This interactive Web site allows the learner to construct factor tree(s) of prime factors for one or two numbers. When learner factors two given numbers then the learner can identify the Least Common Multiple (LCM) and the Greatest Common Factor (GCF) of the two numbers.
The tool helps students break numbers into prime factors. When students master this, they can work on two numbers and progress to finding the least common multiple and the greatest common factor using Venn diagrams and the prime factors they found.
Type of Material:
Drill and Practice
This interactive Web site would provide factorization of numbers practice for upper elementary through high school students. It could be a useful refresher for students in elementary/middle level math methods classes.
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Identify Major Learning Goals:
1. Be able to factor a number into primes.
2. To recognize when the number has been broken down into primes.
3. To be able to find the least common multiple (LCM) and the greatest common factor (GCF) using Venn diagrams.
Target Student Population:
Elementary School, Middle School, High School, and Math Method Courses in Teacher Education Programs in Higher Education
Prerequisite Knowledge or Skills:
Knowing how to multiply and break a number into its factors.
Being able to multiply more than two numbers --associative property.
Understanding how division depends on multiplication
Understanding the difference between factor and multiple
Evaluation and Observation
The NKVM Factor Tree provides a game-like Web site for students to practice the prime factorization of numbers and see how to use the factorization of a pair of numbers to find their Greatest Common Factor and their Least Common Multiple.
The Web site presents valid math concepts. It can be used with students that are just beginning factorization and continue to a higher level of least common multiple.
The model has certain nice features. First, the prime numbers are circled and the composite numbers are in rectangles. Second, the Venn diagram method of illustrating the LCM and GCD is an excellent model because it is easy to see and less confusing than circling the common number with the highest exponent, etc. It is also clear that the student will not end up using the same numbers extra times. The idea that you can make up your own numbers is also good. It is easy to place the numbers correctly in the Venn diagram, although if you make an error you may not know why. GCD and LCM are concepts that most students have difficulty keeping straight. The terms are often confused. This model makes it clear which is which-- and hopefully the students will see that the GCD or GCF is always smaller than the LCM. The words Least and Greatest often get in the way of understanding the concept. The diagram helps with that.
This applet cannot be used without clear instruction since there are no explanations with the applet, just a way to check your answer.
If you click on ? instructions or teacher/parent you will get explanations for the applet.
You are forced to break down the numbers into two factors. There is no way to go directly to primes if a student sees it. For example seeing 8 as 2x2x2.
Potential Effectiveness as a Teaching Tool
This Web site could be used by students at various education levels to practice the factorization of numbers. A teacher could also use the Web site to demonstrate the factorization of numbers to the class. The Venn Diagram is an excellent illustration for the students to visualize the factors of two numbers and the common factors by dragging the numbers in the Venn Diagram.
This applet, with its Venn diagram model, makes it easy to see the difference between the GCF and the LCM. These are two concepts that are difficult for many students. If they are able to understand the Venn diagram it will be clear that the LCM is larger than the GCF. They will eventually get over the confusion over "greatest" and "least" which often leads them to reverse the answers. The differentiation of the shapes for the prime and composite numbers also helps students know if they need to do more factoring. Color coding also makes it clear when factors have been used. The model of the Venn diagram, whether in an applet or just as pencil and paper is most valuable.
This site would also be useful in the mathematics method courses in the teacher education programs at the higher education level. This Web site could be a model for teacher candidates to develop interactive activities.
The applet for factoring should eventually be left behind. Because the applet gives the second factor, students may be tempted to just use 2 or another small number to get started rather than learning the larger factors. Although prime factor trees are a nice model, they don't encourage students to break down numbers more quickly as they get more practice. In fact the applet doesn't allow you to. A student couldn't break down 64 into 8x8 and then 2x2x2 and another 2x2x2.
Ease of Use for Both Students and Faculty
The Web site is easy to use. In case someone does need help, there is a button for instructions for the site. There is also a link for teachers and parents with additional information about the site and the factorization of numbers.
The directions can be found by using the rather small buttons at the top of the page: ? Instructions and Parent/Teacher. Another button called Standards will tie the applet into the NCTM Standards. The directions are complete. Students should like the Venn diagram model, especially if they were taught these concepts in a more traditional way.
It would be helpful for the message for the wrong answer to be closer to the problem. The message does not show up on some screens unless you scroll down. If the message was above the problem, then it would be easy to see.
The feedback basically lets you know whether you are making an error. If you don't know that * means multiplication on the computer and you write 2x2 for an answer, you will be told that you have not written a number.
Other Issues and Comments:
The most powerful part of this tool is the Venn diagram model.
This tool is possibly more useful for students with difficulties in math.