standard_description: Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
problem_activity: Activity
The goal of this activity is for students to recognize rewriting a quadratic expression from standard form to vertex form essentially entails completing the square.
Students had quite a bit of experience completing the square, mostly in the context of solving equations, in which they know to add or subtract the same number from both sides of an equation to keep the equation true. Here students are dealing with expressions and need to be careful to keep each expression equivalent to the original. If they add a number to the expression, they need to remember to subtract the same number to keep the value of the expression unchanged.
Launch
Arrange students in groups of 2. Give students quiet time to think about the first two questions and then time to share their thinking with a partner.
Pause for a class discussion after the second question and invite one or more students to demonstrate their strategy. Highlight approaches that involve completing the square. If no students took that approach, display and discuss the following:
\(\displaystyle \begin {align} &x^2 - 6x + 2\\ &x^2 - 6x + 9 - 9 + 2 \\&(x - 3)^2 - 7 \end {align}\)
Then, ask students to proceed with the rest of the activity. Provide access to devices that can run Desmos or other graphing technology.
Representation: Internalize Comprehension.
Use color and annotations to illustrate student thinking. As students share their reasoning about how to convert standard form back into vertex form, scribe their thinking on a visible display. Use annotations and labels to show the structure of the steps using the form included in the launch. For example, labeling the first step as “standard form,” labeling +9 as “add nine to complete the square,” and drawing an arrow to the -9 and labeling it as “keeping it equivalent.” Continue to add in any key phrases students contributed that illustrate the steps.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Student Facing
Here are two expressions in vertex form. Rewrite each expression in standard form. Show your reasoning.
\((x+5)^2+1\)
\((x-3)^2-7\)
Think about the steps you took, and about reversing them. Try converting one or both of the expressions in standard form back into vertex form. Explain how you go about converting the expressions.
Test your strategy by rewriting
\(x^2+10x+9\)
in vertex form.
Let’s check the expression you rewrote in vertex form.
Use graphing technology to graph both
\(x^2+10x+9\)
and your new expression. Does it appear that they define the same function?
If you convert your expression in vertex form back into standard form, do you get
\(x^2+10x+9\)
?
Student Response
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Anticipated Misconceptions
Some students may wonder why they need to both add and subtract a number from the standard form expression in order to complete the square. In previous lessons, students completed the square while solving an equation, often adding the same number to each side of the equal sign to maintain equality. Here, there is no equal sign. Emphasize that each move must generate an equivalent expression. Show a few expressions such as 5,
\(5+0\)
,
\(5-2+2\)
,
\(5+7-7\)
,
\(5+8\)
, and
\(5+8-3\)
. Ask them which ones are equal to 5. Next, ask them to write 2 more expressions that include the number 5 and also equal 5. Encourage students to notice that the sum of the numbers excluding 5 must be 0.
Activity Synthesis
Invite students to share their graph and response to the last question. Emphasize that while graphing is a quick way to check whether two expressions define the same function, it is not always reliable. If we make an algebraic error, the graph would almost certainly show it. But having two graphs that
appear to be
identical does not
prove
that two expressions are indeed equivalent. We can only be sure that two expressions define the same function by showing equivalence algebraically. For example, when students convert the expression in vertex form back to standard form, does it produce the original expression?
Make sure students understand that whatever operation is performed on an expression to complete the square, it should not change the value of the expression. Adding opposites (for example, 9 and -9, or -25 and 25), or adding and subtracting the same number, has the effect of adding 0, which keeps the original and the transformed expressions equivalent.


aligns: no