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.FFP. t:C}$.:0:1% ?!v?.. . Represent and interpret sequences, patterns and relationships involving numbers and shapes; suggest and test hypotheses; construct and use simple expressions and formulae in words then symbols (e.g. the cost of c pens at 15 pence each is 15c pence)
(Objective repeated in Block B Units 1, 2 & 3)Draw this shape on the board.
Ask the class what the area of the shape is. Ask two volunteers to draw 2 other shapes with the same area. Get the class to find the perimeters of the 3 shapes.Ask the children to try and draw a shape of the same area with the maximum perimeter and a shape with the minimum perimeter.Ask the children how they tried to maximise or minimise the perimeter.
Q What can you say about the areas of these shapes?
Q Do shapes with the same area have the same perimeter?
Show the class three rectangles (as on Activity sheet Y6 32): A (10 cm by 16 cm), B (8 cm by 10 cm), C (5 cm by 8 cm). Ask children to find the area of each rectangle. Use A and B to blutack to the board to form a compound shape such as:
Q What has changed? Why?
Emphasise that while the areas are the same the perimeter is different. With the class find the perimeters of each shape. Show the class how to record this in their books. Using rectangles A, B and C get the children to make compound shapes of their own, record the areas and find the perimeters.
Collect responses and correct any misunderstandings and mistakes.Write 3 numbers on the board such as 10.2, 10.4, 10.6. Tell children these are 3 consecutive terms in a sequence. They are to continue it first forwards then backwards. Take responses.
Choose a starting number and get children to count upwards in steps of 0.25.
Q If my start point is 2 and my steps are 0.25 what is the 6th term in my sequence?
Collect childrens strategies, compare them and discuss other examples using the vocabulary: term, step, sequence.
Ask individual children for start numbers and steps. Practise using these examples.
Model a sequence on board10, 25, __ , __ , 70, __ ,
Q What do you think this sequence is stepping in?
Q How did you arrive at this decision?
Q How would you check your idea?
Ensure children are checking their ideas by matching the given sequence numbers. Use Resource sheet Y6 33 as a reference.
Children to generate their own number sequence on squares of paper.
Arrange sequence in order with some pieces of paper face up and some face down.
103
112
118
Partner to identify the missing numbers and explain the rule.
Each child to generate or solve five number sequences.Generate the numbers 1, 3, 6 by asking questions.
Ask the question:
Q What would be the next number in this sequence?
Explain your reason to your partner.
Take feedback and encourage children to explain their reasoning.
Give children counters and ask them to make a pattern to explore the sequence.
Invite children out to show pattern and explain how the patterns are developing. Ask them to describe what they see.Encourage children to look at what they already know to help them find the next number in the sequence.Ensure the development of the pattern is systematic.e.g.First triangular number1Second triangular number1 + 2 Third triangular number1 + 2 + 3
Q How would the pattern look for the 4th term?
Allow time for children to explore the sequence. Encourage children to record using counters.Show this sequence of squares and circles
1st term 2nd term 3rd term
Q What would the 4th term look like?
Ask the children to describe the sequence. Ask them to describe to a partner what they see.
Q How many squares would there be in the 9th term?
Q How many circles would there be in the 9th term?
Q How did you work that out?
Establish that there is 1 square for the number in the sequence and 3 circles for every square.
Record in a table. Explain how the number sequence extends.
No. of term
1st
2nd
3rd
4th
5th
Squares
1
2
3
4
5
Circles
3
6
9
12
Q How would knowing this help us to work out the 100th term in the sequence?
Give the children circles and squares. Invite them to create their own simple pattern, and repeat to generate a sequence.
Complete a blank table to show their sequence.
No. of term
1st
2nd
3rd
4th
5th
Squares
Circles
Q What would the 50th number in your sequence be?
Generate the first three terms of the sequence using different coloured counters.
Ask children to extend the sequence and explain reasoning (4th term).
Q What is the relation between the 2nd term and the 3rd term?
(Prompt: add one more coloured counter to each arm.)
Q What would the 10th term look like?
Allow time for groups to explore.
Q How many counters are there in each term from 1st to 10th?
Record feedback in a chart.
Number of term
Total number of counters
1
1
2
4
3
7
Explain that now we are going to look at the relationship between the numbers. Look at the term number and the number of counters.
Q Is there a relationship?
Children may suggest x 1 x 2Explore if the pattern is common to all terms. Look at multiplying by 3 and adjusting. Relate this back to prompt questions by adding three to the previous number.
Q If I multiply by 3 how do I need to adjust the number?
Q Does this rule apply to all terms? Look for x 3 and 2 as the rule.
Encourage the children to predict and then test with the results they have generated.
Use the rule to find the 100th term. Encourage children to verbalise rule.Display an enlarged copy of Activity sheet Y6 34.
Work through each pattern in turn with children contributing the rule, the given terms in the sequence and expressing the rule, firstly in words and then as a formula.
Set the problem:
In groups of four, investigate how many Christmas cards there will be if everyone in the group sends and receives a card from every one in the group.Ask the children to investigate how many Christmas cards would be sent. (Prompt what if there were 2, 3, 4, 5, 6 children in the group). Take feedback.
Q How did you keep track of the number of cards?
Discuss the different ways of recording.
Ask the children to investigate how many Christmas cards would be sent if there were 2, 3, 4, 5, 6 children in the group?
Record the numbers of Christmas cards on the board.
Number of people: 2 3 4 5 6
Number of cards: 2 6 12
Encourage the children to predict the number of cards if there were 10 people in the group.
Write 5, 10, 15 on the board. Ask children to describe these numbers. Establish that they are the first three consecutive multiples of 5.
Q What is the sum of these numbers?
Collect answers and discuss the childrens strategies.
Q Can you propose a shortcut to find their sum?
Encourage different suggestions:
Three times 10; two times 15; six times 5.
Q Will these shortcuts always work?
Get children to work in pairs, finding the sums of the first three consecutive numbers in different times tables and testing the shortcuts each time.
Q Did they all work? How would you describe the shortcuts for someone not in your class?
Encourage children to be precise. Record their suggestions and refine them e.g. for the first three consecutive numbers in a times table, their sum is six times the first number.
Q Could we use symbols to describe our shortcuts?
Suggest the three consecutive numbers are f, m, l for first, middle, last. Encourage children to begin to develop formulae. Sum = 6f, Sum = 3m, Sum = 2l. Explain that these are formulae and you now want the children to make up their own formulae for other sums of consecutive numbers.
Encourage some children to explore four consecutive numbers.Think about and discuss how children could have arrived at these answers.
258 6 = 43; 356 4 = 89; 144 3 = 48;
Discuss the strategies the children used.
Q What key bits of information helped most?
Discuss importance of knowing tables and recognising how to use these facts as clues.
Write on the board:
1 3 x = 5535
Q What number facts might help us to solve this problem?
Draw out the following and write them on the board:
knowledge of multiplication facts to look at last digits;
Q Do we know how the second number ends?
the odd and even outcomes of multiplication;
tests of divisibility.
Q Which tests of divisibility could we use? (3, 9)?
Discuss how children might use these and work through the problem with the class. (Answer is: 123 x 45)
Children work on following problem in pairs.
9 6 x 8 = 7 8 Answer is: (96 x 83)
Discuss the clues and strategies used.
Set more puzzles, e.g.:
3 x 7 = 2520 (35 x 72)
6 x 4 = 10 148 (236 x 43)
Collect answers and discuss strategies, correct any errors and misunderstandings.
Ask children to invent a problem for their partner, swap these and solve them.
On the board write the sequences:4, 8, 12, 16 5, 9, 13, 17
Q What are the next four numbers in these sequences?
Correct answers and record them on the board.
Q How would you describe the first sequence?
Draw out the fact that the sequence is the 4 times table.
Q Can we state a rule in words?
Encourage children to provide alternatives; add 4 each time, find the next multiple of 4 etc.
Q Can we write a formula for the sequence?
Write n on the board and say that this stands for any number. Establish that the numbers in the sequence can be represented as 4n.
Q How can we describe the second sequence?
Use the first sequence to agree that the terms can be represented as 4n + 1. Discuss the rotation and emphasise that replacing n by 1, 2, 3, etc. generates the sequence.
Repeat for other pairs of linked sequences that help children to derive formulae.
Write on the board: n + 6 = 8.
Q What number is n representing?
Agree n = 2 and emphasise that this time n stands for a particular number not any number.
Q If n + 3 = 9, what is n + 5?
Collect the childrens answers and discuss their methods. Contrast just adding 2 to both sides with using n = 6 to work out n + 5. Repeat with other questions.
Write on the board:
Q What numbers go in the two boxes?
Collect answers and discuss strategies. Talk through the earlier work and encourage the children to make jottings when they answer questions like these. Repeat using different numbers.
Write s + t = 60.
Q If s and t stand for whole numbers, what could they be?
Collect different cases for s and t and ensure the children can interpret the letters.
Write a + b = 50 and a is 10 less than b.
Q What whole numbers could a and b stand for?
Encourage the children to make jottings. Collect answers and discuss the strategies they need. Offer stories to help them see that a + b = 50.
Q If we were asked to show our working, what would we write?
Establish that the diagram and/or a calculation, e.g. a + a = 40, a = 20 would be enough. Repeat using different letter, equations and conditions.
Introduce a range of test questions involving sequences and use of letters. Discuss the questions with the children. Identify which questions come from a test where calculators are allowed. Encourage the children to make jottings and annotate the questions.Write a number on a whiteboard, which the class cannot see. Say If I multiply this number by 8 the answer is 56.
Q What is the number on my whiteboard?
Collect answers. Repeat using the 4 operations, include double and halve.
Write a number on a whiteboard; say If I increased this number by one quarter the answer is 50. Answer 40.
Q What is the number on my whiteboard?
Collect answers. Repeat using increases and decreases by fractions and percentages.
Set the children to work in pairs posing one another similar questions.
Write two numbers on a whiteboard.
Say, if I add 4 to one number and I subtract 7 from the other, the answers are the same.
Q What are the two numbers on my whiteboard?
Establish there are many answers but that the two numbers are 11 apart. Give the children some properties of the two numbers, e.g. one is a prime number, the other has 3 as a factor. Collect answers and confirm they meet the properties. Repeat using other pairs of numbers and different properties.
Set the children to work in pairs posing one another similar questions.ACTIVITY SHEET Y6 32
RESOURCE SHEET Y6 33
(A) 10, 25, 70,
(B) 1, 4, 36,
61, 42, 23,
(D) 6, 13, 20
(E) 0.9, 2.3, 3.7,
(F) 22, 11, 44
ACTIVITY SHEET Y6 34
Number of term123451020100nDots2581114
Number of term123451020100nDots814202632
Number of term123451020100nDots717273747
Number of term123451020100nDots11357
Number of term123451020100nDots914192429
Choose any two tables
Complete the sequence
Write a rule in words and / or a formula
 PAGE 9 
A
B
Ask for the area of the compound shape. Then make another shape such as:
A
B
n
3n + 1
10
22
50
a
b
a
10
A
10 cm by 16cm
B
8cm by 10cm
C
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