Concrete models for algebraic expressions
Alfinio Flores Peñafiel
University of Delaware
Cut out several small squares of side 1, rectangles of sides x 1 (any length will work, as long as it is not an integer multiple of the length of the side of the small square), and bigger squares of side x. The areas of these pieces represent the algebraic terms x2, x, and 1.
Display 2 bars to represent 2x.
Add five unit squares to represent 2x + 5.
Represent 3x + 4. Take 3 bars and 4 unit squares.
Even though these first steps may seem too simple, they help students prevent the development of strategies that focus on blind manipulation of symbols and lead to wrong answers. A common mistake among students who do not have the opportunity to use concrete representations of the algebraic symbols is to write 7x as a shorter version of 3x + 4, or thinking of 5 as the simplification of the expression 5x - x.
Addition
Represent the expressions in each part of the sum
(2x + 4) + (3x + 2)
What is the sum?
Build the following problems and determine the sums
(2x + 7) + (x + 5)
(3x + 6) + 4x
Subtraction
Represent (6x + 4) - 5x
One meaning of subtraction is take away. You can build (6x + 4), and then take away 5x.
Another meaning of subtraction is finding the difference. You can build both (6x + 4) and 5x and find what the difference is.
Another meaning of subtraction is completion. Start with 5x and find what you need to add to have 6x + 4.
Build the following problems with the corresponding bars and squares and determine the difference.
(4x + 3) - 2x
(3x + 4) - 2
(5x + 3) - (3x + 1)
At some point, students will prefer to use diagrams rather than manipulating the concrete objects.
Perform the indicated operations by drawing a diagram for each.
3x + 5 + 2x
5x + x
4x2 - 2x2
(x2 + 4) + (x + 2)
(x2 + 3x + 3) - (x + 2)
Multiplication
Form a rectangle with three rows of two unit squares each. The length of the rectangle is 3, the width is 2, its area is 6.
Represent the following products as a rectangle
3 4, 1 5, 2 6
We can also form a rectangle that has a length of x, and a width of 2 units.
Represent the following products as rectangles
3 x
5 x
We can also represent as rectangles products like 3(x + 2). The rectangle will have a length equal to x + 2, and a width of 3.
Products that involve x x
We can also represent products that involve a length of x and a width of x. The product xx is the square x2.
The product x(x + 2) can be represented with a rectangle of length x + 2, and width x.
Represent x(x + 3) as a rectangle.
Represent (x+1)(x + 2) as a rectangle. This product is the same as x(x + 2) + 1 (x + 2). Identify the rectangles corresponding to each of these products.
Represent the following products as rectangles.
(x+2)(x + 3)
(x+1)(x + 4)
(x+2)(x + 2)
(2x+1)(x + 2)
In what cases will the rectangle be a square?
Represent the square (x + 3)2
Factoring binomials
Given a number, we can construct rectangles that have two of its factors as length and width. For example, 6 can be written as 2 3, and therefore we can represent this relation with a rectangle of length 3, width 2, and area 6.
In the same way we can look to represent algebraic expressions in terms of their factors.
For example, the expression 3x + 15 could be written as 3(x + 5), or represented by a rectangle of length x + 5, and width 3.
Find the factors of the following expressions and represent them as a rectangle
4x + 2
x2 + 3x
6x + 8
Multiplication of binomials: A geometric representation
Review: Area model for multiplication
When we multiply two two-digit numbers, we take care of multiplying all components to each other (using distributive property) and group those terms that have the same place value. On the right we have “expanded” the usual algorithm to make all steps explicit. Explain what partial products do each of the following numbers represent 6, 40, 30, 200.
23 (20 + 3)
12 (10 + 2)
______
46 40 + 6
230 200 + 30
______
276 200 + 70 + 6
We can also represent the product 23 12 as the area of a rectangle with length 23 and width 12.
Identify in the figure below where are the different partial products 2 3, 2 20, 10 3, 10 20, represented.
Multiplication of binomials.
In the same way, when we multiply two algebraic expressions like (x + 2) (x + 3) we need to use the distributive property to obtain all the partial products, group them according to their kind, and add them.
x + 2
x + 3
_____
3x + 6
x2 + 2x
______
x2 + 5x + 6
Identify what partial product represent the following terms in the above procedure
6, 3x, 2x, x2.
There are several ways to multiply these two terms. The important thing is to make sure we include all partial products. Multiply the first term of the first parenthesis by each of the terms of the second. The multiply the second term in the first parenthesis by each of the terms in the second parenthesis.
(x+2) (x+3) = x2 + 3x + 2x + 6 = x2 + 5x + 6
Explain in your own words how the distributive and commutative properties are used to multiply the two terms in one of the above methods to multiply binomials.
The product of the binomials (x + 2) (x + 3) can also be represented as the area of a rectangle of length x + 3 and width x + 2. Identify in the figure below where are the partial products
xx, x 3, 2 x, 2 3 represented.
In the following section we will see how this geometrical representation of algebraic terms can help us to factor out second degree polynomials.
A concrete model to factor second degree polynomials
Cut out small squares of side 1, rectangles of sides x 1 (any length will work, as long as it is not an integer multiple of the length of the side of the small square), and bigger squares of side x.
The areas of these pieces represent the algebraic terms x2, x, and 1
These models can help us find how to factor out a polynomial like 2x2 + 7x + 3.
Example.
To represent a with the squares and rectangles a polynomial like x2 + 2x + 1 we need to use the corresponding pieces in the appropriate numbers. We need one square that represents x2, two rectangles that represent x, and one square that represents 1.
In order to factor out such an expression, wee need to try to form a rectangle (remember a square is also a rectangle) with all the pieces. The sides of the rectangle will be the factors. Form a rectangle using the pieces shown above.
The square has sides of length x + 1, so that x2 + 2x + 1 = (x + 1)(x + 1) = (x + 1)2
Other examples
Use one x2 square, four x rectangles, and four unit squares to represent x2 + 4x + 4.
Arrange all the terms into a rectangle and find the factors as the sides of the rectangle.
Use one x2 square, three x rectangles, and two unit squares to represent x2 + 3x + 2.
Arrange all the terms into a rectangle and find the factors as the sides of the rectangle.
Use one x2 square, five x rectangles, and six unit squares to represent x2 + 5x + 6.
Arrange all the terms into a rectangle and find the factors as the sides of the rectangle.
Use two x2 squares, seven x rectangles, and six unit squares to represent 2x2 + 7x + 3.
Arrange all the terms into a rectangle and find the factors as the sides of the rectangle.
Use two x2 squares, four x rectangles, and two unit squares to represent 2x2 + 4x + 2.
Arrange all the terms into a rectangle and find the factors as the sides of the rectangle.
References
Flores Peñafiel, A. & Mirabal, F. Prácticas de laboratorio: Algebra. Comunicaciones del CIMAT, 1986.
Ramirez, N. Conceptual Algebra. Author, 1992.
Other algebraic expressions represented as areas
Completing the square
Factoring algebraic expressions with negative terms
With the algebra tiles, blue, green, and yellow pieces represent positive terms.
Red pieces represent negative terms.
Red pieces cancel the corresponding parts of blue, green, or yellow pieces when they overlap. Each of the following combinations represent zero, because a red piece covers another of different color with the same amount of area.
red on top of yellow red on top of green red on top of blue
We can also overlap pieces with different areas. We can figure out the value of the area that has not been cancelled out. For example, superimposing a red rectangle over a blue square, will cancel part of the area.
These pieces represent x2 - x . We can overlap the red piece on top of the blue one. Only one part of the blue area will be cancelled.
The remaining area is thus x2 - x. The width and length of the rectangle that has not been cancelled is x by x - 1. therefore, x2 - x = x (x - 1)
We can form a rectangle with the following pieces: x2, 2x, -x, -2. The net value of these pieces is x2 + x - 2 (because one x cancels with the -x).
The length and width of the rectangle that has not been cancelled are x + 2 by x - 1.
Therefore x2 + x - 2 = (x + 2)(x - 1)
We can also use positive and negative pieces to form rectangles with a given length and width. We can use red pieces on top of pieces of other colors to cancel exceeding areas.
Example. Form a rectangle that is (x - 1) by (x + 3).
We can start with a rectangle x by (x + 3), that is, it has the desired length.
To reduce the width of this rectangle, we can place a red rectangle on top of the blue square, and three little red squares on top of the green rectangles. The area that has not been cancelled is a rectangle (x - 1) by (x + 3)
Dealing with double cancellation.
Form a square with these pieces: x2, -2x, 1
When you overlap a vertical red rectangle and a horizontal red rectangle on top of the blue square, you will see that there are two red square units at the corner and only one blue beneath them. Adding the positive yellow square will cancel the extra red area.
Other representations
Another way to represent the overlap is to show the color of the piece on top. Because we know the dimensions of the pieces, we can tell when a piece has been partly covered. For example, starting with x + 3x, we can overlap -x - 3 on top. The dimensions of the rectangle that has not been cancelled are (x - 1) by (x + 3)
What are all the pieces involved in the next diagram? What is their net value (after cancellation)? What are the dimensions of the area that has not been cancelled?
On top of x2 + 1 we overlap -2x
Express the value of all the pieces involved. What are the dimensions of the remaining square?
On top of x2 + 2, overlap -3x
Express the result as the product of two binomials and as the sum of all the pieces involved.
In the following diagram, red pieces have been placed over positive pieces, so that the non-cancelled area is a rectangle. Determine what are all the pieces involved, both visible and covered. Express their value as a product of binomials.
Exercises.
Form rectangles with the following sets of pieces by overlapping red pieces with pieces of other colors. Determine the area that has not been cancelled and express it as the product of two binomials.
a) x2, -2x, x, -2b) x2, -x, x, -1
c) x2, -3x, 2d) x2, -2x, 2x, -4
e)2 x2, -2x, 2x, -2
Form rectangles with pieces that add to the following values. You may need to add pieces that cancel each other to form the rectangle. Express the non cancelled area as the product of two binomials.
f) x2 -1g) 3x2 -x -2
h) x2 -x -2i) x2 + x - 6
Use positive and negative pieces to form rectangles with the given length and width. Use red pieces on top of pieces of other colors to cancel exceeding areas. Write the value of the area in terms of the pieces used.
j) (x - 3) (x + 1)k) 2x (x - 1)
l) (2x + 1) (x - 1)m) (2x + 4) (x - 2)
Another representation of products of binomials with negative terms
There is another way to use algebra tiles to represent products of binomials involving negative terms. The idea behind is that because the tiles have red color (negative) on the other side of a color representing a positive number, multiplying a tile by -1 is the same as flipping that tile. In this representation, indicators of length are separated from the rectangle that represents the product of binomial to avoid confusion. In this handout the width of the length indicators has also been reduced to make clear they represent a length and not an area.
To represent a product of binomials involving negative terms we start first with the corresponding product of binomials where all terms are positive. For example, if we are interested in multiplying (x -1)(x -1), the corresponding product would be (x +1)(x +1). Represent this product (x +1)(x +1) = x2 + 2x + 1
Now change one yellow length indicator for a red one.
Flip all the tiles in front of a red segment.
(x + 1)[(x +(-1)] = x2 + x + -x + -1 = x2 + -1 = x2– 1
Now change the other yellow length indicator for a red one too.
Flip all tiles below the second red segment.
(x + (-1))(x + (-1)) = x2 + 2(-1)x + (-1)(-1) = (x – 1) (x – 1) = x2 – 2x + 1
Notice that the little yellow unit square was flipped twice, so at the end it is again positive.
Use the algebra tiles to represent the following products. Write first and represent the product of the corresponding binomials with positive terms. Then change the necessary length indicators and flip the corresponding rectangles in the product. Write the value of the pieces (after canceling).
a) (x – 1) (x – 2)b) (x – 1) (x + 1)
c) (x – 2) (x – 3)d) (x – 1) (x + 3)
Products involving -x
We can use the same approach if one or two of the binomials involve -x.
For example, to multiply (-x + 1) (x – 2), we find the corresponding product with all terms positive, (x + 1) (x + 2)
Then we change the green length marker on the side (x + 1) for a red one, and flip all the pieces in front of it. The new rectangle represents (- x + 1) (x + 2).
Now change the two yellow unit markers on the side representing (x + 2) for two red ones, and flip all the pieces below them.
Notice that two of the long rectangles were flipped twice and are positive now.
We have thus (-x + 1) (x – 2) = - x2 + 3x - 2
Use the algebra tiles to represent the following products. Write first and represent the product of the corresponding binomials with positive terms. Then change the necessary length indicators and flip the corresponding rectangles in the product. Write the value of the pieces (after canceling).
a) (-x – 1) (x – 2)b) (-x – 1) (x + 1)
c) (-x – 2) (x – 3)d) (-x – 1) (-x + 3)
Completing the square with negative terms
We can represent the term as as blue square of positive area. Then we overlap four red tiles each of area . Areas of different color cancel out. There is a little red square protuding that is not cancelled.
To cancel this, we add four little unit squares (in yellow). The remaining blue area represents
Concrete representation algebra, Guanajuato 20111