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Algebra and Ratio Unit 1

Whole Class Teaching Input DAY 1

Objectives Y5: Begin to understand and use simple formulae.Y6: Understand and use simple formulae.

Resources ‘Garden sketch’ (see resources)

Teaching Teaching with Y5 and Y6 Show this problem on the board: A company is building houses on

different plots. The back gardens are 5m long, but they will be different widths. Each will have a patio area the width of the plot, and 2m long. The rest of the garden will be turfed (see resources).

Explain that we can use n to stand for the width of the garden, so the patio area is 2 × n metres squared, which we can write as 2n m2

for short. Write a formula for the area of the turf. (3n metres squared or 3n

m2) The company have decided to leave a 1m border along one side for

a flowerbed, so we are going to calculate the new area of turf. Sketch in the border.

What is the width now? Point out that the width is now (n – 1) metres.

So, what is the area of the turf? It is 3 times (n – 1), where 3 is the length of the turf and n – 1 the width. We can write this as 3(n – 1) square metres or 3(n – 1) m2.

Ask children to discuss what would be the area of the turf if they left a border of 1m along each side. It would be 3(n – 2) m2.

© Original resource copyright Hamilton Trust, who give permission for it to be adapted as wished by individual users. teach-activs_alg-ratio_56650

Y5 Group Activity notes DAY 1 Y5 Group Activity notes DAY 1Begin to identify rules for function machines. Write and use a formula to calculate ages. Objectives: Begin to understand simple formulae. Objectives: Begin to understand and use simple formulae.You will need: ‘Function machines 1’ (see resources) You will need: Mini-whiteboards and pensGroups of 6 – with TA Working towards ARE

Ask children to discuss in pairs what they think the first function machine does to each number.

Take feedback and describe this in words. It multiplies each number by 5. We can shorten this using algebra. We can write the output as 5n, which means 5 lots of any number you put in: 5 × n. Write 5n as the output for n.

Children, in pairs, identify what the next function machine does to each input. Agree that it subtracts 2.

Show how we use n to represent any input, and the output as n – 2, i.e. any number you put in, subtract 2.

Ask children to discuss the next function machine in pairs. If children are stuck, say that this is a two-step function machine.

Agree that it multiplies any number by 10, then adds 1. Ask children to check that this rule works for each input.

Show how we can write this using n to represent any input. 10n + 1, i.e. multiply n by 10, then add 1.

Challenge children to identify the two steps of the last function machine and to then to write this using n to represent the input. Agree the function as 2n – 1.

Groups of 6 – with TA Working at ARE/Working at Greater depth

David is 7, his sister Annie is 2 years older than him. How old will Annie be when David is 10? 20? 50? n years old? Agree that we can express Annie’s age as n + 2, where n is David’s age.

David’s brother is 3 years older than Annie. How old will his brother be when David is 10? 20? 50?

Children, in pairs, write a formula to find the brother’s age, where n is David’s age. (n + 5)

David also has a sister, Mai, who is 4 years younger than him. Write a formula to find Mai’s age where n is David’s age. (n – 4)

Say that David was born in September, so at Christmas he is his age in years plus 3 months. This Christmas he will be 7, so how can we calculate his age in months? What will his age in months be next Christmas? What will his age be in months the Christmas after he has his 10th birthday? What about when he is n years old? (12n + 3).

Ask pairs of children to write a formula to calculate their own age in months (preferably pairs children with different ages, i.e. 9 and 10). Compare answers.

Outcomes: I can understand and use simple formulae with support.

Outcomes: I can understand and use simple formulae with support.


Y6 Group Activity notes DAY 1 Y6 Group Activity notes DAY 1

Write formulae for questions. Play ‘Guardians of the rule’ as a group (a game to deduce 2-step rules). Objectives: Understand and use simple formulae. Objectives: Make generalisations. Understand and use simple formulae.You will need: ‘Write a formula’ (see resources) You will need: Mini-whiteboards and pens, ‘Function machines 2’ (see resources)Groups of 6 – with T Working towards ARE

Ask pairs of children to complete the first three questions on the ‘Write a formula’ activity sheet (see resources). Encourage them to describe the relationship in words before using letters.

For question 4, make a sketch showing 3 fence panels to help children see the relationship between the number of fence posts and fence panels. Then children write a formula using n. If there were 3 panels why do you need 4 posts? Why are the number of fence posts always more than the number of panels? So, what if there are n panels?

Together, complete question 6, pointing out that we need to add 20 to 45n, i.e. the formula for the number of minutes to cook the chicken is 45n + 20, where n is the weight in kilograms.

Support pairs of children as they solve other questions.

Groups of 6 – with T Working at ARE/Working at Greater Depth

Draw a table with headings ‘Follows my rule’ and ‘Doesn’t follow my rule’. Write 2, 5 under the 1st heading. Explain that you do one thing to the 1st number, then another to get the

2nd number (× 3, then − 1). Do not tell them this rule. Children try to deduce the calculation steps by suggesting pairs of

numbers which they think follow your rule, e.g. if they think the rule is double, add 1, they might suggest 5, 11 or 10, 21.

Write suggestions in the appropriate column. When children think they have guessed the rule, they must not shout it

out, but instead suggest pairs of numbers to make the rule more obvious to others, e.g. 10, 29, or 100, 299, or 0, −1.

When most children have guessed, ask them to describe the rule using words.

How can we use n to describe the rule? Record 3n – 1 where n is the 1st number.

Working at Greater depth If time, pairs of children could also complete the ‘Function machines 2’ activity sheet (see resources).

Outcomes: I can understand and use simple formulae.

Outcomes: I can make generalisations. I can understand and use simple formulae.



Objectives Y5: Express missing number problems algebraically.Y6: Express missing number problems algebraically. Find pairs of numbers that satisfy an equation with two unknowns; enumerate possibilities of combinations of two variables.

Resources Mini-whiteboards and pens

Teaching Teaching with Y5 and Y6 Write 25 + a = 30 on the board. Explain that this is called an

equation and a stands for a mystery number. Sketch a bar model to show this. What is a?

Write 6b = 42 (explaining that 6b stands for 6 lots of b, or 6 × b). What is b? Sketch a bar model to show this:

If 6 times something is 42, then the something must be 7. Repeat with 35 ÷ c = 7. In this case c represents the number in each

of the 7 bars in 35. We can think of this as 7 lots of something makes 35.

Write 3 × 5 = 17 – d on the board. Give children a chance to think through a strategy, then agree that they need to first calculate 3 × 5, and then, as 15 = 17 – d, we can see that d = 2.

Repeat with 3e + 1 = 18 – 5. Which part can we calculate first? What else do we know?

Sketch a triangle, labelling the angles 45°, 90° and a°. What is the total of the angles inside a triangle? How can we find a?

Repeat with a quadrilateral and angles 80°, 60°, 120° and b°, reminding children that angles inside a quadrilateral add up to 360°.

Further teaching with Y6 (Y5 continue independently or with a TA) Write a + b = 10 on the board, explaining that a and b are two new

mystery numbers. Children discuss what they might be. Draw out that there are lots of possibilities! If a and b are positive whole numbers, ask children to list some pairs of possibilities.

Repeat for c × d = 24, again discussing what the possibilities might be if c and d are whole numbers. Ask children to list these on their whiteboards. This time, children can list all the pairs of possible whole numbers.

Write 2e + f = 8, and draw a bar model to help children visualise what this means:


3025 a

42b b b b b b

Ask children, in pairs, to find a pair of whole numbers that will ‘work’. Encourage them to test out their ideas by substituting for the letters, e.g. if they think 3 and 2 will work, they find 2 × 3 + 2 = 8. So, e could equal 3 and f equal 2. Could e equal 2 and f equal 3? Try it! Draw out other possibilities.


82e f

Y5 Group Activity notes DAY 2 Y5 Group Activity notes DAY 2Deduce the numbers represented by letters in a multiplication grid. Solve equations with missing numbers, then write similar equations. Objectives: Express missing number problems algebraically. Objectives: Express missing number problems algebraically.You will need: Mini whiteboards or paper You will need: Flipchart and pens, sticky notesGroups of 6 – with TA Working at ARE/Working at Greater depth

Display: × 3 ba 21 56c 12 32

Explain how the outside red numbers are multiplied to give the black numbers in the central part of the grid. Our challenge is to deduce what the letters a, b and c represent.

Do you have any thoughts about how to deduce the missing numbers? Give children some time to discuss. What do we multiply by 3 to give 21? Write 3a = 21. So a must be 7. Say that we always write the number before the letter, i.e. 3a not a3.

Explain that we can now write 7b = 56, instead of ab = 56 (explaining that ab stands for a × b).

Children discuss in pairs what b must be. They discuss how they can find the value of c.

Ask children to say the value of b. Then ask pairs to explain their reasoning for how they found the value of c, e.g. b is 8, so 8c = 32, therefore c is 4.

Display the following and ask children, in pairs, to agree the numbers represented by letters d, e and f.

Groups of 6 – with TA Working towards ARE

In advance, write 27 + 3 = 30; 7 × 5 = 35; 100 – 35 = 65 and 45 ÷ 5 = 9 on the flipchart, but cover the red numbers with sticky notes labelled a, b, c and d.

Explain that a, b, c and d represent mystery numbers in these equations. Children’s task is to deduce what numbers the sticky notes are hiding.

Point to the first equation What do we add to 27 to make 30? 3! So, a is 3. Remove the sticky note so children can see.

Pairs of children then deduce the values of b, c and d. Take feedback from one pair. They remove the b sticky

note to check. Another pair give their answer for c, removing the sticky note to check, and a third pair do the same for d.

In pairs, children then write their own equations, one for each operation. They rewrite them with a, b, c and d replacing a number in each of the equations. They swap with another pair to deduce what the letters represent. They swap back to check the other pair’s answers.

Outcomes: I can deduce the numbers represented by letters in multiplications.

Outcomes: I can solve simple equations.


× d ef 60 368 40 24

Y6 Group Activity notes DAY 2 Y6 Group Activity notes DAY 2

Solve a series of equations by finding successive unknowns. Find pairs of numbers which satisfy two equations with two unknowns.Objectives: Express missing number problems algebraically. Objectives: Find pairs of numbers that satisfy two equations.You will need: Mini-whiteboards and pens You will need: Flipchart and pensGroups of 6 – with T Working towards ARE/Working at ARE

Write the following equations on the flipchart:a + 15 = 20 a = ab = 40 b = c ÷ b = 2 c = d – c = 24 d = de = 120 e =

ae = 15 check! Ask children to deduce what a represents in the first equation. Say that a represents the same number in the second equation.

We use 5 instead of a to calculate what b represents, i.e. 5b = 40. Ask pairs to calculate b, use this number in the third equation to

calculate c, use this in the next equation and so on. The last equation is a check. If children’s answers for a and e do

not multiply to make 15, they have made a mistake somewhere.

ChallengeCan children create a similar chain of equations (start with 2 or 3 equations in the chain, before extending to a longer series)?

Groups of 6 – with T Working at Greater Depth

Write a + b = 14 and ab = 40 on the board. Say that a represents the same number in both equations, and b represents the same number in both equations. Ask children to discuss in pairs what a and b might be. What strategies are you using?

Take feedback and draw out that there are lots of possibilities for a + b = 14, but if a × b also equals 40, then only 10 and 4 will ‘work’, so either a = 4 and b = 10 or vice versa.

Write c – d = 4 and cd = 12. Ask children to discuss how they could go about calculating out what c and d represent. Take feedback and suggest that it might be better to start with cd = 12 as if c and d are whole numbers (then there are not many combinations, but there are an infinite number of possibilities for c and d in c – d = 4)

In pairs, children find a pair of numbers which ‘work’ in cd = 12 and in c – d = 4.

Repeat for 2(a + b) = 10 and a – b = 3.

ChallengeCan children create a similar pair of equations with two unknowns?

Outcomes: I can solve simple equations involving unknown quantities.

Outcomes: I can find pairs of numbers which satisfy equations with two unknowns. I can find pairs of numbers which satisfy pairs of equations.



Objectives Y5: Generate and describe linear number sequences.Y6: Generate and describe linear number sequences.

Resources Mini-whiteboards and pens

Teaching Teaching with Y5 and Y6 Write: 2, 4, 6, 8 …. What is the value of the next ‘term’ in this

sequence? What is the 10th term? And the 100th? You didn’t have time to count on in 2s to 200, so how did you calculate it? Together, note that the difference between terms in this sequence is 2, because this is the 2 times table.

Agree we can multiply 10 by 2 to find the 10th term or multiply 100 by 2 to find the 100th term. So, how can we find the nth term? Draw out that nth term is 2n: 2 times the number of the term.

Display 1, 3, 5, 7, 9 …. Look at the terms like this:

number of term in sequence (n)

value

1 12 33 54 75 9n ?

o The values in this pattern also increase by 2 each time, but it’s not the 2 times table. How is it different? Describe what to do in words. [multiply the number of the term by 2, then subtract 1]

o Now, can you write that using n to represent any number?[the nth term is calculated using the formula 2n – 1]

Repeat with 3, 6, 9, 12, 15 …. What is the rule for the sequence? Can you predict the 10th, 100th, then nth terms?

Now write: 4, 7, 10, 13, 16 …. How is this related to the sequence we just looked at? Agree it is 1 more than the values in the 3 times table (3n +1). Talk to your partner. Calculate the value of the 10th term, the, 25th term and the 99th term.

Tell children that the number patterns they are looking at today are all sequences of numbers where the difference between each of the terms is the same. These are known as linear sequences (you may also see them described as arithmetic sequences or arithmetic


progressions). Display these sequences for children to describe an nth term

algebraically: o 5, 10, 15, 20, 25, … [5n]o 4, 9, 14, 19, 24, … [5n - 1] o 9, 16, 23, 30, 37, … [7n + 2]o 1.5, 5.5, 9.5, 13.5, 17.5, … [4n – 2.5]o Challenge: 2, 5, 10, 17, 26, 37, … [n2 + 1] This one is not a linear

sequence, however, the sequence of square numbers is an important one for children to recognise quickly as they move towards secondary school maths (where it will occur frequently at this stage of their learning).


Y5 Group Activity notes DAY 3 Y5 Group Activity notes DAY 3Identify steps and missing numbers in sequences. Identify step increase or decrease, including sequences which pass through zero, or

with decimal steps.Objectives: Generate and describe linear number sequences. Objectives: Generate and describe linear number sequences.You will need: Mini-whiteboards or paper You will need: Mini-whiteboards and pens, flipchart and pens Groups of 6 – with TA Working towards ARE

Write _ , _ , _ , 24, _ , 36, _ , _ , _ , _ . Explain that in this sequence the numbers go up by the same amount each time. Can children figure out the steps and the missing numbers? What clues do they have? How many steps between 24 and 36 in the sequence?

Pairs of children find the missing numbers. What is each step worth? (6) Suggest that they count on, then back to check. Children write the complete sequences on their whiteboards.

Repeat with other sequences, e.g. 2, __, 16, __, __ …, showing only two non-consecutive numbers each time.

Write _ , _ , _ , 21, _ , 15, _ , _ , _ , _. Point out this time the sequence is decreasing rather than increasing.

Ask pairs of children to find the missing numbers. Ask pairs to describe the sequence.

Repeat with another decreasing sequence, e.g. 31, 26, 21, 16, 11 …, again showing only two non-consecutive numbers.

Groups of 6 – with TA Working at ARE/Working at Greater Depth

Write 7, _ , _ , _ , _ , _ , _ , 56 on the flipchart. Explain that this is a sequence which starts at 7 and increases by the same number each time until it reaches 56. Children discuss in pairs what they think the step increase is. Encourage them to test out their idea by writing the sequence from 7 to 56.

Take feedback and agree the step as 7. Repeat with: 8, _ , _ , _ , _ , _ , _, 36. Take feedback. Discuss how the

step this time was 4. Repeat for: 40, _ , _ , _ , _ , _ , _ , 19. What do they notice? Repeat for 10, _ , _ , _ , _ , _ , _ , _ , –6. This passes through zero! Children, in pairs, create their own sequences. Say that the sequence

must increase/decrease by a constant amount. They write the first and last terms, the correct number of spaces, and give to another pair to solve.

Challenge Working at Greater Depth Ask children to calculate the step increase in this sequence: 0.4, __,

1.2, __, 2, __, 2.8. and then create their own sequence which increases/decreases by a number with one decimal place.

Outcomes: I can identify the step in an increasing or decreasing sequence.

Outcomes: I can identify the step in an increasing or decreasing sequence, including

sequences with negative numbers.


Y6 Group Activity notes DAY 3 Y6 Group Activity notes DAY 3Find the 10th term, 100th term, then nth term in linear sequences. Create linear sequences based on 10n, writing an expression for the nth term.Objectives: Generate and describe linear number sequences. Objectives: Generate and describe linear number sequences.You will need: ‘Sequences’ (see resources) You will need: Mini-whiteboards and pens, flipchart and pens Groups of 6 – with T Working towards ARE

As a group, look at the first two ‘Sequences’ (see resources). Point out the similarities, i.e. that the first sequence is the multiples of 4 and the next sequence is the multiples of 4, add 1. The 10th term in the first sequence is 10 × 4, i.e. 40. What is the 10th term in the second sequence?

Children discuss in pairs what the 100th term is in the first and second sequences.

Together, generalise the terms in each: 4n and 4n + 1. Ask children to describe what is the same and what is

different about the next two sequences. (Sequence 3 is the multiples of 5; sequence 4 is the multiples of 5, subtract 1.)

Children, in pairs, deduce the 10th and 100th terms in each. As a group, agree the nth term for each.

Ask children to use what they know about sequences 3 and 4 to calculate the 10th, 100th and nth terms for sequence 5. Take feedback.

Pairs of children solve sequences 6, 7 and 8.

Groups of 6 – with T Working at ARE/Working at Greater Depth Write the following sequence on the flipchart: 10, 20, 30, 40. This is a

sequence we should all know. What is the 10th term? The 100th term? The 1000th term? The nth term? Agree this as 10n, i.e. 10 times the number of the term in the sequence.

Write the following sequences on the flipchart:13, 23, 33, 43, 53 8, 18, 28, 38, 48

These sequences are also based on the 10 times table, but how are they different? Each term in the first sequence is 10n + 3. Ask children to check this by using the expression 10n + 3 to check the 5th term (53), and to find the 10th term.

Children, in pairs, write an expression for the nth term in the second sequence. Agree this as 10n – 2.

Challenge! With a partner, create your own sequences based on 10n, i.e. adding or

subtracting a number to/from 10n. Ask one pair to share their expression. The rest of the group use that to

write the first four terms. The pair checks the group’s work. Now create a sequence based on a different times table. Can the rest

of the group derive the nth term?Outcomes:

With support, I can continue and describe linear sequences and deduce the 10th term without calculating out the all the terms up to that point.

I am beginning to generalise the nth term.

Outcomes: I can continue and describe linear sequences and deduce the 10th term

without calculating out the all the terms up to that point. I can generalise the nth term.



Objectives Y5: Generate and describe linear number sequences.Y6: Generate and describe linear number sequences.

Resources Multilink cubes

Teaching Teaching with Y5 and Y6 Make the following shapes from multilink:

What do you think the next shape in the sequence will look like? Describe it to your partner. And the next shape? Make the next 2 shapes to check. What do you think the 10th shape will look like in the sequence? Describe it to your partner. Agree what the 10th shape looks like and how many cubes are in it (19). How would you describe any shape in this sequence? Talk to your partner. Take feedback. Children are likely to see the shape in two different ways:

1. Some children see it as 2 towers, one with one less cube than the other. How do you know how many cubes will be in each tower? Agree that the number in the 1st tower is the number of the term or shape in sequence, and the number in the 2nd tower is 1 less than this. Record n + n – 1, explaining that n stands for the number of the term in the sequence.

2. Some children will see it as ‘rectangle’ with one cube taken out. How do you know how big the rectangle is? Agree that it is 2 lots of the number of the term, and then 1 is subtracted. Record 2 × n – 1.

Can both these ways of seeing the shapes be correct? Point out that they are actually the same, because n + n – 1 = 2n – 1, and 2 × n – 1 = 2n – 1.

So, let’s use our general expression to check the value of the nth term. If n is 10, 2 lots of 10 is 20, and then subtract 1 leaves 19, and this is what we said. So, how many will be in the 100th shape? What will it look like? How many in the 1000th shape? What will it look like? And the millionth? Point out that once we have a general expression, we can calculate how many cubes will be in any shape in the sequence, without calculating all of the values before it.


Y5 Group Activity notes DAY 4 Use the in-depth problem-solving investigation for this unit ‘Stars and Crosses’ as today’s group activity, or use the activities below.Investigation of a pattern of shapes: identifying/describing shapes further in the sequence without making all the shapes up to that point.Objectives: Generate and describe linear number sequences. You will need: Multilink cubesWhole year investigation in pairs – independent or with T Working towards ARE/Working at ARE/Working at Greater Depth

Ask children to make the following shapes from multilink:

Ask children to describe each shape, and what the next shape in the sequence will look like, e.g. two towers of 4 cubes. So, what will the 5th shape look like? How tall will the 10th shape be? How many cubes will be in it? How tall will the 100th shape be? How

many cubes will be in it? Now ask children to add a cube to each of the shapes they have made. How are these shapes different? If we continued this sequence of

shapes, what would the 5th shape look like? How many cubes would be in it? And the 10th? The 100th?! Now ask children to add an extra ‘tower’ to each shape. In pairs, they sketch what the 5th, 10th and 20th shapes would look like, and write how many cubes would be

in the 5th, 10th, 20th and 100th shapes.

Working at Greater Depth After each sequence, discuss what the 100th shape would look like in each sequence. Together, write a general expression for each shape

in terms of n, i.e. 2n, 2n + 1, 3n + 1.Outcomes:

I can continue and describe linear sequences. I can deduce the 10th term without calculating all the terms up to that point.

Working at Greater Depth I can generalise the nth term.


Y6 Group activity notes DAY 4Use the in-depth problem-solving investigation for this unit ‘Stars and Crosses’ as today’s group activity, or use the activities below.Investigation of a pattern of shapes: identifying/describing the 10th and nth terms.Objectives: Generate and describe linear number sequences. You will need: Multilink cubesWhole year investigation in pairs – independent or with TA Working towards ARE/Working at ARE/Working at Greater DepthAsk children to make the following shapes from multilink:

Pairs of children discuss how the number of cubes is growing and predict how many will be in the 10th shape. They then derive an expression for the nth term, and check this with several different terms. How many cubes were in the 3rd shape? The 10th shape? What would the 20th shape look like? (i.e. one blue cube with two ‘arms’, each

with 20 cubes). So, how many cubes would be in it? (41). How many would be in the 100th shape? 1000th shape? nth shape? (2n + 1).

Working towards ARE Children make the next few terms, then draw what they think the 10th term will look like. They decide which of the following is an expression for the nth term, testing each out: 2n – 1, 2n + 1, n + 2.

Challenge Working at Greater DepthFind the nth term in this sequence using what you have learned:

Outcomes: I can continue and describe linear sequences. I can deduce the 10th term without calculating all the terms up to that point. I can generalise the nth term.


ADDITIONAL RESOURCES

‘Garden sketch’ (see resources)

‘Function machines 1’ (see resources)

‘Write a formula’ (see resources)

‘Function machines 2’ (see resources)

‘Sequences’ (see resources)

Mini-whiteboards and pens

Flipchart and pens

Sticky notes

Multilink cubes

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Therefore, it is your sole responsibility to verify any of the Links which you wish you use. Hamilton Trust excludes all responsibility and liability for any loss or damage arising from the use of any Links.


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