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Short Blocks

Maths Year 6 Autumn Multiplication and Division

Each unit has everything you need to teach a set of related skills and concepts. 'Teaching for Understanding' provides whole-class teaching and fully differentiated adult-led group activities. ‘Problem-solving and Reasoning’ develops these skills, and includes questions to enable you to assess mastery. Practice sheets ensure procedural fluency. Extra support activities enable targeted work with children who are well below ARE.

‘UNIT PLAN’ gives you a text version of all parts of the unit to use in your school planning documentation. ‘DOWNLOAD ALL FILES’ gives you that unit plan plus all of associated documents. These bulk downloads are added value for Hamilton Friends and School Subscribers.

Unit 1 Multiples, factors and prime numbers (suggested as 2 days)

Objectives

Multiples, factors and prime numbers
Unit 1: ID# 6277

National Curriculum
Mult/div (iv) (v)

Hamilton Objectives
9. Know all multiplication and division facts up to 12 x 12; identify common factors, common multiples, square numbers to 144 and prime numbers up to 20.
10. Multiply/divide whole numbers mentally, using facts to 12 × 12 & place value.
14. Perform divisions mentally within the range of tables facts.
18. Perform mental calculations, including with mixed operations.

Teaching and Group Activities for Understanding

Day 1 Teaching
Launch the ITP Number Grid to highlight multiples of 9 and 6. Discuss lowest common multiple. Repeat for multiples of 6 and 8. Reset, ring 18 &24. List all factors of each. Discuss highest common factors.
Group Activities
-- Game to find a number on the grid that is a common multiple of numbers on 2 cards. Then identify facts of numbers on a grid.
-- 2 games: Shuffle cards, take 2 and say a common multiple. Choose a number from a grid and say a factor. Child with highest factor wins.

Day 2 Teaching
Identify prime numbers as numbers with exactly 2 factors, themselves and 1. Identify 1 as not prime. Remind children that other numbers are composite numbers. Find numbers which have a pair of prime factors, e.g. 15.
Group Activities
Use the ‘Magic Multiplication Squares’ in-depth problem-solving investigation below as today’s group activity.
Or, use these activities:
-- Find which numbers from 10 to 30 can be made by multiplying 2 prime numbers together.
-- Write numbers as product of several prime factors.

You Will Need

• ITP: Number grid (see resources)
• Mini-whiteboards and pens
• Grid of multiples (see resources)
• 2–9 cards
• Common factors sheet (see resources)
• Factor tree’ from softschools.com

Mental/Oral Maths Starters

Day 1
Factors (pre-requisite skills)

Suggested for Day 2
Double and halve numbers to 100 (simmering skills)

Procedural Fluency

Day 1
Find common multiples and factors of given numbers.

Day 2
List prime numbers with a given range. Find numbers which can be made by multiplying 2 prime numbers together.

Mastery: Reasoning and Problem-Solving

• Which pair(s) of numbers under 20 have the largest number of common factors? What is the highest common factor?
• Write common multiples of 4 and 6 up to 60. What is the lowest common multiple.
• Use this information to find the lowest common multiple of 8 and 12.
• True or false?
The lowest common multiple of 2 prime numbers, a and b is always a x b.
The highest common factor of 2 multiples of 6 is always 6.

In-depth Investigation: Magic Multiplication Squares
Children complete a magic multiplication square using their knowledge of number properties and relationships. They then explore factors and multiples to create a new multiplication magic square.

Extra Support

Factor Hunt
Finding factors of numbers within times tables

Unit 2 Solve short multiplication problems (suggested as 3 days)

Objectives

Use short multiplication to solve problems
Unit 2: ID# 6287

National Curriculum
Mult/Div (i) (iv)

Hamilton Objectives
9. Know all multiplication and division facts up to 12 x 12.
11. Multiply 2-digit, 3-digit and 4-digit numbers by numbers up to 12 using short multiplication or another appropriate written method.

Teaching and Group Activities for Understanding

Day 1 Teaching
Write 3 x 326 on the board. Have 1 child use grid method on 1 side of the board, whilst you model short multiplication on the other side. Check methods and answers. Model 3 × 7263, estimating the answer first.

Group Activities
-- Help children to move from using the grid method to using short multiplication, using ‘friendly’ digits.
-- Investigation of which arrangement of given digits gives the 4-digit by 1-digit multiplication with the biggest answer, and then the smallest.

Day 2 Teaching
Write: 5 x 2326, 4 × 3943, 6 x 2082 and ask children to estimate which will have the largest answer. Model estimating. Then demonstrate grid method and also short multiplication drawing comparisons.
Group Activities
Use the ‘The Eights Have It’ in-depth problem-solving investigation below as today’s group activity.
Or, use these activities:
-- Support children with practice of short multiplication, estimating which will give the smallest and largest answers.
-- Use the relationship between multiplication and division to work out missing numbers in division.
-- Write multiplications with answers in given ranges. Use short x to find exact answers.

Day 3 Teaching
Show items with prices in an online shop and use these to revise using short multiplication to multiply 4-digit amounts of money by single-digit numbers. Demonstrate this using grid method first, so the place value is clear, then short multiplication.
Group Activities
-- Estimate, and then find the exact cost of buying 3 items (3-digit prices).
-- Target present spend is between £50 and £100. Use rounding to the nearest pound to estimate the cost of 3 of each item, to work out which items are within budget, and then find exact costs.
-- Use given digits to create multiplications of money. Find smallest and greatest possible answers.

You Will Need

• Mini-whiteboards and pens
• Flipchart and pens
• Multiplying 4-digit by 1-digit numbers (see resources)
• Oscar’s online gifts (see resources)
• Greta’s online gifts (see resources)

Mental/Oral Maths Starters

Day 1
Multiply by multiples of 10 (e.g. 7 × 80) (pre-requisite skills)

Day 2
Multiply by multiples of 100 (e.g. 7 × 800) (pre-requisite skills)

Suggested for Day 3
Find the time later using 24-hour clock (simmering skills)

Procedural Fluency

Day 1
Use a written method to work out the answers, but watch out for a few where a mental method could be use instead. Do as many as they can.

Day 2
Use short multiplication to work out the answers to the first 2 questions. Check answers using the grid method, then choose which method to use to work out the remaining questions.
Use short multiplication to multiply 4-digit numbers by 1-digit numbers, first rounding the 4-digit number to the nearest 100 to approximate.

Day 3
Use grid method/short multiplication to find costs of items. Watch out for a few where a mental method could be use instead. Do as many as they can.

Mastery: Reasoning and Problem-Solving

• Maya says that 2578 x 4 gives the same product as 8 x 1289. Is she correct? Demonstrate why/why not.
• Multiply 1386 by 9. Write the product.
Add the same number (1386) to the product.
What do you notice?
Repeat with 2547 x 9, adding 2547 to the product.
Explain what happens.
Could you use this to make finding the product easier?
• Write the missing digits in this multiplication.
3 6 ☐ 2 x 8 = ☐ 9 ☐ 3 6

In-depth Investigation: The Eights Have It
Children multiply numbers starting with 9 by 9 and add single-digit numbers in a decreasing sequence. They identify and describe the patterns and start to explain them.

Extra Support

Greatest Grid Gurus
Using the grid method to multiply 3-digit numbers by single-digit numbers

Unit 3 Use short division to solve problems (suggested as 4 days)

Objectives

Use short division to solve problems
Unit 3: ID# 6299

National Curriculum
Mult/Div (iii)

Hamilton Objectives
9. Know all multiplication and division facts up to 12 x 12.
16. Divide numbers with up to 4-digits by a number up to 12 using short division and giving an appropriate answer.
15. Interpret remainders as whole number remainders, fractions, including decimal fractions where equivalents are known or by rounding up or down.

Teaching and Group Activities for Understanding

Day 1 Teaching
Display a given word problem. Children discuss in 2s what calculation is needed and solve it. Discuss the calculation required. Model vertical layout and short division. Decide whether to round up or down. Repeat with new problem.
Group Activities
-- Use base 10 equipment to support short division, and then decide where to round up or down to solve division problems.
-- Solve division problems, then use multiplication to check division.
-- Investigate dividing 9999 by any number from 2 to 12. Round the answer down to the nearest whole, then divide but the same number again and so on.

Day 2 Teaching
Model using short division for 1248 ÷ 5 and 6453 ÷ 6. Agree answer 249 r 3 and 1075 r 3. Ask children to write the exact answers, simplify fractions. Ask which of 2438 ÷ 3 and 5723 ÷ 4 has an answer > 1000. Model these divisions giving fractional answers.
Group Activities
Use the ‘Awesome Answers’ in-depth problem-solving investigation below as today’s group activity.
Or, use these activities:
-- Choose 3 or 4-digit numbers to divide by 5, 4, 6, 8 then 12 so as to give a mixed number answer.

Day 3 Teaching
Use short division to divide 3-digit numbers by 1-digit numbers. Show how we can write fraction parts of answers (e.g. ½ or ¾) as decimals, e.g. 23¾ as 23.75. Model with 4-digit numbers divided using short division.
Group Activities
-- Choose amounts of money between £100 and £1000 (whole number of pounds) to divide by 4, to give answers which are not whole numbers of pounds.
-- Choose amounts of money between £1000 and £5000 (whole number of pounds) to divide by 4 or 5, to give answers which are not whole numbers of pounds.
-- Write divisions with answers of the form □.5, □.25 and □.125.

Day 4 Teaching
Display some given word problems 1 at a time. Agree the calculation for each, solve it; discuss whether the answer needs to be rounded up or down OR (as in the last 2) answer needs to be exact with a fractional or decimal part.
Group Activities
-- Discuss which problems will need answers rounding up, rounding down, or exact answers.
-- Solve division problems, then make up division word problems with answers 254, 255, 254½, £254.50 and 254.5m.

You Will Need

• PowerPoint presentation (see resources)
• Rounding up or down (see resources)
• Short division practice (see resources)
• Division word problems (see resources)
• Post-it notes
• Base 10 equipment
• Calculators

Mental/Oral Maths Starters

Day 1
12 times table (pre-requisite skills)

Day 2
Mental division (pre-requisite skills)

Day 3
Equivalent fractions, decimals and percentages (pre-requisite skills)

Day 4
Multiplication and division facts (pre-requisite skills)

OR
Place 5-digit numbers on a human number line (simmering skills)

Procedural Fluency

Day 1
Solve division problems, deciding whether answers need rounding up or down.

Day 2
Divide 3-digit or 4-digit numbers by 4, 5, 6, 8, 9, 11 and 12, giving answers with fractions.

Day 3
Divide 3-digit or 4-digit numbers to give answers with fractions. Rewrite each as a decimal using equivalence.

Day 4
Solve division word problems, deciding whether to round up or down, or find the exact answer according to the context.

Mastery: Reasoning and Problem-Solving

• Divide 3666 by 3, 4, 5, 6 and 8 and write exact answers with a fraction part as necessary.
• Write the missing numbers in this division.
6 ☐ 0 7 ÷ 6 = 1 1 3 4. ☐
• Write a division of one 3-digit number by 6 where the answer contains the fraction 1/6. Write a similar division where the answer contains the fraction 5/6.

In-depth Investigation: Awesome Answers
Using a magic square to generate 3-digit numbers, children create divisions with dividends containing specified fractions.

Extra Support

Chunky Jumps
Using chunking to divide (answers less than 60)

Toffee Apples
Using chunking to divide (answers less than 60); round up or down to answer word problems

Unit 4 Long multiplication problems (suggested as 4 days)

Objectives

Use formal written long multiplication to solve problems
Unit 4: ID# 6311

National Curriculum
Mult/Div (i)

Hamilton Objectives
9. Know all multiplication and division facts up to 12 x 12.
12. Multiply numbers with up to 4 digits by 2-digit numbers using formal long multiplication.

Teaching and Group Activities for Understanding

Day 1 Teaching
Children use grid to solve 26 x 345. The first row helps us to work out 20 lots of 345, and the second row helps us to work out 6 lots of 345, then we add both together. Children work in pairs on 345 × 43. Demonstrate on grid.
Group Activities
Use the ‘Stunning Squares’ in-depth problem-solving investigation below as today’s group activity.
Or, use these activities:
-- Support children in using the grid method, with ‘friendly’ numbers.
-- Guess missing numbers in grid method examples.

Day 2 Teaching
Children first use grid method to work out 16 × 258. Then revise and model long multiplication to work it out. Go through the procedure, checking that children understand each stage and where it matches the grid.
Group Activities
-- Use the digits 1 to 6 to create as many 3-digit by 2-digit multiplications as they can. They find the answers using the grid method.
-- Work out 13, 15, 16, 17, and 18 × 485, first discussing in pairs a pair of multiples of 1000 which they think the answer will be in between.

Day 3 Teaching
Ask children how many hours in a year? Write 24 × 365 on the board for a non-leap year. Revise using long multiplication to work this out. Discuss x 20 as double and then x 10 to make it easier to do the first row.
Group Activities
-- Roll a 0–9 dice 3 times to create a 3-digit number of days. Find how many hours are in this number of days using long multiplication. Repeat.
-- Support children in using long multiplication for multiplying numbers greater than 20 and move on to numbers greater than 30.

Day 4 Teaching
Children Working towards ARE use grid method to work out 34 x 436. Model long multiplication to others. To multiply by 30, we can multiply by 10, and then by 3. When we multiply by 10, the digits shift 1 place to the left and we put a 0 in the 1s column. Then multiply by 3 as usual. Children work out the next row, 4 x 436, then add the 2 rows together.
Group Activities
-- Work in pairs to find a multiplication of the form □□ x □□□ with an answer as close to 20,000 as they can. Use the grid method.
-- Multiply 33 by 3-digit numbers with identical digits using long multiplication.
-- Work in pairs to find a multiplication of the form □□ x □□□ with an answer as close to 40,000 as they can. Use long multiplication.

You Will Need

• Mini-whiteboards & pens
• Multiplying 3-digit numbers by 2-digit numbers (see resources)
• Flipchart & pens
• Large piece of paper
• 0 to 9 dice

Mental/Oral Maths Starters

Day 1
Times tables (pre-requisite skills)

Day 2
Multiply pairs of multiples of 10 and 10s by 100s, e.g. 30 × 40, 30 × 400 (pre-requisite skills)

Day 3
Mental multiplication (pre-requisite skills)

Suggested for Day 4
Mental division (simmering skills)

Procedural Fluency

Day 1
Use the grid method to multiply 3-digit numbers by 2-digit numbers.

Day 2
Use grid or long multiplication to multiply 3-digit numbers by numbers between 10 and 20.

Day 3
Use long multiplication to multiply 3-digit numbers by numbers between 20 and 30.

Day 4
Use long multiplication to multiply 3-digit numbers by numbers greater than 30. Aim for an answer close to 20,000.

Mastery: Reasoning and Problem-Solving

• Have a go at doing each of these 3 multiplications mentally.
Explain how you did each one.
280 x 25 =
143 x 21 =
210 x 15 =
Children could check each using long multiplication or grid.
• Multiply 531 by 16 using long multiplication or grid.
Now double 531 four times to check if this gives the same answer.
• True or false?
8 x 572 is the same as 286 x 16
15 x 626 is the same as 6260 add 3130
24 x 333 is half of 12 x 666

In-depth Investigation: Stunning Squares
Children explore patterns in the squares of numbers with reversed digits to find pairs of ‘stunning squares’.

Extra Support

Grid Triplets
Revising using the grid method to multiply 3-digit numbers by single-digit numbers

Unit 5 Formal and informal calculation strategies (suggested as 2 days)

Objectives

Use formal or informal methods as appropriate to solve problems involving addition, subtraction, multiplication and division
Unit 5: ID# 6317

National Curriculum
Mult/Div (vi) (viii)

Hamilton Objectives
18. Perform mental calculations, including with mixed operations & large numbers; carry out calculations using knowledge of the order of operations & brackets.
19. Use estimation to check answers and determine an appropriate degree of accuracy; round answers to multiplications and divisions to a specified degree of accuracy.
20. Solve problems involving all 4 operations

Teaching and Group Activities for Understanding

Day 1 Teaching
Show a set of calculations. Ask children to discuss which to work out mentally and which to work out using a written method and why. The size of numbers is not as relevant as how ‘friendly’ they are. Discuss good mental/written methods
Group Activities
-- Choose mental or written methods to work out a range of calculations. Show working out.
-- Write a calculation in a table that they would solve mentally and a calculation they would solve using a written method for each of 8 categories.
-- Repeatedly add and subtract 80 to/from 5-digit numbers passing through multiples of 10,000.

Day 2 Teaching
Display word problems and discuss how we can elucidate the calculations required. Sometimes these need to be written using brackets. Solve problems by writing clear calculations.
Group Activities
Use the in-depth problem-solving investigation ‘All the Digits’ from NRICH as today’s group activity.
Or, use these activities:
-- Discuss word problems before deciding what calculations are necessary. Sketch simple diagrams (not detailed pictures!), number lines or other jottings to help, or try some initial numbers to get started.
-- Choose a context and measure from vets, school, holiday, shopping, cooking weight, length, money, capacity. Work in pairs to write a 2-step word problem. Swap problems with another pair and solve them.

You Will Need

• Mental & written calculations 1 & 2 (see resources)
• Choosing mental & written strategies table (see resources)
• Word problems sheets 1 & 2 (see resources)

Mental/Oral Maths Starters

Day 1
Add any pair of 2-digit numbers (pre-requisite skills)

OR

Find squares and cubes (simmering skills)

Day 2
Mental subtraction of 2-digit numbers (pre-requisite skills)

OR

Read off a line graph to convert from km to miles (simmering skills)

Procedural Fluency

Day 1
Choose mental or written methods to work out a range of calculations.

Day 2
Solve word problems involving several steps.

Mastery: Reasoning and Problem-Solving

• Solve these using long multiplication or grid
288 x 15 =
386 x 15 =
472 x 15 =
536 x 15 =
Check your answers by multiplying each 3-digit number by 10 (1st answer), halving that (2nd answer), and adding those 2 answers.
• Explain why 348 x (5 + 8) is not the same as 348 x 5 + 8 and find out by how much the answers differ.
• If Sally divides a number by 25 and then adds 359 she gets 382. What was her starting number?
• Tom multiplies his number by 8 and then subtracts 48. He gets 3696. What was his starting number?

In-depth Investigation: All the Digits
Using the information given, can you replace the stars in the multiplication calculation with digits? All the Digits from nrich.maths.org.

Extra Support

RICA
Solving word problems
Choosing how to work out the answers to calculations