Solve problems with or without a calculator Level 4

Interpret a calculator display of 4.5 as £4.50 in context of money
Use a calculator and inverse operations to find missing numbers, including decimals as for example:

6.5 – 9.8 = □

4.8 ÷ □ = 0.96

1/8 of □ = 40

Use inverses to check results, for example,

703/19 = 37 appears to about right because 36 x 20 = 720

What would 0.6 mean on a calculator display if the units were £s, metres, hours, cars?

What is the important information in this problem?

Show me a problem that you would use a calculator to work out the answer. Show me a problem that you wouldn’t use a calculator? How do you decide?
Is it always quicker to use a calculator?
What key words tell you that you need to add, subtract, multiply or divide?
How would you use a calculator to solve this problem?
Choose a number to put into a calculator. Add 472 (or multiply by 26) what single operation will get you back to your starting number?

Will this be the same for different starting numbers? How do you know?

Understand and use an appropriate non-calculator method for Level 5

Show how you could work these out without a calculator:

348 × 27

309 × 44

19 × 423

Explain your choice of method for each calculation.
Find the answer to each of the following, using a non-calculator method.

207 23

976 61

872 55

317 people are going on a school coach trip. Each coach will hold 28 passengers. How many coaches are needed?

611 is the product of two prime numbers. One of the numbers is 13. What is the other one?

Give pupils some examples of multiplications and divisions with mistakes in them. Ask them to identify the mistakes and talk through what is wrong and how they should be corrected.
Ask pupils to carry out multiplications using the grid method and a compact standard method. Ask them to describe the advantages and disadvantages of each method.
How do you go about estimating the answer to a division?

Immediately before Sharon was paid, her bank balance was shown as -£104.38; the minus sign showed that her account was overdrawn. Immediately after she was paid, her balance was £1312.86. How much was she paid?
The temperatures in three towns on January 1^{st} were:

Apton -5°C

Barntown 2°C

Camtown -1°C

Which town was the coldest?

Which town was the warmest?

What was the difference in temperature between the warmest and coldest towns?

The lowest winter temperature in a city in Canada was -15°C. The highest summer temperature was 42°C higher. What was the difference in temperature between the minimum and the maximum temperature?

‘Addition makes numbers bigger.’ When is this statement true and when is it false?
Subtraction makes numbers smaller.’ When is this statement true and when is it false?
The answer is -7. Can you make up some addition/subtraction calculations with the same answer.
The answer on your calculator is -144. What keys could you have pressed to get this answer?
How does a number line help when adding and subtracting positive and negative numbers?
Talk me through how you found the answer to this question.

Apply inverse operations and approximate to check answers Level 5 to problems are of the correct magnitude

Discuss questions such as:

Will the answer to 75 ÷ 0.9 be smaller or larger than 75?

Check by doing the inverse operation, for example:

Do you think your estimate is higher or lower than the real answer?

Explain your answers.

How could you use inverse operations to check that a calculation is correct? Show me some examples.

Calculate percentages and find the outcome of a given Level 6

percentage increase or decrease

Use written methods, e.g.

Using an equivalent fraction: 13% of 48; 13/100 × 48 = 624/100 = 6.24

Using an equivalent decimal: 13% of 48; 0.13 × 48 = 6.24

Using a unitary method: 13% of 48; 1% of 48 = 0.48 so 13% of 48 = 0.48 × 13 = 6.24

Find the outcome of a given percentage increase or decrease. e.g.

an increase of 15% on an original cost of £12 gives a new price of £12 × 1.15 = £13.80,

or 15% of £12 = £1.80 £12 + £1.80 = £13.80

Talk me through how you would increase/decrease £12 by, for example 15%. Can you do it in a different way? How would you find the multiplier for different percentage increases/decreases?
The answer to a percentage increase question is £10. Make up an easy question. Make up a difficult question.

Talk me through the steps you would take to find an estimate for the answer to this calculation?
Would you expect your estimated answer to be greater or less than the exact answer? How can you tell? Can you make up an example for which it would be difficult to decide?

Show me examples of multiplication and division calculations using decimals that approximate to 60.
Why is 6 ÷ 2 a better approximation for 6.59 ÷ 2.47 than 7 ÷ 2?
Why it is useful to be able to estimate the answer to complex calculations?

Use fractions or percentages to solve problems involving Level 8

repeated proportional changes or the calculation of the

original quantity given the result of a proportional change

Solve problems involving, for example compound interest and population growth using multiplicative methods.
Use a spreadsheet to solve problems such as:

How long would it take to double your investment with an interest rate of 4% per annum?

A ball bounces to ¾ of its previous height each bounce. It is dropped from 8m. How many bounces will there be before it bounces to approximately 1m above the ground?
Solve problems in other contexts, for example:

Each side of a square is increased by 10%. By what percentage is the area increased?

The length of a rectangle is increased by 15%. The width is decreased by 5%. By what percentage is the area changed?

Talk me through why this calculation will give the solution to this repeated proportional change problem.
How would the calculation be different if the proportional change was…?
What do you look for in a problem to decide the product that will give the correct answer?
How is compound interest different from simple interest?
Give pupils a set of problems involving repeated proportional changes and a set of calculations. Ask pupils to match the problems to the calculations.