These notes represent one approach to introduce solving equations. I would typically teach this to year 8 students, who have previously studied fractions, decimals, and negatives. Discussion of these problems and their strategies will often yield the following; trial and improvement, collecting terms, cancelling from both sides, and checking solutions by substitution. Things I emphasise are correct use of equals sign, equations as statements of equality, and solutions as values of variables that make statements true. I also think this approach sets up algebra as a construct for solving problems. The lesson numbers are only to give a sense of possible timings, this is not intended to be prescriptive in anyway.

**Lesson 1**

The problem below is posed with no explanation, this works well displayed with students working on whiteboards so they can change values easily. I have found many students will complete with different values in the boxes, I will then pose it as a challenge to complete using same number in each empty box.

Once students have found the solution to the first one, they are given the one below. This time printed as students will require at different times.

Students are encouraged to articulate their thoughts at this stage, prior to group discussion. Whilst facilitating this discussion I would be aiming to attention towards; the meaning of equals sign, the process of checking solutions, and the definition of a solution.

**Lesson 2**

Students work through the following tasks, interspersed with discussions and opportunities for students to write reflections. Again I emphasise are correct use of equals sign, equations as statements of equality, and solutions as values of variables that make statements true. During this lesson I circulate providing extension, this is often in the form of posing problems to introduce another concept (such as fractions, negative coefficients, indices) and support, often by drawing attention to method in previous questions. I will have resources for all lessons printed so students can move on at different paces.

Discussion following these needs to include why boxes are ambiguous (may represent different values), hence use of letter where same letter means same value within a question.

Collecting of terms made explicit at this stage to students.

**Lesson 3**

Mixed questions, often used as self assessment for students, also aids in making the previous links more explicit.

Practice at this stage will be a mixture of students generating questions, questions provided, and students explaining their thinking.

**Lesson 4**

Tasks that encourage students to form equation. Although many students will be able to solve some of these without forming an equation, I ask students to do so.

Here is flipchart for these.

]]>My understanding of roots at this stage was purely operational, √5 to me was not a “number” but rather an instruction “find the number you started with and squared to get 5″. At this stage I don’t think there is anything wrong with that, but at some point(s), my relationship with √5 shifted to that of a position on the number line and this gave the ability to manipulate √5 in surd form.

It is vitally important that students are able to calculate roots (skills are important), but we should also be looking for opportunities to shift the relationship that students have with mathematical objects.

This is why students need a maths teacher! And this is part of the job of a maths teacher, this is where expertise of mathematics is important. Teachers must plan for this and take opportunities when presented, by planning tasks that offer more of these opportunities we are providing students with the opportunity to shift their relationship with mathematical objects and ideas.

Here are some tasks that I believe provide opportunities to shift mathematical thinking.

One thing I am hoping for here is students moving towards an understanding of the composition of a number from factors. Another is using combinatorics and the prime factorisation.

Questions I ask would depend on the students and also what I want to get out of the task. I may be happy for some students to draw the boxes to check number of squares, for some I might pose the question “how many will 2^10 or 3^a x 5^b have?”

In this task I hope for students to develop an understanding of place value, but there might also be opportunity to develop an understanding of commutativity, and also linking back to prime factorisation.

The idea that an infinite sum has a finite limit was not obvious to me at school, the diagram aids this. I may choose to start this task with just finite sums, eg start with 1/2 + 1/4 using the diagram, then 1/2 + 1/4 + 1/8, this would have the benefit of encouraging formal addition of fractions using the diagram as an aid.

Here are a few things I became aware of as a child;

I can double one number and half the other to get the same answer when doing multiplication, this later became ab = (a/c)(bc).

This “trick” doesn’t work with addition, I don’t think this is obvious to students.

I can add any number to one number and take the same away from the other and get the same answer when doing a sum, this later became a + b = (a + c) + (b – c)

This manipulation with numbers made it clear to me why the “rules” of arithmetic with fractions and then algebraic expressions have to be so. I think this task could aid this development as well as be used to introduce completing the square (thanks to Colin Hegarty for that idea).

With this task I would be encouraging students to explore similar triangles and ratio. I may give / suggest some pythagorean triples to ease this investigation. I may draw (to scale to check or not) an enlargement of a student’s rectangle and ask them to predict different lengths. Another question I might ask is “can you draw a rectangle / right angled triangle to represent ….” then give some large field or something.

A lot of students will notice and be able to justify this, with this task I usually try to introduce a more formal method of proof. I remember having a bit of an epiphany myself when being taught proof by induction that all natural number could be thought of as the previous number, plus one. It still fascinates me today that this can’t be done with the reals.

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**Initial contact – September 2014**

Letter sent home introducing teacher and inviting parent in for open evening. Email addresses asked for on reply slip (28 out of 30 emails recieved).

**First open evening – October 2014.** See notebook file “Year 11 parents meeting”

Presentation given to parents with opportunity for questions. Main points made;

- The importance of mathematics GCSE
- Everyone can do maths attitude
- Students must work hard
- How to organise work (use of folders, books and Google spreadsheet)
- Parent support
- Support at school (maths surgery, mymaths, mathswatch, videos)
- How to use videos
- Test results with follow ups to be emailed to parents

Parents given an individual flyer with their child’s personal login details for mymaths, mathswatch and instructions for signing up to hegartymaths.com See flyer “Year 11 flyer”

**Specification and topic list**

All parents emailed course specification with;

- topics studied in year 10- highlighted red
- topics studied in year 11- highlighted in green
- topics to be studied – highlighted in yellow

**Test results**

All parents emailed their child’s test results following topic tests and mocks with links for supporting test follow ups using mail merge. Example email below;

Dear parent,

“Student name” achieved “student’s mark” out of 200 in their recent mathematics mock exam. This is a grade “student grade”.

Your child has been given an individual breakdown of their marks on every question and the topic concerned, students need to set themselves two targets (for example “To calculate the area of compound shapes) and action these. An action should involve them doing some questions on this; students could use mymaths.co.uk, one of my videos, attend maths surgery or come to see me any time. This has all been explained in class. Students have also been emailed link to full solutions to these exam papers. Here they are again;

Paper 1 – http://www.youtube.com/watch?v=IAdpPloVdOU

Paper 2 – http://www.youtube.com/watch?v=2KN1YguHW9M

Please do not hesitate to contact me if you have any questions.

Kind regards,

Sam Hoggard

**Second open evening – November 2014** See notebook file “Year 11 revision”

Focus of presentation was “how to revise for GCSE maths” with opportunity to ask questions. Main points made;

- Students must work hard
- How to organise work (books, folders, online checklist)
- Work area at home
- Equipment needed
- Revision techniques (flash cards, videos, mock papers)
- Visualisation strategy

Parents emailed “Information from revision session”.

**Pre-parents’ evening information**

All parents emailed details of revision information including;

- How students should use checklist

**Survey**

All parents emailed survey to assess impact of this program.

**Summary report**

**Q1. Did you attend first open evening?**

Total responses: 11

Yes – 10

No – 1

Total responses: 10

Very useful – 7

Quite useful – 2

A little – 0

Not at all – 1

Response 1

“A timetable of evening dates – useful for planning”

Response 2

“Make it ‘parent only’ Don’t try to be ‘buddy like’ we r not kids”

**Q4. Did you attend second open evening?**

Total responses: 11

Yes – 10

No – 1

**Q5. Did you find the evening useful?**

Total responses: 10

Very useful – 5

Quite useful – 5

A little – 0

Not at all – 0

**Q6. Any suggestions for improving the evening.**

Response 1

“None”

**Q7. Have you received regular emails?**

Total responses: 11

Yes – 11

No – 0

A lot – 7

Yes – 3

A little – 1

Not at all – 0

**Q9. Any suggestions for improving these emails.**

Response 1

“None”

**Q10. Any other comments.**

Response 1

“The effort you have been putting into your students is fantastic and greatly appreciated. As far as my own individual child is concerned the phrase ‘can lead a horse to water………’ Sadly springs to mind! She is aware of the grade she needs for sixth form do hopefully all your hard work and encouragement will pay off. Many thanks.”

Response 2

“The idea of kids keeping work in an A4 file which is to be bought to each lesson is awful. What is wrong with an exercise book?! Files are too big and bulky – school bags are groaning already especially on PE days. Ditch them or keep them in class.”

Response 3

“Thank you for taking the time to communicate with us parents.”

Response 4

“Very useful to have contact opportunity to find out more. Otherwise info is not fed back from student!”