MA.912.AR.3.1

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

• MA.912.AR.1.2
• MA.912.AR.1.3
• MA.912.AR.1.7

Terms from the K-12 Glossary

• Coefficient 
• Quadratic Equation 
• Real Numbers 
• $x$-intercept

Vertical Alignment

Previous Benchmarks

• MA.8.AR.2.3

Next Benchmarks

• MA.912.AR.3.2

Purpose and Instructional Strategies

• Instruction includes the use of manipulatives, models, drawings and various methods, including Loh’s method. 
• Instruction allows the flexibility to solve quadratics using factoring techniques, taking the square root, using the quadratic formula and completing the square. Students should understand that one method may be more efficient than another depending on the content and context of the problem (MTR.2.1). 
• Instruction emphasizes the understanding that solving a quadratic equation in one variable is the same as the process of determining $x$-intercepts, or roots, of the graph of a quadratic function. 
• While the derivation of the quadratic formula is not an expectation of this benchmark, students can develop the quadratic formula by using completing the square to isolate $x$ in the equation $a$$x$² + $b$$x$ + $c$ = 0; making the connection to solving literal equations. 
• Instruction includes evaluating the discriminant ($b$²– 4$a$$c$) to determine whether there is one real solution (equals zero), two real solutions (equals a positive rational number) or two complex solutions (equals a negative rational number). 
  • Discuss the connection of the number of solutions of a quadratic equation to the graph of a quadratic function. Guide students to see that real solutions result in roots or $x$-intercepts and that quadratic functions that do not produce real roots never touch the $x$-axis (MTR.5.1). 
• Instruction on completing the square includes the use of algebra tiles or area model drawings. Students should understand the visual nature of completing the square and connect it to their prior work with area models. 
  • For example, consider $x$² + 8$x$ = 48 and solve by factoring to show that the two solutions are $x$ = −12 and $x$ = 4. Then, ask the question “If we look at the binomial $x$² + 8$x$ on the left, what would we have to add to it to make a perfect square trinomial?” Write the equation on the board to represent the question. Now, represent the equation using algebra tiles or by drawing an area model as shown below. 
    
    

• Lead student discussion about the quantity of 1 unit by 1 unit tiles needed to “complete the square” on the left hand side. Once they state the need for 16 tiles, point out that 16 tiles must also be added to the right to maintain equivalence then write the number 16 into both blank boxes. Students can then factor the perfect square trinomial to arrive at ($x$ + 4)² = 64. Have student solve using square roots to find the same solutions of $x$ = −12 and $x$ = 4 (MTR.2.1, MTR.4.1, MTR.5.1).

• In many contexts, students may generate solutions that may not make sense when placed in context. Be sure students assess the reasonableness of their solutions in terms of context to check for this (MTR.6.1). 
  • For example, the time it takes for a ball to drop from a height of 28 feet can be modeled by 0 = −16$t$² + 28. Students solve this equation to find that $t$ ≈ ±1.32 seconds. Through discussion, students should see that −1.32 seconds does not make sense in context and therefore should be omitted.
• Enrichment of this benchmark includes determining if a quadratic is a perfect square trinomial. 
  
  • For example, given the equation 0 = 4$x$² − 12$x$ + 9, students can identify $a$, $b$, and $c$ as 4, −12 and 9, respectively. Students should recognize that $a$ and $c$ are perfect squares; where $a$ is 2²= 4 and $c$ is (−3)² = 9. Since the coefficient of $b$ is twice the product of the square roots (−12 = 2(2)(−3)), it can be determined that the given equation is a perfect square trinomial.

Common Misconceptions or Errors

• When completing the square, many students forget to use the addition or subtraction property of equality to add or subtract values from the other side of the equation. Remind these students that additions or subtractions from one side of an equation must be replicated on the other to maintain equivalency. 
• When completing the square and removing a common factor from the $x$² and $x$ term, students may forget to consider that factor when adding/subtracting from the other side of the equation. 
  • For example, when solving 10 = 12$t$² + $t$ + 55, students may ultimately add 4 rather than 48 to the left side. Help students to see that there are 12 sets of 4 ultimately being added to the right and therefore, there must be twelve sets added to the left as well. 
• Some students may see equations such as $x$² – 4$x$ + 6 = 27 and use the number 2, −4 and 6 as $a$, $b$ and $c$, respectively, in the quadratic formula, arriving at an incorrect solutions. In these cases, graph the related function and ask students if the solutions they calculated correspond to the roots of the parabola. Once they see they do not, have students set the equation equal to zero and recalculate.

Strategies to Support Tiered Instruction

• When solving using the quadratic formula, instruction includes separating the two possible solutions to two equations. 
  • For example, when determining the values of $x$ in the equation 6$x$² − 17$x$ + 12 = 0, students can set up the two equations using the quadratic formula: $x$ =  and $x$ = 
• Teacher provides guided notes that show step-by-step directions for solving problems. 
  • For example, teacher can use the directions below when using the quadratic formula.
    
    
    

• Teacher co-creates a graphic organizer, such as the one below, to include different methods used to solve quadratics given specific forms.
  
   

Instructional Tasks

• Given the equation $x$² + 6$x$ = 13, what value(s) of $x$ satisfy the equation? 

• Given the figure below, write an equation that could be used to determine the length and width of the rectangle.

                         

Instructional Items

• Devonte throws a rock straight down off the edge of a cliff that overlooks the ocean. The distance ($d$) the rock falls after t seconds can be represented by the equation $d$ = 16$t$²+ 24$t$. If the ocean’s surface is 16.4 feet below the cliff, to the nearest tenth, how many seconds will it take for the rock to hit the ocean’s surface? 

•  What are the solutions to the equation −0.25$x$² + 4$x$ = 0.75. Round to the nearest tenth if necessary. 

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