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Archive for December, 2009

A triangle determines a unique plane

The vertex points of a specific triangle define one, and only one, Euclidean (that is, flat) geometric plane. A specific Euclidean plane can, however, contain infinitely many different triangles. This is intuitively obvious when you give it a little thought. Just try to imagine three points that don’t lie in the same plane! Incidentally, this principle explains why a three-legged stool never wobbles. It is the reason why cameras and telescopes are commonly mounted on tripods (three-legged structures) rather than structures with four or more legs.

Sum of angle measures
In an triangle, the sum of the measures of the interior angles is 180 degrees (pi rad). This holds true regardless of whether it is an acute, right, or obtuse triangle, as long as all the angles are measured in the plane defined by the three vertices of the triangle.

Theorem of Pythagoras
Suppose we have a right triangle defined by points P, Q, and R whose sides are S, T, and U having lengths s, t, and u, respectively. Let u be the hypotenuse. Then the following equation is always true:

s2 + r2 = u2

The converse of this is also true: If there is a triangle whose sides have lengths s, t, and u, and the above equation is true, then that triangle is a right triangle.

Perimeter of a triangle
Suppose we have a triangle defined by points P, Q, and R, and having sides, S, T, and U of lengths s, t, and u. Then the perimeter, B, of the triangle is given by the following formula:

B = s + t + u

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An embarrassed dad

An embarrassed dad

An embarrassed dad

“Don’t be embarrassed to ask for help dad.”

Points and lines

The fundamental rules of geometry go all the way back to the time of the ancient Egyptians and Greeks, who used geometry to calculate the diameter of the earth and the distance to the moon. They employed the laws of Eculidean geometry (named after Euclid, a Greek mathematician who lived in the 3rd century BC). Euclidean plane geometry involves points and lines on perfectly flat surfaces.

In plane geometry, certain starting concepts aren’t defined formally, but are considered intuitively obvious. The point and the line ar examples. A point can be envisioned as an infinitely tiny sphere, having height, width, and depth all equal to zero, but nevertheless possessing a specific location. A line can be thought of as an infinitely thin, perfectly straight, infinitely long wire.

Naming points and lines
Points and lines are usually named using uppercase, italicized letters of the alphabet. The most common name for a point is P (for “point”), and the most common name for a line is L (for “line”). If multiple points are involved in a scenario, the letters immediately following P are used, for example Q, R, and S. If two or more lines exist in a scenario, the letters immediately following L are used, for example M and N. Alternatively, numeric subscripts can be used with P and L. Then we have points called P1, P2, P3, and so forth, and lines called L1, L2, L3, and so forth.

Two point principle
Suppose that P and Q are two different geometric points on a line. Two distinct points define one and only one line L. The following two statements are always true:

  • P and Q lie on a common line L
  • L is the only line on which both points lie

Distance notation
The distance between any two points P and Q, as measured from P towards Q along the straight line connecting them, is symbolized by writing PQ. Units of measurement such as meters, feet, millimeters, inches, miles, or kilometers are not important in pure mathematics, but they are important in physics and engineering. Sometimes a lowercase letter, such as d, is used to represent the distance between the points.

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Answers in my math book

Answers in my math book

Answers in my math book

“What do you expect? My edition of the math book doesn’t have the answers in it like yours does.”

Helping students to get the most from tutorials

Despite the fact that relatively little real learning happens during most lectures, students tend to regard lectures as more important than tutorials. This is compounded by the fact that many lectures treat tutorials as relatively adhoc occasions. The following suggestions may help you to deliver greater learning pay-off in your tutorials:

Help students to see the purpose of tutorials. Students with no higher education background in their family tradition may think that what is accepted as ‘good behavior’ in school (for example, being quiet!) is what is expected of them in college settings too. Normally, the last thing you want your students to be in tutorials is passive.

Avoid the temptation to use tutorials to elaborate on things that have been covered in lectures. It is all too easy for tutorials to degenerate into an extension of lectures, and for students to be as passive in tutorials as they are in most lectures. Make it worth students’ while to come to the tutorials: ensure they leave having achieved things that they otherwise would have missed.

Have a definite purpose for each tutorial. For example, link at least some tutorials with specific intended learning outcomes. Make it clear to students that there are parts of their programme which will be covered only in tutorials, and that these parts will be assessed in the same way as the lecture content of the programme.

Let students know the agenda. Whenever possible, brief students in advance concerning the topics to be processed in forthcoming tutorials. Give them something specific to prepare for each tutorial, and spend some of (but not all) the time letting them share and discuss what they have prepared. Always have something up your sleeve for students to do or discuss during tutorials for those occasions when none of the students brings questions or problems.

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Politically correct answer in math

Politically correct answer in math

Politically correct answer in math

“Do you want the correct answer or the politically correct answer?”

Helping students to make sense of the learning programme

Almost certainly, one of the first things you were given when you began to prepare to teach your first programme was documentation of one kind or another. This probably contained details of the intended learning outcomes – in other words, what students should be able to show for their learning when they have successfully completed their learning on that part of their studies. Strangely, despite the fact that such things are written to describe what is to be learned, such documents don’t always find their way into the hands of students themselves. Here are some suggestions about how you can use the documentation to help structure students’ learning activities.

Check whether students have actually received the details. Find out whether your students already have their own copies of programme documentation (for example, in a programme or departmental handbook), or whether they know where to find them online. If they do not, give them at least a copy of your own part of their programme, and if possible the rest as well.

Translate the programme information into English! (It might presently be written in ‘academese’!) When students know what they are expected to be learning, they’re in a better position to go about their task of learning it. It helps them if you translate their programme into intended learning outcomes in language that the students themselves can understand, so that they can see exactly what sort of things they are, in due course, going to be expected to achieve.

Focus on evidence of achievement. When you explain exactly what a particular learning outcome means in practice, it is helpful to describe the sort of evidence which indicates that it has been successfully achieved. This helps students to work out what the standards are.

Explain why- build in the rationale. Intended learning outcomes can be very useful for letting students know what they will be learning, but sometimes students also need some explanation regarding why they need to put energy into learning particular things. It is not always obvious to them why they have to learn things – and naturally, if they can’t see a good reason for learning something, they’re not going to invest much mental energy into trying to learn it.

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4×4 is equal to ??

4x4 is equal to ??

4x4 is equal to ??

“4X4? That’s an easy one: truck.”

Finding out what students already know

It has been said that for more than half the time that students spend in formatl teaching-learning situations, they are listening to things they already know. The following suggestions can help overcome such a situation.

Help students to value what they already know. Suggest to them that their existing knowledge is really useful and is worth holding on to, consolidating and building upon.

Find out what students can already do. Give them a numbered list of questions and ask them to privately brainstorm which questions they can already answer. Find out by a show of hands how many in the group feel they can already answer each question in turn. Helping students to see that they already know the answers to atleast some of the questions boosts their confidence, and helps you to see which questions to focus on as you continue to work with them.

Try a visual alternative. Try a ‘public’ brainstorm about what your students already know – and don’t yet know. Put a short set of questions on one or more flip charts and ask students to go round putting ticks beside the questions they can definitely answer, crosses beside those they can’t yet answer, and question marks beside those they are not sure they can answer.

Conduct a group brainstorm. Ask students in groups to list things they already know about particular topics, then allow members of different groups to explain particular things in more detail to other groups.

Try a Post-it knowledge brainstorm. For example, give each student a Post-it note and ask them to jot down on it the most important thing they already know about the sub-topic you’re going to be working with them on next. Ask them to stick the Post-its onto a flip chart or wall, and celebrate what the whole group already knows. Use this display as your agenda for what they need to find out about next.

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Math book ideas!!??

Math book ideas!!??

Math book ideas!!??

“Oh, Wow! You write math books? That’s wonderful! Where do you get your ideas?”