Square Roots on the Number Line
5 min
Teacher Prep
Setup
Narrative
The purpose of this Warm-up is to use the structure of the circle and a rotation to relate the length of the segment to a point on the number line (MP7), which will be useful when students locate square roots on a number line in a later activity. While students may notice and wonder many things about the image, seeing how a decimal approximation can be found by looking at where the circle intersects an axis is an important discussion point.
Launch
Arrange students in groups of 2. Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner.
Student Task
What do you notice? What do you wonder?
Sample Response
Things students may notice:
- The center of the circle is at (0,0).
- There is a point labeled at (1,1) on the circle.
- There are many tick marks between 0 and 1.
Things students may wonder:
- How to find the distance across the circle.
- Where exactly does the circle land on the x- and y-axes?
Activity Synthesis (Teacher Notes)
Ask students to share the things they noticed and wondered. Record and display their responses for all to see without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.
If the length of the radius does not come up during the conversation, ask students to discuss how they could use the image to determine it. While some students may recognize the length from earlier activities, keep the discussion focused on strategies they could use to find the length
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.
- 8.NS.2·Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π²). <em>For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.</em>
- 8.NS.A.2·Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., <span class="math">\(\pi^2\)</span>). <span>For example, by truncating the decimal expansion of <span class="math">\(\sqrt{2}\)</span>, show that <span class="math">\(\sqrt{2}\)</span> is between <span class="math">\(1\)</span> and <span class="math">\(2\)</span>, then between <span class="math">\(1.4\)</span> and <span class="math">\(1.5\)</span>, and explain how to continue on to get better approximations.</span>
15 min
15 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Square Roots on the Number Line
5 min
Teacher Prep
Setup
Narrative
The purpose of this Warm-up is to use the structure of the circle and a rotation to relate the length of the segment to a point on the number line (MP7), which will be useful when students locate square roots on a number line in a later activity. While students may notice and wonder many things about the image, seeing how a decimal approximation can be found by looking at where the circle intersects an axis is an important discussion point.
Launch
Arrange students in groups of 2. Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner.
Student Task
What do you notice? What do you wonder?
Sample Response
Things students may notice:
- The center of the circle is at (0,0).
- There is a point labeled at (1,1) on the circle.
- There are many tick marks between 0 and 1.
Things students may wonder:
- How to find the distance across the circle.
- Where exactly does the circle land on the x- and y-axes?
Activity Synthesis (Teacher Notes)
Ask students to share the things they noticed and wondered. Record and display their responses for all to see without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.
If the length of the radius does not come up during the conversation, ask students to discuss how they could use the image to determine it. While some students may recognize the length from earlier activities, keep the discussion focused on strategies they could use to find the length
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.
- 8.NS.2·Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π²). <em>For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.</em>
- 8.NS.A.2·Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., <span class="math">\(\pi^2\)</span>). <span>For example, by truncating the decimal expansion of <span class="math">\(\sqrt{2}\)</span>, show that <span class="math">\(\sqrt{2}\)</span> is between <span class="math">\(1\)</span> and <span class="math">\(2\)</span>, then between <span class="math">\(1.4\)</span> and <span class="math">\(1.5\)</span>, and explain how to continue on to get better approximations.</span>