Unit 1: Place Value & Ordering

• Course: Pearson Edexcel GCSE Mathematics (1MA1)
• Specification reference: N1
• Tier: Foundation + Higher
• Grade Range: 1-9
• Status: Complete

Lesson

1. The Place Value System

Place value means that a digit changes value depending on where it sits in a number. In the number $4{,}582{,}731.609$, the digit $4$ is worth $4{,}000{,}000$, the digit $5$ is worth $500{,}000$, the digit $8$ is worth $80{,}000$, the digit $2$ is worth $2{,}000$, the digit $7$ is worth $700$, the digit $3$ is worth $30$, the digit $1$ is worth $1$, the digit $6$ is worth $\frac{6}{10}$, the digit $0$ is worth $0$ hundredths, and the digit $9$ is worth $\frac{9}{1000}$.

The columns to the left of the decimal point are millions, hundred thousands, ten thousands, thousands, hundreds, tens and units. The columns to the right are tenths, hundredths and thousandths. Each column is $10$ times the value of the column immediately to its right. For example, $1$ thousand is $10$ hundreds, $1$ hundred is $10$ tens, $1$ unit is $10$ tenths, and $1$ tenth is $10$ hundredths.

A zero can hold a position so that the other digits stay in their correct columns. In $305$, the $0$ shows that there are no tens, so the $3$ is in the hundreds column and the $5$ is in the units column. If you remove the $0$, you get $35$, where the $3$ is only worth $30$. So $305 \neq 35$. The zero is not decoration; it protects the place value.

The same is true for decimals. In $4.07$, the $0$ shows there are no tenths, so the $7$ is in the hundredths column. If you write $4.7$, the $7$ is now in the tenths column, so the number has changed. This is why $4.07 \neq 4.7$.

To read the value of a digit, name its column and multiply the digit by that column value. In $62{,}418$, the digit $2$ is in the thousands column, so its value is $2{,}000$. In $62{,}418$, the digit $1$ is in the tens column, so its value is $10$. In $9.384$, the digit $8$ is in the hundredths column, so its value is $0.08$. In $0.506$, the digit $5$ is in the tenths column, so its value is $0.5$.

Common Error: Saying the digit $7$ in $4.07$ has value $7$ or $0.7$. The digit is in the hundredths column, so its value is $0.07$.

2. Ordering Positive Integers

To order positive integers, compare from left to right, starting with the most significant digit. The most significant digit is the first digit that can decide the size of the number.

For $8{,}431$ and $7{,}999$, compare the thousands digits first. Since $8 > 7$, we know $8{,}431 > 7{,}999$ without checking the other digits.

Sometimes the first few digits match. Compare $54{,}982$ and $54{,}918$. The ten-thousands digits are both $5$, the thousands digits are both $4$, and the hundreds digits are both $9$. The tens digits are $8$ and $1$. Since $8 > 1$, $54{,}982 > 54{,}918$.

For $103{,}450$ and $103{,}405$, the hundred-thousands, ten-thousands, thousands and hundreds digits match. The tens digits are $5$ and $0$, so $103{,}450 > 103{,}405$.

If the numbers have different numbers of digits and both are positive integers, the one with more digits is larger. For example, $12{,}000 > 9{,}999$ because any five-digit positive integer is greater than any four-digit positive integer. This shortcut works for positive integers, but it does not work for decimals, so decimals need their own method.

Common Error: Comparing $9{,}999$ and $10{,}000$ by the first digit only. The number $10{,}000$ has more digits, so it is larger.

3. Ordering Decimals

To order decimals, align the decimal points first. Then add trailing zeros so the decimals have the same number of decimal places. Trailing zeros after the last decimal digit do not change the value, because $5.3 = 5.30 = 5.300$.

Compare $5.29$ and $5.3$. Write $5.3$ as $5.30$. Now compare $5.29$ and $5.30$ column by column. The units are both $5$. The tenths are $2$ and $3$. Since $2 < 3$, $5.29 < 5.30$, so $5.29 < 5.3$.

This is the classic misconception: $5.29$ is not greater than $5.3$ just because $29 > 3$. The digits $29$ are not being compared with the digit $3$ as whole numbers. The digit $2$ in $5.29$ is in the tenths column, and the digit $3$ in $5.3$ is also in the tenths column. Since $2$ tenths is less than $3$ tenths, $5.29$ is smaller.

The number of decimal places has no direct bearing on size. For example, $0.9 > 0.1234$ because $0.9000 > 0.1234$. Also $3.040 < 3.1$ because $3.040 < 3.100$.

To order $4.7$, $4.07$, $4.607$ and $4.67$, write them as $4.700$, $4.070$, $4.607$ and $4.670$. Then compare: $4.070 < 4.607 < 4.670 < 4.700$. So the original order is $4.07$, $4.607$, $4.67$, $4.7$.

Common Error: Treating $0.456$ as bigger than $0.7$ because $456 > 7$. Write $0.7$ as $0.700$; then $0.456 < 0.700$.

4. Ordering Negative Numbers

Negative numbers are best understood on a number line. Values increase from left to right. This means that a number further left is smaller, even if its digits look larger.

For example, $-8 < -3$ because $-8$ is further left on the number line. In temperature, $-8^\circ\text{C}$ is colder than $-3^\circ\text{C}$. In debt, owing $£8$ is worse than owing $£3$, so $-8$ is smaller than $-3$. In altitude, $-80\text{ m}$ means $80\text{ m}$ below sea level, which is lower than $-30\text{ m}$.

The misconception is thinking $-8 > -3$ because $8 > 3$. That compares the sizes without the negative sign. Once both numbers are negative, the order reverses: the number with the larger magnitude is smaller.

Ordering $-2$, $5$, $-7$, $0$ and $3$ from smallest to largest gives $-7$, $-2$, $0$, $3$, $5$. All negative numbers are less than $0$, and among negative numbers the more negative value is smaller.

Common Error: Writing $-1 < -10$. Actually $-10 < -1$ because $-10$ is further left on the number line.

5. Ordering Mixed Forms

Fractions, decimals and percentages cannot be compared directly unless they are in the same form. The safest method is to convert all values to decimals, then compare.

To convert a percentage to a decimal, divide by $100$. For example, $37\% = 0.37$ and $8\% = 0.08$.

To convert a fraction to a decimal, divide the numerator by the denominator. For example, $\frac{3}{4} = 3 \div 4 = 0.75$, and $\frac{2}{5} = 2 \div 5 = 0.4$.

For recurring decimals, keep enough digits to compare safely. For example, $\frac{1}{3} = 0.\overline{3}$, which means $0.3333\ldots$. Since $34\% = 0.34$, we have $\frac{1}{3} < 34\%$ because $0.3333\ldots < 0.3400\ldots$.

Compare $\frac{5}{8}$, $62\%$, $0.61$ and $0.625$. Convert: $\frac{5}{8} = 0.625$, $62\% = 0.62$, $0.61 = 0.61$, and $0.625 = 0.625$. From smallest to largest: $0.61$, $62\%$, $\frac{5}{8}$ and $0.625$ tied. If the question asks for a strict order, equal values must be shown as equal.

Negative mixed forms need extra care. For example, $-\frac{1}{2} = -0.5$ and $-45\% = -0.45$. Since $-0.5 < -0.45$, we have $-\frac{1}{2} < -45\%$.

Common Error: Comparing $\frac{3}{5}$ and $58\%$ by saying $3 < 58$. Convert first: $\frac{3}{5} = 0.6$ and $58\% = 0.58$, so $\frac{3}{5} > 58\%$.

6. Inequality Symbols

The symbol $<$ means "is less than". The symbol $>$ means "is greater than". The symbol $\leq$ means "is less than or equal to". The symbol $\geq$ means "is greater than or equal to". The symbol $=$ means "is equal to". The symbol $\neq$ means "is not equal to".

The symbol opens toward the larger value. For example, $3 < 9$ because the open side points toward $9$. Also $12 > 5$ because the open side points toward $12$.

Use $\leq$ or $\geq$ when equality is allowed. For example, "$x$ is at most $4$" means $x \leq 4$. The value $x = 4$ is allowed. "$x$ is less than $4$" means $x < 4$. The value $x = 4$ is not allowed.

Use $\neq$ when two values are not equal. For example, $0.4 \neq 0.04$ because $0.4 = 0.40$, and $0.40$ is not the same as $0.04$.

To choose the correct symbol, compare the two values first, then place the symbol so it opens toward the larger value. Between $-6$ and $-2$, the larger value is $-2$, so $-6 < -2$.

Common Error: Reading $a < b$ as "the arrow points to $a$". The useful memory aid is that the open mouth faces the larger value.

7. Integer Solutions to Inequality Ranges

An integer is a whole number, including negative whole numbers and $0$. So $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$ are integers. Numbers like $1.5$, $-0.2$ and $\frac{3}{4}$ are not integers.

A combined inequality such as $-3 \leq n < 1.8$ means that $n$ must satisfy both conditions at the same time: $n$ is at least $-3$, and $n$ is less than $1.8$.

List the integers in the range carefully. The lower bound is $-3 \leq n$, so $-3$ is included. Then $-2$, $-1$, $0$ and $1$ are included. The next integer is $2$, but $2$ is not less than $1.8$, so stop. The integer solutions are $n = -3, -2, -1, 0, 1$.

Strict bounds use $<$ or $>$ and do not include the endpoint. Non-strict bounds use $\leq$ or $\geq$ and do include the endpoint. For $2 < n \leq 6$, the integer $2$ is not included, but $6$ is included, so the solutions are $n = 3, 4, 5, 6$.

For $-4.2 < n \leq 0$, the smallest integer greater than $-4.2$ is $-4$, and $0$ is included, so the solutions are $n = -4, -3, -2, -1, 0$.

Common Error: Including the endpoint in a strict inequality. In $n < 5$, the value $n = 5$ is not allowed.

Drills

1. In $73{,}406$, what is the value of the digit $3$? [$3{,}000$]
2. Identify the value of the digit $8$ in $5.286$. [$0.08$]
3. Write $60{,}005$ in words. [$\text{sixty thousand and five}$]
4. Place $47{,}903$, $47{,}309$, $48{,}001$, $47{,}930$ in ascending order. [$47{,}309$, $47{,}903$, $47{,}930$, $48{,}001$]
5. Arrange $0.7$, $0.07$, $0.607$, $0.67$ from smallest to largest. [$0.07$, $0.607$, $0.67$, $0.7$]
6. Sort $3.5$, $3.05$, $3.505$, $3.55$ in descending order. [$3.55$, $3.505$, $3.5$, $3.05$]
7. Complete the comparison: $5.29$ ___ $5.3$. [$<$]
8. Order $-4$, $2$, $-9$, $0$, $7$ from smallest to largest. [$-9$, $-4$, $0$, $2$, $7$]
9. Insert the correct symbol: $-8$ ___ $-3$. [$<$]
10. Convert $37\%$ to a decimal. [$0.37$]
11. Convert $\frac{7}{8}$ to a decimal. [$0.875$]
12. Put $\frac{3}{5}$, $58\%$, $0.62$ in ascending order. [$58\%$, $\frac{3}{5}$, $0.62$]
13. Compare $\frac{1}{3}$ and $34\%$ using $<$, $>$ or $=$. [$\frac{1}{3} < 34\%$]
14. Fill in the symbol: $0.400$ ___ $0.04$. [$>$]
15. Write "$x$ is at most $12$" as an inequality. [$x \leq 12$]
16. List the integers satisfying $-3 \leq n < 2$. [$-3$, $-2$, $-1$, $0$, $1$]
17. Find all integer $n$ such that $1.4 < n \leq 6$. [$2$, $3$, $4$, $5$, $6$]
18. State the integer solutions of $-5 < n < -1$. [$-4$, $-3$, $-2$]
19. Put $-0.6$, $-\frac{1}{2}$, $-55\%$ in ascending order. [$-0.6$, $-55\%$, $-\frac{1}{2}$]
20. Find the largest integer satisfying $-7 \leq n < 4.3$. [$4$]

Worked Examples

Example 1 🔵 Foundation | Grade 3-4

Question: Order $2.4$, $2.04$, $2.404$ and $2.44$ from smallest to largest.

Step 1 - Align the decimal points and pad with zeros: $2.4 = 2.400,\qquad 2.04 = 2.040,\qquad 2.404 = 2.404,\qquad 2.44 = 2.440.$

Step 2 - Compare column by column: All have units digit $2$. Compare tenths: $2.040$ has tenths digit $0$, while the others have tenths digit $4$, so $2.040$ is smallest.

Step 3 - Compare the remaining decimals: Among $2.400$, $2.404$ and $2.440$, the hundredths digits are $0$, $0$ and $4$, so $2.440$ is largest. Between $2.400$ and $2.404$, the thousandths digits are $0$ and $4$, so $2.400 < 2.404$.

$\boxed{2.04,\ 2.4,\ 2.404,\ 2.44}$

Verification: Choose approximate positions: $2.04$ is just above $2$, $2.4 = 2.400$, $2.404$ is slightly above $2.400$, and $2.44$ is above $2.404$.

⚠ Key Error: Saying $2.404$ is largest because it has the most decimal places. Extra decimal places do not automatically make a number larger.

Example 2 🟡 Crossover | Grade 5-6

Question: Order $-\frac{1}{2}$, $-0.6$, $55\%$ and $0.48$ from smallest to largest.

Step 1 - Convert all values to decimals: $-\frac{1}{2} = -0.5,\qquad -0.6 = -0.6,\qquad 55\% = 0.55,\qquad 0.48 = 0.48.$

Step 2 - Order the negative values first: Since $-0.6 < -0.5$, we have: $-0.6 < -\frac{1}{2}.$

Step 3 - Order the positive values: Since $0.48 < 0.55$, we have: $0.48 < 55\%.$

Step 4 - Combine the order: $\boxed{-0.6,\ -\frac{1}{2},\ 0.48,\ 55\%}$

Verification: On a number line, the two negative values lie left of $0$, and $-0.6$ lies left of $-0.5$. The positive values lie right of $0$, and $0.48$ lies left of $0.55$.

⚠ Key Error: Treating $-\frac{1}{2}$ as smaller than $-0.6$ because $\frac{1}{2}$ is smaller than $0.6$. For negatives, the order reverses.

Example 3 🔴 Higher | Grade 6-7

Question: True or false? $-\frac{3}{8} < -0.37$. Justify your answer.

Step 1 - Convert the fraction to a decimal: $\frac{3}{8} = 3 \div 8 = 0.375,$ so $-\frac{3}{8} = -0.375.$

Step 2 - Compare the negative decimals: $-0.375 < -0.370.$

Step 3 - Match the statement: Since $-0.37 = -0.370$, the statement becomes: $-0.375 < -0.370.$

$\boxed{\text{True}}$

Verification: $-0.375$ is further left than $-0.370$ on the number line, so it is smaller.

⚠ Key Error: Comparing $0.375$ and $0.37$ correctly, but forgetting that the negative signs matter. With negative numbers, the value with the larger positive magnitude is the smaller number.

Example 4 ⭐ Stretch | Grade 8-9

Question: Find all integers $n$ satisfying both $-4 \leq n < 6$ and $2n - 3 > 1$.

Step 1 - Solve the second inequality: $2n - 3 > 1.$ Add $3$ to both sides: $2n > 4.$ Divide by $2$: $n > 2.$

Step 2 - Combine with the first condition: The first condition is: $-4 \leq n < 6.$ Together with $n > 2$, this becomes: $2 < n < 6.$

Step 3 - List the integer values: The integers greater than $2$ and less than $6$ are: $n = 3,\ 4,\ 5.$

$\boxed{n = 3,\ 4,\ 5}$

Verification: Substitute each value into $2n - 3 > 1$: for $n = 3$, $3 > 1$; for $n = 4$, $5 > 1$; for $n = 5$, $7 > 1$. All also satisfy $-4 \leq n < 6$.

⚠ Key Error: Including $n = 2$. The inequality is $n > 2$, not $n \geq 2$.

MCQ Bank

1. Foundation | Grade 1 | Spec: N1

What is the value of the digit $6$ in $46{,}209$?

• A) $6$
• B) $60$
• C) $600$
• D) $6{,}000$ ✓

Answer: D) $6{,}000$. The digit $6$ is in the thousands column. Distractor A: reads the digit only, not its place value. Distractor B: uses the tens column. Distractor C: uses the hundreds column.

1. Foundation | Grade 1-2 | Spec: N1

Which number is largest?

• A) $8{,}901$
• B) $9{,}001$ ✓
• C) $8{,}999$
• D) $8{,}910$

Answer: B) $9{,}001$. Its thousands digit is $9$, while every other option has thousands digit $8$. Distractor A: starts with $8$, so it is less than any $9$ thousand number. Distractor C: close to $9{,}000$ but still smaller than $9{,}001$. Distractor D: misreads $910$ as larger than $901$ after the thousands digit.

1. Foundation | Grade 2 | Spec: N1

Select the correct ascending order for $0.5$, $0.05$, $0.505$ and $0.55$.

• A) $0.5,\ 0.05,\ 0.505,\ 0.55$
• B) $0.05,\ 0.5,\ 0.505,\ 0.55$ ✓
• C) $0.05,\ 0.55,\ 0.505,\ 0.5$
• D) $0.505,\ 0.55,\ 0.5,\ 0.05$

Answer: B) $0.05,\ 0.5,\ 0.505,\ 0.55$. Writing them as $0.050,\ 0.500,\ 0.505,\ 0.550$ makes the order clear. Distractor A: keeps the original order without comparing columns. Distractor C: places $0.55$ before $0.505$ because it has fewer decimal places. Distractor D: treats more decimal places as larger.

1. Foundation | Grade 2 | Spec: N1

Complete the statement $4.09$ ___ $4.1$.

• A) $>$
• B) $<$ ✓
• C) $=$
• D) $\neq$ only

Answer: B) $<$. Write $4.1$ as $4.10$; then $4.09 < 4.10$. Distractor A: compares $09$ and $1$ as whole numbers. Distractor C: ignores the hundredths column. Distractor D: true but not the best comparison symbol.

1. Foundation | Grade 2-3 | Spec: N1

Which list shows the numbers from smallest to largest?

• A) $-2,\ -7,\ 0,\ 5$
• B) $0,\ -2,\ -7,\ 5$
• C) $-7,\ -2,\ 0,\ 5$ ✓
• D) $5,\ 0,\ -2,\ -7$

Answer: C) $-7,\ -2,\ 0,\ 5$. On a number line, values increase from left to right. Distractor A: assumes $-2$ is smaller than $-7$. Distractor B: puts $0$ before negative numbers. Distractor D: orders largest to smallest.

1. Crossover | Grade 3 | Spec: N1

Which statement is true?

• A) $-9 < -4$ ✓
• B) $-9 > -4$
• C) $-9 = -4$
• D) $9 < 4$

Answer: A) $-9 < -4$. Of the two negative values, $-9$ is further left on the number line. Distractor B: compares $9$ and $4$ without the negative signs. Distractor C: ignores different values. Distractor D: changes the question to positive numbers and reverses the truth.

1. Crossover | Grade 3-4 | Spec: N1

Convert $18\%$ to a decimal.

• A) $18$
• B) $1.8$
• C) $0.18$ ✓
• D) $0.018$

Answer: C) $0.18$. A percentage is divided by $100$ to convert it to a decimal. Distractor A: leaves the percentage number unchanged. Distractor B: divides by $10$ instead of $100$. Distractor D: divides by $1000$.

1. Crossover | Grade 4 | Spec: N1

Which value is greatest?

• A) $\frac{2}{5}$
• B) $38\%$
• C) $0.41$ ✓
• D) $0.039$

Answer: C) $0.41$. Convert the other values: $\frac{2}{5}=0.4$ and $38\%=0.38$; $0.41$ is greatest. Distractor A: $\frac{2}{5} = 0.4$, which is just less than $0.41$. Distractor B: $38\% = 0.38$. Distractor D: place value error; $0.039$ is much smaller than $0.39$.

1. Crossover | Grade 4 | Spec: N1

What does $x \leq 7$ mean?

• A) $x$ is less than $7$ but not equal to $7$
• B) $x$ is less than or equal to $7$ ✓
• C) $x$ is greater than $7$
• D) $x$ is not equal to $7$

Answer: B) $x$ is less than or equal to $7$. The line under the symbol means equality is included. Distractor A: describes $x < 7$. Distractor C: reverses the inequality. Distractor D: describes $x \neq 7$.

1. Crossover | Grade 4-5 | Spec: N1

List the integer solutions of $-2 \leq n < 3$.

• A) $-2,\ -1,\ 0,\ 1,\ 2,\ 3$
• B) $-2,\ -1,\ 0,\ 1,\ 2$ ✓
• C) $-1,\ 0,\ 1,\ 2$
• D) $-2,\ -1,\ 1,\ 2$

Answer: B) $-2,\ -1,\ 0,\ 1,\ 2$. Include $-2$ because of $\leq$, but exclude $3$ because the upper bound is strict. Distractor A: includes $3$ even though the upper bound is strict. Distractor C: excludes $-2$ even though the lower bound is inclusive. Distractor D: forgets $0$.

1. Higher | Grade 5-6 | Spec: N1

Which comparison is correct?

• A) $-\frac{1}{3} < -0.4$
• B) $-\frac{1}{3} > -0.4$ ✓
• C) $-\frac{1}{3} = -0.4$
• D) $\frac{1}{3} < -0.4$

Answer: B) $-\frac{1}{3} > -0.4$. Since $-\frac{1}{3}=-0.333\ldots$, it lies to the right of $-0.4$. Distractor A: reverses the order after converting $\frac{1}{3}$ to $0.333\ldots$. Distractor C: rounds too roughly. Distractor D: compares a positive number with a negative number incorrectly.

1. Higher | Grade 6 | Spec: N1

All integers $n$ satisfy $-5 < n \leq 2.2$. Which set is correct?

• A) $-5,\ -4,\ -3,\ -2,\ -1,\ 0,\ 1,\ 2$
• B) $-4,\ -3,\ -2,\ -1,\ 0,\ 1,\ 2,\ 3$
• C) $-4,\ -3,\ -2,\ -1,\ 0,\ 1,\ 2$ ✓
• D) $-5,\ -4,\ -3,\ -2,\ -1,\ 0,\ 1$

Answer: C) $-4,\ -3,\ -2,\ -1,\ 0,\ 1,\ 2$. The strict lower bound excludes $-5$, and $3$ is too large. Distractor A: includes $-5$ even though the lower bound is strict. Distractor B: includes $3$, which is greater than $2.2$. Distractor D: both includes $-5$ and misses $2$.

Long-Answer Questions

Q1 — Foundation | Grade 3-4 | Spec: N1

Question: The numbers $6.2$, $6.02$, $6.202$ and $6.22$ are to be arranged in ascending order. Show your working. [4]

Mark scheme: Align decimals using trailing zeros [M1]; identify $6.020$ as smallest [A1]; correctly compare $6.200$, $6.202$, $6.220$ [M1]; final order $6.02$, $6.2$, $6.202$, $6.22$ [A1].

Model answer: Write $6.2 = 6.200,\quad 6.02 = 6.020,\quad 6.202 = 6.202,\quad 6.22 = 6.220.$ Then compare column by column: $6.020 < 6.200 < 6.202 < 6.220.$ So the ascending order is $\boxed{6.02,\ 6.2,\ 6.202,\ 6.22}.$ Check: $6.02$ has $0$ tenths, while the other three have $2$ tenths; $6.202$ is only $0.002$ more than $6.200$, and $6.220$ is larger than both.

Q2 — Crossover | Grade 5-6 | Spec: N1

Question: Order $-\frac{3}{4}$, $-70\%$, $-0.72$ and $-0.7$ from smallest to largest. [5]

Mark scheme: Convert $\frac{3}{4}$ to $0.75$ [M1]; convert $70\%$ to $0.70$ [M1]; correctly handle negative order [M1]; final order [A1]; explicit check or explanation using number line/magnitude [B1].

Model answer: Convert all values to decimals: $-\frac{3}{4} = -0.75,\qquad -70\% = -0.70,\qquad -0.72 = -0.72,\qquad -0.7 = -0.70.$ For negative numbers, the value with the larger positive magnitude is smaller: $-0.75 < -0.72 < -0.70.$ Since $-70\% = -0.7$, the order is: $\boxed{-\frac{3}{4},\ -0.72,\ -70\% = -0.7}.$ Check: on a number line, $-0.75$ lies furthest left, then $-0.72$, then $-0.70$.

Q3 — Higher | Grade 6-7 | Spec: N1

Question: A student says, "$0.304 > 0.31$ because $304 > 31$." Explain the error and write the correct comparison. [4]

Mark scheme: States that decimals must be aligned by place value [M1]; writes $0.31$ as $0.310$ [M1]; gives correct comparison $0.304 < 0.310$ [A1]; explains why comparing $304$ and $31$ is invalid [B1].

Model answer: The error is comparing the digits after the decimal point as if they were whole numbers. Align the decimals: $0.304 \quad\text{and}\quad 0.31 = 0.310.$ Now compare the digits column by column. Both have $3$ tenths. In the hundredths column, $0.304$ has $0$ hundredths and $0.310$ has $1$ hundredth, so: $0.304 < 0.310.$ Therefore: $\boxed{0.304 < 0.31}.$ Check: $0.31 - 0.304 = 0.006$, which is positive, so $0.31$ is larger.

Q4 — Stretch | Grade 8-9 | Spec: N1

Question: Find all integers $n$ such that $-6 \leq n < 5$ and $-2n + 1 \leq 7$. [5]

Mark scheme: Solves $-2n + 1 \leq 7$ to get $n \geq -3$ with sign flip [M1 A1]; combines with $-6 \leq n < 5$ [M1]; lists all integers $-3$ to $4$ [A1]; verifies endpoints or explains strict/non-strict bounds [B1].

Model answer: Solve the second inequality: $-2n + 1 \leq 7.$ Subtract $1$: $-2n \leq 6.$ Divide by $-2$ and reverse the inequality: $n \geq -3.$ Combine this with: $-6 \leq n < 5.$ Both conditions together give: $-3 \leq n < 5.$ So the integer solutions are: $\boxed{-3,\ -2,\ -1,\ 0,\ 1,\ 2,\ 3,\ 4}.$ Check: $n = -3$ gives $-2(-3)+1 = 7$, so it is included. The upper bound is strict at $5$, so $5$ is not included.

Gold Standard Exam Mastery: Place Value & Ordering

Specification mapping

GCSE Mathematics: number, algebra, ratio, geometry, probability, statistics and problem solving across Foundation and Higher tiers.

Exam-board lens for this lesson: All specs. Use this chapter to revise the content, but also to practise how examiners reward marks in real papers.

Assessment objective map

• AO1: use and apply standard techniques accurately.
• AO2: reason, interpret and communicate mathematically.
• AO3: solve problems in familiar and unfamiliar contexts.
• Tier awareness: Foundation rewards secure method; Higher rewards algebraic generalisation, proof and efficient strategy.

Command words to practise

calculate, show, prove, solve, estimate, explain

What examiners reward

• Write the method line before the answer, especially when a calculator shortcut hides the reasoning.
• Use exact values until the final rounding step unless the question asks for an estimate.
• For proof, start from one side or from a general form; never verify with examples only.

Common mistakes to avoid

• Premature rounding in multi-step calculations.
• Using a calculator method in a non-calculator question.
• Dropping units, inequality signs or negative signs in algebraic work.

Answer quality ladder

Grade 4 / basic pass move: Uses a correct standard method with mostly accurate arithmetic.

Grade 7 / strong answer move: Chooses an efficient method, communicates steps clearly and checks reasonableness.

Grade 9 move: Generalises the structure of the problem, proves or models it algebraically and avoids unnecessary numerical trial.

Exam-style practice prompts

• Solve a non-calculator version of this chapter's core skill and show each step.
• Create a calculator method for Place Value & Ordering, then explain what each display value means.
• Write a problem-solving question that combines Place Value & Ordering with algebra or ratio.

Mark scheme guidance

For short answers, make the point precise before adding explanation. For extended answers, build a chain of reasoning, apply it to the named context, then make a judgement only if the command word requires one. A high-mark answer is not just longer; it is more selective, better evidenced and more explicit about why one factor matters more than another.

Topic-specific teaching upgrade

• Mathematics improvement comes from visible method. Students should show the algebraic structure, not just the final numerical result.
• Harder questions usually combine topics: algebra with geometry, calculus with modelling, vectors with proof, or probability with interpretation.
• A proof or modelling answer needs assumptions, definitions and conditions. Checking the domain, sign, determinant, convergence or unit can be the difference between a good method and a complete solution.

Worked example or model move

• Worked-solution routine: identify the method, write the starting equation or theorem, transform one line at a time, check restrictions, then verify the answer.
• Calculator routine: know what the calculator has produced, then write the mathematical interpretation in exact or rounded form as required.

Examiner-method focus for this lesson

• Do not round mid-solution unless explicitly told.
• In 'show that' questions, do not assume the result; work towards it from a valid starting point.
• For modelling, state assumptions and comment on whether the result is realistic.

Original long-answer practice

• Write a full worked solution for Place Value & Ordering, with every algebraic transformation justified.
• Create a harder problem that combines Place Value & Ordering with proof, graph interpretation or modelling assumptions.

Board-aware exam routine

1. Identify the exact method family: algebraic, graphical, numerical, statistical or mechanical.
2. Write the governing equation, theorem, identity or model before substitution.
3. Keep exact working visible and check units, domain, sign and assumptions.
4. Verify the final answer by substitution, dimensional sense, graph behaviour or reasonableness.

Model answer builder

• Opening move: name the exact concept, method, text, process, model or argument being tested.
• Evidence move: add data, quotation, calculation, example, case detail, code trace, source detail or diagram feature.
• Development move: explain the link in a full chain, not a loose comment.
• Precision move: use exam vocabulary from this lesson and avoid vague filler.
• Judgement move: only where the command word requires it, decide which factor, method, interpretation or option is strongest and why.