If you’re struggling with this week’s questions, this walkthrough should help you get the skills you need.

This week, we’re exploring:

- Surface Area of a Prism
- Equation of a Straight Line
- Translation
- Sharing in a Ratio
- Midpoint of a Line Segment

### Question 1: Surface Area of a Prism

A prism is any 3D shape that has a consistent cross-section. To find the surface area of a prism, we simply find the area of each of the faces.

**Let’s find the surface area of the triangular prism below.**

This prism has 5 faces – the cross-sectional triangle at the front and back and 3 different rectangles. We need to find the area of each of these in turn. If you want a reminder on how to find the area of different shapes, have a look at this blog.

Cross-sectional triangle: ½ × 3 × 4 = 6cm^{2}

Base rectangle: 4 × 10 = 40cm^{2}

Top rectangle: 5 × 10 = 50cm^{2}

Left rectangle: 3 × 10 = 30cm^{2}

Now, we simply add these together to find the surface area. Make sure you include two cross-sectional triangles – the front and back.

Surface area = 6 + 6 + 40 + 50 + 30** ** ** ** = 132cm^{2}

### Question 2: Equation of a Straight Line

Every straight line can be written in the form 𝑦 = 𝑚𝑥 + 𝑐 where 𝑚 is the gradient of the line and 𝑐 is the 𝑦-intercept. This week, we are given the gradient of the line and a point the line passes through. With this information, we are asked to find the equation of the line.

**A straight line has a gradient of -3 and passes through the point (2, 10). Work out the equation of the line** **in the form 𝑦 = 𝑚𝑥 + 𝑐.**

We know the gradient is -3 so we can substitute this in for 𝑚 in our equation.

𝑦 = -3𝑥 + 𝑐

Since the line passes through the point (2, 10), we can substitute these coordinates in to find the value of c. Remember, coordinates are in the format (𝑥, 𝑦) so 𝑥 = 2 and y = 10.

𝑦 = -3𝑥 + 𝑐

10 = -3 × 2 + 𝑐

10 = -6 + 𝑐

𝑐 = 16

Putting this into our equation gives us our final answer: **𝑦**** = -3****𝑥**** + 16**.

### Question 3: Translation

Translation keeps a shape the same size and orientation but simply changes its location. Usually, translations are given using vectors.

**Translate the shape below by the vector .**

The top number on the vector tells us how far to move the shape horizontally. If the number’s positive, we move the shape to the right and if it’s negative, we move the shape to the left. The bottom number tells us how far to move the shape vertically. If the number’s positive, we move the shape up and if it’s negative, we move the shape down. In this case, we move the shape 2 units left and 5 units up.

Pick a vertex (corner) and count the number of squares we need. This will give us the new location of the vertex.

Now, we can draw in the rest of the shape. Make sure it’s exactly the same size.

If asked to describe a transformation, we use a very similar method.

**Describe the transformation that maps shape A onto shape B.**

As before, pick a vertex on shape A and count the squares to get to the corresponding vertex on shape B.

So, the shape has been translated **3 units to the right and 4 units down**. We can also write this as a vector:

### Question 4: Sharing in a Ratio

Questions about sharing an amount in a ratio can take many different forms. Take a ratio that splits an amount between two people. You will sometimes be given the total amount and asked to calculate how much each person receives and you will sometimes be given how much one person gets and need to calculate how much the other person gets. You could also be given the difference between how much each person gets and use this to calculate the amounts.

Whatever form the question takes, the easiest way to approach it is using the **bar model** – sometimes called the bucket method. The two methods are based on the same principle and are just set out slightly differently.

**Let’s say Robert and Prav share £136 in the ratio 3:5. Calculate how much each person receives. **

The principle of the bar model is that we set up two bars – one for each person. Robert’s will be 3 squares long and Prav’s will be 5 squares long.

Since we are sharing the total amount of money, we count the total number of boxes – in this case, 8. We share the total amount (£136) evenly across each of the boxes. In other words, we calculate 136 ÷ 8 = 17.

Now, we place £17 in each box and calculate the total amount for each person.

Robert gets 17 + 17 + 17 = **£51**.

Prav gets 17 + 17 + 17 + 17 + 17 = **£85**.

You should always check your answer by adding the values together to see if you get the original amount: 51 + 85 = 136.

If the question gives you one person’s amount, then you simply share this amount between that person’s boxes.

For example, if Robert gets £57, we would find 57 ÷ 3 (= 19) and then write this amount in each box.

If the question gives you the difference between the amounts, you find the difference in the number of boxes and divide by this amount.

For example, if Prav gets £23 more than Robert, as Prav has 2 more boxes than Robert, we would calculate 23 ÷ 2 (= 11.5) and write this amount in each box.

### Question 5: Midpoint of a Line Segment

The midpoint of a line segment is exactly halfway between the two end points.

**Points A and B have coordinates (-2, -7) and (-8, 4) respectively. Find the coordinates of the midpoint of the line segment AB.**

The 𝑥-coordinate of the midpoint will be halfway between -2 and -8.

Similarly, the 𝑦 -coordinate of the midpoint will be halfway between -7 and 4.

So, the midpoint of the line segment AB is at **(-5, -1.5).**

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