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To calculate a single column vector students need to combine their knowledge of substituting known values into expressions, expanding brackets, arithmetic with negative numbers and performing translations on a grid. In my experience, connecting this many skills within a single problem can be challenging for students.

It is not surprising therefore when this topic first appeared on a Foundation GCSE paper few students were able to achieve full marks. Here is a copy of the question and examiners report published by Edexcel.

**Question 30** Here are two column vectors.

a=\left(\begin{matrix}5\\2\\\end{matrix}\right)

b=\left(\begin{matrix}3\\-1\\\end{matrix}\right)

On the grid below, draw and label the vector **a** − 2**b**

**(2 marks)**

**Examiners Report**

“This question was answered poorly with relatively few students showing an understanding of vector arithmetic. Some students scored the first method mark for 5 – 2 × 3 or 2 – 2 × – 1. When

\left(\begin{matrix}5\\2\\\end{matrix}\right)-\left(\begin{matrix}6\\-2\\\end{matrix}\right)was seen it was often not simplified correctly with errors usually occurring in the *y* component. Students that scored the second mark for simplifying to

were rarely able to achieve the final mark for drawing the vector. Some students attempted to find the vector **a** – 2**b** by drawing but most could not manage even the first step. At all stages of the question the drawing of vectors was extremely poor and very few students drew any kind of correct vector.”

In my experience, one the key skills in this question is being able visualise a column vector as a translation. Therefore, at the start of the lesson I remind students they first encountered column vectors when performing and describing translations. As noted in the examiner’s report this will help students later in the lesson to check their algebra.

Students also find it difficult to work out a negative column vector when it is presented algebraically. For instance, working out -2**a** when given +**a**. For this reason, I wanted to focus on negative terms in the second part of the lesson.

At the start of the lesson this problem is presented on the board to remind students how to describe translations using column vectors.

When everyone is settled I explain the task and demonstrate how the red triangle labelled A can be mapped onto triangle B. I then ask the class to match the remaining three vectors to their corresponding images.

When the majority of the class have successfully matched the three remaining vectors I feedback the solutions and begin the main phase of the lesson.

As we move on to the main part of the lesson I demonstrate how to substitute the vectors **a** and **b** into the expression 2**a** – **b**. Some students struggle to understand how to add negative **b** to 2**a**. The algebra becomes much clearer when I demonstrate the translation of –**b** on the grid as the opposite of +**b**.

After working through questions a) and b) I ask students to work out the single column vector for 2c + 3a on their mini-whiteboards.

Most students attempt the algebraic method as shown below.

2\left(\begin{matrix}-2\\1\\\end{matrix}\right)+3\left(\begin{matrix}3\\2\\\end{matrix}\right)\left(\begin{matrix}-4\\2\\\end{matrix}\right)+\left(\begin{matrix}9\\6\\\end{matrix}\right)\left(\begin{matrix}5\\8\\\end{matrix}\right)Some prefer to draw the translation on the grid to work out the final column vector. I encourage students to use whichever approach they are most confident with.

Towards the end of the lesson students need to work out the column vector from a diagram and substitute it into each expression. The five expressions increase in difficulty to make sure all students are challenged. Questions d) and e) are written to challenge the most able as they involve brackets and negative terms.

Most students complete questions the first three questions with little difficulty. However, in questions c) and d) about a quarter of the class make mistakes when multiplying out the negative vectors. Most of these students go back to using the grid to perform the translation.

My name is Jonathan Robinson and I am passionate about teaching mathematics. I am currently Head of Maths in the South East of England and have been teaching for over 17 years. I am proud to have helped teachers all over the world to continue to engage and inspire their students with my lessons.

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