Tag: math lessons

Posted in Elementary School Math, Number Sense

Teaching negative numbers via the numberline with a twist

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One popular way of introducing the negative numbers is through the number line. Most textbooks start with the whole number on the number line and then show to the students that the number is decreasing by 1. From there, the negative numbers are introduced. This seems to be something easy for students to understand but I found out that even if students already know about the existence of negative numbers having used them to represent situations like 3 degrees below zero as -3, they would not think of -1 as the next number at the left of zero when it is presented in the number line. They would suggest another negative number and some will even suggest the number 1, then 2, then so on, thinking that maybe the numbers are mirror images.

Here is an alternative activity that I found effective in introducing the number line and the existence of negative numbers.  The purpose of the activity is to introduce the number line, provide students another context where negative numbers can be produced (the first is in the activity on Sorting Situations and the second is in the task Sorting Number Expressions), and get them to reason and make connections. The task looks simple but for students who have not been taught integers or the number line the task was a problem solving activity.

Question: Arrange from lowest to highest value

When I asked the class to show their answers on the board, two arrangements were presented. Half of the class presented the first solution and the other half of the students, the second solution. Continue reading “Teaching negative numbers via the numberline with a twist” →

Posted in Curriculum Reform

Understanding by Design, one more go

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I have so far written three posts about understanding by design. The first is about my  issues about DepEd’s adoption of understanding by design (UbD), the second is about the information posted about UbD Philippines in WikiPilipinas and the third is about curriculum change and UbD. These posts are very popular especially for readers from the Philippines. This is understandable as our Department of Education wants teachers to implement UbD this June 2010, barely two months from now. I don’t know if there’s a training out there about UbD for our public school teachers. Maybe they will have one, a week before the school year starts this June.

Is this backward or forward design?

Anyway, I am writing this post because some readers land on this blog searching for things like “how to teach algebra using UbD”, “teaching integers the UbD way”, etc. I don’t know if they are just looking for lesson plans using UbD which they will never find in this blog or there’s a misconception out there that UbD is a way of teaching. It is not. It is more a way of planning your lesson rather than how to teach your lesson. In fact the only difference that I see between UbD and the current way of planning the lesson is in the format, not in the way you will actually teach the lesson. UbD says theirs uses backward design. In this model you start with thinking on how you will assess understanding before selecting and organizing your learning activities.  For lack of term, let’s call the traditional method forward design. In this model you think about how you will assess understanding after selecting and organizing your learning activities. In both models of course you start with your learning goals. In UbD it’s called enduring understanding, in the traditional one it is called objectives.

I attended an international conference on science and mathematics teaching a few months ago. One of the parallel session presenter reported her research which compares the use of UbD way of planning the lesson and their so called usual way of planning the lesson for science. She said the class taught using UbD performed better than the one taught using the traditional one. So I asked why is that? She said that it’s because the class taught using UbD used inquiry-based teaching and the class taught using the traditional lesson plan format was taught by lecture method. So I asked further: In your country’s traditional way of planning the lesson, is it not possible to organize the lesson using inquiry-based teaching and teach it that way. She said, “of course we can, and we do. It depends upon the teacher”. There you go. Backward or forward design,  it’s still the teaching and not the format nor the way the lesson plan is prepared that spells the difference in learning.

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.