Alge-Tiles

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Alge-Tiles

• Making the Connection between the Concrete ?
  Symbolic
• (Alge-tiles) ? (Algebraic)

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What are Alge-Tiles?

• Alge-Tiles are rectangular and square shapes
  (tiles) used to represent integers and
  polynomials.

Examples 1?
1x ?
1x2 ?
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Objectives for this lesson

• Using Alge-Tiles for the following

- Combining like terms - Multiplying
polynomials - Factoring - Solving equations
Allow students to work in small groups when doing
this lesson.
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Construction of Alge-Tiles
1 (let the side one unit)
1
For one unit tile (it is a square tile)
Area (1)(1) 1
x
(unknown length therefore let it x)
For a 1x tile (it is a rectangular tile)
1
Side of unit tile side of x tile
Area (1)(x) 1x
x
Side of x2 tile side of x tile
For x2 tile (It is a square tile)
Area (x)(x) x2
x
Other side of x2 tile side of x tile
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Part I Combining Like Terms

• Prerequisites prior to this lesson students
  would have been taught the Zero Property
• Outcomes Grade 7 - B11, B12, B13 Grade 8
  B14, B15 Grade 9
  B8 Grade 10 B1, B3
• Use the Alge Tiles to represent the following
• 3x
• 3
• 2x2

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Part I Combining Like Terms

• For negative numbers use the other side of each
  tile (the white side)
• Use the Alge Tiles to represent the following

-2x ?
-4 ?
-3x - 4 ?
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Part I Combining Like Terms

• Represent 2x with tiles
• Represent 3 with tiles
• Can 2x tiles be combined with the tiles for 3 to
  make one of our three shapes? Why or why not?
• Therefore simplify 2x 3
• 2x 3 cant be simplified any further (cant
  touch this)

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Part I Combining Like Terms
Combine like terms (use the tiles)

4x
2x 2x ?
1x3 (ctt)
1 1x 2 ?


-2x 3x 1?
1x 1(ctt)


Using the zero property
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Part I Combining Like Terms

• After mastering several questions where students
  were combing terms you could then pose the
  question to the class working in groups
• Is there a pattern or some kind of rule you can
  come up with that you can use in all situations
  when combining polynomials.
• In conclusion, when combining like terms you can
  only combine terms that have the same tile shape
  (concrete) ? Algebraic Can combine like terms if
  they have the same variable and exponent.

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Part II Multiplying Polynomials

• Prerequisites Students were taught the
  distributive property and finding the area of a
  rectangle.
• Area(rectangle) length x width
• When multiplying polynomials the terms in each
  bracket represents the width or length of a
  rectangle.
• Find the area of a rectangle with sides 2 and 3.
  Two can be the width and 3 would be the length.
• The area of the rectangle would (2)(3) 6

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Part II Multiplying Polynomials

• We will use tiles to find the answer. The same
  premise will be used as finding the area of a
  rectangle.

Make the length 3 tiles
The width 2 tiles
The tiles form a rectangle, use other tiles to
fill in the rectangle

• Once the rectangle is filled in remove the sides
  and what is left is your answer in this case it
  is 6 or 6 unit tiles

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Part II Multiplying Polynomials

• Try (2x)(3x)?

Side 3x
Side 2x
Therefore (2x)(3x) 6x2
Remove the sides
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Part II Multiplying Polynomials

• Try (1x 2)(3)

Side 1x 2)
Side 3
Therefore (1x 2)(3) 3x 6 (ctt)
Make rectangle, fill rectangle
Remove sides
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Part II Multiplying Polynomials

• Try (1x 2)(1x -1)

Side 1x - 1
Side 1x 2
Tiles remaining x2 2x 1x 2
Simplify to get x2 1x 2 (ctt)
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Part II Multiplying Polynomials

• Pattern After mastering several questions where
  students were combing terms you could then pose
  the question to the class working in groups
• Is there a pattern or some kind of rule you can
  come up with that you can use in all situations
  when multiplying polynomials.
• This can lead to a larger discussion where
  students can put forth their ideas.

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Part III Factoring

• Outcomes Grade 9 B9, B10, Grade 10 B1, B3,
  C16
• Take an expression like 2x 4 and use the
  rectangle to factor.
• You will go in reverse when being compared to
  multiplying polynomials. (make the rectangle to
  help find the sides)
• The factors will be the sides of the rectangle

1. Construct a rectangle using 2 x tiles and 4
   unit one tiles. This can be tricky until you get
   the hang of it.

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Part III Factoring

• Now make the sides width and length of the
  rectangle using the alge-tiles.

(1x 2)
Side 1
Side 2
(2)
2x 4 (2)(1x 2)
Remove the rectangle and what is left are the
factors of 2x 4
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Part III Factoring

• Try factoring 3x 6 with your tiles.

1x 2
First make a rectangle
Make the sides
Remove the rectangle
3
The sides are the factors
Factors ? (1x 2)(3)
3x 6 (3)(1x 2)
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Part III Factoring

• Try factoring x2 5x 6 (make rectangle)

(1x 3)
Hint when the expression has x2, start with
the x2 tile.
Next, place the 6 unit tiles at the bottom right
hand corner of the x2 tile. You will make a small
rectangle with the unit tiles.
(1x 2)
3
2
Then add the x tiles where needed to complete the
rectangle
x2 5x 6 (1x 3) (1x 2)
When the rectangle is finished examine it to see
if the tiles combine to give you the original
expression ? x2 5x 6
Next make the sides for the rectangle
Remove the rectangle and you have the factors.
(1x 3) (1x 2)
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Part III Factoring

• What if someone tried the following

Factor x2 5x 6 (make rectangle)
Start with the x2 tile, now make a rectangle with
the 6 unit tiles.
Now complete the rectangle using the x tiles.
1
When the rectangle is finished examine it to see
if the tiles combine to give you the original
expression ? x2 5x 6
6
When the tiles are combined, the result is x2
7x 6, where is the mistake?
The unit tiles must be arranged in a rectangle so
when the x tiles are used to complete the
rectangle they will combine to equal the middle
term, in this case 5x.
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Factoring

• Have students try to factor more trinomials
• (refer to Alge-tile binder Factoring
  section F 3b for additional questions)

After mastering several questions where students
were factoring trinomials you could then pose the
question to the class Is there a pattern or
some kind of rule you can come up with that you
can use when factoring trinomials?
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Part III Factoring (negatives)

• Try factoring x2 - 1x 6

Start with x2 tile, then fill in the unit tiles
in this case -6 which is 6 white unit
tiles. Remember to make a rectangle at the bottom
corner of the x2 tiles where the sides have to
add to equal the coefficient of the middle term,
-1.
1x - 3
1x 2
-3
Next fill in the x tiles to make the rectangle.
2
Now the rectangle is complete check to see if the
tiles combine to equal x2 - 1x
6.
Therefore x2 - 1x 6 (x 3) (x
2)
Fill in the sides and remove the rectangle to
give you the factors.
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Part IV Solving for X

• Outcomes Grade 7 check, Grade 8 - C6, Grade 9
  C6, Grade 10-C 27
• Solve 2x 1 5 using alge-tiles
• Set up 2x 1 5 using tiles

1x 2
Using the zero property to remove the 1 tile you
add a -1 tile to both sides On the left side -1
tile and 1 tile give us zero and you are left
with 2 x tiles On the right side adding -1 tile
gives you 4 tiles Now 2 x tiles 4 unit
tiles, (how many groups of 2 are in 4) Therefore
1 x tile 2 unit tiles
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Part IV Solving for X

• Solve 3x 1 7

1x 2
Add a -1 tile to both sides Zero Property takes
place Whats left? 3 x tiles 6 unit tiles
(how many groups of 3 are in 6) Therefore 1x tile
2 unit tiles
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Part IV Solving for X

• Solve for x 2x 1 1x 3

1x 4
Now add 1 tile to both sides zero property
You are left with 2x 1x 4
Add -1x tile to each side zero property
Leaving 1x 4
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Alge-Tile Conclusion

• Assessment While students are working on
  question sheet handout, go around to each group
  and ask students to do some questions for you to
  demonstrate what they have learned.
• For practice refer to handout of questions for
  all four sections
• Part I Combining Like Terms
• Part II Multiplying Polynomials
• Part III Factoring
• Part IV Solving for an unknown
• (P.S. the answers are at the end)

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