Overview
Curator: Darrell Glover
Brief Description of Tech Tool: This tool is a combination of a lesson and practice questions on the concept of completing the square to solve quadratic equations. The lesson fully covers the topic, from the steps taken to accomplish the procedure to the applications of completing the square of a quadratic equation.
Technical & Cost considerations: This tool is free of cost from the popular Math is Fun website. The pages are simply normal websites that are accessible from any internet-enabled device.
Evaluation
Description of Learning Activity
The lesson portion on completing the square fully introduces the concept making it easy to use in a variety of ways. A teacher can assign it as pre-reading prior to classroom instruction on the topic or simply step through the page as part of an initial lesson on the subject. Users can refer back to it for review of the topic v. the distraction that sometimes occurs from taking notes. The practice questions can also be used throughout the spectrum of the subject. A teacher introducing the topic might display a question in order to give the class (or small groups) practice or a user needing to review for a test can use the questions for study.
1. Learning Activity Types
- LA-Practice – the automated practice questions can be used to build fluency in many parts of the procedure for completing the square (they are not all straight solve for x type questions)
- LA-Present – the lesson portion is designed such that it can be used to introduce the concept of completing the square
- LA-Present-Demo – the lesson walks through the process step-by-step in order to demonstrate the procedure
- LA-Present-Explain – the lesson also has diagrams that explain the idea behind completing the square and why each step must be accomplished
- LA-Present-Demo – the lesson walks through the process step-by-step in order to demonstrate the procedure
- LA-Explore – a user could read the lesson and complete practice questions on their own in order to personally explore the mathematical ideas behind completing the square
- LA-Apply – the lesson notes the applications of completing the square for a quadratic equation and the user is able to apply their abilities to the practice questions
2. What mathematics is being learned?
Standards
NCTM:
- Generalize patterns using explicitly defined functions
- Understand relations and functions and select, convert flexibly among, and use various representations for them
- Understand the meaning of equivalent forms of expressions, equations, inequalities, and relations
- Write equivalent forms of expressions… and solve them with fluency…
- Use symbolic algebra to represent and explain mathematical relationships
- Judge the meaning, utility, and reasonableness of the results of symbol manipulations…
Common Core (non-specific):
- Interpret the structure of expressions
- Write expressions in equivalent forms to solve problems
- Perform arithmetic operations on polynomials
- Create equations that describe numbers or relationships
- Understand solving equations as a process of reasoning and explain the reasoning
- Solve equations and inequalities in one variable
- Represent and solve equations and inequalities graphically
Proficiency Strands
- conceptual understanding – the tool uses diagrams and graphs to expand on the ideas behind completing the square
- procedural fluency – there are automated questions available for practice
- strategic competence – the lesson explains applications for completing the square, allowing the user to identify useful problems that can be solved by the procedure
- adaptive reasoning – the lesson touches on how completing the square plays into the proof of the quadratic equation allowing users to explore justifications of its use.
3. How is the mathematics represented?
The representation utilized by this tool is completely numerical/symbolic. It is a written lesson on the topic with steps and examples incorporated into it and practice questions that can be answered/graded. While that may seem like the lesson is simply a substitution for a teacher, the detail allows it to cover the topic more completely than a classroom session could. Additionally, the questions are graded in real time and if the user is wrong then the tool will walk the user through the correct way to solve the problem. This allows the user to instantly learn from their mistakes instead of, like in a text, potentially know the correct answer but not know the process to obtain it.
4. What role does technology play?
The fact that the question section doesn’t just say right/wrong and the correct answer but will actually walk the user through the correct steps needed to arrive at the right answer is a key advantage of this tool. This instant feedback is very useful for allowing users to quickly fill gaps in their knowledge and integrate the new material into their overall abilities with the subject.
Affordances of Technology for Supporting Learning
- Computing & Automating – the tool automates the correction of practice questions and explaining wrong answers
- Accessing Information – there are many links available throughout the lesson that a user can utilize to view topics that apply to the completing the square lesson but that he/she might need a refresher on
5. How does the technology fit or interact with the social context of learning?
This tool can be used by a teacher for a whole classroom discussion, utilized in small group discussions, or by an individual learner. The lesson is written, probably making it most useful for individual learning (since people often read a different speeds) but the practice questions can be viewed and worked on by small groups also.
6. Additional Comments
Another useful addition to this tool is the abbreviated “Footnotes” section at the bottom. It is a quick reference guide on the ideas/equations from the lesson that a user can review instead of having to search through the entire page.