Key visualizations: Algebra II

As students begin to explore polynomial operations, concrete models are invaluable tools. This is especially true as students begin to learn the algebraic technique of “completing the square.” By building a model of the square and examining concretely the relationships between the coefficients in the expression x2 + bx + c and the dimensions of the square, students develop a conceptual foundation for the algorithm they will use to complete the square. As students continue to build proficiency with the algorithm, they should no longer need the concrete model. However, the experience of working with the model should help them each time they apply the algorithm.

 Connections to the TEKS [−]
This animation addresses the following Algebra II TEKS:
(1) Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
(C) select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;
(D) communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate;

(4) Quadratic and square root functions, equations, and inequalities. The student applies mathematical processes to understand that quadratic and square root functions, equations, and quadratic inequalities can be used to model situations, solve problems, and make predictions. The student is expected to:
(D) transform a quadratic function f(x) = ax2 + bx + c to the form f(x) = a(x - h)2 + k to identify different attributes of f(x);

In this animation, students use algebra tiles to explore the technique of completing the square to understand different forms of a quadratic function. They build understanding about the different attributes of a quadratic function by connecting the expression that results from completing the square to the vertex of the graph of a quadratic function.

As students build their models, they must reason about the relationships among the areas in the models and the numbers in the quadratic expression and communicate their reasoning with symbols, making connections between the two representations of the problem. They use mental math in completing their square model, noticing they must “split” the middle term of the quadratic expression, and that they must then square this “split” number to complete the model.

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Here is the model for the quadratic expression x2 + 5x + 7.
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Correct! Can you form a square using the tiles in your model?
Correct! Can you form a square using the tiles in your model?
First, use tiles to model the general form of the quadratic expression x2 + 6x + 11. Click the Check button to check your model.
Correct! You have used one x2 tile, six x tiles, and nine unit tiles to form the square and you have two unit tiles left over. What quadratic expression does your model represent?
Form a square in the workspace. Use the x2 tile, the x tiles, and as many unit tiles as possible. Click the Check button to check your model.
Now that you have completed the square (with two unit tiles left over), you see that you can write the algebraic form of the quadratic as y = (x + 3)2 + 2.
Notice that the square you have built has area (x+ 3)2. This square represents only part of the original quadratic expression x2 + 6x + 11.
First, use tiles to model the general form of the quadratic expression x2 + 6x + 11. Click the Check button to check your model.
Form a square in the workspace. Use the x2 tile, the x tiles, and as many unit tiles as possible. Click the Check button to check your model.
Notice that the square you have built has area (x+ 3)2. This square represents only part of the original quadratic expression x2 + 6x + 11.
Click the Check button to check your model.
Correct! Do you see any zero pairs? If so, click the tiles in the workspace to select the zero pairs, and then click the Remove button.
Try again. Look at the coefficients of the variables and the constant term in the expression x2 + 6x + 11 to decide how many of each tile you need. You can remove the extra tiles by dragging them from the workspace.
You have correctly removed all opposites from the workspace. Now, in the text box below enter an expression that matches the tiles. (For x2 terms, type x followed by the Exp button.) Check when you are done.
The tiles you selected don’t form a zero pair. Try again.
There are still more zero pairs in the workspace.
The workspace tiles are correct, but the answer you typed does not match the workspace. Look at the tiles in the workspace. What value do they represent?
Correct! Notice that there are no zero pairs. Do you understand why? Now type in the answer to the addition problem and click the Check button.
Correct!
Correct! Now model -(7x - 8) in the workspace on the right. Check your model after you have finished building it.
Nice work! You have correctly used algebra tiles and the concept of zero pairs to demonstrate that -3 + 3 = 0.
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Tip: you may drag or click on the tiles to add them to the active workspace.
Click the Show expression button to see the algebraic expression these tiles represent, or move more tiles into the workspace to continue building your expression.
The expression represented by the tiles in the workspace is shown here. Move more tiles into the workspace, or click the Clear button to start over. Keep building expressions with the tiles until you understand what each tile represents.
This is the correct model for the expression x2 + 6x + 11. Can you form a square using the tiles in this model?
You need to finish the square. Use as many of the unit tiles as you can.
Try again. You need to use all tiles representing x2 and 6x to form a square.
The square is formed with one x2 tile, six x tiles, and nine unit tiles. There are two unit tiles left over. What quadratic expression does this model represent?
Consider only those tiles that represent x2 and 5x.
Using these tiles, form the beginning of a square.
What should you do with the extra tile?
“Split” the tile so that you can still build a square.
What are the dimensions of this square? How can you use the 7 unit tiles to complete it?
You need 6$\frac{1}{4}$ of the unit tiles to complete the square. This leaves you with $\frac{3}{4}$ of a tile left over. You can write the function in vertex form as y = (x + 2.5)2 + 0.75.