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Multiplication in Year 5

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Ed Pickering

on 2 May 2014

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Transcript of Multiplication in Year 5

Progression: Year 6
They need to be secure when multiplying 4 digits by 2 digits using long multiplication

They need to be able to perform mental calculations with mixed operations and large numbers

They should be secure when identifying common multiple, factors and prime numbers

Solve problems involving addition, subtraction, multiplication and division
Multiplication in Year 5
Lesson 1
Ticketing the Turnstiles
Learning Intentions
Key Questions & Misconceptions
Lesson 2
Running the Rides
Learning Intentions
Key Questions & Misconceptions
Lesson 3
Stocking the Stalls
Learning Intentions
Key Questions & Misconceptions
Lesson 4
Arranging the Area
Learning Intentions
Key Questions & Misconceptions
Working with squared paper, children forget that 1 sq cm is one square on the paper

Can children distinguish between area and perimeter? Some may add lengths rather than multiply when calculating areas

Can children distinguish between the units of measure of area and perimeter?

Do children know that perimeter is one-dimensional where area is two-dimensional? Is this area an abstract concept for them?

Curriculum Links
Problem solving (using multiplication, measures and other operations), recognizing and using square numbers and notation, calculating areas
Link to Division
Multiplying Decimals
Curriculum Map
National Curriculum requirements
Key Concepts and Big Ideas
Strategies & Models to use
The context for our Medium-Term Plan
4 Linked Lessons
Concepts & Big Ideas
Medium Term Plan: 4 Linked Lessons
Placed into
- the funfair
Builds on children's real- life experiences
Gives them ownership (of fair, individual rides etc.)
Allows them to input to the lesson content

Solving problems involving money
Link to
in NC (non-statutory)

Emphasis on sharing and comparing strategies
Entry Point: Year 4
Pupils have been taught to:

Recall multiplication and division facts up to 12 x 12

Use place value, known and derived facts to multiply and divide mentally (inc. x 0 and x 1), A x B x C

Recognise and use factor pairs and commutativity

Multiply 2- and 3-digit numbers by a 1-digit number using formal written layout

Solve problems involving multiplying and adding (inc. using distributive law and integer scaling)

Solve simple measure and money problems involving fractions and decimals to 2 d.p.
Multiply whole numbers and those involving decimals by 10, 100 and 1 000
Multiples & Factors
Identify multiples and factors, including finding all factor pairs of a number, and common factors of two numbers
Prime Numbers
Know and use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers

Establish whether a numbe up to 100 is prime and recall prime numbers up to 19
Formal Written Methods
Multiply numbers up to 4 digits by a one- or two-digit number using a formal written method

Including long multiplication for two-digit numbers
Mental Methods
Multiply numbers mentally drawing upon known facts
Problem Solving: Factors & Multiples
Solve problems involving multiplication - including using their knowledge of factors and multiples, squares and cubes
Other Problem Solving
Solve problems using +, -, x, / and a combination of these, inc. understanding the meaning of the equals sign

Solve problems involving multiplication inc. scaling by simple fractions and problems involving simple rates
Squares & Cubes
Recognise and use square numbers and cube numbers, and the notation for squared and cubed

Link to area & volume (measurement)
: Must be able to create a plan for their funfair to required size and within budget (calculating spend and savings)
: Should be able to include all essential items in their plan
: Should be able to include a coloured key of their funfair (Extension: and scale?)

Assessment Criteria
Children can add the essential items to their plan (calculating areas) and work out their remaining space and budget.
Children can place their attractions and facilities accurately onto squared paper.
Children can choose attractions to fill the rest of their space and calculate how much they have spent.
: Mental maths questions relating to multiplication with money and area.
Planning your plot
: Design and draw a funfair on squared paper in pairs. Impose a spending limit and stipulate essential items. Attractions are categorised into different prices and areas. They must fit the items into a 50 sq cm plan.
: To help the children visualize the task, show an already completed plan as a model and explain.
: Children will need to use squared numbers and multiplication strategies when calculating their area. They will use a range of operations (e.g. + and -) to calculate their remaining area to fill and budget.
: In pairs, children present their plan to the class and it is peer assessed – have they completed the success criteria?
: Once the plan has been created, children are given a scale and use multiplication and division with powers of 10 to work out ratios / actual size of their funfair and its attractions.
To solve word problems in the context of multiplication (using repeated addition and more); to multiply a 2-4 digit number by a 1-2 digit number; to link multiplication and division

: Must be able to work out the cheapest place to buy stock
: Identify where partitioning may increase efficiency of calculation
: Use known number facts to increase efficiency; begin to link to division and how division could solve the problem

Assessment Criteria
Children can see the link between repeated addition and multiplication
Children can identify which is the cheaper way of buying ingredients
Children can determine the correct combination of cheapest ingredients
: Children create a web of facts about what they already know about multiplication.
Stocking the Stalls
: Children are given shopping lists (word problems) stating which items/ingredients to buy for the funfair's refreshment stalls. They work in pairs to work out where to buy each item, and the best overall combination (e.g. at the lowest cost).
: Outline the cost of items at 2 different shops - 1 wholesaler (where buy packets) and 1 shop that sells items individually. Run through example item.
: Children will be able to use a range of strategies, but encourage use of empty number line and arrays (for wholesaler items). Provide coins to some children as necessary.
: Solver-Recorder. Pairs join up with another and take turns to explain their strategies and answers.
: Children determine how many packs of food they will need for a month if they sell X amount per day (see Weetabix problem)
Difficulties with monetary concepts

Visualising - can children see to solve in different ways (more than just repeated addition)

Make sure the multiplication sentence and number line representation match (even though will give identical products)

Unable to identify, or progress to, more efficient calculation methods

Curriculum Links
Problem solving (using multiplication and other operations), multiplying by a 2-digit number, multiples & factors, link to division
To solve multiplication problems - inc. those involving other operations (e.g. fractions & percentages), to apply knowledge of multiplication to monetary amounts

: Must be able to calculate daily takings using simple multiplication.
: Should be able to record daily data sensibly and use addition to work out a weekly total.
: Could compare rides and rank rides in order of revenue raised each day / over the week.

Assessment Criteria
Children can multiply a monetary value (£x.00 or £x.50) by a 2-digit number
Children can record daily revenue totals (e.g. in a table)
Children can calculate weekly totals, combining multiplication with addition
: Children
‘visit’ the fair with a budget of £10. There are 4 rides to choose from – each with a different price. How many different ways can they spend their £10 exactly?
Running the Rides
: C
hildren are given ownership of 1 ride (working individually, but some people will get the same ride so can compare). They choose (or are assigned) a price for the ride – between £2.50 and £7.50.
: Choose price at £X.00 or £X.50
Children track the number of visitors who attend the fair each day and go on their particular ride. They multiply the number of visitors each day by their price to calculate revenue, then total up for the week. Use whiteboards for calculation, then record totals in table.
: Provide a template for children to record daily totals. Run through example.
Children present findings and rank & compare best earning-rides.
Introduce costs for their ride (maintenance & running costs). How much does it cost to run for X hours? How much profit do you make over the week?
Children have difficulty with monetary amounts (e.g. see £3 as 300, struggle with any d.p.)

Children do not keep track of daily data sensibly.

Children struggle with simple fractions and percentages.

Unable to solve problems using other (combined) operations (e.g. linking to division for fractions)

Curriculum Links
Problem solving (using multiplication and other operations), multiplying by a 2-digit number, multiplying decimals, mental multiplication, link to division
Concepts & Big Ideas
Concepts & Big Ideas
Concepts & Big Ideas
Skip counting
E.g. 6, 12, 18, kept on track by using fingers (aided by colouring in on hundreds square)
Doubling and halving
5 x 16 = 10 x 8
Using more manageable numbers
Counting stick
– to identify multiples
Repeated addition
Using known facts
Known facts to multiply efficiently rather than using repeated addition
e.g. Doubling:
I know 2 x 16 = 32, so 4 x 16 must equal 64
14 x 3 (I know that 7 x 3 = 21, so 14 x 3 must be 42)

Models are concrete and pictorial representations
of mathematical ideas, and they are used to allow students to make sense of the mathematics.

It is important to develop understanding of a variety of models so they can be used as tools for learning. By using a
variety of models
, it encourages students to solve the problem in a way that makes sense to them.

• Use a model to represent pictorially a mathematical situation/ problem: Students use a model to represent a mathematical problem.

• When children have a strong understanding of the model they are able to apply it in new learning situations.
Models - Examples
Place Value Grid
Use digit cards to make numbers in the grid. Show how each digit in a number moves one column to the left when a number is multiplied by 10 and two columns to the left when it is multiplied by 100.

Ratio Table
1 2 3 4
6 12 18 24

Number Line
Make sure the multiplication number sentence and the number line make sense

Models - Examples
An ordered arrangement of pictures. Find the total without counting each item (cover part up).
Models - Examples
Area Model
Similar to the array model - the 2 numbers being multiplied are the sides of a rectangle. The product is represented by the area of the rectangle.
Useful visual diagram of the commutative and distributive properties, and clear link to 2D area.
Increasing a quantity by a certain amount.
“ The rollercoaster is 5 times taller than the slide”
Scale factor less than one can be introduced

Combination model:
If I have 3 t-shirts and 2 pairs of shorts, how many different outfits do I have?
Draw an array to match the number sentence

Write an array to match the array
Explore factors - how many ways?
Good for making the commutative property visually clear.
Misconceptions - Entry Point (NNS)
Misconceptions - Exit Point (NNS)
Other Misconceptions (Hansen, 2005)
• Writing tens (T) and units (U) digits in the
wrong columns
when multiplying (or dividing)

Place value errors

– multiplication & division:
Failing to understand the position of a digit determines its value and:
• Carrying a digit into the “answer” in the T column
• Not creating a hundreds (H) column – simply adding the number carried to the value in the T column already there
• Treating the T digit as if it were a U
Lack of understanding of the role of zero as a place holder

Over-generalisation errors
– multiplication and division:
Over-generalising the ‘rule’ for short multiplication, and when dividing working from R to L, instead of L to R.
When multiplying, over-generalising the rule of ‘putting down a zero’ when multiplying by a T digit, and assuming it applies to all multiplication problems (e.g. a U x U calculation).
MTP Lessons
Key Concepts
Understanding the connection between multiplication and division is critical to understanding the part/whole relationships in the multiplicative structure (Fosnot & Dolk).

Using them interchangeably is powerful as long as children are clear what (the whole or which part) they are trying to determine (Fosnot & Dolk).

NC: Pupils use multiplication and division as inverses to support the introduction of ratio in Year 6, by multiplying and dividing by powers of 10 in scale drawings.
To solve multiplication problems, to multiply 2-3 digit numbers by 1-digit numbers, to link division and multiplication

: Will use multiplication to calculate daily money taken; are able to accurately compare money taken on different days
: Can identify efficient multiplication strategies; can apply the concept of a 'half-price' discount to ticket sales
: Will use number facts they already know to increase efficiency

Assessment Criteria
Children can establish how much money is taken per day
Children can identify the day where most money was made
Children understand word problems and what the question is asking
Children are able to apply knowledge of the 5 times table to multiply by £5 ticket prices
: Children create a table showing days of the week and add in sensible visitor numbers for each day.
Ticketing the Turnstiles
: Children work in pairs and and are given a series of problems to solve that involve visitor numbers and ticket prices.
Starting Point
: Give the same initial questions to all pairs: If we charge £5 to enter, how much money do we make each day? On which day do we make the most money?
: Further questions to solve: How much do we make if we charge more on a weekend? If the fair is closed on Monday, how much do we need to alter our Saturday ticket price by to make the same money? What happens to our takings if we halve ticket price and triple the visitors attend.
: Children share answers (tables in poster format) and explain how they calculated.
: Compare with costs incurred for running the fair. Children to come up with own ideas of ways to make money to cover costs.
Children have difficulty with concept of money

Children are not used to solving word problems

Multiplication calculations will involve 3 digit numbers - some children may need simpler figures

Simple calculation errors when working out daily / weekly totals

Difficulty with the concept of 'half-price' - linked to knowledge of fractions and percentages

Curriculum Links
Problem solving, , mental multiplication, link to division
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