Multiplication and division

By Year 6, children’s understanding of multiplication and division (and recall of facts) should be secure enough to enable them to make use of formal written methods such as long multiplication and long division, applying them within a variety of problem-solving contexts.

The ability to do this relies on many layers of learning and multiplicative understanding. In this blog, I will share ideas and strategies for teaching concepts within multiplication and division, set in the imaginary world of a beautiful picture book.

Teaching multiplication and division with beadstrings

Beadstrings are a useful manipulative to use when exploring equal groups for multiplication and division. 

For example, to explore 4 multiplied by 3, equal groups of 3 can be pulled out along the beadstring to show multiples of 3.

The language of ‘equal groups’ and ‘product’ and the formal equation are all connected to the visual representation.

Fact families

The beadstring is a useful resource for exploring the grouping structure of division and the link to multiplication.

(For exploring the sharing structure, counters or cubes would be more appropriate).

If dividing 12 by 3, we can ‘pull’ out groups of 3 beads to found out how many groups we can make.

The beadstring is a staple in many primary school classrooms.  If you are less familiar with them, take a look at this blog to get you started: Step-by-step guide to using the 0-100 bead string from Year 1 to Year 6.

Whole school maths through stories

World Book Day is fast approaching and this year, we shine a spotlight on the work of Elinor J Pinczes, and the book, One Hundred Hungry Ants.

More ideas for incorporating maths into World Book Day here: Maths through stories

Multiplication and division concepts linked to scenarios in this story:

• Equal groups (EYFS and KS1)
• Sharing and grouping (KS1 and KS2)
• Constructing arrays and commutativity (KS1 and LKS2)
• Relationship between multiplication and division facts (KS1 and KS2)
• Factors (Y4 and UKS2)
• Heuristics – build a model, act it out, draw a picture, find all possibilities, solve a simpler but related problem (all phases)

Using a beadstring to support exploration of multiplication and division

Some of the reasons I might use a beadstring to explore this story:

• There are 100 hungry ants and 100 beads on the beadstring!
• Units of 10 are clearly visible and can support skip counting.
• Beadstrings can be easily ‘folded’ to create arrays.

We could use counters or multilink but with larger numbers, this could get messy!

• We can quickly revisit the structure of multiplication and division on the beadstring with lower numbers before exploring the story.
• It’s an easy manipulative to use to support development of heuristics like building a model and drawing a picture.

Teaching ideas for EYFS and KS1 classrooms

Read the story to the children under a visualiser and share the images as the story progresses.

Drawing attention to equal groups by making a model

Draw the children’s attention to the lines / equal groups being formed by the marching ants.

Model and explore this.

For example:

Watch out for children who do not see these lines as equal because of the colours of the beads. Bring their focus back to the value / number of beads in each line.  

The beadstrings could be presented with or without the labels as the focus is on the equal groups.

Solve a simpler but related problem

Children could be tasked with a simpler but related problem.

How many different formations could be made by 20 hungry ants?

0-20 beadstrings would be particularly useful here.

For example:

Children could build or act out the problem using other resources such as counters, cubes or themselves!

Children might draw out their formations to show the multiplication. 

Further exploration and challenge:

• How will the children know they have found all possible formations?
• Can they prove that a given formation will not work? For example, why can the 20 hungry ants not march in lines of 6?
• Would the ants get to the picnic quicker if they marched in 2 lines/groups of 10 or 10 lines/groups of 2?

Linking multiplication and division – fact families

Use the models and formations drawn out through a simpler but related problem to link multiplication and division calculations. Children could use speaking frames to describe the relationships. 

Teaching ideas for KS2 classrooms

Exploring arrays and linking multiplication and division

Read the beginning of the story to set the scene and stop after the ants arrange themselves into 2 lines of 50. As discussed previously, this can then be modelled using a 100 bead beadstring.

Children could then be tasked with finding alternative formations the ants could march in.

• Can children create their formations/arrays using beadstrings?
• Can they find all possible formations the ants could march in?
• Can they predict which order the formations will appear in the story? Why do they think this might be?
• Using the speaking frame mentioned previously, can children list all of the multiplication and division facts for their formations? What is the same and what is different about 2 lines/groups of 50 ants and 50 lines/groups of 2 ants?

Read the children the rest of the story. Do their formations appear? Would the ants be quicker marching in 2 lines of 50 or 50 lines of 2?

Solve a related multiplication and division problem

1. Use the story as a stimulus but change the number of ants marching to the picnic and explore the possible formations.
2. Compare the number of formations possible for different numbers of ants by exploring the validity of a theory, such as:

There are 2 groups of ants.

One group is made up of 30 ants and one group is made up of 35 ants.

Because 35 is a larger number, I think that means the ants could march in a greater number of formations.

Exploring factors

This story also acts as a great stimulus for exploring factor pairs of a number. And instead of factor bugs or beetles, why not draw factor ants!

Here are my efforts of drawing and completing a factor ant inspired by the illustrations in the book! 

I appreciate that these ants won’t necessarily have the ‘standard’ six legs.

As above, the number of ants could be adapted and the number of total formations listed alongside the factors, making links back to commutativity; considering what is the same and what is different. For example, 4 x 25 and 25 x 4 – the same numbers are involved but the formations for these would look different!

As always, we love to see work children produce so please do email the team at primarymaths@hfleducation.org to share any examples that this blog may inspire!

If you have been inspired by ‘One Hundred Hungry Ants’, you may also enjoy, ‘A Remainder of One’, by Elinor J Pinczes. It tells the story of a bug called Joe who is part of the 25th squadron but always finds himself without a partner or group during parades. Poor Joe!

Spoiler alert! There is a happy ending for Joe! 

In the evolving landscape of education, flexible working has emerged as a crucial strategy for attracting and retaining talented staff. Lucy Makins, a recruitment manager at HFL Education, has been a vocal advocate for flexible working in schools. Her insights, shared through various platforms including HFL Education and LinkedIn, provide a compelling case for why school governors and trustees should consider implementing flexible working practices in their schools.

The case for flexible working

Flexible working is not just a trend; it is a response to the changing needs of the workforce. The COVID-19 pandemic accelerated the shift towards flexible working, highlighting its benefits for both employees and employers. For schools, adopting flexible working can lead to increased staff satisfaction, improved retention rates, and a more diverse workforce.

Lucy emphasizes that flexible working can help schools attract a wider pool of candidates, including those who might otherwise be unable to commit to traditional full-time roles due to personal or family commitments. By offering flexible working options, schools can tap into a broader talent pool, ensuring they have the best possible staff to support their students.

Practical implementation of flexible working

One of the main concerns for school leaders is how to implement flexible working without disrupting the continuity of teaching and learning. Lucy provides practical examples of how schools have successfully integrated flexible working into their operations. For instance, Howe Dell Primary School in Hatfield has introduced job shares, flexible hours, and remote working options for staff. This approach has not only helped retain talented staff but also built loyalty and commitment among the team.

Key strategies for implementation:

1. Job sharing: Allowing two teachers to share a full-time role can provide flexibility while ensuring that students receive consistent teaching.
2. Flexible hours: Offering flexible start and finish times can help staff balance their work and personal commitments.
3. Remote working: Enabling staff to work from home for certain tasks, such as planning and professional development, can enhance productivity and job satisfaction.

Addressing challenges

While the benefits of flexible working are clear, there are challenges that need to be addressed. One of the main concerns is the potential impact on workload and the need to ensure that all classroom hours are covered. Lucy suggests that schools need to be creative and willing to compromise. For example, teachers may need to be flexible about their preferred working patterns to ensure that the needs of the school are met.

Another challenge is the perception that flexible working might lead to a reduction in the quality of education. However, evidence suggests that flexible working can actually enhance the quality of teaching by reducing stress and burnout among staff. When teachers are happier and more balanced, they are better able to focus on delivering high-quality education.

The role of governors and trustees

School governors and trustees play a crucial role in shaping the policies and practices of their schools. By championing flexible working, they can help create a more supportive and inclusive working environment. Here are some steps that governors and trustees can take to promote flexible working in their schools:

1. Policy development: Develop and implement policies that support flexible working. This includes setting clear guidelines on how flexible working requests will be handled and ensuring that all staff are aware of their rights and options.
2. Budgetary considerations: Are there additional costs to consider with flexible working opportunities ie with shared roles? It will always be a balancing act between what is desirable and what is affordable in terms of what your school budget can support.
3. Training and support: Provide training for school leaders on how to manage flexible working arrangements effectively. This can include workshops on time management, communication, and team collaboration.
4. Monitoring and evaluation: Regularly review the impact of flexible working on staff satisfaction, retention, and student outcomes. Use this data to make informed decisions and adjustments to the flexible working policy.
5. Communication: Foster open communication with staff about the benefits and challenges of flexible working. Encourage feedback and be willing to make changes based on staff input.

Benefits for students

Ultimately, the goal of flexible working is to create a better working environment for staff, which in turn benefits students. When teachers are less stressed and more satisfied with their work, they are better able to focus on their students' needs. Flexible working can also help schools attract and retain high-quality teachers, ensuring that students receive the best possible education.

Conclusion

The work of Lucy highlights the importance and feasibility of flexible working in schools. By embracing flexible working, school governors and trustees can create a more supportive and inclusive environment for staff, leading to improved retention, satisfaction, and ultimately, better outcomes for students. The key is to approach flexible working with an open mind, a willingness to compromise, and a commitment to continuous improvement.

For school governors and trustees, the journey towards flexible working may present challenges, but the potential benefits make it a worthwhile endeavour. By following the insights and strategies shared by Lucy, schools can successfully navigate the transition to flexible working and create a more positive and productive environment for all.