Tag: algebraic expressions

Posted in Algebra, High school mathematics

PCK Map for Algebraic Expressions

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When I design instruction or plan a lesson I always start with making a map of everything I know about the subject. The map below is an example of a map I made for algebraic expressions. I won’t call it a conceptual map because it’s only the left part of it (the ones in black text) which deals with the concept of algebraic expressions. Those at the right (in red texts) describe what I know about the requisites of good teaching of algebraic expressions including my knowledge about students’ misconceptions and difficulties in this topic. Maybe, I should just call this kind of map, PCK Map, for pedagogical content knowledge map.

Pedagogical Content Knowledge (PCK) Map for Algebraic Expressions

I find doing the PCK Map a useful exercise because it helps me link concepts, synthesize my teaching knowledge about the topic, not leave out important ideas in the course of the teaching and of course in planning the details of the lesson especially in the selection of activity/tasks and in framing questions for discussions.  I also find it useful in evaluating my teaching of the unit.

There are two ways a PCK Map can be enriched: (1) use Google (alright, go to the library and see what experts think are important to cover in the topic, they’re also outlined in the Standard) and (2), after each lesson or at the end of the unit, write your new knowledge about the topic especially students misconceptions and difficulties and how it can be addressed next time.

Click this link to see a the lesson plan I made based on the PCK Map. The lesson is about teaching combining algebraic expressions via a mathematical investigation activity.

Posted in Elementary School Math, Geometry, Math investigations

Math investigation lesson on polygons and algebraic expressions

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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.

To understand mathematics is to make connections among concepts, procedures, contexts. A lesson that has a very good potential for learning a well-connected mathematical knowledge is one which is organized around a mathematical investigation. This is because of the divergence nature of this task which revolves around a single tool or context.

Here is a simple investigation activity about polygons. Change the x by x unit to 1 by 1 unit if you will give this to Grade 5-6 students.

Investigate polygons with area 5x^2 units on an x by x unit grid.

Some initial shapes students could come up with may look like the following:

different shapes, the same area
Figure 1. Polygons with the same area

Note: This is a mathematical investigation so the students are expected to pose the problems they want to pursue and on how they will solve it. It will cease to be a math investigation if the teachers will be the one to pose the problems for them. The following are sample problems that students can pose for themselves.

  • What is the same and what is different among these polygons? How can I classify these polygons?

Possible classifications would be

a. convex vs non-convex polygons

b. according to the number of sides

  • What shapes and how many are there if I only consider polygons made up of squares?

Students will discover that while they can have as many polygons with an area of 5, there are only 12 polygons made of  squares.  This is shown in Figure 2. These shapes are called pentominoes because it is made up of 5 squares. I have arranged it here for easy recall of shapes. It contains the last seven letters of the english alphabet (TUVWKXZ) and the word FILIPINO without the last 2 I’s and O in the spelling.

Figure 2. Pentominoes
  • Is there a way of constructing different triangles or any of the polygons with same area?

Figure 3 shows this process for triangle.

Figure 3.Triangle with same area

Click this or the  figure below to see this process in dynamic mode using Geogebra.

Fig 4 – Preserving area of triangle in Geogebra

Possible extension of this investigation is to consider polygons with areas other than 5x^2.

Click this link to see some ideas on how you can use this activity to teach combining algebraic expressions.

Posted in Elementary School Math, Number Sense

Subtracting integers using numberline – why it doesn’t help the learning

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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. In this post, let’s focus on the first culprit – subtracting integers.

One of the most popular tools for teaching addition and subtraction of integers is the number line. Does it really help the students? If so, why do they always look like they’ve seen a ghost when they see -5 – (-3)?

Teachers introduce the following interpretations to show how to subtract integers in the number line: The first number in the expression tells you the initial position,  the second number tells the number of ‘jumps’ you need to make in the number line and, the minus sign tells the direction of the jump which is to the left of the first number. For example to subtract 3 from 2, (in symbol, 2 – 3), you will end at -1 after jumping 3 units to the left of 2.

taking away a positive integer

The problem arises when you will take away a negative number, e.g., 2- (-3). For the process to work, the negative sign is to be interpreted as “do the opposite” and this means jump to the right instead of to the left, by 3 units. This process is also symbolized by 2 + 3. This makes 2 – (-3) and 2 + 3 equivalent representations of the same number and are therefore equivalent processes.

But only very few students could making sense of the number line method that is why teachers still eventually end up just telling the students the rule for subtracting integers. Here’s why I think the number line doesn’t work:

taking away a negative integer
our mind can only take so much at a time

The first problem has to do with overload of information to the working memory (click the link for a brief explanation of cognitive load theory). There are simply too many information to remember:

1. the interpretation of the operation sign (to the left for minus, to the right for plus);

2. the meaning of the numbers (the number your are subtracting as jumps, the number from which you are starting the jumps from as initial position);

3. the meaning of the negative sign as do the opposite of subtraction which is addition.

To simply memorize the rule would be a lot easier than remembering all the three rules above. That is why most teachers I know breeze through presenting the subtraction process using number line (to lessen their guilt of not trying to explain) and then eventually gives the rule followed by tons of exercises! A perfect recipe for rote learning.

The second problem has to do with the meaning attached to the symbols. They are not mathematical (#1 and #2). They are isolated pieces of information which could not be linked to other mathematical concepts, tools, or procedures and hence cannot contribute to students’ building schema for working with mathematics.

But don’t get me wrong, though. The number line is a great way for representing integers but not for teaching operations.

Click link for an easier and more conceptual way of teaching how to subtract integers without using the rules.