Tag: polygons

Posted in Math blogs

Top 20 Math Posts and Pages in 2012

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The thinker

I blog in order to organise what I think. And I don’t think I’m succeeding judging from the range of topics that I have so far written since I started Math for Teaching blog in 2010. Here’s the twenty most popular math posts and pages in this blog for the year 2012. It’s a mix of curricular issues, lessons, and teaching tips.

  1. What is mathematical investigation? – Mathematical investigation refers to the sustained exploration of a mathematical situation. It distinguishes itself from problem solving because it is open-ended….
  2. Exercises, Problems, and Math Investigations – The quality of mathematics students learn depends on the mathematical tasks or activities we let our students engage in….
  3. What is mathematical literacy? – Mathematical literacy involves more than executing mathematical procedures and possessions of basic knowledge that would allow a citizen to get by. Mathematical literacy is mathematical knowledge, methods,…
  4. My issues with Understanding by Design (UbD) – Everybody is jumping into this new education bandwagon like it is something that is new indeed. Here are some issues I want to raise about UbD…
  5. Curriculum change and Understanding by Design, what are they solving? – Not many teachers make an issue about curriculum framework or standards in this part of the globe. The only time I remember teachers raised an issue about it was in 1989, when the mathematics curriculum moved …
  6. Math investigation lesson on polygons and algebraic expressions – Understanding is about making connection. The extent to which a concept is understood is a function of the strength of its connection with other concepts. An isolated piece of knowledge is not powerful…
  7. Mathematics is an art – Whether we are conscious of it or not, the way we teach mathematics is very much influenced by what we conceive mathematics is and what is important knowing about it…
  8. Mathematical habits of mind – Learning mathematics is not just about knowing, understanding, and applying its concepts, principles and all the associated mathematical procedures and algorithms. It’s not just even about  acquiring the capacity to solve problem,  to reason, and to communicate…
  9. Subtracting integers using numberline – why it doesn’t help the learning – I have reasons to suspect that for a good percentage of students, the end of their mathematics career begin when they are introduced to subtracting integers. Well, for some, it’s when the x‘s start dropping from the sky without warning…
  10. Teaching positive and negative numbers – Here’s my proposed activity for teaching positive and negative numbers that engages students in higher-level thinking…
  11. Trigonometry – why study triangles – What is so special about triangles? Why did mathematicians created a branch of mathematics devoted to the study of it? Why not quadrinometry? Quadrilaterals, by its variety are far more interesting….
  12. Teaching the concept of function – Mathematics is not just about the study of numbers and shapes but also about the study of patterns and relationships. Function, which can define some of these relationships, is an indispensable tool in its study…
  13. Algebraic thinking and subtracting integers – Part 2 – Algebraic thinking is about recognizing, analyzing, and developing generalizations about patterns in numbers, number operations, and relationships among quantities and their representations.  It doesn’t have to involve working with the x‘s and other stuff of algebra….
  14. Patterns in the tables of integers – Mathematics is said to be the science of patterns. Activities that involve pattern searching is a great way to engage students in mathematical thinking. Here are some of my favorites …
  15. Making generalizations in mathematics – Making generalizations is fundamental to mathematics. Developing the skill of making generalizations and making it part of the students’ mental disposition or  habits of mind …
  16. Teaching with GeoGebra: Squares and Square Roots – This post outlines a teaching sequence for introducing the concept of square roots in a GeoGebra environment. Of course you can do the same activity using grid papers, ruler and calculator….
  17. Algebra vs Arithmetic Thinking – One of the solutions to help students understand algebra in high school is to start the study of algebra earlier hence the elementary school curriculum incorporated some content topics traditionally studied in high school. However,…
  18. Teaching with GeoGebra – Educational technology like GeoGebra can only facilitate understanding if the students themselves use it. This page contains a list of my posts …
  19. Teaching combining algebraic expressions with conceptual understanding – In this post, I will share some ideas on how the simple investigation of drawing polygons with the same area can be used as an introductory lesson to teach operations with algebraic expressions with meaning and understanding.
  20. Mistakes and Misconceptions in Mathematics – Misconceptions are very different from the mistakes students make. Mistakes are not consciously made. Misconceptions are. Mistakes are usually one-off, while misconceptions, the gods forbid, could be for keeps….
Posted in Geometry

Regular Polygons Problems

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One  of my favourite lesson design is a sequence of problem solving tasks that requires repetition of same reasoning and analysis by varying the ‘mathematical context’ of the problem in increasing complexity. However the variation in the context of the problem should be such that they still share some properties. In the examples below, the number of sides of the polygons is varying but they are all regular polygons.It is also important that the problems can be solved/ explained in different ways – algebraically, geometrically, arithmetically or a combination of these.

Here is a sample sequence of problems. This lesson is good from Grade 5 up. If you are handling different grade levels and they all reason in the same way as your fifth graders reason, you have a big problem.

Problem 1

The segments in the figure below form equilateral triangles with the dotted line segment. Compare the total lengths of the red segments to the total lengths of the blue segments. You must be able to explain how you arrive at your conclusion or give a justification to it.

equal perimeter

Problem 2

What if the segments form squares instead of equilateral triangles with the dotted line segment? Compare the total lengths of the red segments to the blue segments. Which is longer?

perimeter problem

Problem 3

What if it the line segments form regular pentagons instead of squares? Do you think your conclusion will hold for any regular polygon? Prove.

Problem 4

What if instead of regular polygons, you have a semicircle? Click link to see the problem and solution.

Encourage students to use algebra and geometric constructions to justify their answers. This lesson is not about getting the correct conclusion. That’s the easy part. It is about explaining/ proving it.

You may want to view another similar lesson on quadrilaterals.

 

Posted in GeoGebra worksheets, Geometry

How to scaffold problem solving in geometry

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Scaffolding is a metaphor for describing a type of facilitating a teacher does to support students’ own making sense of things. It is usually in the form of questions or additional information. In scaffolding learning, we should be careful not to reduce the learning by rote. In the case of problem solving for example, the scaffolds provided should not reduce the problem solving activity into one where students follow procedures disguised as scaffolds. So how much scaffolding should we provide? Where do we stop? Let us consider this problem:

ABCD is a square. E is the midpoint of CD. AE intersects the diagonal BD at F.

  1. List down the polygons formed by segments BD and AE in the square.
  2. How many percent of the area of square ABCD is the area of each of the polygons formed?

Students will have no problem with #1. In #2, I’m sure majority if not all will be able to compare the area of triangles ABD, BCD, AED and quadrilateral ABCE to the area of the square. The tough portion is the area of the other polygons – ABF, AFD, FED, and BCEF.

In a problem solving lesson, it is important to allow the learners to do as much as they can on their own first, and then to intervene and provide assistance only when it is needed. In problems involving geometry, the students difficulty is in visualizing the relationships among shapes. So the scaffolding should be in helping students to visualize the shapes (I actually included #1 as initial help already) but we should never tell the students the relationships among the geometric figures. I created a GeoGebra worksheet to show the possible scaffolding that can be provided so students can answer question #2. Click here to to take you to the GeoGebra worksheet.

 

Extension of the problem: What if E is 1/4 of its way from C to D? How many percent of the square will be the area of the three triangles and the quadrilateral? How about 1/3? 2/3? Can it be generalized?

Please share with other teachers. I will appreciate feedback so I can improve the activity. Thank you.

More Geometry Problems:

  1. The Humongous Book of Geometry Problems: Translated for People Who Don’t Speak Math
  2. Challenging Problems in Geometry

 

Posted in Geometry

Twelve definitions of a square

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How does mathematics define a math concept?

Definitions of concepts in mathematics are different from definitions of concepts in other discipline or subject area. A definition of a concept in mathematics give a list of properties of that concept. A mathematics object will only be an example of that concept if it fits ALL those requirements, not just most of them. Further, a definition is also stated in a way that the concept being defined belongs to an already ‘well-defined’ concept. On top of this, economy of words and symbols and properties are highly observed.

Does a math concept only have one definition? Of course, not. A concept can be defined in different ways, depending on your knowledge about other math objects. In a study by Zaskin and Leikin, they suggested that the definitions students give about a concept mirrors their knowledge of mathematics. Below are examples of definitions of squares from that research. Do you think they are all legitimate definitions?

What is a square?

A square is

  1. a regular polygon with four sides
  2. a quadrilateral with all the angles and all the sides are equal
  3. a quadrilateral with all the sides equal and an angle of 90 degrees
  4. a rectangle with equal sides
  5. a rectangle with perpendicular diagonals
  6. a rhombus with equal angles
  7. a rhombus with equal diagonals
  8. a parallelogram with equal adjacent angles and equal adjacent sides
  9. a parallelogram with equal and perpendicular diagonals
  10. a quadrilateral having 4 symmetry axes
  11. a quadrilateral symmetric under rotation by 90 degrees
  12. the locus of all the points in a plane for which the sum of the distances from two given perpendicular lines is constant. Click this link to visualize #12.

 

Making (not stating) definitions is a worthwhile assessment task.

Here’s three great references for definitions of mathematical concepts. The first is from no other than Dr. Math (The Math Forum Drexel University). The middle one’s for mom and kids – G is for Google and the third’s a book of definitions for scientists and engineers.

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