Author: pierckat

Online Resources for Math CCSS

pierckat Assessment of CCSS-Math, Technology Teaching Issues

Illuminations

The first website that had good activities that can be integrated into a mathematics curriculum is Illuminations.  Since this site uses the NCTM Standards, not the CCSS standards, I will mostly use this site for interesting lessons and activities.

There is one activity in particular that I can see myself using in my Geometry classes.  This is an interactive activity relating to Triangle Congruence.  You are asked to construct congruent or non-congruent triangles and only given 3 pieces (sides and/or angles).  For example, given two sides and the angle between them, you try to construct a triangle that is not congruent, and it is not possible! Using this will help students see the significance of our Congruence Theorems and how they are all connected.

Here is the link to the Illuminations website: http://illuminations.nctm.org/ActivityDetail.aspx?id=4

Here is the link to the activity described above from this website: http://illuminations.nctm.org/ActivityDetail.aspx?id=4

 

IXL

The next website that I found interesting is IXL.  This website is organized into grade levels K-8 (separated), then high school in one group.  In the High School section, the Math CCSS are posted and below standards are example problems that go with each standard.  I looked mostly at the Geometry Standards, and there are activities that go with most of the standards, though not with the proof standards.  After you click on a link, there is an interactive problem on the screen that students can complete.  I really like the idea of using this in my classroom since students need the experience of completing math assessments on the computer.  I could make this a benchmark assessment and use each student’s “smart score” to determine how students are doing compared to their peers.  This site is also very helpful for teachers to see examples of what each standard really means.  This can help teachers ensure that their curriculum is aligned to the Common Core.

Here is the link to the high school section on the IXL website: http://www.ixl.com/standards/common-core/math/high-school

Geometry Learning Progression Relating to Triangle Congruence

pierckat Assessment of CCSS-Math, HS Geometry

Attached is a learning progression for CCSSM Critical Area #1 as it relates to Chapter 4 (Triangle Congruence) in the Holt Textbook.  Also attached are sample benchmark assessments and classroom assessments.  The learning progression and two assessments can be used to communicate with colleagues some key areas to focus on in this unit and ideas to present important concepts.  If teachers collaborate using these things, especially throughout the entire curriculum, there will be more consistency in the mathematics program.  This can be greatly beneficial for both students and teachers.  Teachers will have a better idea of a student’s background knowledge since all classes are consistent in content.  Also, students will be receiving the same high quality instruction regardless of which teacher they have.

Triangle Congruence Learning Progression– This learning progression would fit into the Holt Curriculum in Chapter 4 – Triangle Congruence.  In Chapter 1, we talk about the transformations, so students should have a good understanding of reflections, translations, and rotations.  Dilations aren’t typically covered until later in the textbook, so there will need to be some time for a discussion on dilations and non-rigid motions.  This learning progression fits together in the big picture because the CCSS is very focused on transformations.  Since this learning progression involves talking about triangle congruence through transformations, it fits in with both the standards and the curriculum very well.

Triangle Congruence Benchmark Assessment– I would use this benchmark assessment as a quick  summative assessment at the end of the unit.  Since it is so short, you could always include another couple problems to make this more like a test.

Triangle Congruence Classroom Assessment– I would use this classroom assessment as a quiz to check for student understanding and help the teacher determine what areas need to be re-covered.  This would not be a big summative assessment, more of a formative assessment to make sure the students “get it”.

Using Prezi in the Classroom

pierckat Technology Teaching Issues

There are numerous benefits to using multimedia activities in the classroom.  These activities encourage students to work collaboratively in groups, express their understanding in multiple ways, solve problems, communicate solutions, and check their own work. However, there are some challenges to using multimedia in the classroom, including access to resources and time to plan and work on these activities. 

Though these activities are time consuming, teachers should incorporate more of these activities into a curriculum because students can benefit so greatly.  This can be done many ways, including using videos as mini lessons, as a hook, to demonstrate a problem’s application to real life, etc.  Another idea is to make multi-media the end result of a project.  So instead of students writing an essay or making a poster, they would present their solution using a video.

One way for students to present their solution is through a “Prezi”, which is an online program that “opens up a new world between whiteboards and slides”.  Prezi is very interactive and can be a great tool for both teachers and students to make mathematics exciting.  The viewer gets to set the pace of the presentation, choosing autoplay, or advancing “slides” in their own time.  Here’s a pretty cool one: http://prezi.com/vgj5kiy-3zsd/the-magical-theory-of-relativity/

Number and Quantity Domain

pierckat Assessment of CCSS-Math, CCSS-Math Learning Progressions, HS Number and Quantity

Teaching Standards Washington State – Secondary Mathematics

Preparation for teaching the Common Core State Standards – Mathematics 5-12

Number and Quantity Domain

Overview

Mathematics teacher candidates must be able to use and describe multiple number systems and operations.  They must be able to use the number system to model real world situations.  They must be able to describe the problem solving process and justify their solution.  They will develop this understanding through successful completion of Linear Algebra, Abstract Algebra, Discrete Mathematics, and the Calculus series (Lab-based Sciences – Chemistry, Physics).

Numbers & Number Systems

Use, explain, and operate on integers, rational numbers, real numbers, and complex numbers.  Understand operations include addition, subtraction, multiplication, and division and that the communicative, distributive, and associative properties are consistent with their previous meanings.  Use, understand, and explain properties of exponents, including rational exponents and represent in radical form.  Understand the workings of matrix, vector, and complex number algebra.  Use technology, including calculators, spreadsheets, and computer algebra programs to manipulate these number systems.

 Quantities

Make sense of real world problems by reasoning quantitatively.  Use and explain measurements, unit conversions, and label solution with correct units.  Justify the problem solving process and solution in the context of the problem.

 Learning Targets (Indicators)

Teaching candidates will be able to:

  • Use, explain, and operate on integers, rational numbers, real numbers, and complex numbers
  • Model operations including addition, subtraction, multiplication, and division
  • Demonstrate connections between their previous knowledge of the communicative, distributive, and associative properties and how they are consistent in more complex number systems
  • Apply and explain properties of exponents, including rational exponents and represent in radical form
  • Use matrices, vectors, and complex number algebras to solve problems
  • Use and explain quantitative reasoning to solve problems including units of measure
  • Use structures from many branches of mathematics to represent and manipulate number systems
  • Use technology to explore and represent number systems

 

-Katelyn, Jenn, Casey

Using iPad Clinometer App to Solve Problems

pierckat Technology Teaching Issues

I think using iPads in the classroom is a great way to spark students’ interest and engage them in everyday activities.  Overall, most of the math apps that I found were more directed towards lower level mathematics, for use in elementary or middle school grade levels.  However, I wanted to find an application that I could actually use in a high school mathematics classroom that would improve learning and make it fun for students.

I found a great clinometer app, which allows the user to measure slope using percentages, rise over run, or degrees.  Clearly, this application could be used in several mathematics courses, from measuring slopes of lines in Algebra 1 to measuring angles in Geometry.

I particularly like this application for use in a problem solving activity I described for a Geometry class.  I could pose the question, “How high up is the score board in the Gym?”.  Clearly, this would be difficult to measure directly.  Using trigonometry, we can solve for the height.  Originally, I would have explained how to make a clinometer out of paper and string so that students could measure the angle of elevation.  However, using this app, we can skip the rote step-by-step process of making a clinometer and get to the important mathematics concepts.  Since we are using this awesome iPad app to solve a problem involving higher level mathematics, students would be more intrigued and excited to solve the problem.

The Angle Game

pierckat HS Geometry, Technology Teaching Issues

This video could easily work into a Geometry course, after discussing angle relationships with parallel lines and transversals.  Since this game is not simply a one step problem (like just using alternate interior angles), students will have to problem solve and explore different angle pairs to find the measure of the indicated angle.  By using this Khan Academy Video to open a lesson, you spark students’ interests (technology, games, etc)  which will help engage them in the activity.

After opening the lesson with this video, I would give a couple more examples, including some that are more challenging.  While working through these examples, I would expect students to use proper vocabulary to identify angle pairs, such as corresponding angles, alternate interior angles, same side interior angles, etc.  To reinforce the content, I would also ask students to explain the relationship between these angle pairs, indicating whether they are congruent or supplementary.  By doing this, we support their conceptual understanding as well as their understanding of the basic language and vocabulary.

As the main activity, students would create their own angle game, preferably using a computer program.  Students could then print off several copies of their game and would then swap with others for an informal assessment.   During this time, the teacher could monitor students’ understanding and determine progress towards mastery.

http://www.khanacademy.org/math/geometry/angles/v/the-angle-game

Balancing Depth and Territory

pierckat Improve Mathematical Thinking

Mathematics education is often described as “an inch deep and a mile wide”.  Most teachers would agree that mathematics courses often lack depth.  As teachers, we are expected to cover a prescribed amount of territory throughout a course.  However, covering such a large amount of material rarely allows students to learn and understand the concepts at a deeper level.  Feeling the pressure to try to teach all standards assessed on state exams, teachers often settle for basic student understanding rather than deeper student understanding.

This has several negative consequences.  Since few students really had a good initial understanding of the material, too much time is being wasted at the beginning of the year to review what should have been “previously learned”.  Also, teachers often focus on transferring information (and hoping it’s received!) rather than having students problem solve and use higher order thinking to make sense of it.  This can lead to a much lower level of understanding.

I find there are several teacher characteristics that contribute to this problem.  One is that teachers (myself included!) too often teach procedures.  When teachers lay out a step by step guide to solving problems, all higher level thinking is thrown out the window!  We need to have students think about strategies to solving these problems and then determine whether their strategy works.

Clearly, this is a quite a problem in the world of mathematics education.  If we expect to see change from student performance, I believe teachers must first be willing to make the change.  This leads to a couple big questions:

  • How do you balance eliciting deeper student understanding while still covering all of the content?
  • What additional common teaching characteristics can prevent students from using their higher order thinking skills?  How can these characteristics be changed to elicit deeper understanding?

 

-Katelyn