Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Quadratic Equation
- Real Numbers
Purpose and Instructional Strategies
In grade 8, students solved quadratic equations in the form of 2
. In Algebra I, students
solve quadratic equations in one variable over the real number system. In later courses, students
will solve quadratic equations in one variable over the real and complex number systems.
- 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 -intercepts, or roots, of the graph of a
- 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 in
the equation 2 + + = 0; making the connection to solving literal equations.
- Instruction includes evaluating the discriminant (2– 4) 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 -intercepts and that quadratic functions that do not produce real roots
never touch the -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 2 + 8 = 48 and solve by factoring to show that the
two solutions are = −12 and = 4. Then, ask the question “If we look at the
binomial 2 + 8 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 ( + 4)² = 64. Have student solve using square roots
to find the same solutions of = −12 and = 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 = −162 + 28. Students solve this equation to find that ≈ ±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
- For example, given the equation 0 = 42 − 12 + 9, students can identify , ,
and as 4, −12 and 9, respectively. Students should recognize that and are
perfect squares; where is 22= 4 and is (−3)2 = 9. Since the coefficient of
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 2 and term,
students may forget to consider that factor when adding/subtracting from the other side of
- For example, when solving 10 = 122 + + 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 2 – 4 + 6 = 27 and use the number 2, −4
and 6 as , and , 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 in the equation 62 − 17 +
12 = 0, students can set up the two equations using the quadratic formula: = and =
- 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
- Teacher co-creates a graphic organizer, such as the one below, to include different
methods used to solve quadratics given specific forms.
Instructional Task 1 (MTR.2.1)
- Given the equation 2 + 6 = 13, what value(s) of satisfy the equation?
Instructional Task 2 (MTR.3.1)
- Given the figure below, write an equation that could be used to determine the length and
width of the rectangle.
Instructional Item 1
- Devonte throws a rock straight down off the edge of a cliff that overlooks the ocean. The
distance () the rock falls after t seconds can be represented by the equation = 162+
24. 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?
Instructional Item 2
- What are the solutions to the equation −0.252 + 4 = 0.75. Round to the nearest tenth if
Instructional Item 3
- What are the exact solutions to the equation 52 − + = 0?
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.