Using Connections to Build Understanding

I am teaching a Geometry & Spatial Reasoning course for Drexel this semester for their math masters program for teachers. Absolutely love it because I am learning so much from my students/peers, but because it really is bringing home the importance of prior knowledge to help build connections and real-world connections in helping students learn versus memorize, and construct and reconstruct based on their ability to make connections.

My students, who are a mix of very new math teachers, experienced teachers, and even some career-switchers still in the early stages of teaching, are having this great discussions on the importance of using prior knowledge to help student make their own connections. Some have been doing this all along, but others, as they themselves struggle with some of the geometric concepts we are ‘learning’ (relearning in some cases), are coming to understand the value in helping students use what they know to build on and connect to new information. Makes it easier to recall, and builds a confidence in students that when faced with an unknown situation/problem, they have the skills and confidence to look at it, break it down or add in things to make the unknown familiar and then look for and make use of structure (see what I did there….Common Core Math Practice #7!) to help reach a solution or develop a new conjecture/conclusion.

As an example, we’ve been doing a lot of work with inscribed angles in circles and how do you help students use prior knowledge to build the idea that an inscribed angle is half the measure of it’s intercepted arc if you don’t want students just memorizing formulas? Basically, the conversations revolve around constantly using prior knowledge to make connections, which might mean you need to add in an auxiliary line to a given shape to ‘see’ something familiar (i.e. a linear pair or a triangle, as examples). A strategy that really helps students look for and make use of the structures they are familiar with to help them make sense of a problem.  Here’s an example of just one way to explore inscribed angles, using previously knowledge about triangles:

inscribed-angle

  • In Fig 1, we have an inscribed angle and its intercepted arc a. How could you show that angle 1 (the inscribed angle) is half the measure of it’s intercepted arc? Here’s where students need to make sense of this structure – what prior knowledge can they use to help them?
  • In Fig 2, they add in a radius (auxiliary line), because they know all radii in this circle (any circle are equal – doesn’t change the original inscribed angle….but now – we have a triangle and a central angle (angle 2).  What do they already know? Well, they know the central angle 2 is the same as the measure of the intercepted arc, which is the same intercepted arc as angle 1 (inscribed angle).
  • In Fig 3, students are looking at the triangle created and using prior knowledge – we can mark the two radii equal, making this triangle an isosceles triangle, which they already know from prior knowledge has two base angles that measure the same (angle 1 & 3). Angle 2 is an exterior angle to the triangle, and angles 1 & 3 are remote interior, which they know from prior knowledge sum to the measure of angle 2. Since angle 1 & 2 are equal (isosceles triangle), that makes them each half of angle 2 (Sum divided by 2). Angle 2 is equal in measure to the intercepted arc, so angles 1 & 2 are each half of that, so the inscribed angle 1 is half the intercepted arc.
  • Fig 4 shows that the relationship holds true even if you change the size of the inscribed angle.

This is of course just one example for an inscribed angle, but they can then use this to show that inscribed angles that are not going through the center of the circle have the same relationships – ie add in auxiliary lines, use linear pairs, or triangles or other known things to help make sense and show new things. Prior knowledge, connections – they really matter.

As teachers, it is our duty to make sure we are modeling and helping students use what they know to build these connections and see the relationships. It takes deliberateness on our part, it requires modeling, it requires setting expectations for students till it becomes a habit (habits of mind) to look for and make sense by pulling in previous knowledge.

Another thing we need to do is make connections to real-world. My students are sometimes struggling with this idea of relating prior knowledge and new ideas to real-world applications, but if you get in the habit, its not so hard to do. Since I am focused on circles and the lines that intersect them now with my class, I pulled up a ready-to-use lesson from Casio’s lesson library that is a great example of a real-world connection to circle concepts that would force the use of previous knowledge.  The lesson is briefly described below:

The Perfect Glass Dome: (here’s a link to the complete, downloadable activity)

  • – Use coordinate geometry to represent and examine the properties of geometric shapes.
  • – Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

This activity uses the Prizm Graphing calculator and picture capability to help build understanding.

dispcap1 The kinds of questions and connections to prior knowledge that can be asked of students just by looking at the image are pretty endless. What relationships do you see (i.e. lots of diameters, or straight angles, lots of central angles, all the angles are 360, are their auxiliary lines we could add to find the areas or relationships or angle measures, etc.).

If you look around, you can probably find a real-world example of most math concepts your are working on with your students. Show them pictures, show them real objects they can get their hands on. Start asking questions. Ask them what they recognize or think they already know. Ask them if they could add something or take away something to see a familiar object/concept. How does that help them? What relationships and connections help them get to something new or interesting?

My Drexel course and student are reemphasizing for me (and them) the importance of prior knowledge to help build connections on a continuous basis, all the time, every day. It helps students think mathematically and consistently use vocabulary and math concepts to deepen and create new understanding and relationships. It also promotes logical reasoning and problem solving – win-win!

 

 

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Conics – Casio Prizm vs. TI-84+CE

I am currently teaching a course at Drexel University and we are starting a unit on circles. I loved using Sketchpad when teaching because it allowed for dynamic manipulation of objects (shapes, functions) so that students could visually see the impact of variables to the shape, size, position of the object. Unfortunately, my students (math teachers in a Masters Math Teaching Program) do not have access to Sketchpad, though one does use Geogebra, and as this is a course focused on teaching, they need to use what they have access to in their own classrooms with their students. For many of them this does not involve any technology at all, which is sad, but for some, they do have access to graphing calculators.

Naturally, this got me exploring what the graphing calculators could do, and surprise, surprise, I noticed quite a difference between the Casio Prizm and the TI-84+CE graphing calculators, which are the ones my class seems to have. I was investigating conics, and in particular circles, and what options the graphing calculators gave me, especially when thinking about dynamically modifying the variables to see how each impacts the graph of the circle. Here’s is a quick summary of what I found:

  1. Both TI-84 & Casio Prizm can graph conics (circles, ellipses, hyperbola, and parabola, though how to access these conic graphs is different on both.
    • It is more apparent/easy to find on the Casio (there is a Conic Graph menu).
    • TI requires knowing that there is a Conic app in the app menu, which is a button on the calculator. It is not seen from the main screen, and if you don’t know it exists, you won’t know it’s available.
  2. Both provide more than one equation form for each conic.
  3. Both show the graph of the conic, but how is very different.
    • Casio shows the graph on the coordinate grid, where you can see the whole grid, see values on each axis, and identify quadrant and key points on the graph
    • TI shows the graph in the entire window with a weird yellow frame around it. It is difficult to determine where on the coordinate grid the graph appears – there are axis marks, but no values, not origin, making it difficult for students to understand where on the coordinate grid the graph is. Very difficult to identify quadrant and key points on the graph.
  4. Both allow you to enter different values for each coefficient variable,
    • Casio has a modify feature that allows you to see the equation, graph, and coefficient variables on one screen. You can then modify one coefficient at a time and see it dynamically change on the graph, allowing students to visually see how each impacts the graph and see the conic change shape, location, and/or size.
    • TI84+CE only shows the graph or the equation/coefficients – never together.  You have to go back and forth between them when changing values. The TI does not clearly show where on the grid the graph is, does not show a size change (all conics look the same size, but the grid scale is changing). It’s actually very confusing and would be difficult to help student visually see the impact of changing coefficient variables on the size, location and shape.

Below is a video I made showing how to graph a circle and modify the coefficients on both calculators so you can see the differences I am talking about.

Hopefully you will come to the same conclusion I did – Casio Prizm is far superior when graphing conics than the TI84+CE.

Pi Day 2016 – 3.1416 Rounding out to a good year!

In celebration of Monday’s Pi Day, which is a pretty cool one this year, since 3/14/16, or 3.1416 is Pi rounded to the nearest 10,000th, I am devoting this post to Pi.  I will probably do another one on Monday, but for those teachers out there who want some ideas for things to do with their students Monday, a few days advance notice is always a good thing!

There are Pi Day activities everywhere, so I thought I would share a few ideas, links, and an activity from Fostering Geometric Thinking with Casio Technology (Dr. Sonja Goerdt) to support those of you looking for things to do in the classroom.

Here is a couple things I did with my students (middle school and high school alike!)2016-03-10_15-05-29

  1. Have students bring in different circular food objects (i.e. Little Debbie’s cakes, Moon Pies, actual pies, cookies of all sorts, crackers, etc.).  We would take this variety of circles, find the diameter and circumference (using string) and then calculate ratios to “confirm” the magical number Pi. Then we’d eat!
  2. Have a contest of who could recite the digits of Pi.  Winner was the one who could go the furthest – always fun.  This does require students to prepare ahead of time, so if you want to do this, make sure you tell students who are interested to get studying.

Here are some links to activities I’ve found just searching the web.  There are a lot out there.

  1. Pi Day http://www.piday.org/million/ This site lists a million digits of Pi, and then, if you click on the links to the right, you can search the digits of Pi (for special sequences, like your birthdate), Pi puzzle (New York Times), or Einstein Rap.  There are lots of other links, so explore away.
  2. Exploratorium http://www.exploratorium.edu/pi/pi_activities/index.html has a whole list of lessons/activities that explore Pi in many ways. One is the search digits one as well. Another one I think sounds very interesting is the Tossing Pi (scroll down the list to find this) – calculating Pi tossing toothpicks. Kids would love that!
  3. Project Mathematics http://www.projectmathematics.com/storypi.htm This has videos you can choose about the history of Pi, uses of Pi, people explaining what they think Pi is.  Might be good to warm up your class with.
  4. Joy of Pi – http://www.joyofpi.com/pilinks.html#articles  This page has lot of links to interesting articles about Pi, history, etc.  Lots of resources.
  5. Live Science – http://www.livescience.com/29197-what-is-pi.html Has a video and other resources.
  6. Edutopia – http://www.edutopia.org/blog/pi-day-lesson-plans-matthew-davis  Lessons and activities for elementary students.
  7. Education World – http://www.educationworld.com/a_lesson/lesson/lesson335.shtml  Lessons and activities around Pi – multi-grade level.
  8. Teachπ.org – http://www.teachpi.org/  Lots of everything about Pi – books, activities, history, etc.

Finally, we have a fun graphing calculator (Prizm) activity that helps students discover Pi by exploring the circumference/diameter ratio using the picture of clock faces of multiple sizes.  It’s a great activity, especially if you want to integrate technology.  I’ve attached a PDF of the lesson, Investigation 4.3 Investigating Pi.  The activity includes sample answers that go through the calculator actions as well. Download PDF here: Investigating Pi FGTwCT Fostering Geometric Thinking – Investigation 4.3

Have fun planning for Pi Day!