Rounding estimation and standard form Grades F to

SUCCESS CRITERIA: WHERE ARE WE NOW? Level Learning outcomes: F 2 E 1 I

Decimal places and significant figures Rounding Estimation Hyperlinks! Estimation of multi-stage calculations Calculating in

Rounding Learning Objective: Can I round numbers to the nearest 1, 10, 1000 etc?

Why do we round? We round numbers to save writing too much. We also

How do we round? It is often useful to use a short number line

Using a number line: Let’s round 865 to the nearest 100. 865 is between

Using a number line: This time, let’s round 865 to the nearest 10. 865

Round these to what’s in the brackets: 1. 487 [nearest 100] 500 2. 2346

Rounding to the nearest whole number: On the number line put the whole numbers

Round these to the nearest whole number: 1. 4. 8 5 2. 13. 4

Decimal places and significant figures Learning Objective: Can I round a decimal to a

What does this mean? We can round numbers to a certain number of decimal

Decimal places Remember that the decimal point does not move! The number needs to

Decimal places example: Round 83. 7649 to 1 decimal place. 8 3. 7 6

Another decimal places example: Round 83. 7649 to 2 decimal places. 8 3. 7

Round these numbers to the given number of decimal places 1. 2. 365 [1

Significant figures All numbers have significant figures. 1 st significant figure 6 th significant

Significant figures – careful with decimals Decimals are slightly different. 1 st 5 th

Rounding to significant figures Round 4534 to 2 sf. 2 nd significant figure 4

Rounding to significant figures 2 Round 0. 00576 to 1 sf. 1 st significant

Have a go at rounding these: 1. 7238 [1 sf] 7000 2. 926706 [2

Estimation Learning Objective: Can I estimate a difficult calculation? Grade D

Why do we estimate? It is easy to make a mistake when typing calculations

How do we estimate? You want the calculation to be as simple as possible

Two examples Example 1 Estimate 23 × 18 Example 2 Estimate 32. 3 ÷

Estimate these: 1. 7. 6 × 4. 93 8 × 5 = 40 2.

Estimating difficult calculations Learning Objective: Can I use estimate multi-stage calculations? Grade C

Remember bidmas! Sometimes you may be asked to estimate slightly more difficult calculations. Remember

Standard form Learning Objective: Can I read and write numbers in standard form? Grade

Writing large numbers in standard form • 7 0 0 0. How many places

Writing small numbers in standard form • The decimal point is currently here 0.

Things to remember The decimal point moves left or right until the number lies

Have a go at writing these in standard form: 1. 3000 2. 230000 3.

Now try writing these as “normal” numbers: • 800, 000 42, 600, 000 0.

The other type of question you might see: •

Standard form - calculations Learning Objective: Can I perform calculations in standard form? Grade

On your calculator try these, giving your answers in standard form: •

Two examples of multiplying Example 1 • Example 2 •

Two more examples Example 3 • Example 4 •

Calculate these, giving your answers in standard form: •

Error in measurement Learning Objective: Can I calculate the upper and lower bound of

What do you mean? Every time a measurement is made there is the possibility

What does it look like? If a measurement has been taken, it could actually

Some examples: 50 metres measured to the nearest 10 metres: Half of 10 Upper

Find the upper and lower bounds of these measurements: 1. 2300 kg measured to

One last thing… The measurement may have been rounded to a certain number of

Two examples: Example 1 Example 2 Give the upper and lower bound of 3.

Bounds and calculations Learning Objective: Can I perform calculations by taking errors in measurement

What you’ll be asked to do All measurements have an upper and lower bound.

An example: A floor needs carpeting. The floor measures 4. 2 m by 3.

Another example - be careful! • Upper bound Lower bound Upper bound

Have a go at this: A bike that is 1. 5 metres long (to

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Rounding, estimation and standard form Grades F to A

SUCCESS CRITERIA: WHERE ARE WE NOW? Level Learning outcomes: F 2 E 1 I can round to the nearest 10, 1000 etc. D 3 C 1 B 3 B 1 I can estimate a calculation. I can estimate difficult calculations by rounding. I can read and write numbers in standard form. I can calculate with numbers in standard form. I can calculate taking into account errors in measurement. I can use bounds to calculate whether a calculation is possible. A 2 I can round to a given number of decimal places or significant figures. R A G

Decimal places and significant figures Rounding Estimation Hyperlinks! Estimation of multi-stage calculations Calculating in standard form Standard form Error in measurement Calculations using bounds

Rounding Learning Objective: Can I round numbers to the nearest 1, 10, 1000 etc? Grade F

Why do we round? We round numbers to save writing too much. We also round numbers so that they are easier to work with. When a number is rounded, it must be close to the original number.

How do we round? It is often useful to use a short number line when rounding. Whichever end of the number line the number you are rounding is closer to, that is your answer. Remember: halfway always rounds up.

Using a number line: Let’s round 865 to the nearest 100. 865 is between 800 and 900. It is about here on the number line below. It is closer to 900. 800 850 Answer: 900

Using a number line: This time, let’s round 865 to the nearest 10. 865 is between 860 and 870. It is about here on the number line below. It’s halfway so we round up to 870. 860 865 Answer: 870

Round these to what’s in the brackets: 1. 487 [nearest 100] 500 2. 2346 [nearest 1000] 2000 3. 6325 [nearest 100] 6300 4. 6325 [nearest 10] 6330

Rounding to the nearest whole number: On the number line put the whole numbers at either end. Eg. Round 7. 43 to the nearest whole number. 7. 43 7 7. 5 Answer: 7 8

Round these to the nearest whole number: 1. 4. 8 5 2. 13. 4 13 3. 23. 45 23 4. 134. 69 135

Decimal places and significant figures Learning Objective: Can I round a decimal to a given number of decimal places or significant figures? Grade E

What does this mean? We can round numbers to a certain number of decimal places and significant figures. Decimal places refer to how many digits we want after the decimal point. Significant figures refer to how many digits aren’t zero and not sandwiched between nonzeroes.

Decimal places Remember that the decimal point does not move! The number needs to be “cut off” at the given number of decimal places. We then round up or down, depending on what the following number is.

Decimal places example: Round 83. 7649 to 1 decimal place. 8 3. 7 6 4 9 Up until the first decimal place, everything stays the same, so we can write everything up to that point. We may need to round the number in the first decimal place though. The following number is “over halfway” so we round up. Answer: 83. 8

Another decimal places example: Round 83. 7649 to 2 decimal places. 8 3. 7 6 4 9 Up until the second decimal place, everything stays the same, so we can write everything up to that point. We may need to round the number in the second decimal place though. The following number is “under halfway” so we round down. Answer: 83. 76

Round these numbers to the given number of decimal places 1. 2. 365 [1 dp] 2. 4 2. 6. 739 [1 dp] 6. 7 3. 13. 7328 [2 dp] 13. 73 4. 9. 9999 [1 dp] 10. 0

Significant figures All numbers have significant figures. 1 st significant figure 6 th significant figure 7 6 0 2 3. 9 8

Significant figures – careful with decimals Decimals are slightly different. 1 st 5 th significant figure 0. 0 0 2 3 7 0 8 These zeroes aren’t significant This zero is significant

Rounding to significant figures Round 4534 to 2 sf. 2 nd significant figure 4 5 3 7 The 5 is the 2 nd significant figure, so anything before that stays the same. The 3 rd significant figure is less than 5 so we round down. Fill up to the decimal point with zeroes. Answer: 4500

Rounding to significant figures 2 Round 0. 00576 to 1 sf. 1 st significant figure 0 . 0 0 5 7 6 The 5 is the 1 st significant figure, so anything before that stays the same. The 2 nd significant figure is more than 5 so we round up. Answer: 0. 006

Have a go at rounding these: 1. 7238 [1 sf] 7000 2. 926706 [2 sf] 930000 3. 0. 0435 [1 sf] 0. 04 4. 0. 00076498 [3 sf] 0. 000765

Estimation Learning Objective: Can I estimate a difficult calculation? Grade D

Why do we estimate? It is easy to make a mistake when typing calculations into a calculator. We estimate to check that the answer we have sounds or looks sensible. The answer we get when we estimate won’t be exact but close to the right answer.

How do we estimate? You want the calculation to be as simple as possible so that you can do it in your head. Round to 1 significant figure or an easier number. Use numbers that work nicely. You must show your working or you will get no marks!

Two examples Example 1 Estimate 23 × 18 Example 2 Estimate 32. 3 ÷ 5. 8 Round: 20 × 20 Round: 30 ÷ 6 Calculate: 400 Calculate: 5 Answer: 400 Answer: 5 Showing the workings!

Estimate these: 1. 7. 6 × 4. 93 8 × 5 = 40 2. 14. 29 × 4. 41 15 × 4 = 60 3. 48. 13 ÷ 9. 7 50 ÷ 10 = 5 4. 48. 13 ÷ 7. 6 48 ÷ 8 = 6

Estimating difficult calculations Learning Objective: Can I use estimate multi-stage calculations? Grade C

Remember bidmas! Sometimes you may be asked to estimate slightly more difficult calculations. Remember the rules of bidmas when you do these. Use numbers that work nicely!

An example • Show the workings!

Another example • Show the workings!

Estimate these: •

Standard form Learning Objective: Can I read and write numbers in standard form? Grade B

What is standard form? •

Writing large numbers in standard form • 7 0 0 0. How many places did the decimal point move? The decimal point is currently here The decimal point needs to move left until the number is between 1 and 10

Writing small numbers in standard form • The decimal point is currently here 0. 0 0 2 4 The decimal point needs to move left until the number is between 1 and 10 How many places did the decimal point move?

Things to remember The decimal point moves left or right until the number lies between 1 and 10. Count how many places the decimal point has moved – that’s the power of 10. Large numbers – positive power of 10 Small numbers – negative power of 10.

Have a go at writing these in standard form: 1. 3000 2. 230000 3. 0. 0004 4. 0. 00000762

Now try writing these as “normal” numbers: • 800, 000 42, 600, 000 0. 02 0. 000008913

The other type of question you might see: •

Standard form - calculations Learning Objective: Can I perform calculations in standard form? Grade B

How you calculate with standard form: •

On your calculator try these, giving your answers in standard form: •

Without a calculator: •

Two examples of multiplying Example 1 • Example 2 •

Two more examples Example 3 • Example 4 •

Calculate these, giving your answers in standard form: •

Error in measurement Learning Objective: Can I calculate the upper and lower bound of a measurement? Grade B

What do you mean? Every time a measurement is made there is the possibility of an error either way. It all depends to what accuracy the measurement is made. So each measurement has an “upper bound” and a “lower bound”.

What does it look like? If a measurement has been taken, it could actually be half of that measure either side. Each value rounds to the measurement given. A measurement of 5 cm could be anything between 4. 5 cm and 5. 5 cm and round to 5 cm.

Some examples: 50 metres measured to the nearest 10 metres: Half of 10 Upper bound: 55 metres is Lower bound: 45 metres 14 kg measured to the nearest kg: Upper bound: 14. 5 kg Lower bound: 13. 5 kg Half of 1 kg is 0. 5 kg

Find the upper and lower bounds of these measurements: 1. 2300 kg measured to the nearest 100 kg. Upper bound: 2350 kg Lower bound: 2250 kg 2. 60 metres measured to the nearest metre. Upper bound: 60. 5 m Lower bound: 59. 5 m 3. 7. 58 m measured to the nearest centimetre. Upper bound: 7. 585 m Lower bound: 7. 575 m

One last thing… The measurement may have been rounded to a certain number of decimal places or significant figures. Your answer just needs to go “on one further”.

Two examples: Example 1 Example 2 Give the upper and lower bound of 3. 4 which has been rounded to 1 dp. Give the upper and lower bound of 5700 which has been rounded to 2 sf. Upper bound: 3. 45 Lower bound: 3. 35 Upper bound: 5750 Lower bound: 5650

Bounds and calculations Learning Objective: Can I perform calculations by taking errors in measurement into account? Grade A

What you’ll be asked to do All measurements have an upper and lower bound. This means that you can get maximum and minimum values of a calculation. Think carefully about which bound you use!

An example: A floor needs carpeting. The floor measures 4. 2 m by 3. 7 m to the nearest 0. 1 m. What are the upper and lower bounds of the floor’s area? The two upper bounds To find the upper bound: 4. 25 × 3. 75 = 15. 9375 m² To find the lower bound: 4. 15 × 3. 65 = The two lower 15. 1475 m² bounds

Another example - be careful! • Upper bound Lower bound Upper bound

Have a go at this: A bike that is 1. 5 metres long (to the nearest 0. 1 m) needs to go in a shed that is 2 metres long (to the nearest metre). Will the bike definitely fit in the shed? (Explain your answer) No, because the upper bound of the length of the bike is 1. 55 metres and the lower bound of the length of the shed is 1. 5 metres.