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# Maths Year 6 Summer Puzzles and Patterns

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 Calculator patterns (suggested as 5 days)

### Objectives

Calculator patterns
Unit 1: ID# 6709

National Curriculum
Fr (including decimals and percentages) (vi) (xi)

Hamilton Objectives
18. Perform mental calculations, including with mixed operations and large numbers; carry out calculations using knowledge of the order of operations and brackets.
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.
20. Solve problems involving all four operations.
24. Associate a fraction with division; calculate decimal fraction equivalents, e.g. 4/5 is 0.8 and 1/8 is 0.125.
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 Teaching
Children enter 1 ÷ 3 = into a calculator. What’s the answer? What fraction is this decimal equivalent to? If we divide 1 into 3 equal parts, each is a one third; we write this as 0.333333; it repeats 3 forever. We call it a recurring decimal. Repeat for 2 ÷ 3 and 1 ÷ 7. Use a calculator to discover other unit fractions (denominator <10) that have recurring decimals.
Group Activities
-- Use a calculator to convert fractions to decimals by division; place the fractions on a 0–1 (or 0–2) line. Aim to get three fractions in a line without an opponent’s fraction in between.

Day 2 Teaching
Children enter 1 ÷ 3 into the calculator but don’t clear it. If you multiply this answer by 3, what should you get? Try it. What happens? The calculator is wrong! This is called a rounding error. Children predict whether the same thing will happen when dividing then multiplying by 4, 5 and 6.
Group Activities
-- Investigate divisors 2 to 10 (or 20) that will give rounding errors on the calculator.
-- Given the answer 1.181818 (13 ÷ 11), use a calculator to work out the mystery division. Set further similar problems for the group.

Day 3 Teaching
Write on the board the following questions: 675 + ☐ = 1000, 47.2 × ☐ = 283.2, 586.1 ÷ ☐ = 24.7, 84 ÷ ☐ = 7, ☐ − 6.795 = 4.615, 25 × ☐ = 1000; Find two consecutive numbers with a product of 2070; Find two numbers greater than 1000 with a difference of 75. Children discuss each question, in pairs, and decide whether or not they would use a calculator. Share reasoning. Set other problems.
Group Activities
-- Use a calculator to solve multiplication problems.
-- Use a calculator, trial and improvement and reasoning skills to aid problem solving.

Day 4 Teaching
Learn how to use the calculator’s memory button. Write the key sequence: 3 + 2 = M+ 3 × 4 = M+ MR. Children enter it. What did the calculator do? Did you notice the little M in the display? Repeat with similar key sequences, using the M+ and M− buttons to add and subtract to/from numbers in the calculator memory. How do we solve calculations with brackets?
Group Activities
-- Practise using the memory functions in multi-step calculations, including word problems.
-- Use the memory function to help solve a shopping-related multi-step calculation.

Day 5 Teaching
Write on the board this sequence of keys: 6 + 4 = M+ 4 + 2 = × MR = and ask children to work out what happens. Repeat with other sequences including division, e.g. 5 − 3 = M+ 6 + 4 ÷ MR =. Use the memory buttons to work out (32 + 35) ÷ (42 + 19).
Group Activities
-- Discuss given calculator key sequences. Compare estimates of answers with actual answers.
-- Use the memory function to help solve an area-related multi-step calculation.

### You Will Need

• Calculators
• IWB calculator
• Whiteboards and pens
• ‘Three in a row’ sheets 1, 2 and 3 (see resources)
• Flipchart and pen
• Coloured pens
• Cards
• ‘Using the memory function’ (see Practice worksheets)
• ‘Using the calculator’s memory’ sheet 1 (see Practice worksheets)

### Mental/Oral Maths Starters

Day 1
Equivalent fractions and decimals (pre-requisite skills)

Day 2
Convert improper fractions to mixed numbers (pre-requisite skills)

Day 3
Approximating (pre-requisite skills)

Day 4
Order of calculations (pre-requisite skills)

Suggested for Day 5
Division facts (simmering skills)

### Procedural Fluency

Day 1
N/A

Day 2
Enter the key sequences into your calculator. Which calculations does the calculator get wrong?!

Day 3
Choose the calculator or other ways to solve other problems.

Day 4
Practise using the memory functions in multi-step calculations, including word problems.

Day 5
Practise using the memory functions by entering given sequences.

### 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 puzzles (suggested as 3 days)

### Objectives

Number puzzles
Unit 2: ID #6719

National Curriculum
Alg (i) (iv)

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.
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 Teaching
I have three different coloured boxes – one is blue, one is red and one is yellow. Inside each box is a cube and a counter, which are a different colour to each other and to the box. There is one blue cube, one blue counter, one red cube, one red counter, one yellow cube and one yellow counter. Write on the board: The red cube and the yellow counter are not in the blue box. Believe it or not, this one clue is enough to solve the whole puzzle! Draw a table of the information and solve the puzzle together.
Group Activities
-- Work together to solve ‘cube, counter and box’ logic problems.
-- Solve colour Sudoku-like problems.

Day 2 Teaching
Draw a triangle diagram (per teaching activities download). Ask children how many triangles they can see, warning them that not all the triangles are the same size! Model a way of drawing coloured triangles over the top of the diagram to keep track.
Group Activities
Whole class investigations:
-- What numbers of triangles can be left by removing 2, and then 3 straws from the given arrangement?
-- Investigate the numbers of triangles or squares (all different sizes) that can be made by adding successive layers to a diagram.

Day 3 Teaching
Display a 3 × 3 grid. Explain how this magic square (15) works. Agree that if 5 is placed in the middle, the numbers on either side of it must total 10. Write 9 in the top left-hand corner. Continue to work through and agree that having 9 here doesn’t work. Discuss where else the 9 could be placed.
Group activities
-- Find solutions to a magic square, using digit cards. Children discuss what they notice about each square.

### You Will Need

• A red counter and yellow cube in a blue box (or box with blue on the lid), a blue cube and yellow counter in a red box, and a blue counter and red cube in a yellow box
• Blue, green, red and yellow cubes
• ‘Logic puzzles’ (see Practice worksheet)
• ‘Logic puzzles’ pre-drawn tables (see resources)
• ‘Triangle teaser’ (see resources)
• Chopped straws or spent matches
• ‘Magic squares’ (see resources)
• Number cards 1–9

### Mental/Oral Maths Starters

Suggested for Day 1
Spotting relationships (simmering skills)

Suggested for Day 2
Factors (simmering skills)

Day 3
Number facts (pre-requisite skills)

### Procedural Fluency

Day 1
Colour cube, counter and box logic puzzles.

Day 2
How many rectangles can you see in the diagram? Make sure to count those of different sizes!

Day 3
N/A

### 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 Number patterns (suggested as 2 days)

### Objectives

Number patterns
Unit 3: ID # 6737

National Curriculum
PofS (iv)

Hamilton Objectives
11. Multiply 2-, 3- and 4-digit numbers by numbers up to 12 using short multiplication or another appropriate written method.
16. Divide numbers with up to four digits by a number up to 12 using short division and giving an appropriate answer.
17. Divide numbers with up to four digits by 2-digit numbers using a formal written method of long division and giving an appropriate answer.
53. Identify, illustrate and name parts of circles, including diameter, circumference and radius, understanding that the radius is half the diameter.
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 Teaching
Children draw a circle with a 6cm radius, then divide the circumference into 24 equal parts, numbering the points, 1 to 24. They join each point to its double. When they get to 13, ask children to join 13 to 2, because 26 is 2 more than 24. When they get to 14, they join 14 to 4 (as 28 is 4 more than 24) and so on until they get back to where they started. Discuss the resulting pattern.
Group Activities
-- Investigate patterns made by joining multiples around the circumference of a circle. Reason about observations.

Day 2 Teaching
Revise the skills of multiplication and division that will be useful in today’s investigations:
Write on the board 555, pointing out that this 3-digit number has all digits the same. Ask children to add the three digits, then work out 555 ÷ 15, estimating first. Show, using long division. Remind children that they can use short division to divide by small 2-digit numbers such as 11 and 12, or by multiples of 10.
Calculate 1089 × 6, using grid or short multiplication.
Group Activities
-- Use multiplication by 1089 to generate a sequence of answers; reason about patterns in the sequence.
-- Use short or long division to investigate properties of divisibility.

### You Will Need

• ‘Times table patterns’ (see resources)
• Rulers
• Whiteboards and pens
• ‘Amazing 1089’ (see resources)
• ‘Amazing 6720’ (see resources)

### Mental/Oral Maths Starters

Day 1
Division facts (pre-requisite skills)

Day 2
Factors and products (pre-requisite skills)

### Procedural Fluency

This unit has no separate practice sheets.

### 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.