KS2 Mathematics 2024 Paper 1 (Arithmetic)

Why we do this: We align the digits by their place value (ones under ones, tens under tens) to add them correctly.

Action: \( 9 + 8 = 17 \). Write down 7 and carry the 1 to the tens column.

Action: \( 8 + 3 = 11 \). Don’t forget the carried 1: \( 11 + 1 = 12 \). Write down 2 and carry the 1 to the hundreds column.

Action: \( 6 + 0 = 6 \). Don’t forget the carried 1: \( 6 + 1 = 7 \).

Why we do this: Short division (bus stop) helps us divide larger numbers digit by digit.

Action: How many 3s go into 7? \( 3 \times 2 = 6 \). So 3 goes into 7 2 times, with a remainder of 1.

Action: Place the remainder (1) in front of the next digit (2) to make 12.

Action: How many 3s go into 12? \( 3 \times 4 = 12 \). Exactly 4 times.

Action: \( 3 \times 6 = 18 \). Write 8 and carry the 1.

Action: \( 2 \times 6 = 12 \). Add the carried 1: \( 12 + 1 = 13 \).

Why: Align digits to the right to subtract correctly.

Action: \( 2 – 9 \). We cannot do this, so we borrow from the tens (the 3 becomes 2, the 2 becomes 12).

\( 12 – 9 = 3 \).

Action: Tens: \( 2 – 1 = 1 \). Hundreds: \( 5 – 0 = 5 \). Thousands: \( 4 – 0 = 4 \).

Tip: It is easier to put the larger number on top.

Ones: \( 1 \times 3 = 3 \)

Tens: \( 9 \times 3 = 27 \) (Write 7, carry 2)

Hundreds: \( 3 \times 3 = 9 \). Add carry: \( 9 + 2 = 11 \)

Ones: \( 1 – 0 = 1 \)

Tens: \( 7 – 3 = 4 \)

Hundreds: \( 1 – 5 \). Impossible. Borrow from thousands.

Hundreds (continued): \( 11 – 5 = 6 \)

Thousands: \( 8 – 0 = 8 \)

Method: We can partition the number.

\( 6,600 = 6,000 + 600 \)

\( 6,000 \div 6 = 1,000 \)

\( 600 \div 6 = 100 \)

\( 1,000 + 100 = 1,100 \)

First, calculate \( 99 \times 10 \). To multiply by 10, we shift digits one place to the left (add a zero).

\( 99 \times 10 = 990 \)

Any number multiplied by 1 stays the same.

\( 990 \times 1 = 990 \)

Critical: The decimal points must line up perfectly. We can add a placeholder zero to 7.68 so they are the same length.

Thousandths: \( 0 + 3 = 3 \)

Hundredths: \( 8 + 9 = 17 \) (carry 1)

Tenths: \( 6 + 4 = 10 \). Add carry: \( 10 + 1 = 11 \) (carry 1)

Ones: \( 7 + 3 = 10 \). Add carry: \( 10 + 1 = 11 \) (carry 1)

Tens: \( 0 + 1 = 1 \). Add carry: \( 1 + 1 = 2 \)

\( 2 \times 9 = 18 \) (carry 1)

\( 5 \times 9 = 45 \). Add carry: \( 45 + 1 = 46 \) (carry 4)

\( 7 \times 9 = 63 \). Add carry: \( 63 + 4 = 67 \)

We know that \( 64 \div 8 = 8 \).

Since \( 640 \) is 10 times bigger than \( 64 \), the answer will be 10 times bigger than 8.

\( 640 \div 8 = 80 \)

Ones: \( 7 – 9 \). Borrow from 5. \( 17 – 9 = 8 \).

Tens: Now we have 4. \( 4 – 8 \). Borrow from 3. \( 14 – 8 = 6 \).

Hundreds: Now we have 2. \( 2 – 0 = 2 \).

\( 5 \div 3 = 1 \) remainder 2. (Put remainder by the 6).

\( 26 \div 3 = 8 \) remainder 2. (Put remainder by the 1).

\( 21 \div 3 = 7 \).

Rule: Multiply the top numbers together and the bottom numbers together.

Top: \( 4 \times 1 = 4 \)

Bottom: \( 6 \times 8 = 48 \)

Answer: \( \frac{4}{48} \)

Both numbers divide by 4: \( 4 \div 4 = 1 \) and \( 48 \div 4 = 12 \).

Simplified: \( \frac{1}{12} \).

We know that \( 63 \div 7 = 9 \).

\( 630 \) is 10 times bigger than \( 63 \), so the answer is 10 times bigger than 9.

\( 9 \times 10 = 90 \).

Let’s look at the digits in \( 6,020,070 \):

• The 6 is worth \( 6,000,000 \) (Six Million).
• The 2 is in the ten-thousands place.
• The 7 is worth \( 70 \) (Seventy).

We have the \( 6,000,000 \) and the \( 70 \). The missing part is the 2.

Since it is in the ten-thousands column, it represents \( 20,000 \).

Rule: To multiply a whole number by 1,000, we move the digits 3 places to the left, which is the same as adding three zeros to the end.

\( 2,671 \) becomes \( 2,671,000 \).

We cannot add fractions with different denominators (12 and 3). We need to make them the same.

We can turn thirds into twelfths because \( 3 \times 4 = 12 \).

Multiply the top and bottom of \( \frac{1}{3} \) by 4:

Now add the numerators:

\( \frac{9}{12} \) can be simplified by dividing by 3: \( \frac{3}{4} \).

Multiples of 3: 3, 6, 9, 12, 15…

Multiples of 5: 5, 10, 15…

The lowest common denominator is 15.

\( \frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15} \)

\( \frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15} \)

This is an improper fraction. As a mixed number: \( 1 \frac{7}{15} \).

Calculate \( 6312 \times 4 \).

\( 2 \times 4 = 8 \)

\( 1 \times 4 = 4 \)

\( 3 \times 4 = 12 \) (2, carry 1)

\( 6 \times 4 = 24 \). Add carry: 25

Important: Place a 0 placeholder first because we are multiplying by 10.

Calculate \( 6312 \times 1 \).

\( 8 + 0 = 8 \)

\( 4 + 2 = 6 \)

\( 2 + 1 = 3 \)

\( 5 + 3 = 8 \)

\( 2 + 6 = 8 \)

Indices (powers) come before Addition.

\( 3^3 \) means \( 3 \times 3 \times 3 \).

\( 3 \times 3 = 9 \)

\( 9 \times 3 = 27 \)

\( 2 + 27 = 29 \)

To divide by 100, we shift the digits 2 places to the right (making the number smaller).

15.5 (start)

1.55 (divide by 10)

0.155 (divide by 100)

Division (D) comes before Subtraction (S).

\( 56 \div 8 = 7 \)

\( 90 – 7 = 83 \)

63 does not show a decimal point, but it is \( 63.0 \). We need the placeholder zero to subtract.

\( 0 – 8 \) is impossible. Borrow from 3.

\( 10 – 8 = 2 \).

\( 2 – 5 \) is impossible. Borrow from 6.

\( 12 – 5 = 7 \).

\( 5 – 0 = 5 \).

\( 8 \times 7 = 56 \) (6, carry 5)

\( 1 \times 7 = 7 \). Add carry: \( 7 + 5 = 12 \) (2, carry 1)

\( 4 \times 7 = 28 \). Add carry: \( 28 + 1 = 29 \)

Add placeholder zero.

\( 8 \times 4 = 32 \) (2, carry 3)

\( 1 \times 4 = 4 \). Add carry: \( 4 + 3 = 7 \)

\( 4 \times 4 = 16 \)

21 is a multiple of 7. So the common denominator is 21.

\( \frac{6}{7} = \frac{6 \times 3}{7 \times 3} = \frac{18}{21} \)

\( \frac{7}{21} \) divides by 7.

\( \frac{1}{3} \).

Dividing by 2 is the same as multiplying by \( \frac{1}{2} \), or doubling the denominator.

OR: Keep the numerator (7) and multiply the denominator (10) by 2 = 20.

\( 2 \frac{3}{8} = \frac{2 \times 8 + 3}{8} = \frac{19}{8} \)

\( \frac{14}{8} = 1 \frac{6}{8} = 1 \frac{3}{4} \)

Calculate \( 7 \times 26 \).

\( 6 \times 7 = 42 \) (2, carry 4)

\( 2 \times 7 = 14 \). Add carry: \( 14 + 4 = 18 \)

Result: 182

The original question was \( 0.7 \) (1 decimal place).

So we put 1 decimal place into the answer.

18.2

We need to see how many 34s fit into 986.

Useful multiples of 34:

• \( 1 \times 34 = 34 \)
• \( 2 \times 34 = 68 \)
• \( 3 \times 34 = 102 \)

34 into 98?

It goes 2 times (\( 68 \)).

Remainder: \( 98 – 68 = 30 \).

Bring down the 6 to make 306.

34 into 306?

Let’s estimate. \( 34 \approx 30 \). \( 306 \approx 300 \). \( 30 \times 10 = 300 \). So try 9.

\( 34 \times 9 = 306 \). ( \( 30 \times 9 = 270 \), \( 4 \times 9 = 36 \), \( 270+36=306 \) ).

It fits exactly 9 times.

\( 100\% = 600 \).

\( 1\% = 600 \div 100 = 6 \).

\( 99\% = 100\% – 1\% \)

\( 99\% = 600 – 6 = 594 \)

If you have half a pizza and share it between 3 people, each person gets a smaller slice.

Rule: Multiply the denominator by the whole number.

\( 1\% \text{ of } 900 = 9 \).

We need 43 lots of 1%.

\( 43 \times 9 \).

\( 3 \times 9 = 27 \) (7, carry 2)

\( 4 \times 9 = 36 \). Add carry: \( 36 + 2 = 38 \)

Answer: 387

\( 2 \frac{1}{6} = \frac{2 \times 6 + 1}{6} = \frac{13}{6} \).

We are subtracting \( \frac{2}{3} \). Change thirds to sixths.

\( \frac{2}{3} = \frac{4}{6} \).

\( \frac{9}{6} \) divides by 3 = \( \frac{3}{2} \).

As a mixed number: \( 1 \frac{1}{2} \).

\( 1 \frac{1}{4} = \frac{5}{4} \).

\( \frac{5}{4} \times 39 = \frac{5 \times 39}{4} \).

\( 5 \times 39 \): \( 5 \times 40 = 200 \), subtract 5 = 195.

So we have \( \frac{195}{4} \).

How many 4s in 195?

\( 195 \div 4 \).

\( 160 \div 4 = 40 \). Remainder 35.

\( 32 \div 4 = 8 \). Remainder 3.

Total: 48 remainder 3.

Answer: \( 48 \frac{3}{4} \).

Estimate 73 as approx 75 or 70.

• \( 1 \times 73 = 73 \)
• \( 2 \times 73 = 146 \)
• \( 5 \times 73 = 365 \) (Half of 730)

Let’s check if \( 8 \times 73 = 584 \).

\( 8 \times 70 = 560 \)

\( 8 \times 3 = 24 \)

\( 560 + 24 = 584 \). Yes.