Welcome to our maths blog!
Each blog will feature a snapshot of learning that has taken place in school, explanations of calculation strategies we teach, top tips for helping your child at home, a look at famous mathematicians and book recommendations.
The maths blog is now posted half-termly.
I enjoyed sharing our maths provision with our Vice Chair of Governors this half term. Mr Shah was invited into school to see all of the great work that is going on in the subject and learn about about how we teach maths at Margaret Wix. As part of this process, Mr Shah and I visited EYFS and Key Stage One classrooms.
It was wonderful to see our littlest learners immersed in a maths lesson involving number bonds to ten that was contextualised through a wonderful picture book: 'The Snail and The Whale'. The children were enjoying counting the number of seals in the pictures and working out how many more they needed to make ten. We also took a look at the learning environment which provides ample opportunities to practice maths each day. We saw children were able to make patterns, recognise numerals, practise counting and so much more!
In Hummingbirds, we were treated to a high-quality lesson in which the children used concrete resources alongside pictures to construct understanding of counting in multiples of two. It was great to hear the children echoing mathematical vocabulary, that the staff had modelled to them, as they confidently worked with their learning partners in the beginning stages of learning their two times table.
Emperors classroom showcased the early stages of children reasoning and working systematically. The children had an engaging task involving robots and were using their communication and pattern-seeking skills to find all the possible combinations of their robot's body parts. We could also clearly see how the children are able to utilise the environment to support them with independent learning - the children have access to manipulatives and a working wall, with clearly displayed vocabulary and modelled examples, that helps them to succeed.
I ended the visit feeling proud of the quality of maths provision that we offer at Margaret Wix and that sense of pride only increased the following day when our year 3 maths challenge team, who had previously come 1st out of 200 schools in the heats, took part in the grand final of the county-wide competition. The four participants faced two rounds of general maths questions, a tricky shape round and an estimation round that even made Mr Milbourn say, 'Wow! They had to do that? They are only in year 3!" Becca, Eli, Davina and Hin did not let the complexity of the questions phase them though as they tried their hardest to work together to solve all of the questions under timed conditions. They did amazingly well and we were so proud to hear that they came in 4th place overall with a great score of 154/200.
Admirals have been working hard to master a range of different mathematical skills this half term including: finding percentages and fractions of amounts; translating and reflecting shapes on a co-ordinates grid; and naming, measuring and drawing different angles. What a lot of learning they have managed to squeeze into a short six weeks!
After focusing on SATs at the beginning of this half term, Monarchs have latterly been engaged in a young enterprise project that has brought maths to life! The children have learned various financial maths skills including budgeting and managing income and expenditure. They put what they had learnt into practice when developing their own small businesses. Each group was given a £5 budget and had to develop a product to sell. The children were responsible for calculating their break-even points and enjoyed selling their wares to the rest of the school on the last day of the half term.
Telling the time is a vital life skill but is also something that children often find difficult to grasp, especially as it relies on a base 60 number system rather than our usual base 10.
What do children learn when?
In year one, children learn to tell the time to the hour and half past.
In year two, children learn to tell the time to five minutes including quarter past/to. They are also expected to know the number of minutes in an hour and hours in a day.
In year three, children learn to tell the time using roman numerals, 12-hour and 24-hour notation. They also estimate and read time to the nearest minute and use vocabulary such as a.m. and p.m. to compare time.
In year four, children are taught to read, write and convert time between analogue and digital 12 and 24-hour clocks.
Before children can begin to tell the time, they need to understand how it relates to their own life experiences - draw attention to the time by mentioning it at significant points in the day, for example, 'It's five o'clock; it's time for dinner.' Having a variety of clocks at home - both analogue and digital - is important, but try to draw your child's attention to the analogue one more often as this is what they will use at school to learn to tell the time.
Being able to count to 60 will make telling the time easier, as will knowing the five times table. You can make practising counting, telling the time and measuring time fun by playing games that involve a sand timer, using a timer for baking or by timing races or challenges on a stopwatch.
As your child becomes more confident with telling the time, using timetables to solve problems is also a great way to practise for example, you could ask your child to work out how many more minutes until the next bus arrives or how long it has been since you put your cake in the oven.
A number line is a useful tool for calculating differences in time, such as in the following question: You arrived at school at 8:30 am. Mum collected you for an appointment at the dentist at 12:10 pm. How long were you at school this morning?
We encourage the children to 'jump' along their number line to the nearest hour, then jump as far as possible using whole hours, before finally making any last jumps needed using minutes. Once you have drawn all of your jumps along the number line you can add up the hours and minutes to solve the problem.
Maria Gaetana Agnesi was born in Milan, Italy and lived between 1718-1799. She was the first woman to write a book on Mathematics and also the first to be appointed as a professor of the subject at university level. Her first book was published in 1748, when Maria was just 30 years old. The book provided an extensive account of algebra and theology. One of her specialisms was the cubic plane curve, from two diagrammatically opposite points of a circle. This curve is now described as the ‘witch of Agnesi’.
This is a classic story that is perfect for practising using time vocabulary. Accompany Mr Wolf as he goes about his daily routine from breakfast to bedtime - and get to know the real Mr Wolf! The story includes familiar faces from a plethora of nursery rhymes, including Little Red Riding Hood (masquerading as the post girl), three cheeky little pigs (who make prank calls), a cat who's a dab hand at the fiddle, plus four and twenty blackbirds...
It is an enchantingly original story inspired by the well-known playground game. The stunning artwork reveals something new with every reading and the many fabulous details will keep you turning the pages again and again. Don't forget, if you choose to purchase this text, please use Amazon Smile to donate to the PTFA.
Another half term has flown by and children have continued to work hard in maths lessons across the school. Early Years have continued learning through stories such as Farmer Duck Paddington and The Very Hungry Caterpillar. They have been learning how to group and compare amounts, recognise money, use positional language, and much more!
Maths lessons in Hummingbirds this half term have focused on part-part-whole models, word problems and measures. In Emperors, the children have explored written strategies for addition and subtraction as well as developing their understanding of properties of shapes, including symmetry. What a busy half term it has been in Key Stage One!
Cardinals have had the opportunity to work with a maths teaching and learning advisor from Herts for Learning, who has supported Mr Milbourn in developing new small group initiatives. We look forward to reviewing the impact of, and continuing to build on, these next half term.
Four pupils from Admirals represented Margaret Wix in the Herts Year 5 Maths Challenge - this was actually an international event with schools from other countries competing too. We are so proud of Isabella, Joy Ye, Allie and Thomas who tackled a variety of problem solving and reasoning questions confidently and scored a whopping 165 points. This was 20 points above average. unfortunately, they just missed out on making the grand final, but we're looking forward to competing again next year!
Monarchs have been preparing for their SATs tests, focusing on the core knowledge that they will need to answer many of the questions. They taken practice tests in their stride and have shown our learning powers of self-belief and resilience in tackling the tricky questions.
As we've seen, many classes have been (or will soon be) focusing their learning on money. Research indicates that our spending habits are formed by the time we reach 7 years old, so it is important that we think carefully about how we can pass on money skills to children to ensure that they are financially literate. Below are a few top tips, activities and games that can be used at home to teach children about money in a fun and engaging way.
1) Develop understanding of the value of coins. Show your child some real coins and explain to them what the value of each one is. A good game for recognising coins is to put a pile of coins on the table and then give them five seconds to find a certain coin. You could also give them one of each of the different coins and ask them to put them in order, from smallest value to largest.
2) Encourage children to count coins - choose an amount and ask them to work out how many of each coin would be needed to make it e.g.
(of course the columns for 1p - 10p would need to be extended to allow many more coins than we can see in this image to fit in!)
3) Go shopping together and show them the cost of different items. Explain your choices to them while you shop. Let them count money and hand it over to the cashier.
4) Give them a piggy bank to store their pocket money or money they have earned from doing jobs around the house - this helps to teach children that money is earned. It also encourages children to save rather than act on impulse and spend immediately.
5) Take a trip to the bank and open a savings account in their name. This will help to teach the difference between digital and paper currency. You may be able to set up a pre-paid debit card for your child.
Shakuntala Devi (4 November 1929 – 21 April 2013) was an Indian mental calculator and writer, popularly known as the "Human Computer". Her talent earned her a place in the 1982 edition of The Guinness Book of World Records. She demonstrated her remarkable talent by answering questions such as multiplication of two 13-digit numbers – 7,686,369,774,870 × 2,465,099,745,779. The numbers were picked randomly and she solved the problem mentally in just 28 seconds. Devi strove to simplify numerical calculations for students. She wrote a number of books in her later years, including novels as well as texts about mathematics, puzzles, and astrology.
A witty and mischievous tale inspired by The Wind in the Willows, from renowned poet Roger McGough. This story teaches children to be responsible with money. Mr Toad has a shiny gold coin. He gives it to Miss Mole to pay for a room in her hotel. Now Miss Mole can pay Sam Stoat for painting the hotel, and Sam Stoat can pay back Mr Badger...until the coin might just end up back where it started!
Well this half term has been marvellous - full of marvellous maths in fact! We thoroughly enjoyed opening our doors and welcoming parents and carers into school for a week of maths learning and fun. The week began with a session during which Mrs Gibbs and I demonstrated a number of simple card games that could be played to help develop mental fluency. Fluency in maths is about developing number sense, being able to the select the most efficient method for the task at hand and being able to apply a skill to multiple contexts. The games are easy to play and can be made simpler/more challenging to suit your child. The slides from this session can be found linked below if you would like to play some of these card games at home too.
Throughout Marvellous Maths Week, parents and carers joined their children in maths lessons. We aimed for this to provide you with the opportunity to understand the expectations within lessons, see how your child learns, recognise how maths is taught in primary schools and, hopefully, identify ways in which you could support your child at home. We know the children loved being able to share their learning with you and we hope that the experience was valuable for you too.
Strategies for partitioning or regrouping allow children to understand how to break up larger numbers into smaller ones. It is important that children are aware that numbers can be partitioned or regrouped – both along the place value boundaries (canonically) and in other ways (non-canonically) - and how to use this to perform calculations.
These are the ways you can help your child to progress with partitioning or regrouping:
Born in present day Iran, in 940, Abu al-Wafa' Buzjani was a mathematician and astronomer who contributed to the development of trigonometry. His arithmetic work for Baghdad merchants and others included the first use of negative numbers in medieval Arabic texts. In honour of his astronomical work, a crater on the moon was named after him.
This story is a wonderful and unique look at counting and friendship. It is book is about two friends, Sue and Nick, who like very different things but are still best friends. With a focus on counting both forwards and backwards, they introduce us to all their favourite things from one boy called Nick and one girl called Sue to ten cakes for tea, and then all the way back down to one yellow moon shining in the night sky. The artwork is beautiful with each artist contributing alternate pages in their own inimitable style. The simple text is perfect for children learning to read and learning to count. if you'd like to purchase this text, consider using Amazon Smile in order to donate to the PTFA in the process.
Across the school, the children have continued to put a great deal of effort into their maths learning this half term. In Firetips, the staff were pleased to welcome parents to join them for a morning focussed on mathematics. When parents were asked about the session, the response was very positive:
"Really great to see her in her environment, learning and playing so well."
"Seeing how she is learning so we can continue at home. It's also nice seeing how confident she is in class with other children."
"It was very educational and age appropriate."
"You're amazing teachers - thank you!"
"Lovely to see what the children are up to."
"Thank you for all you do. My son loves going to school and I'm so impressed with how far he's come."
Whilst other classrooms have not opened their doors for maths lessons yet (this is something you can look forward to next half term), the children have still been making great strides in their learning. Hummingbirds have been busy making ten and more. Emperors have mastered a range of mental strategies for addition and subtraction. Cardinals have begun working on the tricky topic of fractions, using lots of pictorial representations as they tried to master unit and non-unit fractions. As well as all their work in maths lessons, Admirals and Monarchs have been applying their skills in other areas of the curriculum too: year 5 had to ensure they measured accurately in DT when making their moving volcano pictures and year 6 have created graphs in both science and geography lessons.
Below, you can see all of the basic addition facts that children should know by the end of Key Stage 1. They are colour coded to show which strategies are useful for calculating them all.
One strategy you may not be familiar with is compensating and adjusting. This involves adding more than you need and then subtracting the extra. This is useful for adding numbers that are close to a multiple of 10, such as those that end in 7, 8 or 9. The number is rounded to a multiple of 10 and before adding and then the extra must be subtracted. For example, adding 9 is carried out by adding 10, then subtracting 1. To add a number to 18 we would add it to 20 instead before subtracting 2. A similar strategy works for adding decimals that are close to whole numbers in Key Stage Two.
As a parent of a primary school aged child, you are likely to have encountered concrete resources at some point during your child’s maths education. We utilise a mastery approach to maths teaching at Margaret Wix, a key feature of which is the concreate pictorial abstract (CPA) method. This means that children are initially introduced to new concepts through the use of concrete resources or manipulatives. These are objects that children can use to aid their understanding of different maths concepts. A mastery teaching approach encourages all children to use these concrete resources. The abstract nature of maths can be challenging for children so, through the use of these concrete, practical resources, they are able to ‘see’ the maths and make sense of what is actually happening.
It isn't just at school that children can benefit from using concrete resources; they are equally important at home.
Children often begin to learn about place value by representing numbers using base ten equipment. These are excellent resources for bringing to life the relationship between ones, tens and hundreds. Their relative sizes really enable children to visualise the numbers and see the relationship between them. Straws, lollipop sticks or anything else that can be bundled together are a fantastic alternative to base ten equipment. For example, individual straws can represent the ones and bundled together into tens to represent the tens.
Lego is another good alternative, using individual bricks for the ones and joining together to make tens. The children are able to visually see the link between the size of the individual straws/bricks and the bundles of ten straws/bricks.
Fractions are an area of maths children can find particularly difficult, due to their abstract nature. For this reason, it is essential they have practical resources to use. Fraction cubes or circles and Numicon are very useful for aiding understanding of the fraction of a whole, add to that using place value counters or multi-link cubes, and you can demonstrate a real range of fraction concepts: from the basics of what a fraction is; to recognising equivalent fractions; understanding what happens when a fraction is greater than one and for adding/subtracting fractions. Finding fractions of amounts is another concept children can have difficulty grasping, particularly when the numerator is greater than one. Using resources, such as cubes or counters and physically sharing them out really helps children understand. There are lots of cheap/free alternatives to buying fraction resources. Fraction circles can be made out of paper plates, or downloaded off the internet and printed. Lego and printable fraction strips are a good alternative to buying fraction cubes. For fractions of amounts, anything that can be shared out can be used. Food works particularly well for this. For example: raisins, grapes, sweets etc…
When children are first introduced to decimals, tens frames are useful for helping them to understand tenths, whilst hundred squares support children with recognising hundredths.
Tens frames are a particularly useful resource for enabling children to see the link between decimals and fractions, whilst also helping children to understand concepts such as rounding decimals. Place value counters positioned on a place value grid are useful for helping children to understand place value of decimal numbers. Children can visually see how many tenths and/or hundredths a number has and they can help them understand how to compare and order decimals. All these resources can be found/created at home. Tens frames, hundred squares and decimal strips can be printed off the internet whilst decimal place value counters can be made by writing 0.1 on one set of counters and 0.01 on another.
If you haven’t tried using concrete resources to support your child at home before, then give it a go! This is by no means an exhaustive list of topics and resources. Hopefully these suggestions give you some ideas and a starting point though, whilst providing an insight into how useful concrete resources can be in helping children to understand maths.
Benjamin Banneker was a distinguished African American mathematician born in 1731. His father has been enslaved but Benjamin was free. He had little schooling but developed a love for books and learning. Banneker attracted a lot of attention for his work in his twenties when he designed and developed the first ever fully wooden clock. He taught himself astronomy and became known for his successful prediction of a solar eclipse. He was also part of a team that surveyed and planned the capital of the United States. When the city plans were lost, Banneker was amazingly able to reproduce them from memory.
Striking illustrations and an empowering story combine to introduce young readers to the world of maths, creative thinking and problem-solving. This is a brilliant picture book that unlocks a love of numbers and creative thinking, and celebrates women in STEM.
Setting off on a camping adventure with her cousin, Aliyah soon discovers that numbers are everywhere, whether it's counting out money at the shops, planning trips on the train or even stargazing in a forest. As Aliyah solves some sums of her own, she learns about the brilliant mathematicians who have helped us understand our world. Soon she can't wait to become a maths whizz too!
If you choose to purchase this book, don't forget to use Amazon smile to support the Friends of Margaret Wix PTFA.
It's been a brilliant start to the school year at Margaret Wix, with all children focused on developing their knowledge and understanding of number and place value. They have been learning about the value of each digit in a number, for example the 5 in 530 represents 5 hundreds, or 500; however the 5 in 50,306 represents 50 thousands or 50,000. It is important that children undersand that while a digit can be the same, its value depends on its position within a number.
It has been a pleasure for me, as maths lead, to be able to visit every classroom in the school and see a maths lesson. Our littlest learners, in Firetips, have been practising counting within ten. I saw children developing their mathematical vocabulary through songs and watched them play games that helped with developing a strong grounding in number.
I have joined Hummingbirds for lessons involving using part-whole models to show how a number can be partitioned in different ways. It was fantastic to see the children developing an early understanding of the concept of commutativity in addition, meaning that the numbers in an addition calculation can be swapped around and the answer will remain the same e.g. 3 + 4 = 7 and 4 + 3 = 7.
Emperors impressed me with their use of number bonds when solving problems involving calculating beyond ten. The children were able to spot which numbers could easily be grouped together to make ten before then adding more e.g. 6 + 4 + 3 = 13. I saw children challenging themselves to use the same technique with number bonds to twenty that enabled them to calculate with larger numbers too.
I had the pleasure of teaching a maths lesson in Cardinals this half term. We built on the children's knowledge of commutativity for addition and investigated whether this law applied to subtraction too. We found that, unlike addition, subtraction must be completed in a particular order: the largest number must come first.
Admirals have practised rounding numbers to varying degrees. They have used various pictorial representations to help them identify the nearest 10 or 100 before applying what they have learnt to problems in a variety of contexts.
Monarchs explored the value of decimals numbers recently, developing the understanding that everything after a decimal point represents a fraction of a whole. The children practised ordering and comparing decimal numbers as well as rounding them to varying degrees. Their knowledge of decimals has helped them to become more adept when multiplying or dividing numbers by 10, 100 and 1000 too.
It has been a great start to the term, with all children developing firm foundations from which they can build upon this year.
Number bonds have been called, at various points in the recent past, addition and subtraction facts, facts families, number stories and doubtless other names too. Parents are often unfamiliar with the names which sometimes causes concern, but they are simply combinations of numbers that fit together.
Children must know the number bonds that make 10.
10 can be made up out of:
If your child has instant recall of each of these number bonds and all of the ways they can fit together, they’ll be well on the way to dealing with arithmetic confidently and quickly. Ideally, they’ll also know their number bonds to 4, 5, 6, 7, 8 and 9.
This video shows number bonds being created using the concept of 'parts and wholes'. These models are regularly used in school.
Knowing number bonds fluently up to ten is one of the foundations to more complex arithmetic such as adding and subtracting with the ‘column methods’ and mental addition and subtraction to 100 and beyond.
They also help with addition and subtraction bridging 10. If you want to work out 16 – 8, for example, you can use the following thought process:
Parent may find the prospect of helping their children with maths quite daunting - even if they are pretty good at maths. With a little confidence, parents can make a big difference though. Extra help at home can have a big impact and, at primary age, there are lots of ways you can help in a fun and rewarding way.
Top tips for helping at home:
Age 3 & 4 is a great time to start learning to count - ideally at the kitchen table with real objects. Things you can do at home:
At ages 5 & 6 the focus moves from counting to addition and subtraction. Things you can do at home:
Ages 7 & 8 see the introduction of multiplication which leads on to division. Arrays are an important visual way to understand multiplication - here are two arrays illustrating that 4 x 3 and 3 x 4 are equivalent.
4 x 3 = 12 3 x 4 = 12
By 9 your child should know their times tables fluently – so it’s worth checking to ensure this is in place. They will also be dealing with decimals, fractions, percentages and money.
Florence Nightingale - The Compassionate Mathematician
Florence Nightingale is most famous for her role as a nurse. During the Crimean war, she worked at the military hospitals where British troops were treated. She became known as the 'lady with the lamp', who made her rounds at night to look after injured soldiers. Although Florence Nightingale is famous as a nurse, you might not know that the main tool she used in her campaign to improve hospitals was statistics. Statistics is an area of mathematics in which numbers that describe the real world are collected and then analysed to see what we can learn from them. Florence carefully recorded statistics such as numbers and causes of deaths, and found that the unsanitary conditions killed more soldiers than actual war-wounds. After the war, Florence Nightingale set about persuading people that a hospital reform was necessary, using her statistics. It can be hard to get people to look at and understand long lists of numbers - and this is where one of Florence's brightest ideas came in. Like other statisticians at the time, she realised that the best way to get across statistical information is to use pictures. She invented a new type of graph called polar area graphs — you can see an example below:
This is a Where's Wally? style counting book. Follow the boy as he searches for his dragon and see if you can find and count all the objects that he encounters along the way. The simple black and white illustrations are striking against the coloured objects to spot and the map on the endpapers is useful for discussing number sequence. This is an enjoyable book to share and re-share that helps to develop early counting in EYFS and KS1. You can buy the book here - make sure that you use Amazon Smile to donate to the Friends of Margaret Wix PTA
June has seen children across the school learning about maths in cross-curricular and real-life contexts:
Hummingbirds “raced to 24” by rolling a dice and then doubling the answer, competing against their partner to reach the target number first. Lots of fantastic maths vocabulary could be heard while the children were working.
Emperors had fun pretending to be a compass needle and followed instructions to face N, E, S and W in their classroom. They made a quarter, half, three quarters and whole turns either clockwise or anti-clockwise, and enjoyed rotating! Afterwards, they played a compass game in pairs to reinforce their learning. They have have loved having the opportunity to apply this learning in other subjects, including creating their own maps for the Beebots in their computing lessons. They used directional and positional language when testing their algorithms (set of instructions) to get from one place on the mat to another. Then they debugged the algorithm if the Beebot ended up somewhere else on the mat! Great computational and mathematical thinking!
In design and technology lessons, Cardinals used their knowledge of shape when exploring how to make 3D structures stable (in preparation for designing their own greenhouses).
Admirals explored nets of cubes and cuboids this month, using polydron. Manipulatives are helping them to bring maths to life! They predicted whether the net would work and then tested their predictions out!
Monarchs have immersed themselves in financial maths this month, learning how to budget and manage income and expenses. They even worked in teams to set up their own small businesses with just £5 investment. Having sourced products, conducted market research, created business plans and advertised, I was proud to see them successful sell a range of fantastic products and services, raising just over £150 to put towards their leavers' party!
Cardinals, Admirals and Monarchs also enjoyed their first visit from Ash and Marianna from Metro Bank UK this month. The children are taking part in a series of sessions with them, learning how important maths is in daily life. These real-life maths sessions focus on finances, budgeting and saving. Each child received a fun workbook that we will continue working through and our sessions will culminate with a behind the scenes tour of the Metro Bank in St Albans! What a fantastic way to learn life-long skills!
Finally, even Rise and Shine Club got in on the action and have been putting their measuring skills to good use by baking delicious cookies!
This month, we are looking at two division structures: grouping and sharing.
Presented with the equation 12 ÷ 3 = 4, most will explain this as 12 shared into 3 equal groups, will give us 4 in each group. However, we could also explain it as 12 split into groups of 3 will give us 4 equal groups. This is a subtle, but important, difference.
Sharing happens where we know the number being divided and the number of groups but we don’t know the size of each group.
Grouping happens where we know the number being divided and the size of each group but we don’t know how many groups.
Below are two very similar division problems - one involves sharing and the other involves grouping. This will affect the calculation that is required in order to answer it effectively, even though the numbers involved will be the same.
James has 12 stickers. He wants to share them equally amongst his 3 friends. How many will each friend receive? This is a classic sharing question and children may do this physically, with concrete resources in younger year groups, or utilising multiplication facts in older year groups.
James has 12 stickers. He gives 3 to each friend. How many friends get 3 stickers? This time, children will need to work out how many groups of 3 make 12.
Can you identify which of the following problems involve sharing and which involve grouping?
a) 28 people travel to a pop concert in taxis. 4 people can travel in each taxi. How many taxis will be needed?
b) Three judges award 21 marks overall in a dancing contest. They all gave the same score. What score did each judge give?
Children experience the concept of dividing wholes into parts at home on a daily basis (though these may not be equal), but practicing both styles of division problem is essential to develop secure understanding.
Games are a wonderful way of practicing important mathematics in an engaging, low stakes way. Parents often see the value of this with young children, where games such as Snakes and Ladders can obviously be used to develop counting skills. Although, there is no denying that there are multiple advantages to playing maths games and these reasons maintain their relevance regardless of a pupil's age. The games linked below are aimed at KS2 pupils. As you can see in these demonstrations, they require minimal resources, such as pencil and paper, dice, dominoes or cards, and can be played at home.
Risky - Mental addition and subtraction
Crooked Rules - Place value and number magnitude
Reroupy - Column addition and subtraction
The Remainder Game - Building arrays, finding equal groups and remainders
Dorothy Vaughan - Expert in Mathematics, Computer Programming, Aeronautics
Born in 1910, in the USA, Dorothy Vaughan is best known for:
• making important contributions to the early years of the United States space program;
• being the first Black manager at the National Advisory Committee for Aeronautics (NACA), which later became part of NASA. She was also one of the only female managers;
• being an expert in the computer programming language FORTRAN, which was used for scientific and algebraic equations;
• teaching and influencing other women in the West Computing group in Langley, including renowned mathematician Katherine Johnson.
When Ant receives a special invitation to dine with Lion, she is ready to be on her best behavior. During dessert, the other guests do not mind their manners, each one taking half of the remaining cake as it is passed around. By the time it reaches Ant, barely a crumb is left for her to share with the King! She promises to make up for it by baking another cake for the King, and not to be outdone, all the other animals in turn offer to make twice as many cakes as the next. By the time the hippo speaks up, he's to bake 256 peanut butter cakes! With fantastic humor and wonderful mixed-media art, the math concepts of halving and doubling have never been so much fun. Don't forget to use Amazon Smile if purchasing a copy of this text and donate to the Friends of Margaret Wix PTA in the process!
May has been a busy month with lots of great maths going on in every class, as you can see in the pictures below.
Hummingbirds have continued learning about money,
Emperors have been exploring fractions of shapes and quantities. The children found equal parts of a whole, focusing on halves, quarters and thirds. They have made super progression from practically finding fractions using concrete resources to formal recording involving shapes and amounts! Super understanding Emperors, well done!
Cardinals had visitors from KPMG, a global accounting organisation, who explained their roles and how they use maths in their everyday lives; maths is everywhere after all! The children had chance to ask their visitors lots of questions and found out that even though their jobs involve using numbers now, not everyone at the organisation always enjoyed maths or felt that they were good at it. The children also learned how mathematics can be important in a wide range of jobs other than accounting too - this linked well with a display we have in school showing how everyone from hairdressers to fighter pilots uses maths! I wonder what jobs Cardinals would like to have in the future and how they’ll be using maths!
have mastered translation and reflection this month. They finished their learning by playing Flip and Slide: a two player game which requires secure translaation skills! They love a maths game - what a fun way to learn!
Monarchs finished revising everything they have learned this year and then showed off all of their knowledge during their SATs tests. We are very proud of each child and how they approached the three, tricky maths tests with positivity and determination to succeed. Monarchs then had chance to apply maths skills on their residential trip - they probably didn't even think about how angles were involved when firing arrows in archery, for example!
Following last month's blog about the extremely useful rebalancing strategy 'equal sum', this month I am highlighting a similar tool to help us with subtraction: 'equal difference'.
Let's explore this strategy with single digits to begin with. The calculation 8 - 5 is represented by the image below. The arrow shows the difference between the number 8 and 5. What is the difference? It's 3.
What would happen if we add another cube to each of the numbers? We would have a new calculation, 9 - 6, but the answer (or the difference between the cubes) would remain 3.
This would also be the case if we subtracted 1 from 8 and 5, creating a new calculation 7 - 4. The answer would still be 3. Changing a calculation in this way, keeping the difference between the numbers the same, is known as equal difference.
What is the point in using the equal difference strategy though? It allows us to manipulate the calculation in order to make it simpler for us to complete. So long as we manipulate both numbers in the same way - by adding or subtracting the same amount - we can easily create a simpler calculation that uses friendlier numbers. Let's look at it again with slightly more challenging numbers:
28 - 13 could be rebalanced by adding 2 to both numbers. This would give me a new calculation of 30 - 15. These numbers are much friendlier and therefore the calculation is easier for me to now perform with a reduced possibility of making errors. Similarly, I could have subtracted 3 from both numbers to give a new calculation of 25 - 10. Again, this is much easier than the original calculation and means I do not need to use any kind of formal, written method.
So how would you use equal difference to rebalance 96 - 48? Would you add 4 to both numbers and make a new calculation of 100 - 52? Would you add 2 to both number to change the calculation to 98 - 50? Or would you subtract 6 from both sides to create 90 - 42? Whichever way you would choose, the answer would be the same as in the original question; this is the beauty of equal difference!
The use of mathematics vocabulary is a strong indicator of student success. Language skills and comfort with mathematics vocabulary can have great impact on pupils' achievement. Quite simply, one has to understand what the question is asking in order to answer it correctly! In addition, many everyday words have different meanings when presented in a mathematical context, and understanding and having meaningful mathematics discussion is key to deepening understanding.
With this in mind, we have developed a document that outlines the progression in vocabulary acquired and used in each year group. It is important that we work together and make sure that we are all using the same vocabulary when talking to children about mathematics. You can find it linked here:
We are aware that there will be some changes since many of you were at school. Borrowing or purchasing a maths illustrated dictionary, such as the one pictured, may be of use. Hopefully this monthly blog helps to unpick some terminology too. Of course, our doors are always open to helping you to support your children at home as well!
Albert Einstein - Physicist and Mathematician
Albert Einstein was a scientist and mathematician whose work changed our understanding of time, space, gravity and the universe. Einstein grew up in Germany, where it took him four years to formulate his first word. When he was young, his father gave him a compass. The compass inspired him to try to find scientific explanations for what happened in the world around him. He was fascinated by how objects worked and, although he didn’t love school, he loved physics and maths books. One teacher said, “He will never amount to anything.” However, as time went on it became clear that Albert Einstein was no ordinary student. He excelled at maths and physics
At the age of 16, Albert moved to Switzerland. Eventually, he was allowed to study at the institution now known as the Swiss Federal Institute of Technology. His brilliant mind contemplated space and time, and he eventually came up with the theory of relativity. He shared his knowledge with the rest of the world, becoming the most original mind of the twentieth century.
He also invented the equation E=mc². This equation explained that energy and mass (matter) are different forms of the same thing. They are interchangeable and can be converted from one to another. This is thought to be the most famous equation in the world. In 1921, Einstein received the Nobel Prize for physics.
This book, which celebrates diversity, maths and storytelling, is about three young sisters who are looking at the same seven stars from different perspectives. While lying on the ground, they can see different things: Aarti sees a dipper, Usha sees a digger, and Gloria a kite. They are arguing about what they can see until they realise that by changing their perspective, they can now see the objects the other siblings have described. At the most basic level, the story provides an opportunity for very young children to notice shapes. However, the storyline also provides a nice introduction to the mathematical concept of transformation, particularly rotation. If purchasing the story online, don't forget to support the Friends of Margaret Wix by buying using Amazon Smile.
Despite only being in school for 10 days in April due to the Easter Holidays, the children at Margaret Wix have continued to wow us with their determination to succeed and willingness to give new learning a try. Here’s a snapshot of the great work that has taken place this month:
Hummingbirds investigated the value of different coins and tried grouping them according to their size, colour and shape. They enjoyed playing a ‘guess the coin’ game. Also, they tried matching Cuisenaire rods to the coins and using them to work out the total value of different groups of coins. Some of the children cut out different coins and put them in order of ascending value.
Emperors have mainly been working on halving and doubling in April. They used arrow cards to make 2 digit numbers, represented them with the dienes and then halved them. They thought about whether even/odd numbers could both be halved and some of them even had to regroup a ten when they couldn't halve the number of tens exactly.
Admirals have worked hard on percentages this month. They began by learning about where we would see and use percentages in real life such as the battery level on our phones, on labels when a sale is on in a shop, and on food packaging showing the amount of salt in a certain food. They then studied how to convert percentages to fractions and decimals before learning how to scale fractions up or down to make them accurate percentages. The children can now can find 10%, 25%, 50%, and 75% of numbers too! Finally, they successfully applied this knowledge to real life problems too.
Monarchs have been beavering away this month preparing for their SATs tests. After completing mock tests, Mrs Gibbs and Mrs Walters used the results to target and recap a few key areas of learning to ensure that all children are feeling more confident before the real ones next month. Nearly there, Monarchs!
Below are pictures from some of our classes showing maths in school this month.
This month I am putting a strategy in the spotlight that, once mastered, enables the children to become much more efficient. The strategy in question is used for addition and is known as equal sum.
Equal sum focuses on the closeness of landmark or ‘friendly’ numbers and utilises the skill of rounding. It ensures pupils think about what a reasonable response to a question is and develops number sense.
Children must first have a secure understanding of the concept of sum as well as the fact that numbers can be partitioned in many ways yet will remain the same number (look back at February’s blog that shows how we can use the part-whole model to demonstrate this).
To demonstrate the equal sum strategy I will be using numbers that we can all easily manipulate, but this strategy works with larger numbers and decimal numbers too.
When we add two amounts together, the answer is called the sum. In the calculations below, we can see that the sum remains the same even when the addends (parts being added) change.
25 + 13 = 38
30 + 8 = 38
28 + 10 = 38
It is clear that the first calculation is more challenging than the other two and that is because the other two use ‘friendlier’ numbers. These are numbers that we are confident in using. Multiples of ten are a treat as when we add to them or add them to something else, there is rarely any need to exchange anything.
In order to solve 25 + 13, many pupils would have to perform calculations involving a number of steps or even carry out a formal, written method. We can use equal sum to change the calculation and make it simple and efficient to perform mentally.
Firstly, we must think about which addend is closest to a friendly number. In class, this is often shown using a beadstring:
This also means that the children can physically move the beads from one side of the calculation to the other. In this case, we could move 3 beads from the right side of the calculation to the left. We have not added any new beads or taken any away so our sum will remain the same. The new calculation will be 28 + 10 though, which is much easier for me to perform quickly and accurately in my head than 25 + 13. Some of you may have looked at the calculation and wanted to move 5 beads from the left to the right, creating a new calculation of 20 + 18. You’ve also used a friendly number, I just had to move a few more beads! Once again though, the sum remains 38.
Alongside the physical resource, we encourage children to draw what they are doing. Representing the calculations pictorially will prove to the children that the sum doesn’t change when they move an amount from one side of the calculation to the other and, if it does, something has gone wrong! After mastering this strategy and becoming confident in sliding beads from one side of the calculation to the other, the children will begin to be able to do it mentally, thus developing an efficient strategy for calculation.
Below is an example of how an older child has used equal sum with larger numbers to change the calculation. They have needed to use a formal written method, but in finding friendlier numbers first, they have drastically reduced the likelihood of errors being made when exchanging in the formal method.
As teachers, there is nothing we love more than seeing children enjoying learning, but unfortunately some children face feelings of worry, panic, anxiety and frustration during maths lessons. The signs and symptoms vary from flushed faces and heads in hands, to making excuses to go to the toilet and avoid tasks. Some children try to mask it by copying others, whilst others are vocal about feeling “not good at maths” and saying that they “hate it”. These negative feelings impede on working memory and stop children from being able to think clearly and understand maths. This response is known as maths anxiety.
This is a fantastic video from the team at Childline that explains what anxiety is in a child-friendly way:
Before we can help children with their maths anxiety, we need to know the cause. This isn’t always obvious, but it may stem from a fear of failure (maths involves answers being right or wrong and that can be tricky for some children) or they might have overheard adults speaking negatively about maths (if you are someone who talks openly about how much you disliked maths at school you might want to try to keep that from the children around you as they soak everything up, like sponges).
2018 Ipsos MORI poll
There are a number of things that we do in our classrooms to try to reduce maths anxiety, such as ensuring that children work in mixed ability groups and with different learning partners. This is so that children of different abilities, with different strengths, weaknesses and learning styles, can also help each other overcome obstacles and see new ways of tackling number problems. We also try to instil growth mindsets in our pupils, that is believing that mistakes are learning opportunities and understanding the power of the word ‘yet’.
My top tips for you to help reduce maths anxiety at home are:
Alan Turing - War Hero
Born in London, in 1912, Alan Turing’s mathematical skills were noticed early on in his life, whilst at school. From equations to tough concepts, he managed to understand things which even adults found challenging.
After finishing school, he attended Cambridge University to study Maths, before inventing the Universal Machine. This can be understood as one of the world’s earliest computers, which managed to read simple codes.
During the Second World War, Turing then worked at Bletchley Park. This was the home of the Government Code and Cipher School (GC&CS). Thanks to Alan’s understanding of code and his Universal Machine, he was able to decipher secret messages of enemy forces, such as Germany. With his team, Alan could work out where and when the enemy was planning to attack. He shared this information with the government and British army, so they could prepare.
Without Turing’s amazing discoveries and understanding of code, the Second World War could have lasted much longer, and many more people could have died. In this sense, he can be seen as an inspiring figure who used Maths and Science to save lives.
Sir Cumference and Lady Di of Ameter discover fractions while purchasing cloth and cheese at the fraction faire. While two-fourths may seem the same as one-half, in truth it denotes both two parts of one-half or two quarters of the whole. But the real mystery is the fact that items at the fair keep disappearing, and Sir C, Lady Di, and the Earl of Fraction must set a numeric trap for the thief, teaching an important lesson along the way about the comparative size of fractions. There are puns, both literal and visual, abound in this fun adventure story with beloved characters and a solid mathematical foundation. If buying this book online, do not forget to support the Friends of Margaret Wix by using Amazon Smile.
March has seen our littlest learners immersed in mathematics! Firetips have been making patterns, baking, playing maths games with their parents and carers, and choosing their own mathematical activities in child initiated learning. Foundations for counting are so important so it is brilliant to see so many of our youngest children enjoying mathematics.
Hummingbirds have also been working hard - they have been representing numbers in different ways, using numicon, tens frames and multi-link. They selected 2 digit numbers and then, with a partner, tried to make that number using the different concrete resources. They found out that there were many ways to make each number!
Emperors have been enjoying a variety of maths games including 'Race to Zero' to help them learn how to exchange one ten for ten ones. This will help them to understand how to do this when they move onto the formal, written method. As well as number, the children have also explored 2D-shapes, grouping them according to the number of vertices and sides, and according to whether they are regular or irregular. They also looked at how they could sort shapes using a Carroll diagram.
At the other end of the school, Monarchs have seen how lots of their prior learning has been helpful when tackling algebra. As well as learning about the order of operations, variables and formulae, they played a fun loop game in order to practise calculating unknown values. You can see images of this, and learning from other classes in the photo slideshow below.
Bar modelling is a strategy used to pictorially represent a problem in which there are some known and some unknown values. Bar models in themselves do not solve problems but they do help children to visualise and decide which operations to use. They are especially useful when solving worded problems. Let’s look at a few examples.
This question might be seen in year one and involves subtraction or finding the difference:
Tom has five sweets and James has three sweets. How many more sweets does Tom have than James?
You can see that this bar model has been built using cubes to represent the sweets. Arranging them in bars helps children to clearly see the difference. They can then discuss what they notice and then think about which operation to use to solve the problem. Writing the abstract calculation is the final step.
In this year two example we can see that building the model with cubes would not be efficient as we would need so many. Drawing the model is still helpful though in order to visualise this problem:
Amber has 27 gel pens. She buys 30 more. How many gel pens does she now have?
Once drawn, the children can derive a number of facts from the model as shown. They can think about which is the correct calculation to solve the problem.
This year three example shows how multiplication and division problems can be represented using bar modelling too:
Leigh is helping in the school library. She is packing books into two boxes. The first box has 5 books in it. The second box has five times as many books in it as the first box has. How many more books does the second box hold than the first?
Once the children have drawn the bar model and have seen that they need to multiply five by five to work out how many books are in the second box, they can then choose an efficient strategy to calculate the difference between the two totals.
This model could be adapted to solve further, related problems such as:
How many books have been packed into the two boxes in total?
And, how many books need to be taken from box 2 and placed in box 1 to make them equal?
Bar modelling is an excellent strategy to use to visualise fraction problems, such as this year four problem:
Sally buys four fifths of the shop’s apples. If the shop had 30 apples, how many apples did she buy?
Here we can see how building the problem with concrete resources can then be transferred into a pictorial representation. Children can then see that they need to divide 30 by 5 to work out what each smaller piece (fifth) is worth, before multiplying that answer by 4.
Problems continue to increase in complexity as we move into year five. The bar model is still an excellent tool for visualising problems however, as demonstrated with this percentages problem:
Twelve is 40% of a number. What is the number?
Representing the problem in this way means that children can clearly see that to work out 10% they must divide 12 by 4. Once they know what 10% is worth, they can then see they just need to multiply this answer by 10 to calculate 100%.
Finally, in this year six example, we can see how bar modelling can be used to solve complex ratio problems:
In a survey, the ratio of the number of people who preferred ‘ready-salted’ to ‘cheese and onion’ crisps was 5:3. Forty-six more people preferred ready-salted. How many people took part in the survey?
Modelling the ratio problem in this way means that children can clearly see that two squares on the model are worth 46. From this, they can work out what a single square is worth. Then they are able to solve the problem quite simply.
Bar modelling is important as children often find it challenging to interpret worded problems in particular. Building or drawing the models in this way enables children to break the problems down and think logically about each individual step. The fact it can be used to help solve problems throughout primary school shows what an excellent tool it is.
Challenge: Send Miss Abbott a photo of how you have used a bar model in your home learning, or to help you solve one of her following challenges, and see your maths featured on our maths blog!
In a football match, the red team scored two goals and the blue team scored three goals. How many goals were scored altogether?
Use this form to send Miss Abbott photos of your completed maths challenge. She will add them to our maths blog!
Use technology to support the development of mental fluency at home. You don’t have to do all the work yourself; there are a number of excellent online resources that you can utilise to make your child’s screen time really worthwhile!
White Rose Maths have brought out a new one-minute maths app. It's free to download the app and it helps children build greater number fluency and confidence. It's all about targeted practise in engaging one-minute chunks! We would recommend starting with the subitising section (the skill of instantly recognising the number of items in a group without counting) in early years and building from there.
Times Table Rockstars is a carefully sequenced programme of daily times tables practice. Children can log in using the username and password provided by their class teacher and successfully use the app to help boost their times tables recall speed and accuracy. This app is suitable for pupils in year two upwards.
NumBots is a highly engaging platform for learning to add and subtract. The app can be used to help children improve understanding, recall and fluency in mental addition and subtraction so that they can move from counting to calculating. This app is aimed at pupils in year one upwards.
Twinkl Mental Maths App offers an engaging way for children to practice mental maths across a range of key maths topics. There are over 100 game modes of varying difficulties covering six key topics: times tables, addition, number bonds, division, doubling and halving. You can try parts of this app for free but to access them all requires a subscription.
Katherine Johnson - space pioneer!
After falling in love with maths at a young age, Katherine studied the subject at university and graduated at just 18. She was one of the first three black women to attend West Virginia University and she challenged the stereotypes that surrounded her, becoming an inspiration to women and people of colour. Katherine joined NASA in 1953 and used her knowledge and mathematical skills to calculate the trajectory for Project Mercury and the Apollo 11 flight to the moon, which means she helped the first spaceship and the first Americans reach the moon! Believing that “everything is physics and math(s)”, she encouraged girls to pursue careers in science, technology, engineering and maths (STEM) and often gave talks on the subject.
This is an excellent picture book to help children understand the concept of big numbers. Pipkin, the smallest penguin, is always asking questions, but what he wants to know most of all is how big is a million? So he sets off to find out, and along the way meets a hundred penguins, sees a thousand snowflakes and meets one new friend before being amazed to finally find out how big a million really is. A special fold-out poster at the end of the book shows Pipkin looking at the sky, which is printed with exactly one million stars. It is beautifully illustrated by Serena Riglietti. If buying this book online, do not forget to support the Friends of Margaret Wix by using Amazon Smile.
In February, we worked with a local artist to explore the use of shape and lines in art. We used the famous artwork below, by Kandinsky and Mondrian, in addition to work by our local artist, as inspiration when creating our own whole-school artwork.
EYFS and KS1 talked about shapes and sizes using key vocabulary such as straight, curved, bigger and smaller. KS2 discussed parallel, perpendicular, horizontal and vertical lines as well as naming a range of polygons and angles.
Isn’t our finished artwork brilliant? You can see our amazing, collaborative project on display on the wall opposite The Hub. Take a look at some of the images below of the work in progress too.
The part-whole (or cherry) model, shows how a whole number can be reduced to smaller parts, or how smaller parts can be combined to make a whole. This model can be used for single or multi-digit numbers, including decimals and fractions, and is an excellent calculation strategy used throughout primary school.
The pictorial representations above help children to visualise multiple calculations. For example, the first model shows:
6 + 4 = 10
4 + 6 = 10
10 - 4 = 6
10 - 6 = 4
It is important that children see these calculations written the other way around too, for example:
10 = 6 + 4
10 = 4 + 6
6 = 10 - 4
4 = 10 - 6
From the second example we can see that:
2 + 3 + 1 = 6
2 + 1 + 3 = 6
3 + 1 + 2 = 6
3 + 2 + 1 = 6
1 + 2 + 3 = 6
1 + 3 + 2 = 6
6 - 2 - 1 = 3
6 - 1 - 2 = 3
6 - 3 - 1 = 2
6 - 1 - 3 = 2
6 - 3 - 2 = 1
6 - 2 - 3 = 1
You could use the second model to begin to write more complex balanced equations too, such as:
6 - 3 = 2 + 1
6 - 2 = 3 + 1
6 - 1 = 3 + 2
Children may use objects (such as counters) or pictures (such as dots) in their part-whole models rather than writing digits. At home you could easily create part-whole models using conkers, pine cones, buttons or anything else you can find! The models can also be made larger, and do not need to be confined to paper!
Finally, it is important that children see part-whole models in a variety of orientations and with multiple parts as shown in the examples above too.
Challenge: Send Miss Abbott a photo of how you have used a part-whole model in your home learning, or one that you have just created for fun! How many calculations can you write from your part-whole model?
Use this form to send Miss Abbott photos of your completed maths challenge. She will add them to our maths blog!
Times tables are essential and the aim is for children to have mastered those up to 12 x 12 by the end of year four. Much of this learning happens before year four, however.
When children are first learning times tables they benefit from seeing the multiplication table build up from the beginning, looking first at one group of the amount then building up by adding another group each time and seeing what the total becomes. This can be done physically using objects you have at home. In the images below we can see one group of three is worth three (1 x 3 = 3), four groups of three are worth 12 (4 x 3 = 12) and seven groups of three are worth 21 (7 x 3 = 21). It would also be important to point out no groups of three before beginning this process of adding another row or group of objects each time.
Children can move on from counting every object in the groups to skip counting. This can begin by whispering some of the numbers and saying the multiples of three more loudly e.g. 1, 2, 3, 4, 5, 6, 7, 8, 9…
Eventually, the children can move on to simple counting in threes.
After this, games are a fantastic way to begin to help the children memorise the facts rather than have to count up each time. An easy game to create would include cards with a question on one side and the answer on the other - involve the children in the process of making them too! Once made, mix them up and the children must choose a question to answer; if they get the answer correct, they can keep the card. They could play against you to make the game more competitive!
There is a logical order which usually works when learning times tables:
- 2s, 5s and 10s come first (usually around Year 2)
- 3s, 4s and 8s are taught next (usually around Year 3)
- 11s, 6s, 9s, 12s and then 7s come later (usually around Year 4)
The importance of developing understanding before memory cannot be underestimated. Recall comes last so take time in the early stages and even once you think your child has mastered one, remember to return to it to keep the facts fresh. Using Times Table Rockstars is a good way to do this; ask your child’s teacher if you are struggling with login information for this. Another way that might help is to practice chanting or singing times table songs - they don't need to be boring though, you can use well known songs that have been turned into times tabl versions. Want to practise the 8 times table to the tune of Adele's 'Rolling in the Deep' or the 6 times table to the tune of 'Shake it Off' by Taylor Swift? If so, take a look at this YouTube Playlist.
Ada Lovelace - the mother of computers!
Born in 1815, Ada was taught maths from a young age, which was very unusual for women at the time. In 1833, she met Charles Babbage who had begun designs for a machine that answered very difficult maths problems. Unfortunately, his ideas were ahead of their time and it was hard to get money to build this machine because most people didn’t understand what it could do. Luckily Ada did! She developed plans and described how codes could be created to handle letters and symbols. She also came up with a method for the engine to repeat instructions - something that Charles hadn’t thought of and something that computer programmes still use today. Ada Lovelace is now celebrated every year; she is recognised as the very first computer programmer who made excellent contributions to the fields of maths, science, technology and engineering. How inspiring!
What do one hundred sunbathing snails have in common with ten crabs? This joyful, award-winning counting book has a funny focus on feet! Children will love this hilariously illustrated introduction to simple counting and multiplication with big feet and small – on people and spiders, dogs and insects, snails and crabs – from one to one hundred! You can buy the book here - make sure you are contributing to the Friends of Margaret Wix by using Amazon Smile.