Maths Year 6 Summer Puzzles and Patterns

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.

• 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)

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)

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.

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

This unit has no separate Extra Support activities.

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.

• 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

Suggested for Day 1
Spotting relationships (simmering skills)

Suggested for Day 2
Factors (simmering skills)

Day 3
Number facts (pre-requisite skills)

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

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

This unit has no separate Extra Support activities.

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.

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

Day 1
Division facts (pre-requisite skills)

Day 2
Factors and products (pre-requisite skills)

This unit has no separate practice sheets.

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

This unit has no separate Extra Support activities.