Number and Place Value Revision Notes
Subject: Mathematics | Level: GCSE | Exam Board: OCR
Master the core of your OCR GCSE Maths exam with this guide to Number and Place Value. We break down everything from rounding and standard form to index laws, giving you the examiner's perspective on how to secure every last mark. This is not just about calculation; it is about developing the mathematical fluency that underpins the entire specification.
Revision Notes & Key Concepts
Revision Podcast Transcript
Hello and welcome to Maths Mastery — the podcast that gets you exam-ready, one topic at a time. I'm your tutor, and today we're diving into one of the most fundamental topics in your OCR GCSE Mathematics specification: Number and Place Value. This is Topic 1.1, and whether you're sitting Foundation or Higher tier, this topic will show up in your exam — often in ways you might not expect. So grab a pen, get comfortable, and let's get into it. Now, I know what some of you might be thinking: 'Place value? I learned that in primary school.' And you're right — the basics are familiar. But here's the thing: OCR examiners use this topic as a gateway to testing your mathematical fluency across the entire paper. Questions on estimation, standard form, and indices can appear in calculator and non-calculator papers alike, and they carry marks that are genuinely within your reach — if you know exactly what the examiner is looking for. So let's make sure you do. SECTION TWO: CORE CONCEPTS Let's start with the absolute foundation: place value itself. Every digit in a number has a value determined by its position. In the number 5,382.47, the digit 5 is worth five thousand, the 3 is worth three hundred, the 8 is worth eighty, the 2 is worth two units, the 4 is worth four tenths, and the 7 is worth seven hundredths. This seems simple, but it becomes critically important when you're ordering decimals — which is a classic source of lost marks. Here's the trap: candidates often look at 0.4 and 0.35 and think 0.35 is bigger because 35 is bigger than 4. But add a placeholder zero: 0.40 versus 0.35. Now it's obvious that 0.40 is larger. The rule is: when comparing decimals, always make them the same length by adding zeros. This is a one-mark fix that candidates regularly miss. The same logic applies to negative numbers. Is minus 0.5 smaller than minus 0.8? Yes — because on a number line, minus 0.8 is further to the left, meaning it's the smaller value. Think of it like temperature: minus 8 degrees is colder — and therefore lower — than minus 5 degrees. The bigger the digit after the negative sign, the smaller the actual value. Now let's talk about rounding, because this is where method marks live. There are two types of rounding you need to master: rounding to decimal places, and rounding to significant figures. For decimal places, you look at the digit immediately after the position you're rounding to. If it's 5 or more, round up. If it's less than 5, round down. So 3.746 rounded to two decimal places: look at the third decimal place, which is 6. Six is 5 or more, so the second decimal place rounds up from 4 to 5. Answer: 3.75. For significant figures, the rule is the same, but you count from the first non-zero digit. This catches people out with small decimals. In 0.00483, the first significant figure is 4 — not the zeros. Those leading zeros are just placeholders. So 0.00483 to two significant figures is 0.0048, because the deciding digit is 3, which is less than 5, so we round down. Now — and this is crucial — estimation questions require you to round to one significant figure before you calculate. Not after. Before. OCR's mark scheme awards a method mark — that's an M1 — specifically for showing the rounded values. If you do the exact calculation and then round the answer, you will not get that method mark, even if your final answer happens to be correct. Write down your rounded values. Always. Next up: standard form. Standard form is a way of writing very large or very small numbers using powers of ten. The format is always A times 10 to the power of n, where A must be at least 1 but strictly less than 10. That constraint — one less than or equal to A, which is strictly less than ten — is the most commonly tested marking point. OCR will award a B1 mark specifically for a correct standard form conversion where the coefficient satisfies that condition. So 4,700 in standard form is 4.7 times 10 to the power of 3. The coefficient 4.7 is between 1 and 10. The power of 3 tells us we moved the decimal point three places to the right to get back to 4700. For small numbers: 0.0047 is 4.7 times 10 to the power of negative 3. The negative power tells us the number is small — less than 1. On non-calculator papers, when you multiply or divide numbers in standard form, handle the coefficients and the powers of ten separately. For example: 3 times 10 to the 4, multiplied by 2 times 10 to the 3. Multiply the coefficients: 3 times 2 is 6. Add the powers: 4 plus 3 is 7. So the answer is 6 times 10 to the 7. But always check: is your coefficient still between 1 and 10? If you got something like 15 times 10 to the 6, you'd need to rewrite it as 1.5 times 10 to the 7. Now let's move to index notation and the index laws. An index — also called a power or exponent — tells you how many times to multiply a number by itself. There are three key index laws. First: when multiplying powers with the same base, add the indices. Second: when dividing powers with the same base, subtract the indices. Third: when raising a power to a power, multiply the indices. Two special cases that are frequently tested. First: any number to the power of zero equals 1. Always. Second: negative indices mean reciprocals. Five to the power of negative 2 is 1 over 5 squared, which is 1 over 25. The negative index does not make the answer negative. It means one over the positive power. For Higher tier only: fractional indices. A to the power of one over n means the nth root of a. So 8 to the power of one third is the cube root of 8, which is 2. And 27 to the power of two thirds: cube root of 27 is 3, then 3 squared is 9. SECTION THREE: EXAM TIPS AND COMMON MISTAKES Tip one: show your rounded values in estimation questions. Write 'approximately 50', write 'approximately 0.2' — those words and those rounded numbers are what earn you the method mark. Tip two: when ordering decimals, add placeholder zeros. Make every number the same length before you compare. Tip three: for standard form, always check your coefficient. After any calculation, verify that your A value satisfies 1 less than or equal to A, strictly less than 10. Tip four: never confuse a negative index with a negative number. Five to the minus 2 is positive one twenty-fifth. Tip five: in BIDMAS questions, work carefully through the order of operations. In the expression 3 plus 4 times 2, the answer is 11, not 14. Multiplication comes before addition. SECTION FOUR: QUICK-FIRE RECALL QUIZ Okay, it's quiz time! I'll ask the question, give you three seconds to think, then give the answer. Ready? Question one: What is 0.00567 written to two significant figures? ... The answer is 0.0057. Question two: Write 35,000 in standard form. ... The answer is 3.5 times 10 to the power of 4. Question three: What is 4 to the power of negative 2? ... The answer is 1 over 16. Question four: Estimate the answer to 497 divided by 0.048. Round each to one significant figure first. ... 500 divided by 0.05 equals 10,000. Question five: What is 64 to the power of two thirds? ... Cube root of 64 is 4, then 4 squared is 16. The answer is 16. Question six: True or false — when multiplying numbers in standard form, you multiply the powers of 10. ... False! You add the powers of 10. You multiply the coefficients. SECTION FIVE: SUMMARY AND SIGN-OFF Let's bring it all together. Number one: place value underpins everything. When comparing or ordering decimals, add placeholder zeros. Number two: for negative numbers, the larger the digit, the smaller the value. Number three: in estimation, always round to one significant figure before calculating, and write down your rounded values. Number four: standard form requires the coefficient A to satisfy 1 less than or equal to A, strictly less than 10. Number five: the three index laws — add powers when multiplying, subtract when dividing, multiply when raising a power to a power. Number six: negative indices mean reciprocals, not negative numbers. Number seven: for Higher tier, fractional indices link to roots. That's everything for today's episode of Maths Mastery. This topic is genuinely one of the most mark-rich areas of the specification — the concepts are accessible, the rules are learnable, and with a bit of focused practice, you can be picking up marks here consistently. Good luck with your revision. You've got this. See you in the next episode.
Key Terms & Definitions
- Integer
- A whole number; it can be positive, negative, or zero. Examples include -5, 0, 1, 27.
- Decimal
- A number that uses a decimal point to show the part of the number that is less than one.
- Place Value
- The value of a digit depending on its position in the number (e.g., units, tens, hundreds, or tenths, hundredths).
- Significant Figure
- The first non-zero digit in a number, and all the digits that follow it. Used to denote the accuracy of a number.
- Standard Form
- A way of writing numbers as A x 10^n, where 1 <= A < 10 and n is an integer.
- Index (plural: Indices)
- A number that tells you how many times to multiply a base number by itself. Also known as a power or exponent.
- Reciprocal
- The reciprocal of a number is 1 divided by that number. The reciprocal of x is 1/x.
Worked Examples
Worked Example
Question: **Calculate** the value of (2.4 x 10^7) / (8 x 10^3). Give your answer in standard form.
Solution: Step 1: Divide the coefficients. 2.4 / 8 = 0.3 Step 2: Divide the powers of 10 by subtracting the indices. 10^7 / 10^3 = 10^(7-3) = 10^4 Step 3: Combine the results. 0.3 x 10^4 Step 4: Adjust the answer to be in correct standard form. The coefficient must be between 1 and 10. 0.3 = 3 x 10^(-1) So, (3 x 10^(-1)) x 10^4 = 3 x 10^(-1+4) = 3 x 10^3 Final answer: 3 x 10^3
Worked Example
Question: **Estimate** the answer to the following calculation: (sqrt(98.7) + 20.4^2) / 0.49
Solution: Step 1: Round each number in the calculation to 1 significant figure. sqrt(98.7) is approximately sqrt(100) 20.4^2 is approximately 20^2 0.49 is approximately 0.5 Step 2: Substitute the rounded values into the expression. (sqrt(100) + 20^2) / 0.5 Step 3: Perform the calculation in the correct order (BIDMAS). sqrt(100) = 10 20^2 = 400 (10 + 400) / 0.5 = 410 / 0.5 Step 4: Complete the final division. Dividing by 0.5 is the same as multiplying by 2. 410 x 2 = 820 Final answer: 820
Worked Example
Question: **Evaluate** 27^(-2/3), giving your answer as a fraction.
Solution: Step 1: Deal with the negative index first. This means finding the reciprocal. 27^(-2/3) = 1 / 27^(2/3) Step 2: Deal with the fractional index. The denominator (3) means cube root, and the numerator (2) means square. 27^(2/3) = (cube root of 27)^2 Step 3: Calculate the root first as it makes the numbers smaller and easier to work with. Cube root of 27 = 3 Step 4: Calculate the power. 3^2 = 9 Step 5: Combine with the reciprocal from Step 1. 1 / 9 Final answer: 1/9
Practice Questions
Question: Place the following numbers in order of size, starting with the smallest: 0.305, 0.35, 0.053, 0.3, 3.05
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Question: **Estimate** the value of (489 x 0.52) / 9.8. You must show your rounded values.
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Question: Work out (5 x 10^4) + (3 x 10^3). Give your answer in standard form.
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Question: Given that x = 2.5 x 10^4 and y = 5 x 10^5, find the value of x / y, giving your answer in standard form.
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Question: Find the value of 125^(-2/3).
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