Side Lengths and Areas
5 min
Teacher Prep
Setup
Narrative
In this Warm-up, students use what they know about finding the area of a square given its side length to think about the converse: finding the side length of a square given its area.
First, students work with squares whose side lengths and areas can be found by counting length or area units. Next, students find the area of a “tilted” square using strategies from a previous lesson. They reason that if two squares have the same area, their side lengths must also be the same, and verify this using tracing paper.
Launch
Arrange students in groups of 2. Provide access to geometry toolkits, including tracing paper.
Give students 2–3 minutes of quiet work time followed by a whole-class discussion.
Student Task
- What is the side length of Square A? What is its area?
- What is the side length of Square C? What is its area?
-
What is the area of Square B? What is its side length? (Use tracing paper to check your answer to this.)
Sample Response
- 5 units; 25 square units
- 6 units; 36 square units
- 25 square units; 5 units
Activity Synthesis (Teacher Notes)
The key takeaway from this activity is that if two squares have the same area, they must also have the same side length. This can be reinforced by measuring with tracing paper. Invite 1–3 students to share how they determined that Square A and Square B have the same area.
Standards
- 6.EE.1·Write and evaluate numerical expressions involving whole-number exponents.
- 6.EE.A.1·Write and evaluate numerical expressions involving whole-number exponents.
- 6.G.1·Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
- 6.G.A.1·Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
- 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
All skills for this lesson
No KCs tagged for this lesson
Side Lengths and Areas
5 min
Teacher Prep
Setup
Narrative
In this Warm-up, students use what they know about finding the area of a square given its side length to think about the converse: finding the side length of a square given its area.
First, students work with squares whose side lengths and areas can be found by counting length or area units. Next, students find the area of a “tilted” square using strategies from a previous lesson. They reason that if two squares have the same area, their side lengths must also be the same, and verify this using tracing paper.
Launch
Arrange students in groups of 2. Provide access to geometry toolkits, including tracing paper.
Give students 2–3 minutes of quiet work time followed by a whole-class discussion.
Student Task
- What is the side length of Square A? What is its area?
- What is the side length of Square C? What is its area?
-
What is the area of Square B? What is its side length? (Use tracing paper to check your answer to this.)
Sample Response
- 5 units; 25 square units
- 6 units; 36 square units
- 25 square units; 5 units
Activity Synthesis (Teacher Notes)
The key takeaway from this activity is that if two squares have the same area, they must also have the same side length. This can be reinforced by measuring with tracing paper. Invite 1–3 students to share how they determined that Square A and Square B have the same area.
Standards
- 6.EE.1·Write and evaluate numerical expressions involving whole-number exponents.
- 6.EE.A.1·Write and evaluate numerical expressions involving whole-number exponents.
- 6.G.1·Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
- 6.G.A.1·Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
- 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.