Posted in Lesson Study, Number Sense

Patterns in the tables of integers

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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 for teaching positive and negative integers. If you are wondering why most of my posts are about integers it’s because I’m doing a Lesson Study with a group of Mathematics I (Year 7) teachers about this topic. Last week we concluded the first cycle of our research lessons on teaching subtraction of integers.

The first task students need to do with the tables is to list 3-5 observations. From there you can start asking the ‘whys’ for each observation. Sample questions are provided for each table below:

1. Adding integers

Sample questions for discussion:

a) Under what conditions will the sum be positive? negative? zero?

b) Why are there the same numbers in a diagonal?

c) How come that the sum is increasing from left to right, from bottom to top?

2. Taking away integers

Sample questions for discussion:

a) Under what conditions will the difference be positive? negative? zero?

b) Why are there the same number in a diagonal?

c) How come that for each row/column, the difference is decreasing?

3. Multiplying integers

Sample question for discussion:

a) Under what conditions will the sum be positive? negative? zero?

4. Dividing Integers

Sample question for discussion:

a) Under what conditions will the difference be positive? negative? zero?

       b) Does dividing integers still results to an integer? What do we call these new numbers?

Feel free to share your ideas/questions for discussion.

You may also want to share other  math concepts that students can learn with these tables.

Posted in Misconceptions, Number Sense

Technically, Fractions are Not Numbers

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It is misleading to put fractions alongside the sets of numbers – counting, whole, integers, rational, irrational and real. The diagram below which are in many Mathematics I (Year 7) textbooks is inviting misconceptions.

WRONG WAY

Fraction is a form for writing numbers just like the decimals, percents, and other notations that use exponents and radicals, etc.

The fraction form of numbers is used to describe quantities that is 1) part of a whole, 2) part of a set, 3) ratio, and 4) as an indicated operation. Yes, it can also represent all the rational numbers but it doesn’t make fractions another kind of number or as another way of describing the rational numbers. Decimals can represent both the rational and the irrational numbers (approximately) but it is not a separate set of numbers or used as another way of describing the real numbers! Note that I’m using the word number not in everyday sense but in mathematical sense. In Year 7, where learners are slowly introduced to the rigor of mathematics and to the real number system, I suggest you start calling the numbers in its proper name.

I prefer the Venn diagram to show the relationships among the different kinds of numbers like the one shown below:

The Real Number SYSTEM
The Real Numbers

The diagram shows that the set of real numbers is composed of the rational and the irrational numbers. The integers are part of the set of rational numbers just like the counting numbers are members of the set of whole numbers and the whole numbers are members of the set of integers. The properties of each of these set of numbers can be investigated. We do not investigate if fraction is closed or is commutative under a certain operation for example, but we do it for the rational numbers.

You may want to know why we invert the divisor when dividing fractions. Click the link.

Posted in Assessment, Mathematics education

Conference on Assessing Science and Mathematics Learning

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http://www.upd.edu.ph/~ismed/icsme2010/index.html

2nd International Conference in Science and Mathematics Education via kwout

The University of the Philippines National Institute for Science and Mathematics Education Development (UP NISMED) will hold its second international conference in science and mathematics education on October 26-28, 2010.

The conference will feature plenary sessions, symposia, paper presentations, poster presentations, and workshops.

Click link for details or email me: erlines@ymail.com

Posted in Algebra, High school mathematics

Teaching the concept of function

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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. Function is the central underlying concept in calculus. It is also one of the key concepts of mathematics that can model many quantitative relationships.

Textbooks and teachers usually introduce function via a situation with the related quantities already identified. What is required of the  students is to learn how to set up and represent the relationships in tables, graphs, and equation and analyze the properties. In the real world, when function is used as a model, the first thing that needs to be done is to identify the varying quantities. So, it is important to let students identify the quantities and let them determine which of these quantities may be related. This way they get a sense of what function really is and what it is for. The function is not the graph, not the table of values, and not the equation. The function is the relationship between the variables represented by these. The study of function is the study of these relationships and their properties, not finding y  or f(x) given x and vice versa, not reading graphs,  and not translations among the representations. These are important knowledge and skills, yes, but only in the context for investigating or learning more about the relationships between the quantities, that is, the function. Thus, for an introductory lesson for function, I find it useful to use a situation where students themselves will:

  1. identify the changing and unchanging quantities;
  2. determine the effect of the change of one quantity over the others;
  3. describe the properties of the relationship; and,
  4. think of ways for describing and representing these relationships.

These are the ‘big ideas’ students should learn about function. Of course, there are others like looking or dealing with function as a mathematical object and not only as a process or procedure for generating or predicting values. However, for an introductory lesson on function, teachers need not focus on this yet.

Sample introductory activity:

What are the quantifiable attributes or quantities can you see in the figure below? Which of these quantities will change and remain unchanged if GC is increased or decreased? Click the figure and move point C. Are there ways of predicting the values of these changing quantities?

teaching function
Identifying related quantities

Click here or the image above to go to dynamic window for the worksheet.

I like this particular activity because it gives students the opportunity to link geometry/measurement concepts to algebra and learn mathematics through solving problems.

Click link for a synthesis of the evolution of the definition of function and What are the big ideas in function?