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

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:

Allow students to work in small groups when doing this lesson.

- Combining like terms- Multiplying polynomials- Factoring- Solving equations

Construction of Alge-Tiles

1 (let the side = one unit)

1For one unit tile:(it is a square tile)

For a 1x tile

(it is a rectangular tile)

Area = (1)(1) = 1

(unknown length therefore let it = x)

1

Area = (1)(x) = 1x

Side of unit tile = side of x tile

For x2 tile:

(It is a square tile)Area = (x)(x) = x2

Side of x2 tile = side of x tile

Other side of x2 tile = side of x tile

x

x

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 →


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)


= 4x

1 +1x +2 → = 1x+3 (ctt)

-2x + 3x +1→ = 1x +1(ctt)

+

+ +

+ +

Combine like terms (use the tiles):

2x + 2x →

Using the zero property


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


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

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

Make the length = 3 tiles

The width = 2 tiles


Try: (2x)(3x)→

Therefore: (2x)(3x) = 6x2

Side: 3x

Side: 2x

Remove the sides


Try (1x + 2)(3)

Therefore: (1x + 2)(3) = 3x + 6 (ctt)

Side: 1x + 2)

Side: 3

Make rectangle, fill rectangle

Remove sides


Try (1x +2)(1x -1)

Tiles remaining: x2 + 2x – 1x – 2

Simplify to get: x2 + 1x – 2 (ctt)

Side: 1x - 1

Side: 1x + 2


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. : (1x + 2)Side 1

Side 2

Remove the rectangle and what is left are the factors of 2x +4

: (2)

2x + 4 = (2)(1x +2)

Part III: Factoring

Try factoring 3x + 6 with your tiles.

Factors → (1x + 2)(3)

Make the sides

Remove the rectangle

1x + 2

3

3x + 6 = (3)(1x + 2)

First make a rectangle

The sides are the factors

2

3

Part III: Factoring

Try factoring x2 + 5x + 6 (make rectangle)

Next make the sides for the rectangle

Remove the rectangle and you have the factors. (1x + 3) (1x + 2)

x2 + 5x + 6 = (1x + 3) (1x + 2)

**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.

Then add the x tiles where needed to complete the rectangle

(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

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

6

When the rectangle is finished examine it to see if the tiles combine to give you the original expression → x2 + 5x + 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

Next fill in the x tiles to make the rectangle.

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)

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

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

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

1x = 2

=

Part IV: Solving for X

Solve 3x + 1 = 7

=

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

1x = 2

Part IV: Solving for X

Solve for x: 2x – 1 = 1x + 3

Now add +1 tile to both sides… zero property

1x = 4

=

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)

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

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