Alge-Tiles
Making the Connection between the
Concrete ↔ Symbolic
(Alge-tiles)
↔
(Algebraic)
What are Alge-Tiles?
 Alge-Tiles are rectangular and square
shapes (tiles) used to represent integers and
polynomials.
Examples: 1→
1x →
1x2 →
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.
Construction of Alge-Tiles
1 (let the side = one unit)
For one unit tile:
(it is a square tile)
1
Area = (1)(1) = 1
x
For a 1x tile
(it is a rectangular tile)
(unknown length therefore let it = x)
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)
Other side of x2 tile = side of x tile
Area = (x)(x) = x2
x
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
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 →
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 can’t be simplified any further (can’t touch this)
Part I: Combining Like Terms
Combine like terms (use the tiles):
+
2x + 2x →
1 +1x +2 →
+
-2x + 3x +1→
Using the zero property
= 4x
= 1x+3 (ctt)
+
+
+
= 1x +1(ctt)
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.
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
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
Part II: Multiplying Polynomials
 Try: (2x)(3x)→
Side: 3x
Side: 2x
Remove the sides
Therefore: (2x)(3x) = 6x2
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
Part II: Multiplying Polynomials
 Try (1x +2)(1x -1)
Side: 1x - 1
Side: 1x + 2
Tiles remaining:
Simplify to get:
x2 + 2x – 1x – 2
x2 + 1x – 2 (ctt)
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.
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
i. Construct a rectangle using 2 ‘x’ tiles and 4 unit one tiles. This
can be tricky until you get the hang of it.
Part III: Factoring
 Now make the sides; width and length of the
rectangle using the alge-tiles.
Side 1 : (1x + 2)
Side 2 : (2)
2x + 4 = (2)(1x +2)
Remove the rectangle and what is left are the factors of 2x +4
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)
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(1x
with+ 2)
the unit tiles.
Then add the x tiles where needed to
complete the rectangle
3
2
When the rectangle is finished examine
to see
if+the
tiles
combine
to give
you
2it +
x
5x
6
=
(1x
+
3)
(1x
+
2
the original expression → x + 5x + 6
Next make the sides for the rectangle
Remove the rectangle and you have the factors. (1x + 3) (1x + 2)
2)
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.
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?”
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.
1x - 3
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
1x + 2
middle term, -1.
-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.
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
Part IV: Solving for X
 Solve 3x + 1 = 7
=
1x = 2
Add a -1 tile to both sides
Zero Property takes place
What’s left? 3 ‘x’ tiles = 6 unit tiles (how many groups of 3 are in 6)
Therefore 1x tile = 2 unit tiles
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
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)
