Tag: reasoning tasks

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 Algebra, Assessment

Algebra test items – Zeroes of function

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I find the Trends in International Math and Science Study (TIMSS) Assessment Framework useful for constructing test items. TIMSS classified the questions in terms of cognitive domains namely, Knowing facts, procedures, concepts; Applying the facts, procedures and concepts usually in a routine problem solving task; and, Reasoning. Click here for detailed descriptions of each.

Here are three items assessing students’ learning about zeroes of function using each category in the framework. I used the same polynomial function to illustrate the differences among the three. In the real exam use different polynomial for each category so it won’t stand as hint to other questions.

Knowing

Which of the following is a zero of f(x) =6x3 – 17x2 – 5x + 6?

a. -6                b. -3               c. 0           d. 3        e. 6

There is no way apart from luck that students will choose the correct answer in this question if they don’t know what a zero of a function is. There are many ways of getting the correct answer of course (graphical, applying factor theorem, definition of zero of function).

Applying

What is the value of k if 3 is a zero of f(x) = 6x3 – kx2 – 5x + 6?

Questions about applying usually include standard textbook problem like the one shown above. It involves knowledge of a a fact/concept or procedure to complete the task. It does not only involve straightforward application of concepts unlike those under Knowing questions.

Reasoning

If 3 is a zero of a third degree polynomial function f, which of the following statements can never be true about this function?

a. f(0) = 3.

b. f(-3) = 0.

c. (0,0) is a point on the graph of f

d. (-3,3) is a point on the graph of f

e. (3,-3) is a point on the graph of f

Unlike questions under Applying which are standard or routine tasks, tasks under Reasoning category are usually non-routine and involves decision-making.

Click link to view another set of test items about graphs of rational functions.