TEACHING ROOTS AND RADICALS USING BLENDSPACE IN AN 8TH GRADE MATH CLASSROOM

Untitled design (1)
(Cox, 2016)

The Context for My Lesson

This lesson is part of a unit on radicals and integer exponents as outlined by North Carolina Public Schools’ Standard Course of Study for Mathematics (2016, p.22). The lesson is designed for the students I described in my post on teaching scientific notation which is linked here. Prior to completing this lesson the students would have worked on a review of the concepts related to exponents that they had learned in 6th grade, and they also would have completed lessons on standard 8.EE.1: “Know and apply the properties of integer exponents to generate equivalent numerical expressions.” (p. 22). In addition to understanding exponents, they would also need to have a thorough prior understanding of the difference between a rational and irrational number in order to understand that some roots are rational, while the majority are irrational. Student would have seen the square root symbol (√) on calculators and in other contexts during their math education and the lesson capitalizes on any curiosity about the symbol they might have by asking them to brainstorm about its meaning before they begin learning about it.

The unit that this lesson is part of would also include lessons on approximation using powers of ten and scientific notation. The students would be assigned a project that they would be working on on over the course of the unit and after completing this lesson they would research and investigate real-world connections to roots and radicals as part of that assignment.

The Standard(s) this Lesson Meets

This Blendspace activity is a stand-alone lesson for the North Carolina Public Schools’ Standard Course of Study for Mathematics (2016) standard 8.EE.2: “Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.” (p. 22). Students should be able to progress through the lesson independently, with direct instruction from the teacher only occurring if necessary to help clarify concepts or to assist with solving problems. The lesson is designed to teach the majority of what students need to know to be proficient in this standard, along with the implied foundational skills of evaluating square and cube roots that have both integer and irrational solutions. One portion of this standard, relating to solving equations of the form x² = p and x³ = p, would be taught in a second lesson or in conjunction with a lesson on solving different types of equations. This lesson is meant to meet all other parts of the standard and to lay the foundation for a thorough understanding of solving equations of the type mentioned above.

The Media or Technology I am Integrating

Blendspace Lesson
Screenshot of the Blendspace Lesson on Roots and Radicals

Blendspace Lesson – Roots and Radicals

The Blendspace lesson I created is a complete lesson from a warm up to start, instruction and practice in the middle, and an exit ticket at the end. It is designed for a class that has a 1-to-1 student to technology ratio so that the students can work and progress independently through the lesson while the teacher circulates to work individually with students who may need help. The lesson begins with a link to a brief, 10-question warm up that reviews properties of exponents and asks the students to evaluate simple expressions like 2 to the 0 power, and 1². Then the students are shown a simple visual that prompts them to think about if they have seen the square root symbol (√)  before and what it might mean. Then a slide showing the goals and standards for the lesson is shown which will connect to the exit ticket at the end of the lesson.  After looking over the goals for the lesson, the students begin learning new material and practicing by watching a theatrical video and taking a 10-question quiz about the material they saw in the video. The online quiz provides them with immediate feedback about the accuracy of their answers. They are then asked to write down some more complex problems, use an article that outlines square roots and the steps for estimating them to answer the problems, and then complete another online quiz.  The example-article-quiz sequence is used a second time in the slideshow to teach the same skills for cube roots. The last part of the instruction is a video that demonstrates another method for estimating square roots and a short practice quiz.  Students are then shown a simple visual asking them to reflect on their learning, followed up by an exit ticket that asks them to solve three simple problems from what they learned in the lesson, and to self-evaluate their learning according to the standards presented to them at the beginning.  Throughout the lesson, the students are directed to take written notes on the videos and to write down the practice problems to create a written record of their interaction with the material and to help them learn to practice the skills presented in the video independently.

The Rationale for Integrating the Media or Technology into this Lesson

I believe that one of the most powerful arguments for use of video in the classroom is that it provides access to lectures from some of the most engaging and entertaining teachers in the country, and possibly the world. I feel like that is true of the MathAntics video (2015) that I chose as the first video in my Blendspace lesson. The instructor is funny, thorough, clear, and uses master storytelling techniques to create a video that holds the viewers attention. Videos like this can be a beneficial instructional tool when they are presented in an engaging way. Miller, Morneo, Willcockson, Smith, & Mayes (2006) discuss the ways that they utilized technology to make teaching a challenging topic–neuroscience–more appealing to middle school students.  They emphasize “…the value of narrative or storytelling as aids to the learning process” in their article as a way to engage with students as they learn a challenging topic. (p. 138)  The instructor in the MathAntics video merged lecture, images, and animations, to explained roots in a way that was more engaging than a lecture in a traditional classroom.

Research by Smith and Suzuki (2015) suggests that teaching that embeds multi-media in a traditional teacher-guided classroom creates a learning environment that is more effective and more favorable to students. This type of instruction is called “blended learning” and is the type of learning that is created when a Blendspace lesson like the one I created is used in the context of a normal math class. The researchers did an experiment that compare the performance of students in a traditional “live lecture” classroom to students who were taught using a blend of technology based instruction and teacher interaction. They found that “the great majority (80%) of students in the treatment group preferred the embedded blended learning over traditional live lectures for future learning of math.” and that student surveys suggested that “students in the treatment group appreciated the: (a) ability to control the pace of instruction; (b) new role of the classroom teacher; (c) lack of distraction in the blended learning environment; and (d) accessibility of the embedded multimedia lessons outside the classroom.” (p. 133)  In addition to liking the class more, students in the blended classroom group also showed greater academic gains.

The creator of the popular website http://www.KhanAcademy.org, Salman Khan, outlines powerful positive influences that technology can have in the classroom in his TED talk Let’s Use Video to Reinvent Education (2011). He argues that the use of videos can help individualize lessons, can encourage peer tutoring, allow for more time for the teacher to help individual students, and can connect students with people and ideas across the globe. Though Khan was specifically discussing the flipped classroom model, the same ideas can be applied to a multi-media, interactive lesson like the one I created on Blendspace.

The Integration of the Media or Technology Into the Lesson

The Blendspace lesson is complete from start-to-finish and is designed to take up a class period with minimal direct instruction to the class as a whole. Nearly all of the steps in the lesson are contained within the Blendspace lesson. The lesson would be used as follows:

  1. Students are instructed to turn on their device (computer or tablet), get out their note-taking tool (a notebook or whatever they have chosen to use to document their learning for the class) and to visit the Blendspace link the teacher has posted on the class website for the day.
  2. Students work independently to complete the Blendspace lesson while the teacher circulates to answer questions, check on progress, and give individualized instruction. The steps for the lesson in Blendspace are as follows:
    1. Complete brief warm-up review of properties of exponents using an online quiz
    2. Think about if they have seen the root symbol, “√,” and what it might mean.
    3. Review the goals for the lesson and the standard that it is aligned with.
    4. Watch a high-energy video from YouTube channel MathAntics that explains what square and cube roots are, what perfect squares and perfect cubes are, how to approximate roots of squares and cubes that are not perfect, and the radical notation used to express roots.
    5. Practice questions on finding square roots of perfect squares
    6. Write down questions about square roots to be answered in the notes using an article that the student will read next
    7. Read the article on square roots and work on the problems for the notes
    8. Take a 10 problem practice quiz on square roots, and refer to the website and notes to help with any issues
    9. Write down questions about cube roots to be answered in the notes using an article that the student will read next
    10. Read the article on cube roots and work on the problems for the notes
    11. Take a 10 problem practice quiz on cube roots, and refer to the website and notes to help with any issues
    12. Watch a Khan Academy video on estimating square roots
    13. Practice estimating square roots
    14. Think about what they learned in the lesson and how they would explain it to someone else
    15. Complete an exit ticket that the teacher can use as a summative assessment and that also asks the student to self-evaluate their own learning based on the standard.
  3. If students finish before the class is over they can be paired with another student who may be stuck or need extra help, or they can work independently on ongoing projects or portfolios.
  4. The teacher will review the exit tickets at the end of class or after class and do any review or reteaching activities the following class period.

My Evaluation of the Media or Technology Integration

The multi-media and interactive aspect of the Blendspace lesson are appropriate for this diverse 8th grade class because young adolescents benefit from independent learning and the ability to progress through material at their own pace. This lesson is designed to be as independent from the teacher as possible so that students would have control over the pace of their own learning and that learning would be more student-centered than a traditional classroom. When the students in a class are engaged in independent learning, the teacher has more time and flexibility to assist students that have accomodations and that need extra help learning concepts.

One of the pitfalls of this type of media integration is that the lesson is a one-size-fits-all approach to instruction. Though the research suggests that integrating video and technology engages more students, not all learners will be as interested in learning independently from the videos, readings, and quizzes. To be truly individualized, there must be another instructional option for students who may not respond to the materials and method of presentation in the Blendspace lesson. In addition, the use of the online quizzes and standard practice problems does not allow the teacher to vary the levels of difficulty of the practice problems.

Overall, integrating Blendspace is a wonderful way to bring together a variety of media into a lesson that will be benefit middle-grades students, keep them engaged, give them immediate feedback on their work, and to teach a standard that necessitates some direct instruction in a more interesting way than would be possible with the traditional teacher-centered lecture. It lays the groundwork for the skills needed to reach higher-level understanding and sets the stage for independent research and projects based on roots and integer exponents.

References

Cox, Victoria. (2016). Image of Five Cubes.

Cubes and Cube Roots. (n.d.). Retrieved August 01, 2016, from http://www.mathsisfun.com/numbers/cube-root.html

Khan, Salman (2011, March) Let’s use video to reinvent education. Retrieved from: http://www.ted.com/talks/salman_khan_let_s_use_video_to_reinvent_education

Khan, S. (2011, June 22). Approximating square roots to hundredths | Pre-Algebra | Khan Academy. Retrieved August 01, 2016, from https://www.youtube.com/watch?v=EFVrAk61xjE

Math Antics (2015). Exponents & Square Roots. Retrieved July 17, 2016, from https://www.youtube.com/watch?v=C_iKTTI1E34

Math Games. (n.d.). Retrieved August 01, 2016, from https://www.mathgames.com/skill/8.11-evaluate-exponents

Mathopolis Question Database. (n.d.). Retrieved August 01, 2016, from http://www.mathopolis.com/questions/q.php?id=309

Miller, L, Morneo, J, Willcockson, I., Smith, D., & Mayes, J. (2006). An online, interactive approach to teaching neuroscience to adolescents. CBE-Life Sciences Education, 5, 137-143. Retrieved from: http://asulearn.appstate.edu/mod/page/view.php?id=1394968

Practicing Eighth grade math: ‘Square roots of perfect squares’ (n.d.). Retrieved August 01, 2016, from https://www.ixl.com/math/grade-8/square-roots-of-perfect-squares

Public Schools of North Carolina. Standard Course of Study for Mathematics. Retrieved from: http://www.ncpublicschools.org/docs/curriculum/mathematics/scos/k-12%20Standards.pdf. 22-24

Smith, J., & Suzuki, S. (2015). Embedded blended learning within an Algebra classroom: a multimedia capture experiment. Journal Of Computer Assisted Learning, 31(2), 133-147. doi:10.1111/jcal.1208

Squares and Square Roots. (n.d.). Retrieved August 01, 2016, from http://www.mathsisfun.com/square-root.html

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