Reflecting Powers- Open Middle Problem (maybe?)

When working with a student today, she noticed that:

power example.pngand wondered if that pattern works for any other numbers.

 

So, for any integer a,b when does

power example 2.png

 

Would changing the constraint on a,b change the problem?  What number types would produce more values for a and b?

Advertisements

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.

PowerBlocks -3Act Math

ACT 1:

This video was inspired from comments made in Dan Meyer’s My Opening Keynote for CUE 2014.

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?
  • Is this an Arithmetic Pattern?
  • Is this a Geometric Pattern?
  • How did you make a cube?

ACT 2:

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

  • Patterns
  • Rules of Exponents

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, and then a cube with each side the length of the number of cubes in the square.  The cubing part is where most classes will struggle, many will just try to create a cube out of the preceding square.
  2. Rules of Exponents: This is what I designed this video for, in an attempt to create a visual representation of (x^a)^b = x^(a*b).  Using the unifix cubes created a quick, easy way for students to quickly see and do the mathematical calculations.  The powers I started with were easily recognizable visually: (x^2)^3.  One thing my students start to see is how the base figures into all of this, we normally pause the video and use the SMARTBoard to draw lines to create the pattern of multiplying the base.  For example on slide 3: we circle the bottom left 2 blocks, one stack represents 2^2, two stack represents 2^3, four stacks represent 2^4, etc.

ACT 3:

  • Frame 9: (4^2)^3 = 4096 => 4^6
  • Frame 12: 5^2 = 25
  • Frame 13: (5^2)^3 = 15625 => 5^6

 

Extensions:

  • Patterns: I ask students to predict what a Step 4 figure would look like
  • Rules of Powers: Have the students determine a third power and sketch what their figure would look like & how many blocks it would take to create it.