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MULTIPLICATION & DIVISION – Stage 2 Division Number Sentences

Each year Bev Dunbar analyses the Year 3 and Year 5 NAPLAN Numeracy papers, looking for where students appear to have a major blockage to their understanding. For example  the results for Question 21 in the 2017 Year 3 paper  demonstrate that many Stage 2 students are insecure about how number sentences work.

This Stage 2 activity (Years 3/4), Match my Group (Division Number Sentences), helps these selected students realise that each part of a number sentence has meaning. If you rearrange the position of digits and symbols, then your number sentence may not make any sense. You can’t just put them wherever you like.

It also helps students realise that they need time to think about the actions in a number sentence. If they see the number 42 and the number 6, for example, do you add, subtract, multiply or divide them? You can’t just rush in with the first thing that comes into your head.

Each of the 4 stories in this activity is a single step story. Your students need to be able to manage linking their story and number sentence before tackling more complex events. You need to make time for your students to talk to each other, explain their thinking strategies, to work co-operatively on each activity. You can’t rush their understanding. You need that “aha” moment when they work out that actions matter, the placement of digits and symbols matter. Maths matters!


NAPLAN NUMERACY Year 3 and Year 5 NSW Results

For the last 15 years Bev Dunbar has been analysing the NSW NAPLAN results, looking for the “big picture” information to help all teachers better understand student misconceptions and blockages to understanding. When you look at your NAPLAN results, you are looking at individual students in your class and whether your school achieved at, below or above State averages. This information helps you plan targeted strategies for student improvement.

However here at Maths Matters we look at the whole state results, in our case that’s NSW. Fortunately NSW results are almost identical to the Australian results so we are able to see how Australian students as a whole are tracking in their mathematical progess. But unfortunately … this is not a pretty picture. It appears that our students are not able to adequately demonstrate competency on key test questions. In other words, they are not able to demonstrate that they have achieved key content outcomes.

We can argue about the nature of the test and whether we agree or disagree about its continuation. That is another story. The fact is that our government collects yearly data and that yearly data tells us something.

We analyse every test question and categorise it by both Stage and also by Core or Advanced. We call a test item a Core Stage question if we think it is suitable for most students to answer. For example, in Year 3 a Core Question is one that covers Early Stage 1 or Stage 1 concepts that we expect most Year 3 students to answer. We have a cut off point of 80% for our definition of “most”. So one in five students might answer this Core Question incorrectly. These may be students who are anxious about the test, may be working at an even earlier Stage, may have language and reading difficulties and so on. But we believe the question is an effective one to ask this cohort. Other questions in the test paper may be at the same Stage but may have twists and turns or too many steps in their solution. We call these test items Advanced. We do not expect 80% or more students to answer these questions correctly. These questions help identify more advanced mathematical thinkers in your class, school, state or country.

The strange thing is though, the questions that make up a particular NAPLAN Numeracy test more often than not do NOT test Core Stage below content. Some even test our students on content that is two Stages above. We are not sure what this data is supposed to tell us. We are perplexed as to why these questions are included. If you study William Shakespeare’s plays at University, for example, you would not expect to be tested on the plays of Anton Chekhov, just to see if someone has that knowledge.

  • In the 2017 Year 3 NAPLAN Numeracy paper, only 44% of the questions were Early Stage 1 or Core Stage 1. That’s 16 out of 36 questions. And only 50% of these 16 questions were answered by 80% or more Year 3 students.

 

  • In the 2017 Year 5 NAPLAN Numeracy paper, 50% of the questions were Core Stage 1 or Core Stage 2. That’s 21 out of 42 questions. And only 43% of these 21 questions were answered by 80% or more Year 3 students.

If you would like to see our one page summary of this year’s Year 3 results, click here.

If you would like to see our one page summary of this year’s Year 5 results, click here.

The summary is compact, with test question numbers on the left and the NSW test results for each question in matching spaces on the right. We also include an analysis of key concerns, based on these results. These concerns highlight the “big picture” view. Your individual students may have been successful in answering a question, but the overall results might indicate misconceptions you need to be aware of. We then develop resources that help your students overcome these blockages to their understanding.

 


ADDITION & SUBTRACTION Photographs – Blue Sharks

Our website is packed full of resources for you to use in your lesson planning. And even though we know you can manipulate our images yourself, we like to help make your life just that little bit easier by providing you with resources that are ready to use immediately. This new series of 1-10 blue sharks is such an example. We’ve repeated the shark image and saved it as 10 separate png images. You can now use these as part of your early addition and subtraction discussions. Here are 6 plus 8 sharks. That’s 14 sharks altogether.

 

 

 

 

 

We have also included the blue sharks on our MULTIPLICATION & DIVISION Photographs page as you could also reproduce each image to create as many multiples as you like. Here are 4 groups of 3 sharks, for example.

 


POSITION PHOTOGRAPHS – Road Signs

Learning to read and interpret road signs is an important part of helping your students to locate themselves in 3D space. Our resources include a wide range of road signs for class discussions with any age group. For example, here is an animal crossing sign. What does this sign mean? Why has it been put up on this road? What do you have to do? Will the animals come from the left, the right or straight ahead? Why do you think this? Where might you see a sign like this one? Do you only look out for cows and sheep? Why were these two animals selected for the sign?

If you know individual students are travelling along country roads, encourage them to take their own photographs of unusual road signs. Have they seen one about emus? Deer? Kangaroos? Koalas? Use these for related class discussions about the importance of road signs and the information they tell us.


ADD & SUBTRACT – NAPLAN Follow-up – Match my Story

The first of many NAPLAN 2017 follow-up activities to help your students remove blockages.  Only 48% of one large Year 3 cohort correctly matched a 2-digit addition and subtraction number story to the correct number sentence. The format of the question was slightly unusual and this seemed to have thrown the students off!!

 

Match my Story provides follow-up examples to discuss together with your Stage 1 students. It is also suitable for high and medium block Stage 2 students. Each set has more than one matching number sentence. It is vital to discuss both the ones that work and those that don’t. Help your students think more deeply about what they are learning.

 


PROBLEM SOLVING – Blake’s Guide to Maths Problem Solving

This guide is a vital tool for all middle and upper primary students who want to be successful maths problem solvers. The book is divided into three sections.

  • Why Solve Problems? Explains which personal skills help you solve problems and what a problem solving cycle looks like.
  • Common Problem Solving Strategies The 8 suggested strategies are: Visualise it, Make a table or graph, Guess and check, Break it into smaller parts, Work backwards, Look for a pattern, Eliminate possibilities.
  • Types of Problems More than 140 sample problems include 1-step, 2-step, more than 2-step, multiple choice and open-ended problems.

The book includes detailed, suggested strategies, as well as TRY THIS and CHALLENGE activities to test your understanding. Answers are at the back of the book. It is suitable for students, teachers and parents working with Stage 2 or 3 students. This Guide is packed full of easy-to-understand explanations, real-life photographs and graphics. The book includes curriculum correlation charts. The book also showcases 7 outstanding male and female problem-solvers from across the world.

Click here to buy a copy directly from the publisher or to view sample pages.


NUMBER & ALGEBRA – PLACE VALUE – Hooray for Wombat

Our Maths Matters Creative Director, John Duffield, has just published his first iBook. It’s called Hooray for Wombat.

What a wonderful way to investigate the numbers 1 – 9 and then 10 – 90 using the delightful animal characters Duffy created originally for our website’s Place Value Graphics.

We look forward to many more books to come. Congratulations Duffy!


CHANCE GRAPHICS – Paper, rock, scissors

Paper/rock/scissors is a game of chance between two people. On a signal you each make one of 3 shapes with your hand – rock (a closed fist), paper ( a flat hand) or scissors (a fist with two fingers making a V). The outcomes are that you make the same shape or a different shape. If the shapes are different then paper covers rock (paper wins), rock crushes scissors (rock wins) and scissors cut paper (scissors wins).

This game began in China and Japan and is a simple way to select who goes first, for example, in another game. It is like flipping a coin, throwing a die or drawing a straw from a pile. It is a selection method.

There is even an official website for discussing the rules of this game.


STATISTICS & PROBABILITY – Chance Graphics

Duffy has just created a sweet set of children to help you talk about Chance with your students. What’s the chance of having a boy or a girl? Twins? Triplets? Quadruplets?

Your chance for having a boy is about 50% and 50% for having a girl. But many of you will know families that are all boys or all girls. Having all boys or all girls is almost always due to simple chance. Scientists have actually researched this topic and find that boys are 52% more likely than girls, so you really have a very slight increased chance of having a boy.

Did you know that Nigerian women enjoy the highest rate of twin pregnancies in the world? About 1 in 22 Nigerian women have twins. In 2013 there were 4475 sets of twins born in Australia, out of a total of 298 984 births. This represented about 15 in 1000 births.  And of these 15 sets of twins, about 30% are identical. So fewer than 5 in 1000 births in Australia are identical twins.

In 2015, there were 84 sets of triplets born in Australia, out of a total of 305 377 births. That’s a 26 in 100 000 chance of having triplets. And there has been a 400 percent increase in the rate of triplet births over the last 20 years. To create identical triplets, the original fertilised egg splits and then one of the cells splits again. Identical triplets occur in about one in a million pregnancies, a rare event indeed.

Did you know that the odds of conceiving quadruplets is predicted to be one in 729 000? To create identical quadruplets, one fertilised egg needs to split into 4 separate embryos. Phew.

 


MEASUREMENT – VOLUME & CAPACITY – Stage 1 Estimating fifths

Regular practice estimating volumes will help your students develop effective proportional reasoning. We use this skill when we think about multiplication and division. We also use this skill when we think about fractions. I am working on a whole set of these activities but this is the first in the STAGE 1 set. It will help your students think about fifths. It will help them understand how to mentally divide a length into 5 equal parts. Later sets will focus on dividing a given volume into halves, thirds, quarters and tenths.

The theme of  FILL IT TO THE TOP (fifths) is orange juice. Even though you are just discussing a picture card, encourage your students to imagine it is a container of real orange juice. You want to share this juice equally between 5 people.

Develop a simple language routine to discuss the fraction parts. If this is one fifth, how many more fifths do you need to fill this container to the top? If this is two fifths, how many more fifths do you need to fill this container to the top? If this is three fifths, how many more fifths do you need to fill this container to the top? If this is four fifths, how many more fifths do you need to fill this container to the top?

Remember to talk about the full container as five fifths or one whole container full.

This activity is also part of our STAGE 1 Mental Warm-ups.


MEASUREMENT – VOLUME & CAPACITY – Stage 2 Volume Mental Warm-up

For some reason Volume and Capacity is one of the mist difficult sub-strands in Primary Mathematics to both teach and learn. Too many Year 3 students are not able to answer Stage 1 questions in National or International Assessments.

  • Yes it is a language problem.
  • Yes it is a lack of whole class resources problem.
  • Yes it is a messy water problem.
  • Yes there may be more important strands or sub-strands to focus on.

But … that doesn’t mean we ignore Volume and Capacity. There are still important concepts for students to take with them into adult life. The ability to estimate amounts in a given container, to think about litres and millilitres, for example.

We have just uploaded How much water is that? for your Stage 2 students. It involves using real containers and water to establish the Volume concepts in real life. You then switch to 2D representations to help students internalise their experiences using the following 4 cards:

Your goal is for your students to use proportional reasoning to estimate how many cups of water they need to fill each container. It sounds simple, but only 52% of year 3 students in the Year 3 2016 NAPLAN Numeracy Paper were able to do this successfully.

This activity is also part of our STAGE 2 Mental Warm-ups.

 


MATHS PHOTOGRAPHS – 2D PATTERNS – Ikat Textiles

Ikat textiles are amongst the most beautiful geometric patterns in the world. And they are so complicated to design and create that it is a wonder that they exist at all. I have just uploaded lots of textile photographs, including about 10 ikat ones. “Ikat” means “to tie” or “to bind”.

Ikat is a resist dyeing technique where you dye the warp yarn before you weave it. Different bundles are tightly bound then dipped in the dye. These might be unbound and then rebound in a different way before being dipped in a different colour dye. All the bindings are removed at the end of the dye process then the yarn is used to weave the cloth. The weft yarns are dyed in a solid colour. Once you set up the warp yarn you can see the pattern before you start the weft design weaving. Once woven of course some of the yarn is not in exactly the right position so you get a slightly blurred effect. Famous ikat textile villages can be found in Kalimantan, Sulawesi and Sumatra. The tribal groups are Dayak, Toraja and Batak.

You can also create weft ikats. This time the weft threads are bound and dyed but it is much more difficult to weave as the design only shows once it is actually woven!

And you can even get double ikats. These textiles are created by first binding and dyeing patterns into both the warp and the weft yarn. Utterly astounding! Very few places still create these complex patterns. You can see them in Tengenan in Bali and Puttapaka and Bhoodan Villages in India.

You can find ikats also in Central and South America, Central Asia and Japan. In the 19th century, Bukhara and Samarkand in Uzbekistan were famous centres for silk ikats.


NUMBER & ALGEBRA – MONEY PHOTOGRAPHS – Afternoon Tea

What a delicious way to think about adding or subtracting, multiplying or dividing money with 2-place decimals. Here are 4 shots from the beautiful Hopetoun Tearooms in Melbourne.

My favourites were the raspberry tarts! If they cost $6.75 each, how much for 10? What change would you get from $100? If you bought two tarts and shared them between 4 people, how much would each person pay? If you want to buy the raspberry tarts for a party and give everyone half a tart each, how many will you need to buy if you expect 10 people at your party? How much will this cost? How much change from $50? From $100?

I am sure you and your class will be able to create many more money stories like these ones. Perhaps you could even think about how you might  cook some tarts together, just like these ones?


MATHS PHOTOGRAPHS – CREATURES – Tropical coral and fish

Did you know what the water temperature has to be for tropical fish to survive comfortably? It is 25 – 27 degrees centigrade. Imagine how global warming will affect our sea creatures.

Here are 6 new photographs of tropical fish to inspire your next maths discussion with your students. How many different tropical fish can you find in Australia? What do they eat? How long do they live? What are key facts you need to know to keep them in a tank in your home? How many eggs do they lay at one time? What are their predators? Where can you find them along the Australian coast. How many might you find in a school of fish? How far might they travel in a day?


NUMBER & ALGEBRA – PLACE VALUE to 100 PHOTOGRAPHS – Floor numbers

I love seeing numbers in real life. But it gets difficult to see quite large numbers doesn’t it. We have just included 4 new photographs of floor numbers in high rise buildings (Floor 27, 28, 29 and 80). The larger of these numbers is a little blurred but it is Floor 80 in the Empire State Building in New York.  It was taken by our gym buddy, Costa, on a recent trip to New York. Imagine all the class discussions that relate to floor levels. For example, the Empire State Building has 102 floors. But is this the most in the world? It is 381 m tall or 443 m from the ground to the tip at the very top. What does that tell us about the height of each floor? It was built way back in 1932 so imagine all the buildings that must be even taller than this today.

Currently the Burj Khalifa in Dubai is the world’s tallest skyscraper with a height of 828 m. It was built in 2010 and has 163 floors. The Empire State Building is now only the 34th tallest building in the world. But in 2020, the Jeddah Tower in Saudi Arabia will become the world’s tallest building. It will be 1008 m tall. That’s more than a kilometre!!!!! And it will have 167 floors at a crazy cost of about $1.5 billion. It was originally planned to be 1.6 km tall – that’s one mile in the old measures. It was designed by the architect Adrian Smith, who also designed the Burj Khalifa.


HISTORY of MATHEMATICS – Medieval and Renaissance PHOTOGRAPHS – Money Stainglass Windows

What a different way to study the history of money. This set of Money Stainglass Windows photographs shows how money was made and distributed in the Middle Ages. They were taken in the Palazzo Massimo alle Terme (National Roman Museum). Each stain glass picture can be the basis of a fascinating discussion with your students. What do you think the people are doing? Why? How many people might be involved? How might we do this today? What did money look like in the Middle Ages? Why do we have money? What might we do if we did not invent money as a system?


GEOMETRY – 2D Patterns PHOTOGRAPHS – Tribal textile patterns

What a beautiful way to investigate patterns and design. People have been creating magnificent geometric patterns for 1000s of years. Wandering nomads weave camel and sheep hair to produce rugs, and those closer to big towns spin silk and weave intricate designs – clothes, decorations for the house, cloths for rituals. Here are plenty of examples of tribal textiles  to start a conversation with your students.

Are the designs symmetrical? How many different colours are used? Are the designs simple or complex? Who might create such a textile? How long might it take them to weave or embroider?

Follow up in your own classroom by decorating some placemats or decorated cloths. Try creating some simple weaving frames and weave coloured wool.


MEASUREMENT – TIME PHOTOGRAPHS – iphones, wristwatches and digital clocks

It’s always a good idea to discuss time concepts daily – whether you teach Kinder or Year 6. Digital time is everywhere and most of the latest gadgets only show digital examples of time passing.We just uploaded 6 new photographs to help with your class discussions.

Elect one student to be your weekly time monitor. As a class, decide on a  list of times for your focus this week. Ask the time monitor to let everyone know when this time is approaching e.g. afternoon for Kinders, 1:30 pm for Year 1, a quarter to 11 for Year 2 and so on. We also have a great range of quick daily mental warmups for time. I love the Numberless Clockface cards. Try them today with your class. We include teacher instructions to get you started.


NUMBER & ALGEBRA – PLACE VALUE GRAPHICS – Skateboard digits

To start the year off we have just uploaded a brand new set of 1 – 9, 10 – 90, 100 – 900 and 1000 – 9000. They are all based on Duffy’s new set of Skateboard Robots. Heavens knows where he gets all his crazy ideas from but they look brilliant and should be a great way to focus on Place Value and Whole Number concepts in Term 1.

How many different ways can you talk about 3000?

Skateboard 3000 John Duffield duffield-design

 

e.g. 1 more than 2999, 1 fewer than 3001, 3000 ones, 300 tens or 30 hundreds. It is a multiple of 2, 5 and 10. It is also a multiple of 3 and 6. It is also a multiple of 4 and 8. But it is NOT a multiple of 9, 11 or 12. Half of it is 1500, Double 3000 is 6000. Ten times more would be 30 000. 3000 mL of orange juice is the same as 3 litres. 3000 m is the same as 3 km. 3000 square metres is the same as 3 tenths of a hectare.

What else will your students think of? Use a 1 minute time limit and challenge them to discuss their ideas with a partner.

Create a class book to share at the next school assembly.

Create a financial plan to raise $3000 over the year for your favourite school charity. e.g. We have 40 weeks so that’s $75 per week or $15 per day. What can we do to earn that much money?


NUMBER GRAPHICS – NUMBERS & SYMBOLS – Robot Skateboarder Digits 0 – 9

Wow, Duffy has really gone crazy. He has created this delightful set of robots that are all skateboard fanatics. They look like they are in a frantic skateboard race with their bodies forming the digits  0 – 9.

Use them in a wide variety of classroom activities. Make 3, 4, 5, 6-digit numbers to discuss as part of your daily Maths Mental Warmups. Add them to an activity card.

e.g. Did you know that 4840 is the number of square yards in an acre? This is an old imperial measurement from the time of the British. In Australia we now measure area in hectares, as part of the SI measurement system. An acre is now 0.405 hectares.

4840-john-duffield-duffield-design