2018 Key Stage 2 Mathematics Paper 1: Arithmetic

💡 Why we do this:

When adding numbers, it is safer to write them in columns, aligning the units, tens, and hundreds. We put the larger number on top.

🗣 What we did:

1. Units: \( 3 + 9 = 12 \). Write down 2, carry 1.

2. Tens: \( 7 + 3 + 1 \text{ (carried)} = 11 \). Write down 1, carry 1.

3. Hundreds: \( 6 + 1 \text{ (carried)} = 7 \).

💡 Why we do this:

When subtracting fractions, the bottom numbers (denominators) must be the same. Here, both are 11, so we can simply subtract the top numbers (numerators).

🗣 What this tells us:

The denominator stays as 11. We just subtract 4 from 9 to get 5.

💡 Why we do this:

Multiplying by 2 is the same as doubling the number. We can do this mentally or by partitioning.

💡 Why we do this:

Dividing any number by 1 leaves the number unchanged. Think of it as sharing 838 sweets among 1 person. That person gets all 838 sweets.

💡 Why we do this:

We need to find how many 11s fit into 99. We can count in 11s or use our times table knowledge.

💡 Why we do this:

When multiplying three numbers, we can do it in any order. It is often easiest to multiply the first two, and then multiply the answer by the third.

💡 Why we do this:

We must align the digits correctly in columns (units under units, tens under tens). The larger number goes on top.

🗣 What we did:

1. Units: \( 4 – 2 = 2 \).

2. Tens: \( 6 – 0 = 6 \).

3. Hundreds: \( 0 – 5 \) we cannot do. Borrow from the 7 (thousands). The 7 becomes 6, and the 0 becomes 10. \( 10 – 5 = 5 \).

4. Thousands: \( 6 – 0 = 6 \).

💡 Why we do this:

We must calculate the squared number first. \( 6^2 \) means “6 squared”, which is \( 6 \times 6 \).

💡 Why we do this:

When adding decimals, the most important rule is to line up the decimal points. We can add a zero to 24.7 to make it 24.70 so both numbers have the same number of decimal places.

🗣 What we did:

1. Hundredths: \( 8 + 0 = 8 \).

2. Tenths: \( 3 + 7 = 10 \). Write 0, carry 1.

3. Units: \( 6 + 4 + 1 \text{ (carried)} = 11 \). Write 1, carry 1.

4. Tens: \( 5 + 2 + 1 \text{ (carried)} = 8 \).

💡 Why we do this:

We are starting with a mystery number, taking away 10, and ending up with 298. To find the mystery number, we need to do the opposite (inverse). The opposite of subtracting 10 is adding 10.

💡 Why we do this:

Dividing numbers with zeros can be made easier by ignoring the zero first, doing the simple division, and then putting the zero back.

🗣 What this tells us:

Because 270 is 10 times bigger than 27, the answer must be 10 times bigger than 9.

💡 Why we do this:

Just like the previous question, we can look for a simple division hidden inside the large number.

💡 Why we do this:

Dividing by a 2-digit number like 15 can be tricky, but 15 is a nice number to count in. We can just add 15s until we reach 60.

💡 Why we do this:

The empty box is just asking for the answer to the subtraction. We set this up as a column subtraction.

🗣 What we did:

1. Units: \( 6 – 5 = 1 \).

2. Tens: \( 7 – 5 = 2 \).

3. Hundreds: \( 7 – 8 \) we cannot do. Borrow from the 5 (thousands). The 5 becomes 4, and the 7 becomes 17. \( 17 – 8 = 9 \).

4. Thousands: \( 4 – 0 = 4 \).

💡 Why we do this:

The question breaks the number down into its parts (millions, thousands, tens, etc.). We need to identify which part is missing.

💡 Why we do this:

The number 10 doesn’t show a decimal point, but we can write it as 10.0 to help us subtract 5.4. We must line up the decimal points.

🗣 What we did:

1. \( 0 – 4 \) we cannot do. We need to borrow.

2. We go to the 10. It becomes 9, and the 0 becomes 10.

3. \( 10 – 4 = 6 \).

4. \( 9 – 5 = 4 \).

💡 Why we do this:

We cannot add fractions with different denominators (7 and 21). We need to change them so they are the same. Since 21 is in the 7 times table (\( 7 \times 3 = 21 \)), we can use 21 as our common denominator.

💡 Why we do this:

When dividing by 100, the digits move 2 places to the right (or the decimal point moves 2 places to the left).

💡 Why we do this:

To find a fraction of a number, we first split the number into equal parts (divide by the denominator) and then take the number of parts we need (multiply by the numerator).

Here, we divide 1,000 by 4, then multiply by 3.

💡 Why we do this:

We are multiplying by a 2-digit number (23). We split this into two simpler multiplications:

1. \( 785 \times 3 \)

2. \( 785 \times 20 \)

Then we add the two answers together.

🗣 Step-by-step:

Row 1 (\( \times 3 \)):

• \( 5 \times 3 = 15 \) (write 5, carry 1)
• \( 8 \times 3 = 24 \). Add 1 = 25 (write 5, carry 2)
• \( 7 \times 3 = 21 \). Add 2 = 23 (write 23)
• Row 1 = 2,355

Row 2 (\( \times 20 \)):

• Write the Magic Zero first!
• \( 5 \times 2 = 10 \) (write 0, carry 1)
• \( 8 \times 2 = 16 \). Add 1 = 17 (write 7, carry 1)
• \( 7 \times 2 = 14 \). Add 1 = 15 (write 15)
• Row 2 = 15,700

Total: \( 2,355 + 15,700 = 18,055 \)

💡 Why we do this:

Percentages like 20% are easy to find if we find 10% first. To find 10%, we just divide by 10.

💡 Why we do this:

Dividing by 43 is hard to do in our heads! We use long division. It helps to write down a few multiples of 43 first.

Useful list: \( 1 \times 43 = 43 \), \( 2 \times 43 = 86 \), \( 10 \times 43 = 430 \).

🗣 Step-by-step:

1. Into 64: How many 43s go into 64? Only 1. (Write 1 on top).

2. Subtract: \( 1 \times 43 = 43 \). \( 64 – 43 = 21 \).

3. Bring down: Bring down the 5 to make 215.

4. Into 215: How many 43s go into 215? Let’s estimate. \( 40 \times 5 = 200 \). Let’s try 5.

5. Check: \( 43 \times 5 = (40 \times 5) + (3 \times 5) = 200 + 15 = 215 \). Exactly 5.

💡 Why we do this:

\( 0.5 \) is the same as one half (\( \frac{1}{2} \)). Multiplying by 0.5 is the same as finding half of the number.

💡 Why we do this:

We cannot add halves and fifths directly. We need to turn them into the same type of fraction. The number 10 is in both the 2 times table and the 5 times table.

💡 Why we do this:

The denominators are already the same (4). We can add the fractions first, then deal with the whole number.

💡 Why we do this:

The number 6 doesn’t look like a decimal, but we need to subtract decimals from it. We write 6 as 6.000 so it has the same number of decimal places as 5.738.

🗣 What we did:

1. We can’t do \( 0 – 8 \), so we borrow across from the 6.

2. The 6 becomes 5. The zeros become 9s, and the last zero becomes 10.

3. \( 10 – 8 = 2 \)

4. \( 9 – 3 = 6 \)

5. \( 9 – 7 = 2 \)

6. \( 5 – 5 = 0 \)

💡 Why we do this:

Multiplying by 30 is the same as multiplying by 10 and then by 3. This makes the numbers easier to handle.

💡 Why we do this:

It is usually easier to subtract mixed numbers if we turn them into improper fractions first. Also, we need a common denominator (15).

💡 Why we do this:

We need to split 86 into 80 and 6. We multiply 5413 by 6, then multiply 5413 by 80, and add the totals.

🗣 Key details:

Row 1 (\( \times 6 \)): \( 3 \times 6 = 18 \) (carry 1), \( 1 \times 6 = 6 + 1 = 7 \), \( 4 \times 6 = 24 \) (carry 2), \( 5 \times 6 = 30 + 2 = 32 \).

Row 2 (\( \times 80 \)): Put the zero first! Then \( 3 \times 8 = 24 \) (carry 2), \( 1 \times 8 = 8 + 2 = 10 \) (carry 1), \( 4 \times 8 = 32 + 1 = 33 \) (carry 3), \( 5 \times 8 = 40 + 3 = 43 \).

💡 Why we do this:

99% is very close to 100%. If we find 1% and subtract it from the total (100%), we get 99%.

💡 Why we do this:

We have one quarter and we need to share it between 2. If you cut a quarter of a pizza in half, the pieces get smaller. Dividing a fraction by an integer makes the pieces smaller, so the denominator (bottom number) gets bigger.

💡 Why we do this:

We must follow the rules of BODMAS:

1. Order/Indices (powers) first.

2. Division next.

3. Subtraction last.

💡 Why we do this:

A mixed number like \( 1 \frac{1}{2} \) is made of a whole number (1) and a fraction (\( \frac{1}{2} \)). We can multiply each part by 40 separately and add the answers.

💡 Why we do this:

28% is not easy to calculate in one go. We can break it down into chunks we can find easily:

10%, 10%, and 8% (or 20% and 8%).

To find 8%, we can find 1% and multiply by 8.

💡 Why we do this:

Denominators are 3 and 7. The common denominator is \( 3 \times 7 = 21 \).

💡 Why we do this:

Dividing by 97 is tricky. It helps to write down a “cheat sheet” of multiples of 97 first.

Since 97 is very close to 100, we can estimate easily.

• \( 1 \times 97 = 97 \)
• \( 2 \times 97 = 194 \)
• \( 10 \times 97 = 970 \)
• \( 9 \times 97 = 970 – 97 = 873 \)

🗣 Step-by-step:

1. Into 88: 97 doesn’t fit into 88. Write 0.

2. Into 882: How many 97s in 882? \( 9 \times 97 = 873 \). It fits 9 times.

3. Subtract: \( 882 – 873 = 9 \).

4. Bring down: Bring down the 7 to make 97.

5. Into 97: How many 97s in 97? Exactly 1.