• Hamilton's Flexible Maths Blocks Explained Open our sample 'Year 5 Autumn Unit 1' to find out how the blocks of units work.

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# Maths Year 6 Autumn Algebra

Each unit has everything you need to teach a set of related skills and concepts. 'Teaching for Understanding' provides whole-class teaching and fully differentiated adult-led group activities. ‘Problem-solving and Reasoning’ develops these skills, and includes questions to enable you to assess mastery. Practice sheets ensure procedural fluency. Extra support activities enable targeted work with children who are well below ARE.

‘UNIT PLAN’ gives you a text version of all parts of the unit to use in your school planning documentation. ‘DOWNLOAD ALL FILES’ gives you that unit plan plus all of associated documents. These bulk downloads are available to friends and School Subscribers. These bulk downloads are added value for Hamilton Friends and School Subscribers.

## Unit 1 Generate and use simple formulae (suggested as 2 days)

### Objectives

Generate and use simple formulae
Unit 1: ID# 6323

National Curriculum
Alg (i)

Hamilton Objectives
36. Use simple formulae, including formulae expressed in words.

### Teaching and Group Activities for Understanding

Day 1 Teaching
Show the garden problem about area. Show how we can use a letter, n, to stand for the width of the garden so the area of the patio is 2n or (2 × n) metres squared. Children then use this to write a formula for the area of the turf. Then alter the width to (n – 1) metres. So, the area of the turf is 3 times (n – 1) or 3(n – 1) square metres. Alter the width again so the area is 3 times (n – 2) or 3(n – 2) square metres
Group Activities
-- Together, write and use formula to work out ages.
-- Support children in writing formulae.

Day 2 Teaching
Choose ‘pairs to fit own rule of the machine’ from topmarks.co.uk. Ask children to close their eyes while you select × 2, + 1 for the first machine, and × 10, – 4 for the second. Each group choose several inputs and study the outputs. Show children how to record a generalisation using n, e.g. 2n + 1.
Group Activities
Use the ‘Stars and Crosses’ in-depth problem-solving investigation below as today’s group activity.
Or, use these activities:
-- Practise inputting numbers into a formula, then guess inputs for given outputs.
-- Play ‘Guardians of the rule’ as a group. Pairs make up 2-step rules for others to guess.

### You Will Need

• Garden sketch (see resources)
• ‘Write a formula’ sheet (see resources)
• Function machine from topmarks.co.uk
• Flipchart and pens
• 1–10 cards

### Mental/Oral Maths Starters

Day 1
1-step function machines (pre-requisite skills)

Suggested for Day 2
Times tables (simmering skills)

### Procedural Fluency

Day 1
Write and use simple formulae.

Day 2
Identify functions and write outputs in terms of n.

### Mastery: Reasoning and Problem-Solving

• If the perimeter of a regular shape is 5 × n, where n is the length of a side, what is the shape? Find the perimeter when n = 6.5cm.
• Darren draws a function machine. It trebles a number and then subtracts 6. Sophie sees that one of the outputs is 15. What was the input? Then Darren inputs 11. What output will Sophie see? Write the formula for Darren’s machine.
• Formula A is 3n.
Formula B is n + 6.
What number can n represent which will make these two formulae equal the same amount?

In-depth Investigation: Stars and Crosses
Children find totals of the numbers in shapes on a 1-100 grid, make generalisations and then write a formula to find the total of any similar shape on the grid.
The PowerPoint can be used to guide your class through this investigation.

### Extra Support

This unit has no separate Extra Support activities.

## Unit 2 Solve equations with two unknowns (suggested as 3 days)

### Objectives

Solve missing number problems and equations with two unknowns
Unit 2: ID# 6329

National Curriculum
Alg (iii) (iv) (v)

Hamilton Objectives
37. Solve missing number problems, including where letters are used to replace constants.
38. Find pairs of numbers that satisfy an equation with two unknowns and list, in order, the possibilities of combinations of two variables.

### Teaching and Group Activities for Understanding

Day 1 Teaching
Write 25 + a = 30 on the board. This is called an equation and ‘a’ stands for a mystery number. Sketch a bar model to show this and find a. Write 6b = 42 explaining that 6b stands for 6 × b. Use a bar model to solve. Repeat with 35 ÷ c = 7 and other equations.
Group Activities
-- Cover numbers with Post-its labelled a, b, c and d to form equations. Children solve the equations, then write similar ones for others to solve.
-- Support children in solving harder equations where some arithmetic needs to be done first to isolate the unknown, e.g. 2a + 4 = 10, 5 + 3b = 17.

Day 2 Teaching
Display:

 x 3 b a 21 56 c 12 32

Explain how the outside numbers are multiplied to give the numbers in the central part of the table. Our challenge is to work out what numbers ‘a’, ‘b’ and ‘c’ represent. What do we multiply by 3 to give 21? So ‘a’ must be 7. Children discuss in pairs what ‘b’ and ‘c’ must be. Repeat with another display.
Group Activities
-- Solve algebraic puzzles from mathplayground.com.
-- Solve a series of equations by finding an unknown and substituting in the next equations.

Day 3 Teaching
Write a + b = 10, explaining that a and b are two new mystery numbers. What could they be? There are lots of possibilities! If a and b are whole numbers, ask children to list the pairs of possibilities. Repeat for c × d = 24. Write 2e + f = 8, and draw a bar model diagram to help children. Children find a pair of whole numbers that will work.
Group Activities
Use the in-depth problem-solving investigation ‘This Pied Piper of Hamlin’ from NRICH as today’s group activity.
Or, use these activities:
-- Use algebra to express word problems, then list possible answers.
-- Find pairs of numbers which satisfy two equations with two unknowns.

### You Will Need

• Mini-whiteboards and pens
• Flipchart and pens
• Post-it notes
• Algebra puzzle from www.mathplayground.com

### Mental/Oral Maths Starters

Day 1
Equivalence (1) (pre-requisite skills)

Day 2
Equivalence (2) (pre-requisite skills)

Suggested for Day 3
Pairs with a total of 10m (simmering skills)

### Procedural Fluency

Day 1
Solve equations, including bar model and geometry examples.

Day 2
Solve algebra puzzles and word problems.

Day 3
List pairs of solutions for pairs of equations with two unknowns.
Find a pair of numbers which satisfy two equations, each with two unknowns.

### Mastery: Reasoning and Problem-Solving

• Both a and b are whole numbers. How many possibilities are there for values of a and b if a + 2b = 13.
• 2a is 5 more than 3b. If a and b are both whole numbers and a < 10, what are the possible values for a and b?
• A number less than 10 is multiplied by itself. The answer is equal to a different number multiplied by 9. What are the possible numbers?

In-depth Investigation: This Pied Piper of Hamelin
Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether! This Pied Piper of Hamelin from nrich.maths.org.

### Extra Support

Mathematical Mysteries
Solving equations using the bar model

## Unit 3 Generate and continue linear sequences (suggested as 3 days)

### Objectives

Generate and continue linear sequences
Unit 3: ID# 6337

National Curriculum
Alg (ii)

Hamilton Objectives
39. Generate, describe and continue linear sequences.

### Teaching and Group Activities for Understanding

Day 1 Teaching
Write _, _, _, 24, _, 36, _, _, _, _. Children work in pairs to find the missing numbers. What is each step worth? (6) Suggest that they count up/down to check. Children write the complete sequences on their whiteboards to show you. Repeat with other sequences.
Group Activities
-- Use Number sequences from topmarks.co.uk. Identify missing terms and the step increase/decrease.
-- Given first and last terms in a sequence, identify step increase/decrease, including sequences which pass through zero, and which increase/decrease by numbers with 1 decimal place.

Day 2 Teaching
Start with 2, 4, 6, 8 ... Show how we can use the relationship between the number of the term and its value in the sequence. E.g. the 10th term is 2 × 10 = 20. Repeat with 3, 6, 9, 12 … then 4, 7, 10, 13, 16 … then 5, 9, 13, 17, 21.
Group Activities
-- Support children in finding the 10th term, 100th term, then nth term in related sequences.
-- Create sequences based on the sequence which follows the rule 10n, adding/subtracting a constant, e.g. 12, 22, 32, 42 … and writing an expression for the general term, e.g. 10n + 2.

Day 3 Teaching
Make a line of shapes from multilink and use these to generate a linear sequence. Find the 10th term, and then discuss how we can describe what is happening and then find the general rule.
Group Activities
Use the in-depth problem-solving investigation ‘Sticky Triangles’ from NRICH as today’s group activity.
Or, use these activities:
-- Whole class investigation of a pattern of shapes, identifying/describing the 10th and nth terms.

### You Will Need

• Mini-whiteboards and pens
• Number sequences from www.topmarks.co.uk
• Flipchart and pens
• ‘Sequences sheet’ (see resources)
• ITP: Co-ordinates

### Mental/Oral Maths Starters

Day 1
Sequences (pre-requisite skills)

Suggested for Day 2
Co-ordinates in 4 quadrants (simmering skills)

Suggested for Day 3
Parts of circles (simmering skills)

### Procedural Fluency

Day 1
Work out the step increase/decrease in sequences in order to complete them. Sequences include those which pass through zero, and which increase/decrease by numbers with 1 decimal place.

Day 2
Children work in pairs to find the 10th, 100th then nth term for sequences.

Day 3
Draw the next spinner/staircase in a sequence. Find a relationship between the number of shapes in the spinner/staircase and its position in the sequence.

### Mastery: Reasoning and Problem-Solving

• Three matches are arranged to make a triangle. How many more need to be added to make 2 triangles? How many more to make 3 triangles? Continue this sequence for 6 triangles. Write the number of matches that it takes to have 12 triangles.
• The fourth term of a sequence is 17, the fifth is 21 and the sixth is 25. Write the next four terms. Then write the first three terms. What is the general rule?
• Write the 10th term in this sequence: 1.5, 4.5, 7.5, 10.5 …

In-depth Investigation: Sticky Triangles
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new 'layer'? Sticky Triangles from nrich.maths.org.

### Extra Support

Spot the Rule
Counting sequences