I first used Algebra Tiles, back in the 90’s, when I was teaching 8th grade and then continued to use them when I switched to 7th grade.  Now that I’m a math coach, I’ve been helping teachers learn how to integrate Algebra Tiles into their lessons.  This post and accompanying Desmos Activity Builder is a result of a recent attempt to introduce  Algebra Tiles to a 6th grade class.

I traditionally begin by establishing a pattern.   We discuss dimensions, exponents and area.  We draw lots of squares.  I then drew a square in between the grid lines.  This was my attempt to explain the variable concept.  If we don’t know exactly what the length is, then we could use the variable, x.

At first the 6th graders bought into the side of the square being, x.  But things went south quickly when the conversation turned to the rectangle.

• Me:  What are the dimensions of the small blue square?
• Students:  1 unit by 1 unit
• Me:  What are the dimensions of the big blue square?
• Students:  x units by x units
• Me:  What are the dimensions of the rectangle?

At this point, Mrs Shea and I circulated the room to check on students.  The following interaction happened multiple times.

• Me:  What have you decided on for the rectangle’s dimensions?
• Student(s):  1  by 4
• Me:  How did you determine that?  Students would show me the following picture.
• Me: Are you positive that the length is exactly 4 units?
• Student(s):  No.
• Me:  (Based on the conversation about the big square, I’m anticipating they would then say it has to be, x.  I was so WRONG!)
• Student(s):  Oh, the length is 3.5.
• Me:  How can you be positive?
• Student(s):  Look (directing my view to the above picture proving their answer of 3.5). 
• Me:  Could the length be 3.6 units?
• Student(s):  N0.

I stepped away to regroup and find a new line of questioning.

• Me:  What are the dimensions of the big blue square?
• Students:  x by x
• Me: (sliding the big blue square and the rectangle together) Knowing the the side of the square is x, what do you think the length of the rectangle is?
• Student(s):  Oh, it (the square) has to be 3.5 units like the rectangle.
• Me:  (On the inside)  AAAAHHHhhhhhhh!

6th graders struggled much more, with the concept of a variable, than their 7th and 8th grade counterparts.   This is understandable. We’re giving them 3 tiles and telling them that 1 is a constant and never changes and the other 2 involve variables, therefore can change.  But the physical x and x² pieces don’t grow or shrink depending on the value of x – very confusing for our concrete thinkers.

I vowed to NEVER repeat these conversations again. There had to be a better way for students, especially our concrete thinking 6th graders to understand why 2 of the algebra tiles were named with a variable.

**********************************************************

DESMOS to the Rescue

The Goal:  The purpose of the Desmos/Algebra Tile connection was to

1. Develop the concept of a variable
2. Explain why the pieces were named 1, x & x²

**********************************************************************

Algebra Tiles:  

Why are they named 1, x and x²? 

I had the opportunity to teach this lesson several times – each time refining the delivery. Thank you to Tiffany, Kim, Michelle, Peytra and Carmen for inviting me into their rooms to teach/co-teach this lesson.  Here’s the Desmos Activity: Algebra Tiles:  Why are they named 1, x & x²  I collaborated on the Desmos Activity Builder with J.J. Martinez.

Slide 1:  Algebra Tiles get their name from their area.  Therefore, the conversation had to include vocabulary such as dimensions, length, width and area from the beginning. After students, submit their answers for slide 1, I documented their responses on the white board. It was the perfect time to discuss units vs square units

Slide 2:  I gave the students a short period of time to move the rectangle. Then counted down from 10.  Once I said, “1”, students had to enter the dimensions of their rectangle.  Students really enjoyed the interactive aspect and many ignored the question – which is why I ended up giving a time limit.

Again, I documented their responses on the white board.  When I recorded the dimensions and area, I called on students.  Yes, I’ve read through the submitted answers, but requiring students to share out loud allows them to verbalize the math vocabulary.  I, also, prepared the quieter students ahead of time.

• Me:  I’m going to call on you to share your response.  You have a little bit of time.  Practice what you’re going to say.

Students noticed the following:

• The value of the length changed
• The value of the width stayed the same
• Apart from the units vs square units, the value of the area & length were equal.

Questions:

• Me:  Why did the value of the length change?  Why did the value of the area change?
• Students:  We could change the value of the length.
• Students:  The rectangle grew. The rectangle got longer.  We could make the rectangle shorter.
• Me:  What symbol do mathematicians use to represent something that can change?
• Students: (after discussion in their groups)  A variable!
• Me:  What symbol could we write to describing the changing length?
• Students:  x
• Me:  What can we write to describe the width column?
• Students:  1
• Me:  I have a few questions, that I expect everyone to answer in unison.  When I point to the class, that’s your cue to answer my question.
  • Me:  1 times 5?   Students:  5
  • Me:  1 times 11?   Students:  11
  • Me:  1 times 32?   Students:  32
  • Me:  1 times 100?   Students:  100
  • Me:  1 times 427?   Students:  427
  • Me:  1 times x?   Students:  x   (quick reference to the Identity Property)

Slide 3:  This slide gives students the chance to reflect on and write about the concept of a variable.

Slides 6 – 8:  These slides repeat the questioning series completed w/ the rectangle with the big blue square.  

The tricky part was leading students to the area being x².  Some students made the connection of x by x means there are 2 of the same number being multiplied, hence, x²  Eventually, I incorporated the following:

Me:  (recapping).

• The orange square didn’t have any moveable/changeable sides.  So the area is 1 square unit.
• The green rectangle has 1 changeable side.  The area is x square units.
• The big blue square has 2 changeable sides.  What do you think is the area?
  

*******************************************************************

The Next Day

Activity 1:  We started the next class by passing out this worksheet.  Students a few minutes to fill out what they could within their groups before talking about it whole class.

This picture documents the first time, I used this worksheet.  At first, we used the word “moveable”.  Since variables describe situations that change, the word “moveable” was substituted out for “changeable”

********************************************************************Activity 2: We then moved onto a quick kinesthetic vocabulary activity

• Me:  Lift up the shape known as x
• Me:  Show me the shape known as 1.  etc…

Mrs. Becker leading her class in the “Show Me” activity

Clues for students to lift the small square: Show me:

• The shape known as 1.
• The shape with the algebraic name of 1
• That shape that has an area of 1
• The constant
• The shape that doesn’t change

Clues for students to lift the rectangle.  Show me the shape:

• known as x.
• with the algebraic name of x
• that has an area of x
• that has one side that changes
• has one dimension that changes

Clues for students to lift the big square: Show me the shape:

• known as x²
• with the algebraic name of x²
• that has an area of x²
• that has 2 side that change
• that has 2 changing dimensions 

Combination lift ups.  Lift the shapes

• that have a width of 1 unit
• that has at least one changing dimension
• that involves a variable.

********************************************************************

Transitioning to the Algebra Tiles

After the “Show Me” activity, we transitioned to using the actual algebra tiles. 🙂

********************************************************************