
Shape & Measurement A
Years 2  10


A Mathematics Centre Resource
You are encouraged to contribute to these notes. Comments from teachers or students, photos and student work samples will enrich the professional experience for all of us.
Picture Puzzles have multiple levels of success. They do not have to be 'finished'. They can be revisited and continued.
Area and perimeter don't have to be measured by squares and the side lengths of squares. We do so as a culture because a choice was made some time in history, but that choice can cloud the concept behind the measurement. Simply put, perimeter is measuring the boundary of a shape and area is measuring the space inside the boundary. For both of these, if we agree on the same unit of measure to use in every case, then we can meaningfully compare different shapes. This menu explores the same area and perimeter concepts using squares as the standard unit in some puzzles and triangles as the standard in others.

Summary
Measuring With Sphinx uses triangles as the measuring unit to highlight the concepts of area and perimeter. It begins with a few straightforward examples and then asks the equivalent of Can I check it another way?, which is the question that leads to perimeter and area formula in the world of square units. The puzzle goes on to present several challenges related to finding perimeters for fixed areas and includes a final challenge that indicates what mathematical formulas for area might look like if the original choice had been to measure area with triangles rather than squares.

Answers


The shape is made of two sphinxes, each of which has an area of 6 triangles, so its area is 12 triangles. The top and bottom sections of the perimeter are the same length (2 triangle sides each), as are the left and right sections (3 triangle sides each). So, perimeter = (2 x 2) + (2 x 3) = 4 + 6 = 10 triangle sides.
Encourage students to record their answers in their journal.




Area is still 12 triangles.
Perimeter = 14 triangle sides 



Area = 12 triangles.
Perimeter = 12 triangle sides 

As they explore (and record) 2Sphinx shapes of their own, students are likely to find short cuts for calculation. For example for the previous shape, its rotational symmetry means only one half of the perimeter needs to be counted and then doubled to find the total. Different shapes suggest different shortcuts.

However the last question on this slide is asking students to find an alternative to counting that will work for any 2Sphinx shape. One approach, and the students may well find others, is to realise that:
 two separate Sphinxes have a total perimeter of 16 triangle sides
 every side, or part side, match removes double that length from the total perimeter.
For example, again in the previous shape, two side lengths of 2 are touching, so 4 lengths are removed from the total of 16, hence the answer of 12.



Area = 18 triangles.
Perimeter = 18 triangle sides
Area could be counted triangle by triangle or it could be a multiplication of 3 x 6 = 18. Which is more efficient  additive thinking or multiplicative thinking? Perimeter could be counted side by side, or we could use the multiplicative approach of (3 x 8)  6. 



Area = 24 triangles.
Perimeter = 20 triangle sides
And your two ways are?



Given that putting a fixed number of sphinxes together means that all the shapes have the same area, but perimeters vary, there must be a minimum and maximum perimeter. This slide is for 2Sphinx shapes and the next two are for 3Sphinx and 4 Sphinx shapes. In this case the reasoning to find maximum and minimum must be:
 Maximum perimeter = minimum matching sides = 2 only for 2Sphinxes, but is that true for 3Sphinxes and 4Sphinxes?
 Minimum perimeter = maximum matching of sides = ...depends on the number of sphinxes involved.
In each case there may be more than one way.


For 2Sphinx shapes, the maximum perimeter is 14 and the minimum is 10. So, is it possible to make a 2Sphinx shape with a perimeter 11? ...12? ...13?
Miss I can't make one with 11 ... Me either Miss.
Can you make one with 12?
Yeah. that's easy Miss.
Well could there be a reason why you can't make one with 11?

Similar reasoning works for 4Sphinx shapes. When two shapes touch an even number of side lengths (2, 4, 6, ...) must be removed from the total perimeter (ODD + ODD = EVEN and EVEN + EVEN = EVEN). The perimeter of one Sphinx is an even number. Therefore the total perimeter of any number of Sphinxes is an even number (EVEN x EVEN = EVEN and ODD x EVEN = EVEN). Subtracting an even from an even is always even. So, no odd perimeter measures are possible. Isn't it wonderful how playing with shapes gives meaning to numbers?

Sliding a piece of paper down the screen row by row reveals at least two interesting patterns:
 1, 4, 9, 16, 25, 36
 1 + 3 + 5 + 7 + 9 + 11
The first indicates A_{ET} = N^{2}, where A_{ET} means the area of the equilateral triangle and N is the size of the triangle.
The second indicates that A_{ET} = sum of the first N odd numbers, where n is the size of the triangle. This also turns out to be equal to N^{2} (see Pattern & Algebra A).

So, to find the area of an equilateral triangle simply square the length of the side. A lot easier than the rule you know for calculating the area when the accepted unit of measure is a square. So, what happens if the world started again and everyone agreed that the unit for area measure was an equilateral triangle?
 What would be the formula for the area of nonright angled triangles? ... squares? ...rectangles? ...rhombii? ...parallelograms? ...other shapes?
There's challenge enough in these questions for senior secondary and university level students; indeed, even mathematicians.

To learn more about this Picture Puzzle click the photo to visit the Task Cameo. 


Summary
Measuring With Trisquares uses squares as the unit of measure for area and sides of the square as the unit for measuring perimeter. Its focus is the concept of measuring area and perimeter. It begins with a few straightforward examples and then asks the equivalent of Can I check it another way?, a mathematician's question. The puzzle goes on to present several challenges related to finding perimeters for fixed areas and includes a final challenge with a link to pattern and algebra.

Answers


Area = 6 squares
Perimeter = 10 square sides.
 How do you know?
 Can you check it another way?
Encourage students to record their answers in their journal.




Area = 6 squares
Perimeter = 12 square sides.
 How do you know?
 Can you check it another way?




Area = 6 squares
Perimeter = 12 square sides.
 How do you know?
 Can you check it another way?




Area = 6 squares
Perimeter = 14 square sides.
 How do you know?
 Can you check it another way?
But recording this one might not be correct.
The shape is made from two Trisquares, but unlike the ones above, it has been rotated so that sides are not parallel to the sides of the screen. Recording on graph paper is emphasised in this question because sometimes students draw it on the paper 'as they see it', as shown in the second picture.



In this case, counting area by squares on paper and counting area by squares on the actual object will produce a conflict. (Check the number of squares in the drawing). In the previous cases the number of squares on the shape and the paper are (almost) certainly the same. Why has this happened?
The answer is to do with the distance between dots chosen as the length unit. A second look at the diagram shows that the shape is certainly correct, but the length chosen as the scale is the diagonal of the square, which is longer than the side length.


For those who know Pythagoras Theorem, the diagonal length is actually square root of 2 if the measuring unit is the side of the square, as we have agreed. So, by (accidentally) choosing the diagonal as the unit, the diagram has been scaled up in each direction by the square root of 2. Therefore the area has been scaled up by:
(square root of 2) x (square root of 2) = (square root of 2)^{2} = 2
Therefore the square count on paper will be twice the square count on the shape.
The perimeter change is not quite so obvious to some students. If you count the perimeter sections, there are still 14 on the drawing. However you are using the diagonal length as the unit, so, on the same scale for both pictures, the perimeter will also be different. In fact perimeter will be square root of 2 longer on the drawing.


Area = 9 squares
Perimeter = 18 square sides.
As in the triangle case in Measuring With Sphinx (above), area could be counted square by square or it could be a multiplication of 3 x 3 = 9. Which is more efficient  additive thinking or multiplicative thinking? Perimeter could be counted side by side, or we could use the multiplicative approach of (3 x 8)  6 = 18, because every side, or part side, match removes double that length from the total perimeter.




Area = 12 squares
Perimeter = 20 square sides.
And your two ways are?



Given that putting a fixed number of Trisquares together means that all the shapes have the same area, but perimeters vary, there must be a minimum and maximum perimeter. This slide is for 2Trisquare shapes and the next two are for 3Trisquare and 4Trisquare shapes. In this case the reasoning to find maximum and minimum must be:
 Maximum perimeter = minimum matching sides = 2 only for 2Trisquares, but is that true for 3Trisquares and 4Trisquares?
 Minimum perimeter = maximum matching of sides = ...depends on the number of Trisquares involved.
In each case there may be more than one way.


For 2Trisquare shapes, the maximum perimeter is 14 and the minimum is 10. So, is it possible to make a 2Trisquare shape with a perimeter 11? ...12? ...13?
 Miss I can't make one with 11 ... Me either Miss.
 Can you make one with 12?
 Yeah. that's easy Miss.
 Well could there be a reason why you can't make one with 11?

Similar reasoning works for 4Trisquare shapes. When two shapes touch an even number of side lengths (2, 4, 6, ...) must be removed from the total perimeter (ODD + ODD = EVEN and EVEN + EVEN = EVEN). The perimeter of one Trisquare is an even number. Therefore the total perimeter of any number of Trisquares is an even number ( EVEN x EVEN = EVEN and ODD x EVEN = EVEN). Subtracting an even from an even is always even. So, no odd perimeter measures are possible. This is the same reasoning as in Measuring With Sphinx above and again it's wonderful how playing with shapes gives meaning to numbers?

Area = N^{2}, where N is the size of the square.
However, there are are least two ways to calculate this. See Square Numbers in Pattern & Algebra A.


To learn more about this Picture Puzzle click the photo to visit the Task Cameo. 


Summary
The slides begin with a visual definition of how to join sphinxes together. Sides touch completely, or if short sides touch longer sides it must be so that the exposed length of the longer side can be counted in unit side lengths (the unit being the shortest side of a sphinx). Students explore 2, 3 and 4Sphinx shapes and then work backwards to discover how given shapes have been made. Eventually, they are challenged to make a Sphinx from sphinxes. How many? In what arrangement? Additional challenges include finding all the 2Sphinx shapes and tessellating with Sphinx shapes.

Answers

Students have already explored their own 2Sphinx shapes, so they may have come upon some that are shown. It doesn't matter. However, it does matter that their journal record shows how they are made. Triangle Dot Paper can help.
These ones are made from two sphinxes.


This shape is where the Sphinx becomes amazing.

It just replicated itself! ... and we have a template for how it does it. We can see how 4 Sphinxes make a Sphinx. So, this shape is a Sphinx ... and if we had four of this shape then ... The link below to Task 166, Sphinx, will reveal a Sphinx Album that opens up many investigation lessons based on the selfreplicating nature of the Sphinx.

How many are there? We don't know ... but we do think it can be found out. And we would love to see your students' work investigating the question.
One way to begin is to make a 2Sphinx display corner. The first students to make a particular 2Sphinx shape, trace it onto coloured paper (black looks fabulous) showing the outline only and pin it to the display. Then each new shape is traced and displayed in the same way. Keep the students on the look out for rotations or reflections. Will these be new 2Sphinx shapes? When the class can't find any new ones, it might mean you have found all the 2Sphinx shapes.

Another approach uses the mathematician's strategy of try every possible case:
 Fix one of the sphinxes to the centre of the table with stickytac.
 Choose one of the sides of the other sphinx (let's say the unit length sloping edge at the end of the longest side).
 Take this 'for a walk' around the fixed sphinx touching it at every possible place.
 Photograph the 2Sphinx made with each touch. (It helps if the camera is in a fixed position for each photo.)
 Flip the sphinx and take the same side for a walk again. Take more photographs as you go.
 Repeat the procedure using each side of the Sphinx in turn and photographing every touch position.
Trying every possible case means you can be certain of finding every possible 2Sphinx shape. However, you might have found some of them more than once, so the photos have to be compared to look for repeats, some of which might be reflections or rotations. Working like this is just like the work forensic scientists do when hunting through databases for finger print or DNA matches. Perhaps you can put this investigation in the context of the class becoming mathematical detectives.

Tessellations are sometimes called tiling patterns, just like on the kitchen wall or bathroom floor. Usually these are the one tile repeated to cover an area with no gaps (grouting doesn't count!) and no overlaps.
One way to tessellate sphinxes is like this...
...which works because any quadrilateral will tessellate (why?) and one of the 2Sphinx shapes is a quadrilateral called a parallelogram.
Another 2Sphinx tessellation begins with
and becomes...



With a little imagination, this might be seen as a tessellation of stylised maps of mainland Australia.

Sometimes tilers make floor patterns with two shapes. This tessellation of 'cogs' made from a sphinx rotated six times and joined with triangle spaces between (perhaps for pot plants) would be a really cool entry foyer for an engineering company. See Sphinx Album for Nickey Harland's idea for another 6Sphinx rotation that might be a tessellation using just sphinxes.
To learn more about this Picture Puzzle click the photo to visit the Task Cameo. 



Summary
Building around student knowledge of the word diamond, the definition of an 'iamond' is illustrated as far as hexiamonds, which are made from 6 equilateral triangles. The meanings of area and perimeter are refreshed, then the search for various polyiamond shapes is linked to counting their perimeters. For a given polyiamond, the area is fixed as the number of triangle units that make it up. Perimeters appear at first to follow a pattern, but all is not as it seems. Extra challenges include shape puzzles using hexiamonds the students cut from paper and tessellation.


One way to find all the tetriamonds is to begin with a triamond and add a single triangle in all possible places.
Area = 4 triangle units
Perimeter = 6 triangle sides

Area = 4 triangle units
Perimeter = 6 triangle sides 


One way to find all the pentiamonds is to begin with each tetriamond and add a single triangle in all possible places, making sure you eliminate repeats.
Area = 5 triangle units
Perimeter = 7 triangle sides

Area = 5 triangle units
Perimeter = 7 triangle sides

Area = 5 triangle units
Perimeter = 7 triangle sides 

Looking back at this data and including the area and perimeter for the triamond (3, 5), diamond (2, 4) and moniamond (1, 3) we could conclude that for polyiamonds, P = A + 2, where P = perimeter counted in triangle sides and A = area counted in unit triangles. That's neat. But what happens if we try hexiamonds?
What would you predict about the perimeter(s) of heptiamonds (occasionally called septiamonds) made from 7 unit triangles. Encourage students to web search these two words. There are some interesting challenges available including using them artistically to make silhouettes of a dog, cat, fish (several ways), butterfly and candle in a candlestick.

Students were asked to draw and cut out the hexiamonds so they could try this puzzle. It may help if they first mark out the frame from the same scale paper as they used to cut their hexiamonds. Also the puzzle will require discussion of perimeter because each side has to have a perimeter of 6. Which parts of which hexiamonds will allow this?
This is one way to do it. The puzzle is also an opportunity to introduce the term rhombus as a special case of a parallelogram.



Other geometric hexiamond puzzles include making:
4 x 9 Parallelogram 
7/11/4 Isosceles Trapezium 
Is it possible for the 6 x 6 rhombus to be coloured in only 3 colours so that pieces touching on a side are different colours??

As the slides show there are many possible results for this investigation. Please send examples of those created by your students and we will be happy to display them with acknowledgement.


To learn more about this Picture Puzzle click the photo to visit the Task Cameo. 


Summary
The slides begin with a visual definition of how to join Trisquares together. Sides touch completely, or if short sides touch longer sides it must be so that the exposed length of the longer side can be counted in unit side lengths (the unit being the shortest side of a trisquare). Students explore 2, 3 and 4Trisquare shapes and then work backwards to discover how given shapes have been made. Eventually, they are challenged to make a Trisquare from trisquares. How many? In what arrangement? Additional challenges include finding all the 2Trisquare shapes and tessellating with trisquares.

Answers

Students have already explored their own 2Trisquare shapes, so they may have come upon some that are shown. It doesn't matter. However, it does matter that their journal record shows how they are made. Square Dot Paper can help.
These ones are made from two trisquares.
Note: See Measuring Trisquares above for an important comment about checking the scale students have chosen to represent the bottom two of these drawings. 

There could be some discussion about whether the last one is allowed because the corners are touching. However it also needs to be considered that by obeying the joining rules the 'hole' has been automatically formed. 

This shape is where the Trisquare becomes amazing.

It just replicated itself! ... and we have a template for how it does it. We can see how 4 Trisquares make a Trisquare. So, this shape is a Trisquare ... and if we had four of this shape then ... Task 238, Growing Trisquares, will reveal more about this template.

How many are there? We don't know ... but we do think it can be found out. And we would love to see your students' work investigating the question.
One way to begin is to make a 2Trisquare display corner. The first students to make a particular 2Trisquare shape, trace it onto coloured paper (black looks fabulous) showing the outline only and pin it to the display. Then each new shape is traced and displayed in the same way. Keep the students on the look out for rotations or reflections. Will these be new 2Trisquare shapes? When the class can't find any new ones, it might mean you have found all the 2Trisquare shapes.

Another approach uses the mathematician's strategy of try every possible case:
 Fix one of the trisquares to the centre of the table with masking tape loops.
 Choose one of the sides of the other trisquare (let's say the unit length sloping edge at the end of the longest side).
 Take this 'for a walk' around the fixed trisquare touching it at every possible place.
 Photograph the 2Trisquare made with each touch. (It helps if the camera is in a fixed position for each photo.)
 Flip the trisquare and take the same side for a walk again. Take more photographs as you go.
 Repeat the procedure using each side of the Trisquare in turn and photographing every touch position.
Trying every possible case means you can be certain of finding every possible 2Trisquare shape. However, you might have found some of them more than once, so the photos have to be compared to look for repeats, some of which might be reflections or rotations. Working like this is just like the work forensic scientists do when hunting through databases for finger print or DNA matches. Perhaps you can put this investigation in the context of the class becoming mathematical detectives.

Tessellations are sometimes called tiling patterns, just like on the kitchen wall or bathroom floor. Usually these are the one tile repeated to cover an area with no gaps (grouting doesn't count!) and no overlaps.
One way to tessellate trisquares is like this...
...which works because any quadrilateral will tessellate (why?) and one of the 2Trisquare shapes is a quadrilateral called a rectangle.



Another 2Trisquare tessellation begins with and becomes...
With a little imagination, you might see it as a school of stylised fish (like the yellow one) swimming down to the bottom right. Add fishlike details inside the yellow one. If you add the same details in the same places inside each of the 'fish' you will have an Escherstyle tessellation.

To learn more about this Picture Puzzle click the photo to visit the Task Cameo. 

