Identifying multiples of an integer (from 1.2
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slide1. Identifying multiples of an integer (from 1.2 Properties of number) KS3 Mastery PD Materials: Exemplified Key Ideas
Materials for use in the classroom or to support professional development discussions Summer 2021<br>
slide2. About this resource These slides are designed to complement the 1.2 Properties of number Core Concept document and its associated Theme Overview document 1 The Structure of the number system, both found in the Secondary Mastery Professional Development pages.
These slides re-present the key examples from the Core Concept document so that the examples can be used either directly in the classroom or with a group of teachers. There are prompt questions alongside the examples, and further clarification in the notes.
These slides do not fully replicate the Core Concept document, so should be used alongside it. Reference to specific page numbers, and to other useful NCETM resources, can be found in the notes for each slide.
This slide deck is not designed to be a complete PD session, rather it is a selection of resources that you can adapt and use as needed when planning a session with a group of teachers.<br>
slide3. About this resource The slides are structured as follows:
The big picture:
Where does this fit in?
What do students need to understand?
Why is this key idea important?
Prior learning
Misconceptions
Exemplified key ideas
Reflection questions
Appendices:
Key vocabulary
Representations and structure
Previous and Future learning
Useful links The exemplified key idea slides have the following symbols to indicate how they have been designed to be used: Into the classroom
The examples are presented on these slides so that they could be used in PD, but also directly in the classroom. The notes feature suggested questions and things teachers might consider when using with students. PD discussion prompts
These slides look at the examples in more detail, with question prompts to promote discussion among maths teachers. The notes feature reference to further information and guidance within the Core Concepts document.<br>
slide4. Where does this fit in? The NCETM has identified a set of six ‘mathematical themes’ within Key Stage 3 mathematics that bring together a group of ‘core concepts’.
The first of these themes is The structure of the number system, which covers the following interconnected core concepts:
1.1 Place value, estimation and rounding
1.2 Properties of number
1.3 Ordering and comparing
1.4 Simplifying and manipulating expressions, equations and formulae<br>
slide5. Where does this fit in? Within this core concept, 1.2 Properties of number, there are three statements of knowledge, skills and understanding.
These, in turn, are broken down into eleven key ideas. The highlighted key idea is exemplified in this slide deck.<br>
slide6. What do students need to understand? What prior knowledge might your students already have?
What language might they use to describe this key idea?
What questions might you want to ask to assess their prior learning? 1.2.1.2 Identify and explain whether a number is or is not a multiple of a given integer
Identify numbers which are and are not multiples of 2, 5 or 10.
Identify numbers which are and are not multiples of 2, 4 or 8.
Identify numbers which are and are not multiples of 3, 6 or 9.
Make connections between multiples of integers that are 10 or less.
Solve problems where there is more than one answer and there are elements of experimentation, investigation, checking, reasoning, proof, etc.
Solve familiar and unfamiliar problems, including real-life applications.<br>
slide7. Why is this key idea important? Students who have a deep understanding of the structure of numbers, who can partition numbers, recombine them and represent them in different ways, are more fluent and flexible when it comes to calculating. As students learn about the mathematical structures that underpin numbers and the number system, the need to generalise these structures will arise.
Students will have been introduced to multiples and factors at Key Stage 2 and will have had the opportunity to find factor pairs for a given number. They should know that prime numbers have exactly two factors and why, therefore, one is not prime. They should also be able to recall prime numbers up to 19 and identify others (possibly using the Sieve of Eratosthenes to find all the prime numbers up to 100).
Students will have found common factors and multiples for pairs of numbers, and it is likely that they will have done this by making lists of factors and multiples and looking for common items.
The focus at Key Stage 3 is on examining the structure of numbers and being able to reason whether numbers are multiples of other numbers or not without the need for creating lists of multiples. For example, students should recognise that 176 is a multiple of eight because it is the sum of 160 and 16, both of which are multiples of eight. Connections can be made here to the rules for divisibility, with students exploring why the rules work and how they can help identify multiples of a number.<br>
slide8. Prior learning What prior knowledge might your students already have?
What language might they use to describe this key idea?
What questions might you want to ask to assess their prior learning?<br>
slide9. Checking prior learning The following slides contain questions for checking prior learning.
What representations might students use to support their understanding of these questions?
What variation might you put in place for these questions to fully assess students’ understanding of the concept?
How might changing the language of each question change the difficulty?
Why are these such crucial pre-requisites for this key idea?<br>
slide10. Checking prior learning Write all the common multiples of 3 and 8 that are less than 50. a) b)<br>
slide11. Checking prior learning Fill in the missing numbers in this multiplication pyramid. c)<br>
slide12. Common difficulties and misconceptions What aspects of this key idea might pupils find challenging?
What misconceptions might pupils have? When teaching this topic, you may find students encounter difficulties with…
Using inefficient strategies to find multiples
Linking to division
Understanding and using the connections between multiplication tables
More information, and some suggestions for overcoming these challenges, can be found on the following slides.<br>
slide13. Common difficulties and misconceptions (1) Students often find multiples of an integer by listing numbers in the specified times table. This strategy is efficient for small numbers of multiples but can lead to misconceptions, such as thinking that numbers have only 12 multiples or that numbers outside of the times tables do not have multiples.
Students need to be able to identify the patterns present in multiples of an integer and explore the structures which generate those patterns.
For example, students should understand that adding two different multiples of the same number results in another multiple of that number.
Similarly, that if a number is a multiple of 15, for example, it is also a multiple of five and of three. By exploring multiples and reasoning in this way, students can decide whether any number is or is not a multiple of a given integer.<br>
slide14. Common difficulties and misconceptions (2) Strategies for identifying multiples usually link to division, especially for larger numbers which are not multiples known from multiplication tables.
Students who find it challenging to make the connection between the idea of multiples (numbers in multiplication tables) and division may benefit from revisiting prior work on factora × factorb = product, and variations of this: product ÷ factora = factorb
The use of partitioning can also be a useful strategy when identifying multiples. As shown in Example 4 (slide 22) 6 132 is not a multiple of eight because 6 132 = 6 000 + 120 + 12 and while 6 000 and 120 are both multiples of eight, 12 is not.
Using divisibility rules to test whether a number is a multiple or not may also be helpful, but if using these, students should be given time to investigate both why they work and how they can be used.<br>
slide15. Common difficulties and misconceptions (3) Students should also understand the connections between multiplication tables. For example, students should know that all multiples of ten are multiples of five but not all multiples of five are multiples of ten.
The use of a multiplication grid may support students to see these connections, consider the structures behind them and, consequently, be able to reason fully.
If students have only experienced multiples as a list of positive integers, defining multiples by a generalised statement, such as, ‘For any integers a and b, a is a multiple of b if a third integer c exists so that a = bc’ will help students understand that 14, 49, 70 and −21 are all multiples of seven because 14 = 7 × 2, 49 = 7 × 7, 70 = 7 × 10 and −21 = 7 × −3.<br>
slide16. Identify numbers which are and are not multiples of 2, 5 or 10 Example 1 Place a tick () in the cell if the number is a multiple of 2, 5 or 10. Explain how you know. What general statements can you make about multiples of 2, 5 and 10?
Find an integer which is a common multiple of 2, 5 and 10. Can you find another? And another? What do you notice?<br>
slide17. Identify numbers which are and are not multiples of 2, 5 or 10 Example 1 How do you define a multiple?
Do some of your students think multiples are the products listed in a times table?
How might they respond if you asked them if 91 is prime?*
What other questions can you ask to uncover any possible misconceptions your students might have? Place a tick () in the cell if the number is a multiple of 2, 5 or 10. Explain how you know. What general statements can you make about multiples of 2, 5 and 10?
Find an integer which is a common multiple of 2, 5 and 10. Can you find another? And another? What do you notice?<br>
slide18. Identify numbers which are and are not multiples of 2, 5 or 10 Example 2 The following four-digit number is a multiple of 2 but not a multiple of 5. What could the 1s digit be? How many possibilities are there? The following four-digit number is a multiple of 5 but not a multiple of 2. What is the biggest four-digit number which is both a multiple of 2 and a multiple of 5? What could the 1s digit be? How many possibilities are there?<br>
slide19. Identify numbers which are and are not multiples of 2, 5 or 10 Example 2 What is the potential effect of asking two similar but different questions here?
Can you construct other ‘all possibilities’ examples that might highlight the importance of the rules for divisibility? The following four-digit number is a multiple of 2 but not a multiple of 5. What could the 1s digit be? How many possibilities are there? The following four-digit number is a multiple of 5 but not a multiple of 2. What is the biggest four-digit number which is both a multiple of 2 and a multiple of 5? What could the 1s digit be? How many possibilities are there?<br>
slide20. Identify numbers which are and are not multiples of 2, 5 or 10 Example 3 Guy says that 543Â 210 is a multiple of 5 because it is a multiple of 10. Is Guy correct? Justify your answer.
Harriet says that 12Â 345 is a multiple of 10 because it is a multiple of 5. Is Harriet correct? Justify your answer.
Is there a similar relationship between other pairs of multiples, for example, 3 and 9? Can you explain why this is the case?<br>
slide21. Identify numbers which are and are not multiples of 2, 5 or 10 Example 3 How might students justify their answers? What language will you what to hear?
Can you construct other ‘what it’s not’ examples that might highlight the importance of the structure of the numbers? Guy says that 543 210 is a multiple of 5 because it is a multiple of 10. Is Guy correct? Justify your answer.
Harriet says that 12Â 345 is a multiple of 10 because it is a multiple of 5. Is Harriet correct? Justify your answer.
Is there a similar relationship between other pairs of multiples, for example, 3 and 9? Can you explain why this is the case?<br>
slide22. Identify numbers which are and are not multiples of 2, 4 or 8 Example 4 Place a tick () in the cell if the number is a multiple of 2, 4 or 8. Explain how you know. How can partitioning the number into known multiples of 4 and 8 help determine whether the number is a multiple of 4 or 8?
Raj says that 264 is not a multiple of 8 because 264 = 100 + 100 + 64 and although 64 is a multiple of 8, 100 is not. Explain why Raj is wrong.
Can you extend this idea to find a divisibility test for 4 and 8? Explain to your partner why it works.<br>
slide23. Example 4 How will you model using partitioning to check whether a number is a multiple? What misconceptions will you need to avoid?
How would you work through this example with a class to ensure students ‘go deeper’ with their thinking?
What prompts could you give them? Can you think of an example which might help? Identify numbers which are and are not multiples of 2, 4 or 8 Place a tick () in the cell if the number is a multiple of 2, 4 or 8. Explain how you know. How can partitioning the number into known multiples of 4 and 8 help determine whether the number is a multiple of 4 or 8?
Raj says that 264 is not a multiple of 8 because 264 = 100 + 100 + 64 and although 64 is a multiple of 8, 100 is not. Explain why Raj is wrong.
Can you extend this idea to find a divisibility test for 4 and 8? Explain to your partner why it works.<br>
slide24. Identify numbers which are and are not multiples of 3, 6 or 9 Example 5 Place a tick () in the cell if the number is a multiple of 3, 6 or 9. Explain how you know.<br>
slide25. Example 5 How can you use this example to support students to derive divisibility rules for 3, 6 and 9?
What questions might you give students next to reinforce this learning? Identify numbers which are and are not multiples of 3, 6 or 9 Place a tick () in the cell if the number is a multiple of 3, 6 or 9. Explain how you know.<br>
slide26. Make connections between multiples of integers that are 10 or less Example 6 If a number is a multiple of the integer indicated *, what else must it also always be a multiple of?
Complete the table below by indicating (with a tick) other numbers it is a multiple of.<br>
slide27. Example 6 What would you hope students might take from this activity?
What questions or follow-up tasks might you use to ensure that they build this understanding? Make connections between multiples of integers that are 10 or less If a number is a multiple of the integer indicated *, what else must it also always be a multiple of?
Complete the table below by indicating (with a tick) other numbers it is a multiple of.<br>
slide28. Solve problems where there is more than one answer and there are elements of experimentation, investigation, checking, reasoning, proof, etc Example 7 Answer these statements using: Always, Sometimes or Never.
If a number is a multiple of 10, it is also a multiple of 5.
If a number is a multiple of 4, it is also a multiple of 8.
If a number is a multiple of 9, it is also a multiple of 2.
Multiples are positive integers.
Is it always, sometimes or never true that adding two consecutive multiples of 5 will give a multiple of 10?
Is it always, sometimes or never true that adding five consecutive multiples of 2 will give a multiple of 10?<br>
slide29. Example 7 How can you build a classroom culture where students feel comfortable and confident to express their opinions, debate and challenge each other?
How do you teach learning behaviours so that students justify their answers as a matter of habit rather than requiring prompting? Solve problems where there is more than one answer and there are elements of experimentation, investigation, checking, reasoning, proof, etc. Answer these statements using: Always, Sometimes or Never.
If a number is a multiple of 10, it is also a multiple of 5.
If a number is a multiple of 4, it is also a multiple of 8.
If a number is a multiple of 9, it is also a multiple of 2.
Multiples are positive integers.
Is it always, sometimes or never true that adding two consecutive multiples of 5 will give a multiple of 10?
Is it always, sometimes or never true that adding five consecutive multiples of 2 will give a multiple of 10?<br>
slide30. Solve familiar and unfamiliar problems, including real-life applications. Example 8 Two lighthouses flash at different intervals. One flashes every 5 seconds and the other every 8 seconds.
At exactly midnight (00:00:00) they flash together. When will they next flash at the same time?<br>
slide31. Example 8 Do your students recognise this question as a multiples problem? How might you support them to make this link?
Can you create some other problems and contexts suitable for your classes where the need to find multiples is relevant? Solve familiar and unfamiliar problems, including real-life applications Two lighthouses flash at different intervals. One flashes every 5 seconds and the other every 8 seconds.
At exactly midnight (00:00:00) they flash together. When will they next flash at the same time?<br>
slide32. Reflection questions What other mathematical concepts will be supported by students’ stronger understanding of this key idea?
What mathematical language will you continue to use to support pupils to make connections with other areas?
Which representations might you continue to use to further develop students’ understanding?<br>
slide34. Appendices You may choose to use the following slides when planning or delivering a PD session. They cover:
Key vocabulary
Representations and structure
Previous learning
Future learning
Library of links<br>
slide35. Key vocabulary<br>
slide36. Representations and structure There are a number of different representations that you may wish to use to support students’ understanding of this key idea. These might include: Multiplication grids
You may find that the use of a multiplication grid supports students to identify the connections between the divisibility rules for different, but related, numbers.<br>
slide37. Previous learning From Upper Key Stage 2, students will bring experience of:
reading, writing, ordering and comparing numbers up to 10 000 000 and determining the value of each digit
rounding any whole number to a required degree of accuracy
using negative numbers in context
identifying the value of each digit in numbers given to three decimal places and multiplying and dividing numbers by 10, 100 and 1 000, giving answers up to three decimal places
using, reading, writing and converting between standard units, converting measurements of length, mass, volume and time from a smaller unit of measure to a larger unit and vice versa, using decimal notation up to three decimal places
using symbols and letters to represent variables and unknowns in mathematical situations that they already understand, such as:
missing numbers, lengths, coordinates and angles
formulae in mathematics and science<br>
slide38. Future learning<br>
slide39. Library of links The following resources from the NCETM website have been referred to within this slide deck:
NCETM Secondary Mastery Professional Development
1 The Structure of the number system Theme Overview Document
1.2 Properties of number Core Concept Document
Using mathematical representations at KS3 | NCETM
NCETM primary mastery professional development materials
NCETM primary assessment materials
There are also references to:
Standards & Testing Agency’s past mathematics papers<br>
Materials for use in the classroom or to support professional development discussions Summer 2021<br>
slide2. About this resource These slides are designed to complement the 1.2 Properties of number Core Concept document and its associated Theme Overview document 1 The Structure of the number system, both found in the Secondary Mastery Professional Development pages.
These slides re-present the key examples from the Core Concept document so that the examples can be used either directly in the classroom or with a group of teachers. There are prompt questions alongside the examples, and further clarification in the notes.
These slides do not fully replicate the Core Concept document, so should be used alongside it. Reference to specific page numbers, and to other useful NCETM resources, can be found in the notes for each slide.
This slide deck is not designed to be a complete PD session, rather it is a selection of resources that you can adapt and use as needed when planning a session with a group of teachers.<br>
slide3. About this resource The slides are structured as follows:
The big picture:
Where does this fit in?
What do students need to understand?
Why is this key idea important?
Prior learning
Misconceptions
Exemplified key ideas
Reflection questions
Appendices:
Key vocabulary
Representations and structure
Previous and Future learning
Useful links The exemplified key idea slides have the following symbols to indicate how they have been designed to be used: Into the classroom
The examples are presented on these slides so that they could be used in PD, but also directly in the classroom. The notes feature suggested questions and things teachers might consider when using with students. PD discussion prompts
These slides look at the examples in more detail, with question prompts to promote discussion among maths teachers. The notes feature reference to further information and guidance within the Core Concepts document.<br>
slide4. Where does this fit in? The NCETM has identified a set of six ‘mathematical themes’ within Key Stage 3 mathematics that bring together a group of ‘core concepts’.
The first of these themes is The structure of the number system, which covers the following interconnected core concepts:
1.1 Place value, estimation and rounding
1.2 Properties of number
1.3 Ordering and comparing
1.4 Simplifying and manipulating expressions, equations and formulae<br>
slide5. Where does this fit in? Within this core concept, 1.2 Properties of number, there are three statements of knowledge, skills and understanding.
These, in turn, are broken down into eleven key ideas. The highlighted key idea is exemplified in this slide deck.<br>
slide6. What do students need to understand? What prior knowledge might your students already have?
What language might they use to describe this key idea?
What questions might you want to ask to assess their prior learning? 1.2.1.2 Identify and explain whether a number is or is not a multiple of a given integer
Identify numbers which are and are not multiples of 2, 5 or 10.
Identify numbers which are and are not multiples of 2, 4 or 8.
Identify numbers which are and are not multiples of 3, 6 or 9.
Make connections between multiples of integers that are 10 or less.
Solve problems where there is more than one answer and there are elements of experimentation, investigation, checking, reasoning, proof, etc.
Solve familiar and unfamiliar problems, including real-life applications.<br>
slide7. Why is this key idea important? Students who have a deep understanding of the structure of numbers, who can partition numbers, recombine them and represent them in different ways, are more fluent and flexible when it comes to calculating. As students learn about the mathematical structures that underpin numbers and the number system, the need to generalise these structures will arise.
Students will have been introduced to multiples and factors at Key Stage 2 and will have had the opportunity to find factor pairs for a given number. They should know that prime numbers have exactly two factors and why, therefore, one is not prime. They should also be able to recall prime numbers up to 19 and identify others (possibly using the Sieve of Eratosthenes to find all the prime numbers up to 100).
Students will have found common factors and multiples for pairs of numbers, and it is likely that they will have done this by making lists of factors and multiples and looking for common items.
The focus at Key Stage 3 is on examining the structure of numbers and being able to reason whether numbers are multiples of other numbers or not without the need for creating lists of multiples. For example, students should recognise that 176 is a multiple of eight because it is the sum of 160 and 16, both of which are multiples of eight. Connections can be made here to the rules for divisibility, with students exploring why the rules work and how they can help identify multiples of a number.<br>
slide8. Prior learning What prior knowledge might your students already have?
What language might they use to describe this key idea?
What questions might you want to ask to assess their prior learning?<br>
slide9. Checking prior learning The following slides contain questions for checking prior learning.
What representations might students use to support their understanding of these questions?
What variation might you put in place for these questions to fully assess students’ understanding of the concept?
How might changing the language of each question change the difficulty?
Why are these such crucial pre-requisites for this key idea?<br>
slide10. Checking prior learning Write all the common multiples of 3 and 8 that are less than 50. a) b)<br>
slide11. Checking prior learning Fill in the missing numbers in this multiplication pyramid. c)<br>
slide12. Common difficulties and misconceptions What aspects of this key idea might pupils find challenging?
What misconceptions might pupils have? When teaching this topic, you may find students encounter difficulties with…
Using inefficient strategies to find multiples
Linking to division
Understanding and using the connections between multiplication tables
More information, and some suggestions for overcoming these challenges, can be found on the following slides.<br>
slide13. Common difficulties and misconceptions (1) Students often find multiples of an integer by listing numbers in the specified times table. This strategy is efficient for small numbers of multiples but can lead to misconceptions, such as thinking that numbers have only 12 multiples or that numbers outside of the times tables do not have multiples.
Students need to be able to identify the patterns present in multiples of an integer and explore the structures which generate those patterns.
For example, students should understand that adding two different multiples of the same number results in another multiple of that number.
Similarly, that if a number is a multiple of 15, for example, it is also a multiple of five and of three. By exploring multiples and reasoning in this way, students can decide whether any number is or is not a multiple of a given integer.<br>
slide14. Common difficulties and misconceptions (2) Strategies for identifying multiples usually link to division, especially for larger numbers which are not multiples known from multiplication tables.
Students who find it challenging to make the connection between the idea of multiples (numbers in multiplication tables) and division may benefit from revisiting prior work on factora × factorb = product, and variations of this: product ÷ factora = factorb
The use of partitioning can also be a useful strategy when identifying multiples. As shown in Example 4 (slide 22) 6 132 is not a multiple of eight because 6 132 = 6 000 + 120 + 12 and while 6 000 and 120 are both multiples of eight, 12 is not.
Using divisibility rules to test whether a number is a multiple or not may also be helpful, but if using these, students should be given time to investigate both why they work and how they can be used.<br>
slide15. Common difficulties and misconceptions (3) Students should also understand the connections between multiplication tables. For example, students should know that all multiples of ten are multiples of five but not all multiples of five are multiples of ten.
The use of a multiplication grid may support students to see these connections, consider the structures behind them and, consequently, be able to reason fully.
If students have only experienced multiples as a list of positive integers, defining multiples by a generalised statement, such as, ‘For any integers a and b, a is a multiple of b if a third integer c exists so that a = bc’ will help students understand that 14, 49, 70 and −21 are all multiples of seven because 14 = 7 × 2, 49 = 7 × 7, 70 = 7 × 10 and −21 = 7 × −3.<br>
slide16. Identify numbers which are and are not multiples of 2, 5 or 10 Example 1 Place a tick () in the cell if the number is a multiple of 2, 5 or 10. Explain how you know. What general statements can you make about multiples of 2, 5 and 10?
Find an integer which is a common multiple of 2, 5 and 10. Can you find another? And another? What do you notice?<br>
slide17. Identify numbers which are and are not multiples of 2, 5 or 10 Example 1 How do you define a multiple?
Do some of your students think multiples are the products listed in a times table?
How might they respond if you asked them if 91 is prime?*
What other questions can you ask to uncover any possible misconceptions your students might have? Place a tick () in the cell if the number is a multiple of 2, 5 or 10. Explain how you know. What general statements can you make about multiples of 2, 5 and 10?
Find an integer which is a common multiple of 2, 5 and 10. Can you find another? And another? What do you notice?<br>
slide18. Identify numbers which are and are not multiples of 2, 5 or 10 Example 2 The following four-digit number is a multiple of 2 but not a multiple of 5. What could the 1s digit be? How many possibilities are there? The following four-digit number is a multiple of 5 but not a multiple of 2. What is the biggest four-digit number which is both a multiple of 2 and a multiple of 5? What could the 1s digit be? How many possibilities are there?<br>
slide19. Identify numbers which are and are not multiples of 2, 5 or 10 Example 2 What is the potential effect of asking two similar but different questions here?
Can you construct other ‘all possibilities’ examples that might highlight the importance of the rules for divisibility? The following four-digit number is a multiple of 2 but not a multiple of 5. What could the 1s digit be? How many possibilities are there? The following four-digit number is a multiple of 5 but not a multiple of 2. What is the biggest four-digit number which is both a multiple of 2 and a multiple of 5? What could the 1s digit be? How many possibilities are there?<br>
slide20. Identify numbers which are and are not multiples of 2, 5 or 10 Example 3 Guy says that 543Â 210 is a multiple of 5 because it is a multiple of 10. Is Guy correct? Justify your answer.
Harriet says that 12Â 345 is a multiple of 10 because it is a multiple of 5. Is Harriet correct? Justify your answer.
Is there a similar relationship between other pairs of multiples, for example, 3 and 9? Can you explain why this is the case?<br>
slide21. Identify numbers which are and are not multiples of 2, 5 or 10 Example 3 How might students justify their answers? What language will you what to hear?
Can you construct other ‘what it’s not’ examples that might highlight the importance of the structure of the numbers? Guy says that 543 210 is a multiple of 5 because it is a multiple of 10. Is Guy correct? Justify your answer.
Harriet says that 12Â 345 is a multiple of 10 because it is a multiple of 5. Is Harriet correct? Justify your answer.
Is there a similar relationship between other pairs of multiples, for example, 3 and 9? Can you explain why this is the case?<br>
slide22. Identify numbers which are and are not multiples of 2, 4 or 8 Example 4 Place a tick () in the cell if the number is a multiple of 2, 4 or 8. Explain how you know. How can partitioning the number into known multiples of 4 and 8 help determine whether the number is a multiple of 4 or 8?
Raj says that 264 is not a multiple of 8 because 264 = 100 + 100 + 64 and although 64 is a multiple of 8, 100 is not. Explain why Raj is wrong.
Can you extend this idea to find a divisibility test for 4 and 8? Explain to your partner why it works.<br>
slide23. Example 4 How will you model using partitioning to check whether a number is a multiple? What misconceptions will you need to avoid?
How would you work through this example with a class to ensure students ‘go deeper’ with their thinking?
What prompts could you give them? Can you think of an example which might help? Identify numbers which are and are not multiples of 2, 4 or 8 Place a tick () in the cell if the number is a multiple of 2, 4 or 8. Explain how you know. How can partitioning the number into known multiples of 4 and 8 help determine whether the number is a multiple of 4 or 8?
Raj says that 264 is not a multiple of 8 because 264 = 100 + 100 + 64 and although 64 is a multiple of 8, 100 is not. Explain why Raj is wrong.
Can you extend this idea to find a divisibility test for 4 and 8? Explain to your partner why it works.<br>
slide24. Identify numbers which are and are not multiples of 3, 6 or 9 Example 5 Place a tick () in the cell if the number is a multiple of 3, 6 or 9. Explain how you know.<br>
slide25. Example 5 How can you use this example to support students to derive divisibility rules for 3, 6 and 9?
What questions might you give students next to reinforce this learning? Identify numbers which are and are not multiples of 3, 6 or 9 Place a tick () in the cell if the number is a multiple of 3, 6 or 9. Explain how you know.<br>
slide26. Make connections between multiples of integers that are 10 or less Example 6 If a number is a multiple of the integer indicated *, what else must it also always be a multiple of?
Complete the table below by indicating (with a tick) other numbers it is a multiple of.<br>
slide27. Example 6 What would you hope students might take from this activity?
What questions or follow-up tasks might you use to ensure that they build this understanding? Make connections between multiples of integers that are 10 or less If a number is a multiple of the integer indicated *, what else must it also always be a multiple of?
Complete the table below by indicating (with a tick) other numbers it is a multiple of.<br>
slide28. Solve problems where there is more than one answer and there are elements of experimentation, investigation, checking, reasoning, proof, etc Example 7 Answer these statements using: Always, Sometimes or Never.
If a number is a multiple of 10, it is also a multiple of 5.
If a number is a multiple of 4, it is also a multiple of 8.
If a number is a multiple of 9, it is also a multiple of 2.
Multiples are positive integers.
Is it always, sometimes or never true that adding two consecutive multiples of 5 will give a multiple of 10?
Is it always, sometimes or never true that adding five consecutive multiples of 2 will give a multiple of 10?<br>
slide29. Example 7 How can you build a classroom culture where students feel comfortable and confident to express their opinions, debate and challenge each other?
How do you teach learning behaviours so that students justify their answers as a matter of habit rather than requiring prompting? Solve problems where there is more than one answer and there are elements of experimentation, investigation, checking, reasoning, proof, etc. Answer these statements using: Always, Sometimes or Never.
If a number is a multiple of 10, it is also a multiple of 5.
If a number is a multiple of 4, it is also a multiple of 8.
If a number is a multiple of 9, it is also a multiple of 2.
Multiples are positive integers.
Is it always, sometimes or never true that adding two consecutive multiples of 5 will give a multiple of 10?
Is it always, sometimes or never true that adding five consecutive multiples of 2 will give a multiple of 10?<br>
slide30. Solve familiar and unfamiliar problems, including real-life applications. Example 8 Two lighthouses flash at different intervals. One flashes every 5 seconds and the other every 8 seconds.
At exactly midnight (00:00:00) they flash together. When will they next flash at the same time?<br>
slide31. Example 8 Do your students recognise this question as a multiples problem? How might you support them to make this link?
Can you create some other problems and contexts suitable for your classes where the need to find multiples is relevant? Solve familiar and unfamiliar problems, including real-life applications Two lighthouses flash at different intervals. One flashes every 5 seconds and the other every 8 seconds.
At exactly midnight (00:00:00) they flash together. When will they next flash at the same time?<br>
slide32. Reflection questions What other mathematical concepts will be supported by students’ stronger understanding of this key idea?
What mathematical language will you continue to use to support pupils to make connections with other areas?
Which representations might you continue to use to further develop students’ understanding?<br>
slide34. Appendices You may choose to use the following slides when planning or delivering a PD session. They cover:
Key vocabulary
Representations and structure
Previous learning
Future learning
Library of links<br>
slide35. Key vocabulary<br>
slide36. Representations and structure There are a number of different representations that you may wish to use to support students’ understanding of this key idea. These might include: Multiplication grids
You may find that the use of a multiplication grid supports students to identify the connections between the divisibility rules for different, but related, numbers.<br>
slide37. Previous learning From Upper Key Stage 2, students will bring experience of:
reading, writing, ordering and comparing numbers up to 10 000 000 and determining the value of each digit
rounding any whole number to a required degree of accuracy
using negative numbers in context
identifying the value of each digit in numbers given to three decimal places and multiplying and dividing numbers by 10, 100 and 1 000, giving answers up to three decimal places
using, reading, writing and converting between standard units, converting measurements of length, mass, volume and time from a smaller unit of measure to a larger unit and vice versa, using decimal notation up to three decimal places
using symbols and letters to represent variables and unknowns in mathematical situations that they already understand, such as:
missing numbers, lengths, coordinates and angles
formulae in mathematics and science<br>
slide38. Future learning<br>
slide39. Library of links The following resources from the NCETM website have been referred to within this slide deck:
NCETM Secondary Mastery Professional Development
1 The Structure of the number system Theme Overview Document
1.2 Properties of number Core Concept Document
Using mathematical representations at KS3 | NCETM
NCETM primary mastery professional development materials
NCETM primary assessment materials
There are also references to:
Standards & Testing Agency’s past mathematics papers<br>