Quadratic Equations with Irrational Solutions
5 min
Narrative
This Warm-up reminds students that we can use the notation A to express the side length of a square with area A and that the value of A may not be a whole number or a fraction.
Launch
Display the entire task for all to see. Give students 3 minutes of quiet think time. Select students to share their responses and how they reasoned about the side length and area of each square.
Student Task
Here are two squares. Square A has an area of 9 square units. Square B has an area of 2 square units.
- What is the side length of Square A?
- How does that side length compare to the solutions to the equation s2=9?
- What is the side length of Square B?
- How does that side length compare to the solutions to the equation x2=2?
Sample Response
- 3 units
- Sample response: The solution to the equation is 3 and -3, so the side length is only the positive value.
- 2 units
- Sample response: There are two solutions to the equation, one positive and one negative. The side length is the positive solution.
Activity Synthesis (Teacher Notes)
Invite students to share their solutions and comparisons.
Emphasize that quadratic equations like x2=A have two solutions, one positive and one negative. The notation A refers only to the positive solution. To refer to the negative solution, we write -A.
Standards
- 8.EE.2·Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
- 8.EE.A.2·Use square root and cube root symbols to represent solutions to equations of the form <span class="math">\(x^2 = p\)</span> and <span class="math">\(x^3 = p\)</span>, where <span class="math">\(p\)</span> is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that <span class="math">\(\sqrt{2}\)</span> is irrational.
15 min
15 min
Knowledge Components
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Quadratic Equations with Irrational Solutions
5 min
Narrative
This Warm-up reminds students that we can use the notation A to express the side length of a square with area A and that the value of A may not be a whole number or a fraction.
Launch
Display the entire task for all to see. Give students 3 minutes of quiet think time. Select students to share their responses and how they reasoned about the side length and area of each square.
Student Task
Here are two squares. Square A has an area of 9 square units. Square B has an area of 2 square units.
- What is the side length of Square A?
- How does that side length compare to the solutions to the equation s2=9?
- What is the side length of Square B?
- How does that side length compare to the solutions to the equation x2=2?
Sample Response
- 3 units
- Sample response: The solution to the equation is 3 and -3, so the side length is only the positive value.
- 2 units
- Sample response: There are two solutions to the equation, one positive and one negative. The side length is the positive solution.
Activity Synthesis (Teacher Notes)
Invite students to share their solutions and comparisons.
Emphasize that quadratic equations like x2=A have two solutions, one positive and one negative. The notation A refers only to the positive solution. To refer to the negative solution, we write -A.
Standards
- 8.EE.2·Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
- 8.EE.A.2·Use square root and cube root symbols to represent solutions to equations of the form <span class="math">\(x^2 = p\)</span> and <span class="math">\(x^3 = p\)</span>, where <span class="math">\(p\)</span> is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that <span class="math">\(\sqrt{2}\)</span> is irrational.