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# Maths Year 6 Summer Exploration in Maths

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 Explore a million (suggested as 2 days)

### Objectives

Explore a million
Unit 1: ID# 6761
National Curriculum
Num/PV (i) (iv)
Meas (i) (ii)

Hamilton Objectives
4. Solve number and practical problems involving place value, rounding and negative numbers.
19. Use estimation to check answers and determine an appropriate degree of accuracy; round answers to multiplications and divisions to a specified degree of accuracy.
40. Use, read and write, and convert between, standard units including miles and kilometres, using decimal numbers with up to three places as appropriate.
41. Solve problems using standard units and converting between them.

### Teaching and Group Activities for Understanding

Day 1
Read the book How Big is a Million, showing the poster of a million stars. Place children into ten groups. If we produce a picture of a million altogether, how many does each group need to produce? Discuss strategies such as copying an array of 10 by 10 dots, or 100 by 100 dots.
Group Activities
-- Whole class activity: Reason about the size of the number 1 million. Use place value and ICT skills to accurately create 10 lots of 100,000 ‘objects’.

Day 2
If a million people were to stand in line, one behind the other, how long might the line be? Discuss how you might work this out, then show the answer of a map of Britain to give children a sense of the distance. How much might a million lentils weigh? How could we find out?
Group Activities
-- Reason about and calculate the height of a stack of 1 million sheets of paper.
-- Complete one of several ‘one million’ reasoning challenges.

### You Will Need

• How Big is a Million by Anna Milbourne and Serena Riglietti
• Computer access to at least ten desktop computers/laptops in the classroom
• Map of Britain: preferably on the IWB
• Calculators, lentils and electronic weighing scales
• Rulers and two reams of paper
• Tape measures and ten £1 coins

### Mental/Oral Maths Starters

Day 1
Place value (pre-requisite skills)

Day 2
Measures (pre-requisite skills)

### Procedural Fluency

Day 1
Work out the area occupied by 1,000,000 dots if 10 dots fit in 1cm². Explore related questions.

Day 2
None

### Mastery: Reasoning and Problem-Solving

This unit has no separate problem-solving activities or investigations.

### Extra Support

This unit has no separate Extra Support activities.

## Unit 2 Number games and puzzles (suggested as 3 days)

### Objectives

Games and puzzles
Unit 2: ID# 6779

National Curriculum

Hamilton Objectives
5. Consolidate: Add and subtract mentally with confidence, where numbers are <100 or it relies upon simple addition/subtraction and place value.
18. Perform mental calculations, including with mixed operations and large numbers; carry out calculations using knowledge of the order of operations and brackets.
55. Begin to reason mathematically making simple generalisations, using mathematical language and making connections between mathematical ideas.

### Teaching and Group Activities for Understanding

Day 1
Four dominoes are arranged in a square so that ends match. The total of all the spots is 20. The dominoes have up to six spots on each side. Work with a partner to draw, on your whiteboard, what the dominoes might be. Ask pairs of children to share their ideas and how they went about solving the problem. Repeat for a total of 30.
Group Activities
-- Children explore a choice of several domino investigations.

Day 2
Use ITP 20 cards. Select eight random cards. Reveal the first three cards. You can use any operation and these single-digit numbers in any order to make 34. Take feedback. If it is not possible, click on the first card to turn it over, and click on the fourth card to reveal it. Children use these three numbers. Keep going until you can make 34!
Group Activities
-- Use four given digits, any operations and brackets to make single-digit answers.
-- Play a game using digit cards, any operation and brackets to make given 2-digit numbers.

Day 3
A pentomino is a shape made from five touching squares. Draw on the board one pentomino as an example. There are 12 different pentominoes. Give each child a piece of cm-squared paper. Work with a partner to see if you can come up with all 12. No pentomino is a reflection or rotation of another. Agree the whole set.
Group Activities
-- Children explore using pentominoes to make rectangles.
-- Children explore area relationships using pentominoes.

### You Will Need

• Whiteboards and pens
• A complete set of 0–6 dominoes per group
• ‘Domino squares’ sheet (see resources)
• 20 cards ITP (see resources)
• Set of four lots of number cards 1–9 (or sets of playing cards with the pictures and tens removed)
• ‘Card games’ (see resources)
• cm² paper
• ‘Pentominoes’ colourful and black and white versions provided (see resources)

### Mental/Oral Maths Starters

Day 1
Number facts and reasoning (pre-requisite skills)

Day 2
Mental calculation (pre-requisite skills)

Day 3
2-D shapes (pre-requisite skills)

### Procedural Fluency

Day 1
None

Day 2
Card games using mental arithmetic, different operations and brackets.

Day 3
None

### Mastery: Reasoning and Problem-Solving

This unit has no separate problem-solving activities or investigations.

### Extra Support

This unit has no separate Extra Support activities.

## Unit 3 History of maths (suggested as 5 days)

### Objectives

History of maths
Unit 3: ID #6791

National Curriculum
Alg (i)

Hamilton Objectives
4. Solve number and practical problems involving place value, rounding and negative numbers.
9. Identify prime numbers up to 20.
19. Use estimation to check answers and determine an appropriate degree of accuracy; round answers to × and ÷ to a specified degree of accuracy.
36. Use simple formulae, including formulae expressed in words.
55. Begin to reason mathematically making simple generalisations, using mathematical language, making connections between ideas.

### Teaching and Group Activities for Understanding

Day 1
Each civilisation came up with its own number system, or number systems spread with trade. Explain and discuss Roman, Hindu and Mayan number systems.
Group Activities
-- Whole class activity: Children explore writing numbers using different systems. Compare the merits of each system.

Day 2
Highlight the prime numbers on the ITP Number grid. Christian Goldbach (1690–1764) thought that every even number greater than 4 could be made by adding 2 prime numbers (and that every odd number more than 5 could be made from adding three prime numbers), but he couldn't prove it. Centuries later, mathematicians all over the world are trying to prove or disprove his conjecture…
Group activities
-- Whole class investigation: Children collaborate to investigate Goldbach’s conjecture.

Day 3
Show children a set of Napier’s rods and demonstrate how they work. A similar method known as the Gelosia method was used in India in the 12th century, and may have been used even before then. Demonstrate drawing the grid for 34 × 26, then solving the calculation. Consider reading about Charles Babbage’s ‘analytical adding machine’, and how Ada Lovelace wrote what is now considered to be the first ever computer program for it.
Group activities
-- Explore the use of Napier’s rods/bones for multiplication.

Day 4
Write the first six terms in the Fibonacci sequence: 1, 1, 2, 3, 5, 8. Children write the next 4 terms. Share the mathematical strategy for finding the total of 10 consecutive numbers in the sequence. You multiply the 7th number by 11. Show how Pascal’s triangle is made. Can you see any patterns in this arrangement of numbers?
Group activities
-- Continue to create a version of Pascal’s triangle. Explore underlying number patterns.
-- Explore Fibonacci-style sequences; test a strategy for adding ten consecutive numbers in the sequence.
-- Shade multiples on Pascal’s triangle to reveal underlying patterns.

Day 5
Who was Pythagoras? Children draw, on cm-squared paper, a (3-4-5) right-angled triangle. Explain Pythagoras’ theorem: a2 + b2 = c2. Children check that the theorem works for their triangles. 3, 4, 5 is a special set of numbers that fit Pythagoras’ theorem because they are all whole numbers. Not many right-angled triangles are made from whole numbers in this way! One of our challenges today asks you to try to find some more.
Group activities
-- Whole class investigation: Children explore right-angled triangles on cm-squared paper, and attempt to discover ‘Pythagorean triples’.

### You Will Need

• ‘Different number systems’ sheets 1 to 3(see resources)
• ITP Number grid
• ‘Goldbach’s Conjecture’ (see resources)
• Large sheets of paper for group work
• Large copies of ‘Napier’s rods/ bones’ (see resources)
• ‘Napier’s rods/ bones’ (see resources)
• ‘Pascal’s triangle: blank’ (see resources)
• ‘Pascal’s triangle: completed’ (see resources)
• ‘Pythagoras’ theorem’ (see resources)
• cm-squared squared paper
• Rulers, scissors and calculators
• Whiteboards and pens

### Mental/Oral Maths Starters

Day 1
Japanese numbers (pre-requisite skills)

Day 2
Factors and prime numbers (pre-requisite skills)

Day 3
Multiplication and division facts (pre-requisite skills)

Day 4
Sequence stick-man (pre-requisite skills)

Day 5
Square numbers (pre-requisite skills)

### Procedural Fluency

Day 1
Work out how to say 3-digit numbers using Japanese and how to write 3-digit numbers using Chinese numerals.

Day 2
None

Day 3
Solve ‘Gelosia’ 2-digit × 2-digit and 2-digit × 3-digit multiplications.

Day 4
None

Day 5
None

### Mastery: Reasoning and Problem-Solving

This unit has no separate problem-solving activities or investigations.

### Extra Support

This unit has no separate Extra Support activities.