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Mathematics Lessons for Remote LearningStrand: Geometry and Measurement Target: Y7, 8, 9, 10 –NZC Level 3/4Topic: GM Transformations and Symmetry - ROTATION

Starter – Odd One Outtwist pinchripple shear

The idea here is to select the odd one .

I choose __________ because _____________________Learn by DoingTHE DEEP UNDERSTANDING OF GEOMETRY

- Being enabled to describe shapes and their properties in the world around us, to reason, to generalise (prove), to make sense of and connect ideas to solve problems.

Transformations – IsometricThe object and image are the same size in an isometric transformation.

• ROTATIONIn any 2D or plane rotation the object rotates about a single point in either a clockwise or anti-clockwise direction. This point of rotation is called the Center of Rotation and along with the Angle of Rotation completely describes the transformation.

Angle is a curious notion. It is just simply a choice and a decision but there are 360 degrees in a full turn and that turning to the left or anti-clockwise is considered positive. This might seem a bit wrong but it is just a decision. Angles can go bigger than 360 so 5 full turns would be 5x360 = 1800 degrees. A half turn is 180 and a quarter turn 90.

A navigational compass is graduated in degrees as well but these number from 0 for due North all the way to 360 which is also due North. A compass reading is called a bearing. N80E means from N turn 80 degress to the East. A bearing of 180 is the same as due South and 225 is SW. The major direction is always mentioned first hence SSW is halfway between S and SW on a 16 point compass. Add 180 to any bearing for the back-bearing to head home.

Task – Construct a 16 point compass. The illustration is a good example to copy.

• Rotating an Object to locate the Image

Draw a shape, the object, and trace it onto some see-through or thin paper. Put a pencil firmly on the chosen rotation point and rotate the tracing paper. Now dot or press through and draw the image of your rotation. See photo below.

• Constructing the Location of the Center of Rotation (C of R)The photo shows a hexagon with a yellow hat, rotated clockwise by 50 degrees. Two corresponding object/image pairs are required here to locate the C of R. Construct the bisector of line segment AA’ and of BB’. The point O is the intersection of these two bisectors and is the Center of Rotation for all points on the object.

Task – Repeat this task for a shape of your own. Here is a BC comic strip talking about wheels. You could choose one of these shapes.

Geometry is making sense of size and shape

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• Task - Which letters of the alphabet have rotational symmetry greater than 1?

ABCDEFGHIJKLMNOPQRSTUVWXYZ• A Calendar ProblemThis problem involves writing numbers on the two cubes so that all the dates needed for a calendar can be made by rotating the cubes. There are three square prisms underneath the cubes on which the months are written. Task – Figure out how to number the two cubes.

• The Earth RotatesThree dimensional (3D) objects such as our planet Earth rotates about a line. This is called the axis of rotation and determines the length of our day.

- How many times does the Earth rotate in one day?- How many times does the Earth rotate in one year?- Does the Earth rotate from West to East or East to West?- What other planets rotate? https://www.exploratorium.edu/ronh/age/ - A pulsar is a very solid remnant of a star explosion and rotates very fast. Many have been measured to rotate at

thousands of times per second. https://www.youtube.com/watch?v=gjLk_72V9Bw for more NASA infomation. You can listen to these planets rotating!

- Here is a great picture of Mars - https://apod.nasa.gov/apod/image/2005/marsglobe_viking_1552.jpg

• Gear Rotation ProblemsHere is a link to some space Math form NASA https://spacemath.gsfc.nasa.gov/engineering/9Page14.pdf

• A Roundabout ProblemA man and a squirrel are on opposite sides of a tree. The squirrel moves so that he remains opposite the man whenever the man moves. Does the man go around the tree? Does the squirrel go around the tree? Does the man go around the squirrel? Does the squirrel go around the man? You could even ask if the tree goes around one or both too.

• Clock ProblemThe hands of a clock are together at Noon. When exactly are they together again?

Geometry is making sense of size and shape

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Rotational SymmetryEvery 2D shape has a rotational symmetry of at least 1. Rotational symmetry is when a shape can be rotated and look the same within one turn. A square can be rotated 4 times until it has completed one turn. An isosceles triangle (and you) have a rotational symmetry of 1. An equilateral triangle has a rotational symmetry of 3.

• Task – What is the rotational symmetry of the other three shapes above?

Total Order of SymmetryThe total order of symmetry of a shape is the sum of the number of axis of reflection and the order of rotation. The Heart above has 1 axis of symmetry (reflection) and 1 order of rotation so has a total symmetry of 2. Very symmetric shapes and designs have a high total order of symmetry.

• Task – What is the total order of symmetry of all the shapes above?

• The Fortessa of Sciena – Four arrowheads.This grand castle or fort was built by the Spanish and completed in 1563. It is a well preserved building and with 4 arrow heads protecting the corners. The fort is about 270m long and 200m wide. The parade ground in the middle and top few floors are now used as a public art, wine and concert venue. It is many layers which could be used to store supplies and house people in times of war. https://en.wikipedia.org/wiki/Fortezza_Medicea_(Siena)

JournallingToday I learned ________________________________________________________________And I would like to know about ___________________________________________________

CommentsMake any comment you feel like making here.

Geometry is making sense of size and shape

Learn by Doing

Math Language: List all the math words you can find in this document and write what you think it means beside the word. Eg subtraction means to take away or to find the difference. Keeping a list of these words is a very good idea.

Geometry is making sense of size and shape

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AnswersStarter – Odd One Out

twist pinchripple shear

The idea here is to select the odd one .

I choose ____shear______ because ______there is no bending involved______

• Task - Which letters of the alphabet have rotational symmetry greater than 1?

HINOSXZ and 018 in the single digit numbers. There have been some dates that have rotational symmetry as well. 18/1/81 for example was of great interest in 1981. This century 1/1/1 worked and 11/8/11 worked. See if you can figure out others. These dates are also palendromic meaning they read the same forwards as backwards. “Madam I’m Adam” - the first words spoken by man were palendromic! Do you know others?

• A Calendar ProblemThe rotation of 6 to make 9 is the trick that allows this to happen. One cube has 0,1,2,3,7,8 and the other has 0,1,2,4,5,6 and I think that is the only solution. Notice, no “9”.

• The Earth Rotates- How many times does the Earth rotate in one day? - Once- How many times does the Earth rotate in one year? – 365 and ¼

There is the solar day and solar year and sidereal day and sidereal year depending on the reference to the sun (solar) or the stars (sidereal)

- Does the Earth rotate from West to East or East to West? – towards to East or the sunrise. A curious question is what is the path of the moon around the sun? Try and figure that out!

• A Roundabout ProblemThe man and the squirrel problem exposes the complexity of language. Of course both man and squirrel go around the tree and this is relative to the tree. If you imagine the man carrying a compass when he goes around the compass does a fullturn from N to E to S to W and back to N depending on which way he goes. So he was N of the squirrel, E of the squirrel, S of the squirrel and W of the squirrel and likewise for the squirrel so man went around the and the squirrel went around the man. This is relative to rest of the trees in the forest or the planet. The tree likewise if it had carried a compass could show it too went around both the man and the squirrel! All statements are correct. A very roundabout problem indeed.

• Clock ProblemThe hands of a clock are together at Noon. When exactly are they together again?

This is a favorite problem. If you look at a clock with hands and have a play you will notice that the minute hand overtakes the hour hand eleven times in a complete 12 hour period. Try it! So exactly 1/11th of 60 minutes after Noon the hands will overtake. This is exactly at 5 minutes and 27.2727... seconds past 1pm. You can now compute all 1 exact times the hands overtake and I am not sure that will ever be useful to you!

Rotational Symmetry

• Task – What is the rotational symmetry of the other three shapes above?From L to R, 4, 1, 3, 1, 8 and 4

• Task – What is the total order of symmetry of all the shapes above?From L to R 8, 2, 6, 2, 16, 8. The octogon is the most symmetric shape.

Geometry is making sense of size and shape

Learn by Doing

• The Fortessa of Sciena – Four arrowheads.The arrowheads extend out on each corner giving the soldiers on the fort a way to shoot back at invaders attacking the walls. This is a very cunning design and is a good reason why this fort was never overtaken by attack. The walls are 30m high and very thick so it will last another 500 years at least! Here is a link to castle design just in case! https://www.historyextra.com/period/norman/how-to-build-a-medieval-castle/

FeedbackStudents and teachers are welcome to email jimhogan2@icloud.com with comments. This was a lesson that could be given to a NZC Level 2, 3, 4, 5 student for some placevalue learning and revision. Students should select a set time each day and perhaps using the timer on a cell phone set 45 minutes or so to learn and practice mathematics. Keep trying on problems and expect to struggle. Persevering and struggling are great competencies to develop. You can learn more about these from https://www.youcubed.org/resource/growth-mindset/. We have a great math website in Nzwith a special resource called e-AKO https://nzmaths.co.nz/information-about-e-ako-pld-360 .

Geometry is making sense of size and shape