KS2 SATs 2023 Mathematics Paper 3: Reasoning

💡 Why we do this: We need to follow all the rules given in the question to get the correct answer.

The rules are:

• Use three cards from: 4, 8, 2, 7.
• The number must be a 3-digit number (Hundreds, Tens, Ones).
• The digit 4 must be in the Tens place.
• We want the lowest possible number.

So our number looks like this: [ ? ] [ 4 ] [ ? ]

We have cards 8, 2, and 7 remaining to use.

💡 How to make it the lowest: To make a number as small as possible, we need the smallest possible digit in the highest place value (the Hundreds column).

The available digits are 2, 7, and 8.

The smallest digit is 2. So, we put 2 in the Hundreds place.

Current number: 2 4 [ ? ]

Now we need to pick the Ones digit. We want the number to be the lowest, but the value of the Ones digit doesn’t change the Hundreds digit. However, we should check if there’s any other choice. We just need to pick the smallest remaining digit for the Ones place? No, actually, for the number 24?, the Ones digit just completes it. Wait, does the order of remaining digits matter for “lowest”?

Yes. We have 7 and 8 left. If we pick 7, we get 247. If we pick 8, we get 248. Since 247 is lower than 248, we use 7.

Wait, could we swap them? If we put 7 in Hundreds, we get 742. That is much bigger. So 2 must be in Hundreds.

💡 Why we do this: Converting words to digits helps us match it exactly to the options.

The number is “eighty thousand, three hundred and six”.

• Eighty thousand: 80,000
• Three hundred: 300
• Six: 6

💡 What this tells us: We need to find 80,306 in the list.

• 8,306 (Eight thousand…)
• 80,036 (Eighty thousand and thirty-six)
• 80,306 (Correct)
• 800,306 (Eight hundred thousand…)
• 80,300,006 (Eighty million…)

💡 Why we do this: Reflecting in the \(y\)-axis means using the vertical line (the \(y\)-axis) as a mirror. Points on the left move to the right, keeping the same distance from the axis.

The rule for reflection in the \(y\)-axis is: The \(x\)-coordinate flips sign (\(-x\)), but the \(y\)-coordinate stays the same.

💡 How we calculate: Let’s find the coordinates of the original points (counting squares from the axis) and reflect them.

• Point C: It is 1 square left of the axis. The reflection will be 1 square right.
• Point B: It is 3 squares left of the axis. The reflection will be 3 squares right.
• Point A: It is 3 squares left (and 4 squares up). The reflection will be 3 squares right.

💡 Why we do this: To find the next numbers, we need to know what we are adding or subtracting each time.

Let’s look at the difference between the numbers:

• \( 1,880 – 1,780 = 100 \)
• \( 1,980 – 1,880 = 100 \)

The rule is: Add 100.

💡 How we calculate: Add 100 to the last number each time.

First missing number:

\( 1,980 + 100 = 2,080 \)

Second missing number:

\( 2,080 + 100 = 2,180 \)

💡 How to round: Look at the digit after the decimal point (the tenths). If it is 5 or more, round up. If it is 4 or less, round down.

• 13.2: Tenths is 2 (less than 5) → Rounds down to 13
• 14.7: Tenths is 7 (5 or more) → Rounds up to 15
• 15.9: Tenths is 9 (5 or more) → Rounds up to 16
• 16.3: Tenths is 3 (less than 5) → Rounds down to 16
• 17.6: Tenths is 6 (5 or more) → Rounds up to 18

💡 What this tells us: We can see that two numbers rounded to 16.

These numbers are 15.9 and 16.3.

💡 Why we do this: The number is broken down into its place value parts (Millions, Hundred Thousands, etc.). We need to find which part is missing.

Let’s look at the target number: 1,300,450

• 1,000,000 (One million) is accounted for.
• 400 (Four hundred) is accounted for.
• 50 (Fifty) is accounted for.

What is left?

💡 How we calculate: Compare the digits.

1,300,450

The digit 3 is in the Hundred Thousands place.

Its value is 300,000.

💡 Why we do this: We need to reconstruct the missing parts of the square.

Looking at the top row:

\( \frac{1}{2}, 1, 1\frac{1}{2}, 2, 2\frac{1}{2} \)

The numbers go up by 0.5 each time.

There are 5 columns in the top row.

Let’s check if the rows continue from each other. The last number in Row 1 is 2.5. The first number in Row 2 is 3.

This fits perfectly: \( 2.5 + 0.5 = 3 \). So it is a 5-wide number square.

💡 What this tells us: The “bottom-left corner” corresponds to the last row, first column.

From the fragment, we can see 5 rows in total:

• Row 1 starts at 0.5
• Row 2 starts at 3
• Row 3 starts at 5.5 (Note: 6 is the second number)
• Row 4 starts at 8 (Note: 9 is the third number)
• Row 5 starts at ?

We need to find the first number in Row 5.

💡 How we calculate: Each row adds 2.5 to the starting number (since width is 5 and step is 0.5, \(5 \times 0.5 = 2.5\)).

• Row 1 start: 0.5
• Row 2 start: \(0.5 + 2.5 = \) 3.0
• Row 3 start: \(3.0 + 2.5 = \) 5.5
• Row 4 start: \(5.5 + 2.5 = \) 8.0
• Row 5 start: \(8.0 + 2.5 = \) 10.5

Alternatively, the visible numbers in the first column shift right by one place each row.

💡 Why we do this: The name depends on the number of sides.

• Pentagon: 5 sides
• Hexagon: 6 sides

💡 The difference: A Regular shape has all sides equal length and all angles equal. An Irregular shape does not.

• Shape 1: Sides are different lengths. Irregular Hexagon.
• Shape 2: Looks like a house. Sides are not equal. Irregular Pentagon.
• Shape 3: All sides look equal. Regular Pentagon.
• Shape 4: All sides look equal. Regular Hexagon.

💡 Why we do this: If you multiply a whole number by 3, the answer must be a multiple of 3 (it must be in the 3 times table).

Let’s check the numbers near 32:

• \( 10 \times 3 = 30 \)
• \( 11 \times 3 = 33 \)

Is 32 in the list?

💡 How to explain: State that 32 is not in the 3 times table, or that it doesn’t divide by 3 exactly.

We can also use the digit sum rule: \(3 + 2 = 5\). Since 5 is not divisible by 3, 32 is not divisible by 3.

💡 What this means: \(8^2\) means \(8 \times 8\).

💡 How we calculate: We know the total is 73. So we subtract 64 from 73 to find what’s missing.

💡 The question asks: “Write the missing square number…”

The equation is \( 8^2 + [?]^2 = 73 \)

We found the value of the missing part is 9.

We need a number that, when squared, equals 9.

\( 3 \times 3 = 9 \)

So the missing number in the box is 3.

Wait, looking at the layout: \( 8^2 + \_\_^2 = 73 \). The box is the base number.

💡 Why we do this: First, let’s find out how many games were sold in total during April and May.

💡 How we calculate: Subtract the total sold (12,542) from the starting amount (15,000).

💡 What shape is it? It is a “square prism”.

• It has 4 long rectangular faces.
• It has 2 square faces (top and bottom).

A correct net must have all 6 faces and they must fold together without overlapping.

Net 1 (Top): The two square faces are both on the left side of adjacent rectangles. If you fold the rectangles into a tube, the squares will try to occupy the same space (overlap) or leave one end open. ❌

Net 2 (Second): 4 rectangles in a row make the tube sides. One square is on top, one is on bottom (spaced apart). This folds perfectly. ✅

Net 3 (Third): The arrangement of large rectangles is 2×2. This does not fold into a tube of 4. ❌

Net 4 (Bottom): This is a more complex layout, but if you visualise folding:

• The middle squares become the ends.
• The top rectangles fold down to form front/back.
• The bottom rectangles fold up to form left/right.

💡 Why we do this: Instead of doing all the big multiplications, we can look at the structure.

Imagine “754” is an object, like an “apple”.

The equation says:

6 apples + 3 apples = ? apples

💡 How we calculate: simply add the multipliers.

\( 6 + 3 = 9 \)

So, \( 754 \times 6 + 754 \times 3 \) is the same as \( 754 \times 9 \).

💡 Convert percentage to fraction:

\( 25\% = \frac{25}{100} = \frac{1}{4} \)

Do we have cards 1 and 4?

Yes, we do!

💡 Convert decimal to fraction:

\( 0.4 = \frac{4}{10} \)

We simplify this fraction: \( \frac{4}{10} = \frac{2}{5} \)

Do we have cards 2 and 5?

Yes, we do!

💡 The Problem: Arrival time – Travel time = Departure time.

Arrival: “Ten past seven” means 7:10.

Travel time: 1 hour 20 minutes.

💡 The Problem: End time – Start time = Duration.

Start: 7:20 pm

End: 9:05 pm

💡 Why we do this: The total mass includes the box. We need to remove the weight of the box to find the weight of just the chocolate.

Total mass = 870 g

Box mass = 30 g

💡 How we calculate: We have 840g of chocolate shared equally between 24 eggs.

Calculation: \( 840 \div 24 \)

💡 The Problem: We need to find the total shaded area.

Shaded Area = Part A + Part B

Part A = \(\frac{1}{2}\)

Part B = \(\frac{1}{3}\)

💡 How we calculate: To add \(\frac{1}{2} + \frac{1}{3}\), we need a common denominator.

The common denominator for 2 and 3 is 6.

• \(\frac{1}{2} = \frac{3}{6}\)
• \(\frac{1}{3} = \frac{2}{6}\)

💡 Identify the data: Look at the table for the year 2012.

Rainfall in 2012 = 700 mm.

💡 Draw the bar: Find 700 on the \(y\)-axis. It is halfway between 600 and 800.

💡 The Problem: Calculate the mean (average) sunshine hours for 2013, 2014, and 2015.

Data:

• 2013: 1,452
• 2014: 1,669
• 2015: 1,508

Price: £3.20 per kg.

Amount: 500g.

500g is half a kg (\(0.5\) kg).

Price: 60p per kg.

Amount: \(1\frac{1}{4}\) kg (which is 1.25 kg).

Cost = 1 kg price + \(\frac{1}{4}\) kg price.

\(\frac{1}{4}\) of 60p = \(60 \div 4 = 15\)p.

Total Cost: Mushrooms + Carrots

£1.60 + £0.75

💡 What is perimeter? Perimeter is the total distance around the outside of a shape.

For a rectangle, you add all 4 sides:

Side 1: 6 cm

Side 2: \(w\) cm

Side 3: 6 cm (opposite side)

Side 4: \(w\) cm (opposite side)

So, Perimeter = \( 6 + w + 6 + w \)

• Option 1: \(w \times 6\) → This is Area, not Perimeter. ❌
• Option 2: \(w \times 2 + 12\) → This simplifies to \(2w + 12\). Since \(6 + 6 = 12\), this is \(2w + 12\). ✅
• Option 3: \(2 \times (w + 6)\) → This means \(2 \times\) (length + width). This is the standard formula. ✅
• Option 4: \(6 + w + 6 + w\) → This is adding all sides directly. ✅

💡 Why we do this: To find percentages, we first need the total amount.

25 classes \(\times\) 34 pupils.

💡 Two ways to solve:

Method A: Calculate 62% (who play sport) and subtract from total.

Method B: Calculate the percentage who do not play sport first.

If 62% play sport, then \(100\% – 62\% = 38\%\) do not play sport.

Let’s use Method B (finding 38% of 850) as it saves a subtraction step at the end.

💡 How we calculate: \(0.38 \times 850\)

We can do \(38 \times 850\) and then divide by 100.

💡 What is an octagon? An octagon has 8 sides and 8 vertices.

So the first box (number of vertices) is 8.

Let’s look at the previous examples:

• Triangle (3 vertices) → \(\times 0\) (\(3-3=0\))
• Quadrilateral (4 vertices) → \(\times 1\) (\(4-3=1\))
• Pentagon (5 vertices) → \(\times 2\) (\(5-3=2\))

The rule seems to be: Multiply by (Vertices – 3).

For an octagon (8 vertices): \( 8 – 3 = 5 \).

So the first operation is \(\times 5\).

The second operation is always \(\div 2\).

Let’s calculate the result:

\( 8 \times 5 = 40 \)

\( 40 \div 2 = 20 \)

💡 How to convert to decimal: We want the denominator to be 10, 100, or 1000.

To turn 20 into 100, we multiply by 5.

\( \frac{3 \times 5}{20 \times 5} = \frac{15}{100} \)

15 hundredths is 0.15.

💡 How to convert: We know some common eighths.

• \(\frac{1}{8} = 0.125\)

So \(\frac{5}{8}\) is \(5 \times 0.125\).