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.

]]>While researching the history of division recently, I came across this old photograph of a Hospital School in Vienna, 1921. If you look carefully you see the students attempting to calculate 6975 ÷ 235.2. Obviously about 100 years ago we did not have calculators available to students to help them work out their division. Notice also that there is no real-life link, just the algorism waiting to be solved. The students have pen and paper ready and they also seem to be listening to the student at the front who may be trying to explain to the rest of the class how he will work it out.

My worry is that this sort of experience still happens today. We no longer need to calculate such an example using pen and paper as we have a calculator on our phone or computer that can do this for us. We also need a real-life context so that this calculation makes sense to our brain and is not just a meaningless number.You could ask your Stage 3 students to brainstorm for 1 minute with a partner as many different examples as they can. For example, there is a length of timber that is 6975 cm long and we need it cut into 235.2 cm lengths. How many lengths will you get? Or you have 6975 mL and you need to pack it into containers that each hold exactly 235.2 mL. How many containers will you need? But notice how this actual calculation is pretty obscure. It is not something you will come across every day.

And we still need to estimate our answer so that we can check we pressed the correct buttons. For example, 6975 is reasonably close to 7000. And 235.5 is closer to 250 than 200, if we round to the nearest 50. There are 4 groups of 250 in 1000 so there will be 7 x 4 = 28 groups of 250 in 7000. So I would expect to get an answer round about 28.

When I press 6975 ÷ 235.2 on my calculator it shows 29.6556122. I know this rounds up to 29.66 as a 2-place decimal or to 30 if I want no remainders. So my estimate was a useful check that my calculation was correct. Division estimates are a little tricky as you don’t need to round to the nearest 100, 1000 or 10 000 but to the nearest easy multiple. In this case 7000 was OK but if your calculation was 4530 ÷ 72 then an effective estimate would be 4200 ÷ 70 which you can then calculate mentally. 72 was rounded to the nearest 10, but then 4530 was rounded to an easy-to-think-about multiple of 7 and 42 is closer to 45 than 49 is to 45 so 4200 was a suitable number to round down to. I can then mentally calculate 4200 ÷ 70 to get 60. The calculator shows 62.9 so my estimate was in the right ball park.

So, what do we want our students to learn?

This is a difficult question to answer today as the world changes so rapidly. We are not sure what the world of our students will look like in 20 years time. But we are pretty sure that, in terms of number calculations, estimation is a vital skill and an understanding of what happens to numbers when you round up or down is a vital part of this skill. Once we go beyond our basic number facts and extended number facts, we no longer need to do the actual calculations ourselves. Those pages and pages of complex number calculations no longer have a central role in our primary classrooms.

We want students who are confident problem solvers, who can work co-operatively and explain processes. We want students who can tackle relevant, real-life problems, who can persist and not give up easily. Is this the focus in your maths sessions?

]]>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?

]]>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?

]]>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.

]]>In fact student blockages can be quite profound. For a start the word volume is commonly linked to sound on a computer, TV or radio. That sort of volume makes sense to students. Here’s what some students write when asked to brainstorm what they already know about volume.

Students also understand volume 1 and 2 in a book such as an encyclopedia. So we need to acknowledge this meaning of volume whenever we start a class discussion about it. But what about volume as a mathematical concept in primary school? Students can go for years without experiencing sufficient volume activities and discussions that help them unravel their misunderstandings. My advice is always to create a block of time rather than ad hoc lessons here and there. If you can only spend 10 hours a year thinking about Volume & Capacity, then two 1-week programs (e.g. Term 1, Term 3) where you can tackle any misunderstandings head on will be the most effective.

Start by asking your students to talk about and/or record what they know. This student thinks volume is all about capacity.

The bowl, jug, can of spray, bath and rubbish bin can all be filled with something. But the desk, computer and book are seen as objects that can’t hold anything else inside them. If you don’t know about this misunderstanding, it would be difficult to unblock this student.

The second student agrees and even records their thinking with words.

Notice how capacity is the dominant concept. There is no consideration that the ruler is made of wood that in itself has a volume. Previous experiences must have focussed on filling empty things up.

This third student tells us what many students think – if there is no capacity then there is no volume.

So, thinking about the large red capsicum at the top of this article, does a capsicum have volume? Many students of course will say DEFINITELY NOT as you don’t put anything in a capsicum, unless you are from a Greek family with a granny who cooks delicious stuffed capsicum of course!

How can you define volume? Volume is a measurement of the amount of space an object or substance takes up. You and I take up a specific amount of space, so does every 3D object around us. Even our earth has volume as it takes up space within space. If you think of that red capsicum, it takes up the same space as say a grapefruit, even though they have different shapes. It takes up more space than an egg, less space than a watermelon. It doesn’t matter if it is hollow or solid. It doesn’t matter whether the outside skin is paper thin or really thick.The volume of the capsicum remains the same. It can be measured in units of volume such as cubes, for example. There are several other ways to measure a capsicum. Mass would measure how heavy the capsicum feels and would be measured in units of mass such as grams. Surface area would tell you how much skin area there is in total and would be measured using units of area such as square centimetres. Length or height or width will tell you different length information measured in units of length such as cm or mm. Time would be a measurement of how long it took to grow from a tiny bud to this size. But, if you squash this capsicum flat, the volume will change. It will become smaller. It will take up less space. That’s because there is also a volume of empty space held inside the capsicum that would not be contained there if it was squashed. So there is actually a specific volume of capsicum flesh that is different to the volume of the capsicum as a whole. Measurement outcomes depend on what it is you want to find out. Do you want to know how much space this whole capsicum takes up, or what volume of flesh there is? If you are packing capsicums into a box, then the whole volume matters. If you are chopping capsicums up to make soup then just the volume of the flesh matters.

Think of a balloon. You know that when it is not blown up it is just a small volume of rubbery substance. This might have the same volume as one cubic centimetre, for example. But if you add air and blow it up quite large and tie it off, the volume of the balloon changes. We now think of the volume as the space taken up by this inflated balloon, not just the small amount of rubbery substance on the outside. The volume of this balloon now includes the volume of the air inside it. So what is it you want to find out? Do you want to know how much space an inflated balloon takes up? For example, you may want to fit 20 inflated balloons in your car, to take to a surprise birthday party. Will they fit? But if you are a balloon manufacturer, you want to know how much volume is in the rubbery stuff that the balloon itself is made from. For example, if it takes 1 cubic centimetre of stuff to make one balloon, you will need 10 cubic centimetres to make 10 balloons and 100 cubic centimetres to make 100 balloons. The answer to “what’s the volume?” depends on which volume you have as your focus.

To help your students understand volume better, you need to brainstorm lots of objects like this until they “get it”. Volume is NOT the same as capacity. I will talk about that later. All 3D objects and substances have volume. And the funny thing is that when the student says a piece of paper has no volume, they need to investigate this further. They need to unblock their thinking. A piece of paper looks like a 2D object but really it is an extremely flat 3D rectangular prism. But it still has height, width and breadth. And that paper 3D rectangular prism definitely has volume. If you are a paper manufacturer you know that you need to order large quantities of pulp and stuff before you can create your thin paper sheets. That stuff is the volume. It may be only a tiny fraction of a cubic centimetre for each sheet but if you scrunch up that one piece of paper you will notice that it creates a more obvious visible 3D object.

Students associate volume with water. For example,we can estimate how many millilitres of water match the volume of our red capsicum and then place it in a tub of water. The amount of space the capsicum takes up will push that water somewhere else. If you place the capsicum into a completely full container, sitting in a larger empty bowl, you can measure the amount of water that overflows. This is the displacement. This displaced water can then be measured informally by marking where it comes up to in a container or formally using a container marked in millilitres. The volume of this displaced water matches the actual volume of the capsicum. You can check this measure against your estimate. The volume of this water looks nothing like the shape of the capsicum even though the volumes match. After repeating this guess and check process with different objects each time, your students will become experts at volume estimation. They will realise that objects with the same volume can differ greatly in their shape.

Or instead of using water to help us measure volume, we can estimate how many one centimetre cubes will match a 3D object. We can build a model that matches the size and shape of the red capsicum. Once we think we have a match we can then count up how many cubic centimetres we needed. This will approximate the volume of the red capsicum too. And we can rearrange these same cubes to create any shape we want. All the shapes we create will have the same volume.

Notice it doesn’t matter whether we measure with water or blocks. Both measures are measures of volume. Volume isn’t just about water or liquids.

But now we need to ask “What is capacity?” and “How does this differ from Volume?”. To answer this we need to go back to our red capsicum. The way it is created (apart from the Greek granny stuffing it with rice …) it doesn’t usually contain anything else, other than air or hollow space inside it. It has a volume, but no capacity to hold another volume inside it.

Now think about this cup.

It is made from plastic. The manufacturer (probably in China) had to order in a whole bag of materials that were melted together perhaps in a mould to create this object. Let’s say the manufacturer uses about 25 cubic centimetres of plastic to make one cup. That means the volume of this cup is 25 cubic centimetres. But because it is a cup, this plastic object now has the capacity to hold the volume of something else inside it. This could be a volume of grapes, a volume of water, a volume of flour if you are making a cake and so on. Even though the cup has a volume of 25 cubic centimetres of plastic, it might hold 250 millilitres of water inside it. In other words, there are now TWO volumes. And I think it is at this point that students get very confused. Again, we need to know what question we really want answered. Do we want to know the exact amount of plastic used to make this cup? Or are we more interested in how much it can hold of another volume? I think for most people, we want to know how much this cup can hold. What is its capacity?

Only things with a shape that can fit other things inside them have capacity. A theatre has the capacity to hold say 1000 people inside for a concert. A bath has the capacity to hold so many litres of water. A box has the capacity to hold so many oranges packed in together to put on a truck to take from a farm to a shop. A supermarket shelf has the capacity to hold a specific number of cereal boxes. A human lung has the capacity to hold a specific amount of air inside it. A jar has the capacity to fit a specific amount of biscuits inside it. Capacity can be measured in many units, including units of water and blocks.

Students seem to understand this idea. In fact it pervades all their thinking.

Most students seem to clearly understand that capacity can relate to measuring with water.

But not only do our students start to think of volume as capacity, they then start to get confused with other measurements too.

This student must have had plenty of experiences measuring how far up a container some water came. They start to confuse this with height itself. It is true you measure the height of the water but only as an indicator of where the total volume reaches to in the container. The whole amount is the volume, not just the height. You need to focus on the whole volume.

And this student confuses volume with mass, or how heavy an object feels. You need regular class discussions to fix this blockage. An object might look really large, such as a gigantic cardboard box that a fridge came in. But just because it is very large does not automatically mean this box will be heavy. Similarly a small metal object might feel really heavy, even though its volume is the same as a ping pong ball which is very light. Are you measuring how heavy something feels (Mass)? Or are you measuring how much space something takes up (Volume)?

Sometimes volume and capacity is invisible. A room has the capacity to hold a specific number of invisible cubic metres, for example. We measure the volume of a room in cubic metres. We learn to estimate this space by imagining stacking invisible cubic metres in rows and columns one on top of another. After many experiments with making models from cubic centimetres, we can start to imagine these as invisible cubes instead of using real objects. This skill uses proportional reasoning to work it all out.

Finally, misunderstandings occur when students are introduced to the formula for calculating the volume of a rectangular prism as L x W x H. As this is relatively easy to explain and model, it becomes the end action. Students then believe that volume only occurs when the criteria of length, width and height are measurable. They come to believe that if something is irregular in shape then it doesn’t have a volume. There is nothing wrong with thinking about the volume of a rectangular prism. but at the same time we need to discuss many other objects that are not rectangular prisms. A bean bag has volume, a cabbage has volume, a car has volume, an elephant has volume. The volumes are not easy to calculate using a formula in the primary school. But they can be estimated using invisible units and proportional thinking. Students need plenty of experiences estimating the volume of weird and irregular shaped objects.

Notice that this Year 6 student clearly believes volume is about what is inside this shape. If this is a model of a room, this is true. If this is a model of a solid block then the whole object has a volume. It s not just about what is inside. This next Year 6 student has probably had experiences calculating the surface area of rectangular prisms. This is a totally different measure. Surface area only concerns the outer 2D surface of this shape, not the volume or amount of space this object takes up.

All 3D objects have volume. Not all 3D objects have volumes that are easy to calculate. We need our students to be experienced imaginers, able to use proportional reasoning to divide up spaces into imaginary cubes. This process is certainly helped by talking about rectangular prisms. We can estimate how many cubes would fit along one side, then lay rows like this side by side to fill the base, then stack layers of these imaginary cubes to reach the top. The end result is that we have a measure of length, width and height. But our concentration when thinking about volume should always be on seeing stacked 3D cubes, not 2D lengths.

Enough for now, I need a nice cup of coffee and a walk around the garden. Perhaps my red capsicums are ready to pick!

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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.

]]>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.

]]>How many different ways can you talk about 3000?

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?

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