Creating Probability Area Models

Directions: Using triangles to partition a square with side length of 6, create the following probability area models:

IMG_7188

1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9.

  1. How many triangles are needed to create each?
  2. If you are unable to create some of the models, explain why.
  3. What modifications to the problem is needed to create all 9 probability area models?
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Binomial Powers

Act 1:

What questions do you have when you watch this video?

Ask students to write down their questions, I normally ask students to find at least 3.  When I observe that most students have questions written, I ask them to share those questions with their neighbor.  I then throw up a Microsoft Word document and start typing down questions students supply.  Students from my classroom came up with all sorts of different questions, some we can easily answer and others that we can’t.  I am looking for a key question or questions to start this lesson.  If students do not ask one of these questions, I tell them that I hope I can answer most of the questions provided, but that I need them to consider one of these questions first.

Any of these type of questions will lead students down the inquiry I hope to explore with them.

  • Is there a pattern?
  • What is changing from one frame to the next?
  • Is the pattern constant?
  • What are the different different pieces called?

How do you square binomials?

ACT 2:

Once again, this video can create a few different paths of exploration.  We can explore:

  • Patterns
  • Squaring Binomials

These are both excellent topics and students generate a lot of classroom discourse discussing each one.

  1. Patterns:  Students will notice that I first make a square, each side the same length as the initial picture.  The thing that switches up is the replacement of a “flats” for  “longs”, “longs” for “cubes” and then fill the square with “cubes.”  Once students get the idea of the first couple of squares, then I will introduce math vocabulary for these: “flats” = x^2, “longs” = x, and “cubes” = constant.
  2. Squaring Binomials:  Students will make the same observations as the pattern students.  Using the math vocabulary, they should create expressions of what they are shown, and what it becomes.  Students may make notes of individual changes; x^2 replaces the x, etc.  This should lead them into the idea of squaring, or raising the binomial to the power of 2.  Discussion should happen about how the constant value is added at the end, wasn’t the constant replaced by x?  Some students will even try to physically measure the cubes versus the long to create a ratio.

ACT 3:

  • 25 cubes (25)
  • 4 longs (4x)
  • flat, 8 longs, 16 cubes (x^2 + 8x +16)
  • 4 flats, 8 longs, 4 cubes (4x^2 + 8x + 4)

Extensions:

  • Patterns: I ask students to create their own binomials, their squares and sketches of diagrams of both.  I will then ask what happens if there is a negative on either the variable or constant and how that effects our pattern.
  • Squaring Binomials: I ask students to try a quadratic and see if they can square it.

Reflection:

  1. Ask students to summarize what the activity was about.
  2. Ask students what mathematics they explored.
  3. Ask students what they understood really well.
  4. Ask students what they felt they needed more instruction on.