Always, sometimes or never?
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Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task requires you to know if equations are always true, only true for some values of x, or never true.
Some equations are always true.
e.g., 2x = x + x is true for all values of x (3, 5.5 etc)
Some equations are only true for some value of x.
e.g., x + 1 = 2 is only true for one value of x (i.e., x = 1)
Some equations are never true.
e.g., x = x + 1 is never true whatever value x takes.
Task administration:
This task can be completed online (with SOME auto marking).
Level:
5
Curriculum info:
Key Competencies:
Keywords:
Description of task:
Students identify whether algebraic equations are always, sometimes, or never true, and explain their reasoning.
Curriculum Links:
Key competencies
This resource involves explaining how many solutions some equations have, which relates to the Key competency: Using language, symbols and text.
For more information see https://nzcurriculum.tki.org.nz/Key-competencies
Answers/responses:
| Y10 (04/2016) | ||
| a) |
"It is true for only one value of x." [Option 2]
Correct explanations
Either 1 of:
Statements including the pronumeral x, e.g.,
Statements just referring to the solution (4), e.g.,
|
easy
difficult
(correct explanation)
|
| b) |
"It is true for only two values of x." [Option 3]
Correct explanations, e.g.,
or
Partial explanation
"It is true for only one value of x" [Option 2] and gives a consistent explanation, e.g.,
|
difficult
very difficult
(correct explanation)
moderate
(correct or partial explanation)
|
| c) |
"It is never true." [Option 1]
Correct explanations
Statements including the pronumeral x, e.g.,
Statements just comparing 7 with 12 (4 × 3) e.g.,
Other correct explanations, e.g.,
|
moderate
very difficult
(correct explanation)
|
| d) |
"It is true for only two values of x." [Option 3]
Correct explanations, e.g.,
or
Partial credit "It is true for only one value of x" [Option 2] and gives a consistent explanation, e.g.,
|
very difficult
very difficult
(correct explanation)
difficult
(correct or partial explanation) |
Based on a representative sample of 106 Year 10 students, April 2016.
Teaching and learning:
This task is about recognising that there can be different numbers of solutions to algebraic equations. It develops beyond the balancing representation of the "=" sign towards the idea that equations (which contain an equals sign) can be either true or false. When an equation is true for a particular value of a pronumeral (here it is x) then that value (of x) is referred to as a solution for the equation.
Read more information about the equals sign in Equations and expressions.
Read more about this in the Meaning of equals and in the Algebraic thinking concept map.
Diagnostic and formative information:
| Common misconceptions / errors | |
|
b) or
d)
|
Student identifies one solution for a quadratic equation e.g.,
or
|
| b) |
Student identifies 1 as a solution to (x + 1)(x – 2) = 0
|
| c) |
Incorrect expansion of an expression
|
| d) |
Student identifies x2 as 2x, e.g.,
|
| a) — d) | Does not give an explanation for their answer |
Next steps:
Student identifies one solution for a quadratic equation
Most students who did this gave only the positive solution, i.e., x = 2 for b), and x = 4 for d). In both cases, challenge the students to find another answer.
- For (x + 1)(x – 2) = 0, students most likely gave x = 2 (or just 2). The reason for this that multiplying any number by zero gives zero, and the student sees that (2 – 2) = 0. If they cannot see how to make the first bracket zero [i.e. (⁻1 + 1) = 0], they may need to do further work with negative numbers (see Working with negative numbers). For more, look at the section on the additive identity section in the Algebraic thinking concept map or in the Negative numbers section of the Addition and subtraction concept map. {to be uploaded}
- For x2 = 16, students easily see that 4 × 4 = 16. If they are not aware that (⁻4 × ⁻4) = 16, they may need to work on multiplication of negative numbers (see, for example, Red and black). Drawing a graph of x2 = 16 would also be useful. First they could plot it with the positive integers. They can be asked "What will the graph look like with negative numbers".
Student identifies 1 as a solution to (x + 1)(x – 2) = 0
The student may not be familiar with negative numbers. They may use the resources such as Positive and negative number line or Ordering temperatures. The section on additive identities in the Algebraic thinking concept map may also be useful.
Incorrect expansion of an expression
Students may need further work on expanding algebraic expressions (see, for example, Expanding expressions).
Student identifies x2 as 2x
Students may need to do more work on the conventions for powers / exponents (see, for example, Working with powers II).
Does not give an explanation for their answer
This was evident in the sample for students of all ability levels, not just students who could not recognise the correct multi-choice options. A sub-sample of students answered the questions with pencil and paper. These students gave far more explanations than those who worked online.
Students need to be encouraged to supply explanations even when the result seems self-evident to them, especially when working online. They may need to overcome the barriers of not having standard conventions easily available to them in the online environment (e.g., x2, ×, ÷, etc.).
For more information read the Algebraic Thinking Concept Map.