Alyson:

In week 8 tutorial you will be allocated one of these Geometer’s sketchpad lessons to evaluate. You are to carry out the lesson and evaluate it (with a link to it included) on your blog. Ask yourself questions like: What part (if any) of the NSW syllabus would this lesson be appropriate for? What is the benefit of using the Geometer’s sketchpad software in this lesson? Could the lesson be taught any other way? If so, how? How could the lesson be improved? Are the instructions for the lesson clear?

Finding Relationships Between Area and Parallel Lines in Triangles

This lesson was alright. I couldn’t find the “Label Options” thing, so I don’t know if it was for a different Geometer’s Sketchpad than the one I was using. Apart from the slight differences, the instructions were very clear.

I guess the lesson is useful in cementing the idea of why area=1/2 the base times the perpendicular height. If students just remember the formula A=1/2bh they may not remember that the “h” stands for perpendicular height and may measure some obscure kind of height. This lesson proves that even if the triangle is incredibly stretched out, the height is still the same, and the formula tells us this because the base is staying the same, therefore if the height changed the area would change, but it doesn’t.

Trying to do this lesson without the help of Geometer’s Sketchpad would be very difficult. I think it is very handy in allowing students to see the ramifications of the formula for the area of a triangle in action. You really do need the software to move the point around and see that it doesn’t change. Drawing a billion different triangles within parallel lines with the same base would be tedious and not worth it.

OK, so I actually did the readings for last week (I read them on Friday actually but today I’m bored hanging around uni waiting for my carpooling buddies – perfect blogging opportunity). I actually found I do have some things to say.

Number1: Zero is not and never will be a counting number. According to me. If you have none of something, you don’t count it. If you’re doing that fence thing, you will have to realise that there is a post at the point 0 and add it on. It isn’t difficult. You definitely should not count the post as 0, because then you are saying that one post is zero posts!!! WHAT!!!!! No. I actually do say babies are 0 years old. Because I’m rounding down. To the nearest 2 years. Not because they are actually 0.

I don’t actually have the readings here with me so I might not remember everything I hated about the readings, which by the way were tedious and I did skim read them because they were dense and mainly the same as every other reading, except worse.

Number 2: North/South, up/down thing. I don’t remember what it actually said, mayhap I will edit this post later. But this ridiculous statement actually went on for like 2 pages, as I recall. It was too much.

This post is dedicated to Sarah P, Pru, Marjorie, Chris, Geoff, Dave, Nick and Ben. Love your work guys!

All of the articles in the course to date are starting to show similar ideas. Teachers need to discuss mathematical applications and use of mathematical language. Students need to see where problems can occur in the use and talk of mathematics and find ways to conquer these problems. Mathematics is a language in itself, and yet we try to explain it using another language. Problems arise when we use words that have relatively straightforward meanings in colloquial use, yet we use these same words to describe quite complex concepts. Conceptual complexity and apparent contradictions in mathematics language, and the artificial symbolic nature of mathematics, with its reliance on many non verbal ways of representing information, often acts to disturb students. Using background knowledge, using multiple teaching methods (for students from different learning styles) is helpful.

We need to be able to describe what we are doing in non-mathematical language as well as mathematical language. This use of students’ own understanding and explanations when dealing with new ideas is important as it connects the new knowledge with established structures that can later be refined to be more in line with conventional definitions. Though we should encourage students to talk in mathematical English, at times it is appropriate (and neccessary) to talk about examples in normal English to aid discussion and the deeper understanding that comes from such a free flowing discussion.

This week’s readings have a similar message to previous weeks; the language of maths conflicts with everyday use of the English language. They also offered similar suggestions for handling the difficulties that can arise for students. Mathematics “borrows words that already exist with everyday meaning” and in doing so changes the context of the word and therefore changes the meaning. cause a lot of difficultly and impatience for the untrained mathematical mind.

For example starting with a number and imagining that it is the area of a square, the square root gives the side length of that square, which is something that gives students a real example of how to view the square root. Teachers need to foster the use of correct mathematical language in the mathematics classroom at all times. Correct mathematical language should be prevelant in any mathmatical discussion. I also disagree that zero should be used as a number. For example, “zero years old” is not a useful description and would be taking things to extremes.

Students need to participate with the language, that is they need to be using it both orally and in its written form. It is usually agreed that mathematics cannot be learnt by watching, therefore it makes sense that using correct mathematical language cannot be learnt by simply reading and listening.

Why does every 2d shape have a perimeter except for the circle and elipse which have a circumference, and would it be of benefit to standardise?

The circumference is the perimeter around a closed curve. A perimeter is the sum of the lengths of the edges of a closed figure.

I would say that because a circle/ellipse only has one edge, and “perimeter” generally implies “add the sides” in school mathematics, the term “circumference” could mean a continuous perimeter (with no corners).

I reckon that when students are asked for the “perimeter” of a shape they just add up the side lengths, but the “circumference” of a shape has a special formula.

Then when you get to doing perimeters of semicircles and such, then you’re mixing them so that to find the perimeter you use half the circumference, plus the straight bit, coz remember you add all the edges to get the perimeter. I think “circumference” is a useful word and it wouldn’t benefit anyone to standardise.

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Ambiguous Words

LENGTH – the longest extent of anything as measured from end to end. Can refer to shapes, time, clothing, as well as meaning long (”remarkable length”) and difficult (”he went to great lengths to get that”). However, I think that a questions such as “Find the length of the line AB” doesn’t bring that much difficulty.

BREADTH – refers to width, or side-to-side length, or sometimes ” the measure of the second largest dimension of a plane or solid figure”, or just size in general. Extremely confusing and ambiguous.

WIDTH – the extent from side to side, or breadth. It would appear that “width” means the same as “breadth”, but in a more precise and well used way. Students would probably be able to guess what the “width” of something was more easily than they could find the “breadth”. Put both these words on a test and heads will explode.

HEIGHT – the distance upward from a given level to a fixed point, the extent or distance upwards, or the distance between the highest and lowest points. Evidently if you have a shape and rotate it, its height will be different. How do you know which is supposed to be the height? The area of a triangle is half the base times the perpendicular height. In this case the height is in relation to the side you use as the base.

DEPTH – “a dimension taken through an object or body of material, usually downward from an upper surface, or from top to bottom of something regarded as one of several layers”. It also applies to seriousness, complexity, emotions, colour intensity, pitch, and an unfathomable space (the depths). I always think of depth as going into the page when shown a drawing of a 3D shape. It is clearly confusing.

THICK – the state or quality of being thick, or the measure of the smallest dimension of a solid figure. One of the more thought-provoking definitions: is it always the smallest dimension? Does it mean the same as width? Or breadth? If the smallest dimension is the height, is that the thickness too?

All these words have slightly different meanings but it could be possible for them all to be the same. It wouldn’t be that much of a stretch.

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Number - 44 definitions, icluding:

Mathematics

1. A member of the set of positive integers; one of a series of symbols of unique meaning in a fixed order that can be derived by counting.
2. A member of any of the further sets of mathematical objects, such as negative integers and real numbers.
3. A symbol or word used to represent a number.
4. A numeral or a series of numerals used for reference or identification: his telephone number; the apartment number.
5. A position in an ordered sequence that corresponds to one of the positive integers: the house that is number three from the corner; ranked number six in her class.
6. One item in a group or series considered to be in numerical order: an old number of a magazine.
7. A large quantity; a multitude: Numbers of people visited the fair.  

Digit – any of the Arabic figures 1-9 and 0. Also any of the symbols of other number systems. Can also mean the breadth of a finger used as a unit of linear measurement.

Numeral - A word, letter, symbol or figure expressing a number.

Figure - A numerical symbol, and amount or value expressed in numbers, or a written symbol other than a letter. Plus 34 other non-mathematical meanings.

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Curve: A collection of points whose coordinates are continuous functions of a single independent variable.

Line: A collection of points whose coordinates are continuous functions of a single independent variable.

Straight line: A line traced by a point traveling in a constant direction; a line of zero curvature

Line segment: A portion of a line delimited by two end points; also, a line described by two sets of coordinates and the shortest path between them

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Invariably

Not able to change, the same every time. Is this what we mean when we use it? Is this what students understand?

Sum

“Sum” redirects straight to “summation” on Wikepedia, which also gives this:

• Sum (category theory), also referred to as “coproduct”
• Sum (country subdivision), an administrative division used in Mongolia and nearby regions
• SUM (interbank network)
• Soccer United Marketing
• Sum (Unix), a program for generating checksums
• Som (also spelt “sum”) meaning “pure”; the name of two national currencies:
  • Kyrgyzstan som
  • Uzbekistan som
• Sum (Mongolia), an administrative subdivision of Mongolia
• Spilka Ukraïns’koï Molodi (СУМ/SUM) Ukrainian Youth Movement
• Sum 41, a Canadian Punk band
• Direct sum of modules

Sum is also an acronym for Surface-to-Underwater Missile.

There are 15 definitions here.

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Order

Order has 56 definitions here.

Wikepedia gives 11 categories for what “order” might refer to. In the category of Mathematics:

Order in arithmetic, analysis and combinatorics

• Orders of approximation in Big O notation
• Orders of magnitude, a class of scale or magnitude of any amount
• Order of convergence, a measurement of convergence
• Order, or degree of a polynomial
• Order, or matrix (mathematics)#Definitions and notations
• Order (differential equation)#Types of differential equations, or order of highest derivative, of a differential equation
• Order in the Josephus permutation
• Weak order of permutations
• Twelvefold way#Ordered selections and partitions of the Twelvefold way in combinatorics
• Ordered set, a permutation, bijection or cyclic order
• Un-ordered subset or combination
• Ordered list, a tuple or sequence
• Long-range aperiodic order, in for instance pinwheel tiling
• Multiplicative order in modular arithmetic
• Z-order (curve) space-filling curve
• List of order topics in mathematics

Order in fractals

• Complexor, or complex order in fractals
• Orders of construction in the Pythagoras tree
• Order of extension in Lakes of Wada
• Order of Rényi dimensions

Order in graphs

• Graph order, the number of nodes in a graph [1]
• Ordered pair, including undirected and directed graphs
• Ordered triple, or mixed graph
• Glossary of graph theory

Order in mathematical theories

• Order (group theory), the cardinality of a group or period of an element
• Order (number theory)
  • Multiplicative partition, Unordered factorization of numbers
• Order in Ramsey theory, uniform structures in consequence to critical set cardinality
• Order (ring theory), an algebraic structure
• Order theory, which studies various binary relations known as orders
  • Dense order of rational and real numbers
  • Order topology, a topology of Total order for totally ordered sets
  • Partially ordered set, or poset
  • Total order, a generalization of the complete ordered field of real numbers
  • Glossary of order theory
• Set theory (music) encompasses ordered pitch and pitch classes (Musical set theory)
• Type theory encompasses: First-order logic, Second-order logic and Higher-order logic

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Net

23 definitions.

Wikepedia:

Net may refer to:

• Net (device), fibers woven in a grid-like structure
• Net (textile), any textile in which the warp and weft yarns are looped or knotted at their intersections
• Net (mathematics), a generalization of a sequence similar to a filter
• Net (polyhedron), an arrangement of polygons that can be folded up to form a polyhedron
• An electronic network connection in a netlist
• New Jersey Nets, a basketball team
• Net Television, a Maltese television station
• The Net (film), a 1995 film starring Sandra Bullock
• Cricket nets a safe netted environment for practicing cricket

Wikepedia also gives other definitions for “net” in computing and communication, for the Internet-related prefix net, .net or .NET, in business and finance, as an acronym, in broadcast and media and in science and psychology.

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Base

46 definitions. Wikepedia gives numerous possible meanings, including

In mathematics:

• Base (mathematics), the number b in an expression of the form bⁿ, hence also the number of digits (known also as the radix) in a positional numeral system, for example, base 10
• Base (topology), a topological space
• Base (group theory)
• One of the sides of a trapezoid or triangle

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Vertex

–noun, plural -tex·es, -ti·ces

1.	the highest point of something; apex; summit; top: the vertex of a mountain.

2.	Anatomy, Zoology. the crown or top of the head.

3.	Craniometry. the highest point on the midsagittal plane of the skull or head viewed from the left side when the skull or head is in the Frankfurt horizontal.

4.	Astronomy. a point in the celestial sphere toward which or from which the common motion of a group of stars is directed.

5.	Geometry. a. the point farthest from the base: the vertex of a cone or of a pyramid. b. a point in a geometrical solid common to three or more sides. c. the intersection of two sides of a plane figure.

And from Wikipedia:

Vertex (Latin: corner; plural vertices or vertexes) may refer to:

Mathematics

• Vertex (geometry), a corner point of a polygon, node in a Triangulated irregular network or TIN
• Vertex (graph theory), a node in a graph
• Vertex (curve), a local extreme point of curvature

Physics

• Vertex (physics), a point where particles collide and interact
• Vertex (optics), a point where the optical axis crosses a lens surface
• Vertex algebra in conformal field theory
• Vertex function describing the interaction between a photon and an electron
• Vertex model in statistical mechanics, a discrete model of a physical system in which weights are associated with vertices of a grid graph.

Companies

• Vertex – Business Process Outsourcing, a business services provider spun off from United Utilities
• Vertex Pharmaceuticals, a biotech company
• Vertex Standard, a radiocommunication products company.

Music

• A song in In The Groove (video game)
• Vertex (album), an album by Buck 65
• Vertex (band), a band formed in 1996

Other

• Vertex (anatomy), the uppermost surface of the head of an arthropod or vertebrate

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Square

54 definitions.

And from Wikepedia, square may mean:

Mathematics

• Square (geometry), a polygon with four equal sides and angles
  • Unit square
• Square (algebra), to multiply a number or other quantity by itself
  • Square root
  • Square matrix
  • Square number
  • Perfect square
• Square wave, a basic kind of non-sinusoidal waveform

Entertainment

• Square (album), an album by Buck 65
• Square (band), a musical trio from Lincoln, Nebraska
• Square Enix, a video game company
  • Square Co., former video game company that merged with Enix to become Square Enix
  • Square Electronic Arts, joint venture between console video game developers Square and Electronic Arts
  • Square Pictures, a computer-animated film division of Square

Language

• Square (slang), either an unhip person or a cigarette
• Square character (■), a written character
• A modifier adjective, used when describing fielding positions in cricket
• An adjective or noun, used when describing two positions parallel with one another relative to the goal in football

Places

• Square Lake, Maine, an unorganized territory in Aroostook County, Maine
• Times Square, popular landmark in New York City
• The Square, a common name for public squares
  • Town square or Market square, an open area commonly found in the heart of a traditional town
• Tiananmen Square in Bejing China, the largest open-urban square in the world.

Other

• Square (building area), an Imperial unit of floor area
• Square (cipher), a cryptographic block cipher
• Square division, a form of military organization
• Square rig, a type of sail and rigging arrangement
• An astrological aspect of 90 degrees
• Carpentry square, a tool
• Infantry square
  • Broken square

ArcytechTABLE 16.2

Math Tools

This site includes resources for every area of maths you could possibly want, categorised into broad subheadings. An easier way to find what you need is to view the whole list of topics, each with how many resources it has written in brackets next to it. Each resource comes up with a small blurb when you place your mouse over it, so you know what you’re getting before you click on it, which is a help to find the better ones faster.

Most resources on this site require a fee for you to be allowed to print them. Many of the resources are simple worksheets but there are some interactive-type sites, eg Working With Fractions. Finding good ones might take a while because there is so much on this site.

There are lesson plans, stories, activities and technological tools available on the site and places to discuss, rate and review. The Getting Started page tells you what kind of resources are available on the site and how to search for them.

National Library of Virtual Manipulatives

This site allows you to download a free trial desktop version of the software. Resources are ordered both in topic areas (Number & Operations, Algebra, Geometry, Measurement and Data Analysis & Probability) and in grade levels (K-12), so browsing through them is easy. It offers interactive resources rather than lesson plans.

Project Interactive

This site offers 90 lessons divided into topics, for students in years 3-12.

It also includes over 100 discussions, which are extremely useful for preparing for students’ questions.

Interactive activities are also categorised into topics, and there are over 100 of those too. You can search for topics within the site, which makes it easy to find whatever you’re looking for.

Educational Software Components of Tomorrow (ESCOT) Project

This site has about 40 interactive problems but is very technological and the site itself is not interesting. The font is small and it is unappealing. The activities are well written with reflections and extra questions so if you were used to using technology the activities might be very good.

Arcytech

This site offers only a few maths interactive activities. Only about three would be relevant for a high school classroom. You can only use these if your system and browser are Java capable.

Count On

This site is brightly coloured but finding your way around is not as easy as with some of the other sites. There are fun games but little is written about the content so finding a relevant one would be tricky. The site offers The Handbook for Leading Mathematics Teachers and a section on misconceptions, so it might prove useful for a teacher, but doesn’t have anything that could be used for classroom activities (not easily found ones anyway!)

-prepositions, homonyms etc

-5 steps to answer – unfair to grade based on word problems

-nonsensical questions – have the skill to know if a numerical answer is erasonable for what they think they’re answering

-recognition and realisation

-calculators hard to use, individuals need help when push the wrong button

Edublogs is way cooler than zunal.

Let us use this one.

It reminds me of my livejournal.