Posted in Number Sense

Teaching absolute value of an integer

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integers and algebraic reasoning

The tasks below are for deepening students’ understanding about the absolute value of a number and provide a context for creating a need for learning operations with integers. You may give the problems  after you have introduced the students the idea of absolute value of an integer and before the lesson on operations with integers.

Tasks (Set 1)

1) Find pairs of integers whose absolute values add up to 12.
2) Find pairs of integers whose absolute values differ by 12.
3) Find pairs of integers whose absolute values gives a product of 12.
4) Find pairs of integers whose absolute values gives a quotient of 12.

These may look like simple problems to you but note that these questions involve equations with more than one pair of solutions. The problems are similar to solving algebraic equations involving absolute values. Problem 1) for example is the same as “Find the solutions to the equation /x/ + /y/ = 12. Of course we wouldn’t want to burden our pupils with x’s and y’s at this point so the we don’t give them the equation yet but we can already engage them in algebraic thinking while doing the problems. The aim is to make the pupils  be comfortable and confident with the concept of absolute values as they would be using it to derive and articulate the algorithm for operations with integers later in the next few lessons.

Tasks (Set 2)
1) Find pairs of integers, the sum of absolute values of which is less than 12.
2) Find pairs of integers the difference of absolute values of which is greater than 12.

Encourage students to show their answers in the number line for both sets of task.
Posted in Elementary School Math

What is an integer?

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Here are some ideas pupils need to learn about integers:

•A number represents a quantity. An integer is a type of number. An integer represents a quantity.

•Integers are useful in representing quantities and includes opposite sense. For example, going up 5 floors and going down 5 floors can be represented by +5 and -5 respectively. The sign ‘+’ represents up and ‘-’ represents  down. The ‘5’ represents the number of floors.

•The integer +5 is read as “positive five” and NOT “plus five”. The integer -5 is read as “negative five” and NOT “minus 5”.

•The words positive and negative are descriptions of the whole number 5 while the words plus and minus describe operation to be done with the numbers. That’s why it doesn’t make sense to read the integer -5 as “minus 5”. From what number are you subtracting it?

•The number 0 is an integer which is neither positive nor negative.

•Integers can be represented in a number line. An integer and its opposite are of the same distance from 0. For example, -4 is 4 units to the left of zero so its opposite must be 4 units to the right of 0. This integer is +4.

integers

Problem: The distance between two integers in the number line is 4 units. If one of the integer is 3 units from zero, what could be these two integers?

•The distance of an integer from zero is called the absolute value of the integer. So the absolute value of -4 is 4 and the absolute value of +4 is also 4.In symbol, /+4/ = 4 and /-4/ = 4.

Of course, merely explaining to students these ideas and giving them lots of exercises will not work. It will never work for many of them. Teachers have to design tasks or activities pupils can work on so that students can construct their own understanding of these ideas. Teachers can help scaffold their learning through problem solving tasks and through the questions and feedback they will provide the students.

Next post on this topic will be about absolute value and operations with integers.

Posted in Algebra

What is algebraic thinking?

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In my post Arithmetic and Algebra, I wrote that it’s how you solve a problem that tells whether you are doing algebra or arithmetic, not the problem itself. Here’s a description of algebraic thinking that I think teachers in elementary school mathematics might find useful especially when they are teaching about numbers and number operations:

Algebraic thinking is about generalising arithmetic operations and operating on unknown quantities. It involves recognising and analysing patterns and developing generalisations about these patterns.(NZCER)

I find the description clear, concise, can easily be committed to memory and can form part of teachers everyday discussion with a little effort. The keywords that should be remembered are patterns and generalizations. You can’t actually separate one word from the other. If you see a pattern, you can’t but make some generalizations. It’s human nature. If you make generalizations it must from the patterns that you recognize.

Patterns about what?

In the description of algebraic thinking above it says patterns about arithmetic operations (add, subtract, etc) or relationships between numbers for example in equations. Of course it could also involve patterns in shapes, colors, positions in sequences. In short, you also use algebraic thinking in geometry.

Here’s my other favorite description of algebraic thinking:

Algebraic thinking involves the construction and representation of patterns and regularities, deliberate generalization, and most important, active exploration and conjecture. (Kaput, NCTM, 1993).

It is similar to the first but added representing patterns and regularities observed and active exploration as important processes. Without these, generating cases needed for making conjectures/generalizations and verifying them would be difficult.

Click algebraic thinking to see the collection of articles and lessons in this blog about this topic. You may also want to check on these book for other lessons..

Posted in Algebra, Assessment, High school mathematics

Levels of understanding of function in equation form

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There are at least three representational systems used to study function: graphs, tables and equations. But unlike graphs and tables that are used to visually show the relationships between two varying quantities, students first experience with equation is not as a representation of function but a statement which state the condition on a single unknown quantity.  Also, the equal sign in the early grades is taught to mean “do the operation” and not to denote equality between the quantities of both sides of the sign. To complicate the matter, when equation is used as a representation of function, it takes an additional meaning, that of a representation of two varying quantities! For example, let y be number of t-shirts and x be the number of t-shirts to be printed. If the cost of printing a t-shirt is 2.50 then the function that defines y in terms of x is y = 2.50x. Understanding function equation form is not an easy concept for many students.

I believe that if mathematics teachers are aware of the differing level of abstraction in students’ thinking and reasoning  when they work with function in equation form then the teachers would be better equipped to design appropriate instruction to lead students towards a deeper understanding of this concept.Failure to do so would deprive students the opportunity to understand other advanced algebra and calculus topics.I would like to share a framework for assessing students’ developing understanding of function in equation form. This framework is research-based. You can download the full paper here or you can view the slides in my post Learning Research Study Module for Understanding Function.

The framework is in terms of levels of understanding. You can use it to design tasks or assess your students understanding of function in equation form. Each of these levels are “big ideas” or schemas in the understanding of function.
Level 1 – Equations are procedures for generating values.
Students at this level can find x given y or vice versa. Some can generate a number of pairs but not really see the equation as a rule for all pairs of values in a situation.
Level 2 – Equations are representations of relationships.
Students at this level understands domain and range, can generate pairs of values and graph it. They also know that that relationship is unique and true for the values in the domain and range.
Level 3 – Equations describe properties of relationships.
Students at this level can interpret the properties of the function like rate of change and intercepts form the equation but can do it by generating values.
Level 4 – Functions are objects that can be manipulated and transformed
This is the highest level. At this point students see the equation as a math object. They can do composition of function, can find its inverse by algebraic manipulation and can interpret the meaning or effect of the parameters to the graph of the function. 

You can reference the above framework:

Ronda, E. (2009). Growth points in students’ developing understanding of function in equation form. Mathematics Education Research Journal, 21, 31-53.