods and Strategies Relating to Numbers 1A_Ch5(15) B) E.g. Estimate 963 ÷ 【 Take 3 as an approximate value of 2.89, then 963 and 3 form a pair of compatible numbers under division. 】 2.Compatible Numbers + Example Example 963 ÷  963 ÷ = = 328

Estimation Strategies Index 5.2Estimation Methods and Strategies Relating to Numbers 1A_Ch5(16) B) 3.Translation + Example Example E.g. Estimate. 【 Change the order of the numbers or the operations in a given expression so that the entire expression can be estimated more easily. 】

Estimation Strategies Index 5.2Estimation Methods and Strategies Relating to Numbers 1A_Ch5(17) B) 4.Compensation + Example Example E.g. Estimate 【 Make adjustment to initial estimates so as to raise the degree of accuracy of the estimation. 】  = 130 ∴   10 ∴ Estimated value after compensation= = Index 5.2Index 5.2

Estimate ( ) ÷ 5. Index 5.2Estimation Methods and Strategies Relating to Numbers 1A_Ch5(18) 【 The approximate value of all the 3 numbers inside the brackets is 200, so 200 can be regarded as the cluster. 】 ( ) ÷ 5  200 × 3 ÷ 5 = 120

Index 5.2Estimation Methods and Strategies Relating to Numbers 1A_Ch5(19) Locate a cluster in each of the following numerical expressions and find an estimated value of the expression. (a) (b)( ) × 3 (a)  80 × 5 = 400 (b) ( ) × 3  20 × 4 x 3 = Key Concept 5.2.3Key Concept 5.2.3

Estimate × 933 – 11. Index 5.2Estimation Methods and Strategies Relating to Numbers 1A_Ch5(20) × 933 – 11 = 311 – 11 = 300 【 If we take as an approximate value of 0.325, then and 933 will form a pair of compatible numbers under multiplication. 】 

Index 5.2Estimation Methods and Strategies Relating to Numbers 1A_Ch5(21) Find an estimated value of each of the following numerical expressions using compatible numbers. (a) 92 × × 2(b) 179 ÷ ÷ 2 (a)92 × × 2 = 18 × 2 = 36  (b)179 ÷ ÷ 2 = 1620 ÷ 2 = 810  + Key Concept 5.2.4Key Concept 5.2.4

Estimate (878 × 7.4) ÷ 63. Index 5.2Estimation Methods and Strategies Relating to Numbers 1A_Ch5(22) 【 (878 × 7.4) ÷ 63 can be written as 878 × (7.4 ÷ 63), then use the approximate values of 878 and 7.4 to do the estimation. 】 (878 × 7.4) ÷ 63 = 878 × (7.4 ÷ 63)  878 × (7 ÷ 63) =  = 100

Index 5.2Estimation Methods and Strategies Relating to Numbers 1A_Ch5(23) Estimate (37 × 4.6) ÷ 18. (37 × 4.6) ÷ 18 = 37 × (4.6 ÷ 18)  37 × (5 ÷ 20) =  = 10 + Key Concept 5.2.5Key Concept 5.2.5

Estimate Index 5.2Estimation Methods and Strategies Relating to Numbers 1A_Ch5(24) 【 At first, the value of the expression is estimated to be 8 using the method of Front-end. Then, we adjust this initial estimate by compensating for the sum of the digits after the decimal place so that the degree of accuracy of the estimate is increased. 】 Front-end :  = 8

Index 5.2Estimation Methods and Strategies Relating to Numbers 1A_Ch5(25) Adjust the digits after the decimal place:    1 Estimated value after compensation = = 11 + Back to QuestionBack to Question

Index 5.2Estimation Methods and Strategies Relating to Numbers 1A_Ch5(26) Estimate Key Concept 5.2.6Key Concept Front-end :  = 19 Adjust the digits after the decimal place:   1 Estimated value after compensation = = 21

Benchmark Strategies Index 5.3Estimation Strategies in Measurement 1A_Ch5(27) A) ‧ In benchmark strategies, the quantity to be measured is estimated by comparing it with a known benchmark. + ExampleExample + Index 5.3Index 5.3

In the figure, the height of the door is known to be around 2 m. Estimate the height of the ceiling above the ground. Index 5.3Estimation Strategies in Measurement 1A_Ch5(28) 【 By observation, we see that AB is higher than the door by about half the door’s height. 】 The height of the ceiling  = (2 + 1) m = 3 m

The figure shows a measuring cup with a 600 mL mark. Index 5.3Estimation Strategies in Measurement 1A_Ch5(29) (a)Estimate the volume of the liquid inside the measuring cup. (b)The measuring cup is now completely filled with liquid and not overflowing. Estimate the volume of the liquid inside the measuring cup. + SolnSoln + SolnSoln

Index + Back to QuestionBack to Question 5.3Estimation Strategies in Measurement 1A_Ch5(30) (a)We see that the liquid inside the measuring cup occupies about of 600 mL. ∴ The volume of the liquid inside the measuring cup  = 200 mL

Index + Back to QuestionBack to Question 5.3Estimation Strategies in Measurement 1A_Ch5(31) (b)When the measuring cup is completely filled with liquid, the volume of the liquid is about 4 times what we presently have. ∴ The volume of the liquid inside the measuring cup  4 × 200 mL = 800 mL Fulfill Exercise Objective  Use benchmark strategies to estimate. + Key Concept 5.3.1Key Concept 5.3.1

Decomposition-recomposition Strategies Index 5.3Estimation Strategies in Measurement 1A_Ch5(32) B) ‧ In decomposition-recomposition strategies, the quantity to be measured is broken down into smaller parts which are estimated first. Then all these smaller parts are recomposed and the original quantity can be estimated. + Index 5.3Index ExampleExample

In the figure, the length of the corridor can be decomposed into the widths of 6 classrooms. It is estimated that the width of each classroom is about 7.5 m. Estimate the length of the entire corridor. Index 5.3Estimation Strategies in Measurement 1A_Ch5(33) The length of the entire corridor  6 × 7.5 m = 45 m

Index 5.3Estimation Strategies in Measurement 1A_Ch5(34) In the figure, a bookcase is decomposed into 8 compartments. Books each about 6 cm thick are now put into the bookcase. Estimate (a)the width of each compartment, (b)the width of the entire bookcase, (c)the number of books when the bookcase is full. Width = ?

Index + Back to QuestionBack to Question 5.3Estimation Strategies in Measurement 1A_Ch5(35) 【 By observation, about 5 books of the same thickness can be put into each compartment. 】 (a)The width of each compartment  5 × 6 cm = 30 cm (b)The width of the entire bookcase  4 × 30 cm = 120 cm (c)The number of books  8 × 5 = 40 Fulfill Exercise Objective  Use decomposition-recomposition strategies to estimate. + Key Concept 5.3.2Key Concept 5.3.2

Measuring Groups of Objects Index 5.3Estimation Strategies in Measurement 1A_Ch5(36) C) ‧ In measuring objects of very small sizes, instead of making the measurement directly, we often measure a large number of such objects. The measurement we need for the small object is then obtained by division. This is called the measuring groups of objects strategy in estimation. + Index 5.3Index ExampleExample

Suppose the total volume of water droplets is measured to be 270 mL, Index 5.3Estimation Strategies in Measurement 1A_Ch5(37) then the volume of a water droplet  = 0.27 mL

Index 5.3Estimation Strategies in Measurement 1A_Ch5(38) Bobby measures that the total weight of 200 rubber bands is 52 g. Estimate the weight of a rubber band. The weight of a rubber band  = 0.26 g Fulfill Exercise Objective  Measure groups of objects to estimate. + Key Concept 5.3.3Key Concept 5.3.3

Using Formulas Index 5.3Estimation Strategies in Measurement 1A_Ch5(39) D) ‧ Using formulas, the required quantity is obtained indirectly when related quantities are measured and the values are substituted into the formula for calculation. + Index 5.3Index ExampleExample

If the length of each side of the cube in the figure is measured to be 3.5 cm, estimate the volume of the cube. Index 5.3Estimation Strategies in Measurement 1A_Ch5(40) By formula, volume of cube = length × length × length. ∴ The volume  3.5 × 3.5 × 3.5 cm 3 = cm 3  43 cm 3

Index 5.3Estimation Strategies in Measurement 1A_Ch5(41) The figure shows a cardboard box for storing canned food. It is known that two layers of cans can be stacked up in the box and each layer contains 12 cans. The height and the diameter of each can have been measured to be 11 cm and 8 cm respectively. Estimate the volume of the cardboard box.

Index + Back to QuestionBack to Question 5.3Estimation Strategies in Measurement 1A_Ch5(42) Fulfill Exercise Objective  Use formulas to estimate. Height  2 × 11 cm = 22 cm Width  3 × 8 cm = 24 cm Length  4 × 8 cm = 32 cm ∴ Volume of the cardboard box  32 × 24 × 22 cm 3 = cm 3  cm 3 + Key Concept 5.3.4Key Concept 5.3.4

Using Graph Papers Index 5.3Estimation Strategies in Measurement 1A_Ch5(43) E) ‧ For figures with irregular shapes, we use graph papers to estimate their areas. + Index 5.3Index ExampleExample

Index Estimate the area of the following figure. + Key Concept 5.3.5Key Concept Estimation Strategies in Measurement 1A_Ch5(44) Full square: counted as 1More than half: counted as 1Less than half: counted as 0 Area = 42 cm 2 1 cm