The Equation of a Circle

Great geometry lesson

Easing the Hurry Syndrome

Expressing Geometric Properties with Equations

G-GPE.A Translate between the geometric description and the equation for a conic section

  1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

How do you provide an opportunity for your students to make sense of the equation of a circle in the coordinate plane? We recently use the Geometry Nspired activity Exploring the Equation of a Circle.

Students practiced look for and express regularity in repeated reasoning. What stays the same? What changes?

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It’s a right triangle.

The hypotenuse is always 5.

The legs change.

What else do you notice? What has to be true for these objects?

The Pythagorean Theorem works.

How?

Leg squared plus leg squared equals five squared.

What do you notice about the legs? How can we…

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CAS with Unit Rates

Using CAS to allow students to investigate unit rates opens up a world of options due to the device recognizing words as variables.  Teachers can actually enter one cup of sugar makes 2 cookies in fraction form on a TI-Nspire CX CAS and get the results shown below.

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Asking questions that promote deep thinking about the topic gets students into the discussion.

  • Why are these true?
  • What stays the same?
  • What changes?
  • What other ratios would work?
  • What other ratios would NOT work?

Asking the right questions is key to starting effective discussions, but what then? Multiple representations help students visually see what the math is doing which in turn leads to better understanding and skill mastery.  Look at the following situation:

Joe can mow 7 lawns in 4 hours.  How many lawns can he mow in 3 days?

Now look how the CAS can investigate this problem using Numerical form, a table, and a graph.

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This allows students to see relationships between “doing the math” and “using the math.”  See how long it takes the students to realize that the data is in hours but the problem is in days.  The number 3 is nowhere in the data which will generate questions from students.  This opens up a broad range of “teachable moments” which is what teachers love to see. The students want you to explain instead of you begging them to listen. Now let’s throw in some geometry.

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This introduces using the unit rates for unit conversion.  Although unit rates are a 6th grade skill, students can see how solving equations will play a part even though they haven’t mastered that particular skill.  Ask the students to investigate how the device got that answer.  You may never have to teach “cross multiply” again. 🙂

10 Yeah But’s I hear in Education with Possible Solutions!

Here are some helpful ideas to overcome come excuses for not using technology.

Inside the classroom, outside the box!

“Progress is impossible without change, and those who cannot change their minds cannot change anything.” By George Bernard Shaw

Over the years, I have heard a lot of what I call “Yeah but’s”….these are the excuses/arguments educators make that usually have a fixed mindset verse a growth mindset. As a leader (you don’t need to be in a leadership position, to be a leader) you need to be able to navigate around the yeah but’s; here are a few I have heard with some possible solutions.

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1. “Yeah But…I don’t have the time to integrate ________ (fill in the blank technology, PBL’s, data analysis etc.)” Solution: Try to find a way to make it so they are saving time, see the value in it and how it connects to the curriculum. For example: start slow by offering to create a Project or Problem Based Learning (PBL) for them that…

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SMP-8: attend to precision #LL2LU

Experiments in Learning by Doing

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We want every learner in our care to be able to say

I can attend to precision.
(CCSS.MATH.PRACTICE.MP6)

But what if I can’t attend to precision yet? What if I need help? How might we make a pathway for success?

Level 4:
I can distinguish between necessary and sufficient conditions for definitions, conjectures, and conclusions.

Level 3:
I can attend to precision.

Level 2:
I can communicate my reasoning using proper mathematical vocabulary and symbols, and I can express my solution with units.

Level 1:
I can write in complete mathematical sentences using equality and inequality signs appropriately and consistently.

 How many times have you seen a misused equals sign? Or mathematical statements that are fragments?

A student was writing the equation of a tangent line to linearize a curve at the point (2,-4).  He had written:  y+4=3(x-2)

And then he wrote:

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He absolutely knows what he…

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CAS with Percentages

As I think about important skills for students, percentages stand out to me as an important skill for both elementary and middle school students that is not understood completely.  The students memorize the algorithms we set before them but never really reach the mathematical understanding of why and how we use them.  Using the TI-Nspire CX CAS handheld and some teacher preparation, students can delve into the why’s and how’s of percentages.  For example, start the lesson by putting the following slide on the board or send to student handhelds.

Percent Pic 1

Ask students the following questions:

  • What stays the same?
  • What changes?
  • Why do you think the last one is false?

Using a Quick Poll in the TI Navigator system or a cooperative learning strategy, facilitate student discussions on their answers.  The students can use their devices to investigate other percentages to see if their theories hold true. Teachers can use the activity Solving Percent Problems from the TI Math Nspired website as a follow up activity or intro activity for the lesson on using percents to solve problems.  As the unit progresses other investigations and discussion starters could look like this:

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Ask questions like, “What do you notice about the numbers?”, “What is the relationship between the numbers?”, “What conclusion can you make based on this pattern?”.  Also show some other percentages such as the slide below asking similar questions.

Percent pic 2


Of course as in any good unit of study, opportunities for practice and hands-on applications are needed throughout the unit to master the skill but getting students motivated to understand the math is the first step.  Investigations such as these will allow students to delve deeper into the math instead of skimming the surface with algorithms only.

SMP-8: look for and express regularity in repeated reasoning #LL2LU

The levels are a perfect way to see student growth using the Math Best Practices.

Experiments in Learning by Doing

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We want every learner in our care to be able to say

I can look for and express regularity in repeated reasoning. (CCSS.MATH.PRACTICE.MP8)

But what if I can’t look for and express regularity in repeated reasoning yet? What if I need help? How might we make a pathway for success?

Level 4
I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

Level 3
I can look for and express regularity in repeated reasoning.

Level 2
I can identify and describe patterns and regularities, and I can begin to develop generalizations.

Level 1
I can notice and note what changes and what stays the same when performing calculations or interacting with geometric figures.

What do you notice? What changes? What stays the same?

Can we use CAS (computer algebra system) to help our students practice look for and express regularity in…

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