Posted in Algebra

Fun with ‘definitions’ in algebra

WARNING:  use the following definitions with great caution.

  • number phrase is a mathematical phrase which does not express a complete thought.
  • An arithmetic expression is any grammatically sensible expression made up of numbers and (possibly) arithmetic operations (like addition, division, taking the absolute value, etc). Notice that it only has to be grammatically sensible; an undefined expression like 5/0 is still an arithmetic expression, but something like ‘5)+/7?’ is just nonsense. You can always work out an arithmetic expression to a specific value, unless it’s undefined (in which case you can work that out).
  • An algebraic expression is any grammatically sensible expression made up of any or all of the following:

– specific numbers (called constants);
– letters (or other symbols) standing for numbers (called variables); and
– arithmetic operations.

  • By an algebraic expression in certain variables, we mean an expression that contains only those variables, and by a constant, we mean an algebraic expression that contains no variables at all.
  • polynomial is an algebraic sum, in which no variables appear in denominators or under radical signs, and all variables that do appear are raised only to positive-integer powers.
  • monomial is an algebraic expression made up only from any or all of these:

– Constants;
– Variables;
– Multiplication;
– Taking opposites (optional);
– Division by nonzero constants (optional);
– Raising to constant whole exponents (optional).

  • An algebraic expression is made up of the signs and symbols of algebra. These symbols include the Arabic numerals, literal numbers, the signs of operations, and so forth.
I got these definitions from where else, www. Of course we just want to simplify things for students but … . Anyway, just make sure that you don’t start your algebra lessons with definition of terms, be they legitimate or not legitimate.
Posted in Algebra

Teaching algebraic expressions – Counting smileys

This is an introductory lesson for teaching the concept of variable and algebraic expressions through problem solving. The problem solving task combines numerical, geometric, and algebraic thinking.  The figure below shows the standard version of the task. Of course some easier versions would ask for the 5th figure, then perhaps 10th figure, then the 100th figure, and then finally for the nth figure. This actually depends on the mathematical maturity of the students.

An alternative version which I strongly encourage that teachers should try is to simply show first the diagrams only (see below).

Study the figures from left to right. How is it growing? Can you think of systematic ways of counting the number of smileys for a particular “Y” that belongs to the group? This way it will be the students who will think of which quantity (maybe the number of smileys in the trunk of the Y or the position of the figure) they could represent with n.The students are also given chance to study the figures, what is common among them, and how they are related to one another. These are important mathematical thinking experiences. They teach the students to be analytical and to be always on the lookout for patterns and relationships. These are important mathematical habits of mind.

Here are possible ways of counting the number of smileys: The n represents the figure number or the number of smiley at the trunk.

1. Comparing the smileys at the trunk and those at the branches.

In this solution, the smileys at the branches is one less than those at the trunk. But there are two branches so to count the number of smileys, add the smileys at the trunk which is n to those at the two branches, each with (n-1) smileys. Hence, the algebraic expression representing the number of smileys at the nth figure is n+2(n-1).

2. Identifying the common feature of the Y’s.

The Y’s have a smiley at the center and has three branches with equal number of smileys. In Fig 1, there are no smiley. In Fig 2, there is one smiley at each branch. In fact in a particular figure, the number of smileys at the branches is (n-1), where n is the figure number. Hence the algebraic expression representing the number of smileys is 1+ 3(n-1).

3. Completing the Y’s.

This is one of my favorite strategy for counting and for solving problems about area. This kind of thinking of completing something into a figure that makes calculation easier and then removing what were added is applicable to many problems in mathematics. By adding one smiley at each of the branches, the number of smileys becomes equal to that at the trunk. If n represents the smileys at the trunk (it could also be the figure number) then the algebraic representation for counting the number of smilesy needed to build the Y figure with n smiley at each branches and trunk is 3n-2, 2 being the number of smileys added.

4. Who says you’re stuck with Y”s?

This is why I love mathematics. It makes you think outside the box. The task is to count smileys. It didn’t say you can not change or transform the figure. So in this solution the smileys are arrange into an array. With a rectangular array (note that two smileys were added to make a rectangle), it would be easy to count the smileys. The base is kept at 3 smileys and the height corresponds to the figure number. Hence the algebraic expression is (3xn)-2 or 3n-2.

The solutions show different visualization of the diagram, different but equivalent algebraic expressions, and all yielding the same solution. Of course there are other solutions like making a table of values but if the objective is to give meaning to algebraic symbols, operations, and processes, it’s best to use the visuals.

A more challenging activity involved Counting Hexagons. Click the link if you want to try it with your class.

Posted in Algebra

Teach for conceptual and practical understanding

Whit Ford left this comment on my post Curriculum Change and Understanding by Design: What are they solving? He makes a lot of sense. I just have to share it.

I believe the method of planning lessons is less important than WHAT you are asking the students to think about. Most Algebra I and II texts I have come across suffer severely from “elementitis” (see “Making Learning Whole” by David Perkins), which makes it very challenging for teacher to convey “the whole game” to students while still following the text. For example…

A teacher who is talking about how to “collect like terms” is not going to motivate the students as much as one who succeeds in relating this to a more interesting and complex problem which is related to student’s daily lives in some way. This is a HUGE challenge when teaching mathematical abstractions, one I am struggling with as I prepare to teach the first semester of Algebra I using a traditional text. However, it does lead to some interesting potential exercises:

– Ask students to give you examples of two objects in their lives (or in the room). Chances are you will get answers like two apples, two desks, two eyes, etc. Note these on the board as students mention them, then ask… so do you ever come across “two” all by itself? The answer is NO – “two” is an abstract concept, one which we apply constantly in our daily lives, but an abstraction nevertheless.

– So how do we come up with “two” of something? By finding them, collecting them, putting them together, etc. The abstraction of this process is what we have called “addition”. But what kinds of things can you add together and have it make sense? A foot and another foot – certainly. An apple and a pear – only if you recast each as “a fruit” – then you have “two pieces of fruit”. A meter and three centimeters – only if you recast each in the same units – then you have “103 centimeters”. So what is to be learned from this process? We can only add “like” things together, or quantities that are measured in the same units, if the answer is to make any sense. Addition certainly lets us add the quantities of one apple and one pear… 1+1=2, but 2 of what? The answer must make sense in the real world, and the abstract process of adding abstract quantities does not always result in a useful answer.

– So what about 2x+3y? We have two of “x” and 3 of “y”. Can we simplify this abstract expression? Until we know what “x” and “y” represent, until we have been given values for each of them (with units), we don’t even know if adding them together will produce an answer that makes any sense (see apples and pears above). Furthermore, since we have differing quantities of each, we will have to postpone combining them until we know values for each variable (since one value must be doubled, while the other must be tripled). On the other hand, if the problem were 2x+3x, we are being asked to assemble two and three of the same quantity “x”… intuitively, this MUST be 5 of the same quantity “x” – no matter what quantity and units “x” represents, since the units of both terms will always be the same.

I am hoping that such approach (extended considerably with more examples and practice) will begin to build both a conceptual and a practical understanding of the mathematical abstraction “like terms”, along with how to combine them when they occur… yet, this is just ONE of the many topics covered at a very procedural level by most Algebra I texts. Our challenge is to get students to understand the forest, when the textbook spends most of its time talking about trees.

His post Learning the Game of Learning is a good read, too.

You may also want to check-out my post on combining algebraic expressions . It links conceptual and procedural understanding and engages students in problem posing and problem solving tasks.

Posted in Lesson Study, Number Sense

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 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.