Short Blocks

# Maths Year 5/6 Spring Algebra and Ratio

Each unit has everything you need to teach a set of related skills and concepts.

First time using Hamilton Maths?

The PowerPoint incorporates step-by-step teaching, key questions, an in-depth mastery investigation, problem-solving and reasoning questions - in short, everything you need to get started.

All the other resources are there to support as-and-when required. Explore at your leisure - and remember that we are always here to answer your questions.

## Unit 1 Algebra (suggested as 4 days)

### Objectives

Algebra
Unit 1: ID# 56650

Y5: Algebra
Y6: Algebra

National Curriculum
Y5: Mult/Div (x) (xi)
Y6: Alg (i) (ii) (iii) (iv) (v)

Y5 Hamilton Objectives
22. Solve problems involving addition, subtraction, multiplication, division and a combination, including understanding the meaning of the equals sign.

Y6 Hamilton Objectives
36. Use simple formulae, including formulae expressed in words.
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.

### Planning and Activities

Day 1
Show how n can be used to stand for the width of a garden in order to calculate different areas of turf.
Group Activities: T with Y6
Y5 -- Begin to identify rules for function machines. Or, write and use formulae to calculate ages.
Y6 -- Write formulae for questions. Or, play ‘Guardians of the rule’ as a group (a game to deduce 2-step rules).

Day 2
Write 25 + a = 30 and sketch a bar model to show this. Calculate what a represents. Repeat with similar equations.
Further Teaching with Y6
Write a + b = 10, explaining that a and b are two new mystery numbers. List possible whole answers. Repeat with similar equations.
Group Activities: T with Y6
Y5 -- Deduce the numbers represented by letters in a multiplication grid. Or, solve equations with missing numbers, and write similar equations.
Y6 -- Solve a series of equations by finding successive unknowns. Or, find pairs of numbers which satisfy two equations with two unknowns.

Day 3
Display 2, 4, 6, 8 .... Show how we can use the relationship between the number of the term and its value in the sequence, i.e. each value is 2 times the term number. Calculate the 10th term and the 100th term. Repeat with other linear sequences, recognising the times table they are based on. Describe an nth term for each sequence algebraically.
Group Activities: T with Y6
Y5 -- Identify steps and missing numbers in sequences.
Identify step increase or decrease, including sequences which pass through zero, or with decimal steps.
Y6 -- Find the 10th term, 100th term, then nth term in linear sequences. Or, create linear sequences based on 10n, writing an expression for the nth term.

Day 4
Make a sequence of shapes from multilink (see download); use these to generate a linear numerical sequence. Find the value of the 10th term; describe what is happening in words, then write an algebraic expression or general rule for the nth term.
Group Activities: T with Y5
Use the in-depth problem-solving investigation ‘Stars and Crosses’ as today’s group activity.
Or, use these activities:
Y5 -- Investigation of a pattern of shapes: identifying/describing shapes further in the sequence without making all the shapes up to that point.
Y6 -- Investigation of a pattern of shapes: identifying/describing the 10th and nth terms.

### You Will Need

• ‘Garden sketch’ (see resources)
• ‘Function machines 1’ (see resources)
• ‘Write a formula’ (see resources)
• ‘Function machines 2’ (see resources)
• ‘Sequences’ (see resources)
• Flipchart and pens
• Sticky notes

### Short Mental Workouts

Day 1
Equivalence

Day 2
Function machines

Day 3
Counting back

Day 4
Sequences

### Worksheets

Day 1
Y5: Identify functions and write outputs in terms of n.
Y6: Write and use simple formulae.

Day 2
Y5: Solve equations, including bar model and geometry examples.
Y6: Solve equations, including bar model and geometry examples; list solutions for equations with two unknowns.

Day 3
Y5: Complete sequences by deducing their step increase/ decrease.
Y6: Find the 10th, 100th then nth term for sequences.

Day 4
Y5: Draw the next ‘Y’ or ‘V’ shape in a sequence. Find a relationship between the number of shapes in the pattern and its position in the sequence.
Y6: Draw the next ‘spinner’/ ‘staircase’ in a sequence. Find a relationship between the number of shapes in the pattern and its position in the sequence.

### Mastery: Reasoning and Problem-Solving

Y5/Y6

• If the perimeter of a regular shape is 5 × n, where n is the length of a side, what is the shape? Find the perimeter when n = 6.5cm.
• Darren draws a function machine. It trebles a number and then subtracts 6. Sophie sees that one of the outputs is 15. What was the input? Then Darren inputs 11. What output will Sophie see? Write the formula for Darren’s machine.
• Formula A is 3n.
Formula B is n + 6.
What number can n represent which will make these two formulae equal the same amount?
• 2a is 5 more than 3b. If a and b are both whole numbers and a < 10, what are the possible values for a and b?
• The 4th term of a sequence is 17, the 5th is 21 and the 6th is 25. Write the next four terms. Then write the first three terms. What is the nth term in this sequence?

In-depth investigation: Stars and Crosses
Children find totals of the numbers in shapes on a 1–1 00 grid, make generalisations and then write a formula to find the total of any similar shape on the grid.
A presentation is available to guide children through the investigation.

### Extra Support

Y5/Y6
Mathematical Mysteries
Solving equations using the bar model.

Spot the rule
Counting sequences

## Unit 2 Ratio (suggested as 3 days)

### Objectives

Ratio
Unit 2: ID# 56666

Y5: Ratio
Y6: Ratio

National Curriculum
Y5: Frac (i)
Y6: Ratio/Prop (i) (ii)

Y5 Hamilton Objectives
24. Compare and order fractions where the denominators are multiples of the same number.

Y6 Hamilton Objectives
23. Identify simple fraction/decimal/percentage equivalents, e.g. 1⁄4 = 0.25 = 25%, 1/3 = 0.33 = 33%.
33. Calculate simple percentages of whole numbers; solve problems involving use of percentages for comparisons.
35. Solve problems involving simple ratios, using tables facts and knowledge of fractions and multiples, e.g. 2 eggs for every 250g of flour.

### Planning and Activities

Day 1
Display a recipe for chocolate chip cookies. Use this to show that if we want to increase the quantity made, we need to increase the quantities in the recipe but keep the ratios the same. Demonstrate how to do this.
Group Activities: T with Y6
Y5 -- Adapt a smoothie recipe, keeping the ratios between ingredients the same. Or, maintain a ratio of red to yellow paint to mix a larger quantity of the same shade of orange.
Y6 -- Maintain a ratio of red to yellow paint to mix a larger quantity of the same shade of orange. Some children may paint a colour scale by mixing two paints in a gradually-changing ratio.

Day 2
Show a stick made of unequal numbers of red and blue cubes. Establish the ratio between the two colours and use this to create different length sticks with the colours in the same ratio.
Group Activities: T with Y5
Y5 -- Investigate what ratios of red to blue cubes are possible for a total of 12 or more cubes.
Y6 -- Explore ratio in shaded rectangles in two colours, deducing how many squares are in each colour. Or, use bar models to help solve ratio word problems.

Day 3
Write ‘swimming’ and ‘cycling’ on large cards. Ask 10 children to stand by their favourite. Y5 write preferences as fractions; Y6 write as a percentage. Say that we can use fractions or percentages to describe a proportion (‘3 in every 10 children prefer cycling’/ ‘30% of children prefer cycling’). Extend to other similar problems.
Group Activities: T with Y6
Use the in-depth problem-solving investigation ‘Calculating Colours’ from The Royal Institution as today’s group activity.
Or, use these activities:
Y5 -- Calculate numbers of tulips in different sized beds, keeping proportions of each colour the same. Or, convert dance competition scores to fractions with the same denominators in order to compare them.
Y6 -- Write equivalent percentages for fractions; then calculate percentages and fractions of amounts. Or, convert dance competition scores to percentages in order to compare them.

### You Will Need

• Red and yellow paint
• Teaspoons
• Mixing palettes; Brushes
• Red and blue multilink cubes
• Squared paper and coloured pencils
• ‘Ratio problems’ (see resources)
• Two large cards
• ‘Dance competition scores’ (see resources)
• ‘Equivalent fractions and percentages’ (see resources)

### Short Mental Workouts

Day 1
Multiply 3 numbers together

Day 2
Find fractions of amounts

Day 3
Equivalent fractions, decimals and percentages

### Worksheets

Day 1
Y5/Y6: Scale a pizza recipe to make different numbers of pizzas.

Day 2
Y5: Identify ratios; shade rows of squares to given ratios.
Y6: Shade squares within rectangles to given ratios.

Day 3
Y5: Divide increasing numbers of flower bulbs, while maintaining the same proportion of each.
Y6: Write equivalent percentages for given fractions; calculate percentages of amounts

### Mastery: Reasoning and Problem-Solving

Y5/Y6

• This recipe (see download) bakes 8 cupcakes. What ingredients will be needed for: 16 cakes, 12 cakes? 20 cakes?
• A total of £60 is split between Ana and Eva in the ratio 3:2. How much do they each get?
• Green paint is mixed using blue and yellow paint in the ratio 2 : 7.
Sam uses 4 litres of blue paint. Assuming she uses the correct amount of yellow, how many litres of green paint will she make?
• If 6 children in a class do not like sport, and there are 30 children in the class, what proportion do like sport? Give your answer as a fraction and as a percentage.

In-depth Investigation: Calculating Colours
Children gain practical experience of combining coloured solutions in varying proportions then describe solutions in terms of fractional (and percentage) composition.
Calculating Colours from The Royal Institution.

### Extra Support

Y5/Y6
Car wash
Exploring dividing money in 3:1 and 3:2 ratios.

Y6
The Key to Percentages
Finding simple percentages of amounts (1%, 10%, 25% and 50%).