Category: Number Sense

Posted in Elementary School Math, Number Sense

Are negative numbers less than zero?

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I found this interesting article about negative numbers. It’s a quote from the paper  titled The  extension of the natural number domain to the integers in the transition from arithmetic to algebra by Aurora Gallardo. The quote was transcribed from the article Negative by D’Alembert (1717-1738) for Diderot Encyclopedia.

In order to be able to determine the whole notion, we must see, first, that those so called negative quantities, and mistakenly assumed as below Zero, are quite often represented by true quantities, as in Geometry where the negative lines are no different from the positive ones, if not by their position relative to some other line or common point. See CURVE. Therefore, we may readily infer that the negative quantities found in calculation are, indeed, true quantities, but they are true in a different sense than previously assumed. For instance, assume we are trying to determine the value of a number x which, added to 100, gives 50, Algebra tells us that: x + 100 = 50, and that: x = –50, showing that the quantity x is equal to 50, and that instead of being added to 100, it must be subtracted from that number. Consequently, the problem should have been stated in the following way: Find the quantity x which, subtracted from 100, gives 50. Thus, we would have: 100 – x = 50, and x = 50. The negative form for x would then no longer exist. Thus, the negative quantities really show the calculation of positive quantities assumed in a wrong position. The minus sign found in front of a quantity is meant to rectify and correct a mistake in the hypothesis, as clearly shown by the above example. (quoted in Glaeser, 1981, 323–324)

Interesting, isn’t it? Numbers are abstract ideas. They get their meanings from the context we apply them to. Of course from the school mathematics point of view we cannot start with this idea.

Here are the different meanings of the negative number that students should know before they leave sixth grade: 1) it is the result of subtraction when a bigger number is taken away from a smaller number; 2) it is the opposite of a counting/ natural number; 3) that when added to its opposite counting number results to zero; and 4) it represents the position of a point to the left of zero.

Likewise for the minus sign which indicates subtraction. Subtraction has three meanings: take away, find the difference, and inverse operation of addition. For further explanation read the post What exactly are we doing when we subtract?

Posted in Elementary School Math, Number Sense

How to scaffold algebraic thinking in teaching integers

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One way to make algebra make sense to students is to show where those mathematics objects (e.g. algebraic expressions/equations/formula) come from. It will even makes sense more to students if they themselves can generate those objects. As John Mason puts it,

The mechanics of algebra (algebraic manipulation) are concerned with studying the effects of combining, undoing and otherwise relating expressions. These make little or no sense unless learners have themselves gained facility in generating expressions so that they know how they arise.  – from Actions and Objects by John Mason.

Generating expressions helps develop algebraic thinking. There are many ways of of embedding this in your teaching. For example with equations, you can ask the students to find as many equations given a solution. Read the post on how to teach the properties of equality on how this can be done. My post about teaching algebraic expressions also shows an example of a task that generates several equivalent algebraic expressions from the same problem situation. Generating formulas by deriving it from other expressions can also be a good activity.  Examples of these is deriving the formula of the area of the triangle from parallelogram/ rectangle and then from these deriving the area of trapezoids. These type of activities help develop students algebraic thinking skills.

Now, how can we do this generating expressions in earlier grades? Let me describe a lesson I taught to a class of year 6 students. This lesson is a continuation of the lesson on teaching integers via the number line with a twist. In that lesson, instead of asking student to arrange numbers, I asked them to arrange number expressions. From there we were able to extend the numbers they know (whole numbers) to now include the negative numbers. The main aim of that lesson is to extend the students’ concept image of negative number from a number that can be used to represent situation (see post on a problem solving approach for introducing integers) to a number that results when you take away a bigger number from a smaller number.

In this lesson with negative numbers, zero, and positive numbers on the number line, I can now proceed to defining integers or perhaps compare integers. But what will the students learn from that except that they are called integers? If I ask them to compare the numbers what good is that at this point? Where will they use that knowledge? So the task that I gave  them in this lesson was to make as many number expressions whose answer corresponds to the numbers in the number line. Note that this task is an open-ended problem solving task. Below is a sample student solution.

When I asked the class to share their answers I was surprised that they did not restrict themselves to addition and subtraction operation.

With the data shown on the board (pardon my handwriting) I asked them to make some observations and generalizations. They gave the following:

1. It is easy to make number expressions when the answer should be positive.

2. You always get a negative if you subtract a bigger number from a smaller number.

3. You always get a zero if you subtract equal numbers. (If you think this knowledge easily transfer to negatives, you’re wrong. I did try my luck when I asked them “is it also true to -7 – (-7)?”. I got blank stares. Clearly the expression it is still beyond them.

If these are not powerful mathematics to you especially #2 and well, #3, I don’t know what they are. Anyway, the point of my story here is that it is good practice to ask students to generate expressions. It is like asking them to think of a problem given the solution, a highly recommended  mathematics teaching practice.

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 Number Sense

Introducing negative numbers

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One of the ways to help students to make connections among concepts is to give them problem solving tasks that have many correct solutions or answers. Another way is to make sure that the solutions to the problems involve many previously learned concepts. This is what makes a piece of knowledge powerful. Most important of all, the tasks must give the groundwork for future and more complex concepts and problems the students will be learning. These kinds of task need not be difficult. And may I add before I give an example that equally important to the kind of learning tasks are the ways the teacher  facilitates or processes various students’ solutions during the discussion.

I would like to share the problem solving task I made to get the students have a feel of the existence negative numbers.

We tried these tasks to a public school class of 50 Grade 6 pupils of average ability and it was perfect in the sense that I achieved my goals and the pupils enjoyed the lessons. This lesson was given after  the lesson on representing situations with numbers using the sorting task which I describe in my post on introducing positive and negative numbers.

Sorting is a simple skill when you already know the basis for sorting which is not case in the task presented here.

Just like all the tasks I share in this blog, it can have many correct answer. The aim of the task is to make the students notice similarities and differences and describe them, analyse the relationship among the numbers involved, be conscious of the structure of the number expressions, and to get them to think about the number expression as an entity or an object in itself and not as a process, that is speaking of 5+3 as a sum and not the process of three added to five. The last two are very important in algebra. Many students in algebra have difficulty applying what they learned in another algebraic expression or equation for failing to recognize similarity in structure.

Here are some of the ways the pupils sorted the numbers:

1. According to operation: + and –

2. According to the number of digits: expressions involving one digit only vs those involving more than 2 digits

3. According to  how the first number compared with the second number: first number > second number vs first number < second number.

4. According to whether or not the operation can be performed: “can be” vs “cannot be”.

5. This did not come out but the pupils can also group them according to whether the first/second term is odd or not, prime or not. It is not that difficult to get the students to group them according to this criteria.

Solution #4 is the key to the lesson:

During the processing of the lesson I asked the class to give examples that would belong to each group and how they could easily determine if a number expression involving plus and minus operation belongs to “can be” or “cannot be” group. From this they were able to make the following generalizations: (1) Addition of two numbers can always be done. (2) Subtraction of two numbers can be done if the second number is smaller than the first number otherwise you can’t. You can imagine their delight when they discovered the following day that taking away a bigger number from a smaller number is possible.

One pupil proposed a solution using the result of the operations but calculated for example 3-10 as 10-3. This drew protests from the class. They maintained that 3-10 and similar expressions does not yield a result. Note that class have yet to learn operations on integers. And obviously they could not yet make the connection between the negative numbers they used to represent situation from the lesson they learned the day before to the result of subtracting a bigger number from a smaller. To scaffold this understanding I ask them to arrange the number expressions from the smallest to the biggest value. This turned out to be a challenging task for many of the students. Only a number of them can arrange the expressions for smallest to the biggest value. My next post will show how the task I gave to enable the class to make the connection between the negative number and the subtraction expressions.