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.
Activity
None
What do you notice? What do you wonder?
Student Response
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Building on Student Thinking
Activity Synthesis
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
5.2
Activity
Standards Alignment
Building On
Addressing
8.EE.A.2
Use square root and cube root symbols to represent solutions to equations of the form and , where is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that is irrational.
The purpose of this activity is for students to connect values expressed using square roots with values expressed in decimal form by determining the length of a diagonal line segment on a grid.
Monitor for students who use the following strategies to find the length of the segment, ordered to show students a reasoning strategy that produces an exact square root length followed by concrete measurement strategies that help students understand the value in a decimal form that they may be more familiar with:
Draw a tilted square and find the area.
Use tracing paper.
Use a compass to make a circle.
Launch
Provide access to geometry toolkits and compasses, but do not provide access to a calculator with a square root button since part of this activity asks students to estimate the value of a square root. Students will be able to use a calculator in later lessons.
Begin by displaying the diagram for all to see. Ask students how this diagram is similar and how it is different from the diagram in the Warm-up. (Both diagrams show a line segment on the coordinate plane. This diagram does not have a circle drawn around it.) Give students 2–3 minutes of quiet work time followed by a whole-class discussion.
Select students with different strategies, such as those described in the Activity Narrative, to share later.
Activity
None
Find the length of the segment.
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is for students to see multiple ways the length of a line segment can be represented. This helps students transition from thinking of square roots as side lengths to thinking about them as values that can be plotted on the number line.
Invite previously selected students to share how they determined the length of the line segment. Sequence the discussion of the methods in the order listed in the Activity Narrative. If possible, record and display their work for all to see.
Since the task did not specify whether students should find an exact or approximate side length, some students will draw a square and use the area to find the exact side length, which is a familiar strategy. Other students may use tracing paper to use the number line as a ruler. Students who use a compass are finding another way to use the number line as a ruler.
Connect the different responses to the learning goals by asking questions, such as:
“Which of these methods gives the most accurate length of the line segment?” (Finding the area of the square and taking the square root will give an exact answer.)
“What do the different methods tell us about the exact value of ?” (It is about 3.1.)
"How could you use the compass method to approximate the side lengths of other squares?" (Use the compass to measure a side length, then set the point at and draw a circle, marking where it crosses the number line.)
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation, such as “First, I because . . . .” and “I tried , and what happened was . . . .” Supports accessibility for: Language, Social-Emotional Functioning
5.3
Activity
Standards Alignment
Building On
Addressing
8.EE.A.2
Use square root and cube root symbols to represent solutions to equations of the form and , where is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that is irrational.
In previous activities and lessons, students used the areas of squares with whole number side lengths to find an approximation for the square root of an integer. In this activity, students start with the square root of an integer and then use a drawn square to explain why a given approximation of the square root is reasonable or not (MP3).
This activity is optional in this course. Use this activity to give students a visual and algebraic strategy for checking the value of square root approximations.
Launch
Arrange students in groups of 2. Provide access to four-function calculators without a square root button.
Display the diagram for all to see. Ask students what is the same and what is different about this diagram and diagrams they have seen in earlier activities. (This diagram also shows line segments on a coordinate plane. This diagram has 2 segments instead of 1. None of the line segments go through the origin.) If not mentioned by students, make sure to highlight how the vertices of the drawn square are not at the intersection of grid lines.
Give students 2–3 minutes of quiet work time followed by a partner then whole-class discussions.
MLR1 Stronger and Clearer Each Time. Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their first draft response to how they identified a point on the line that is closer to . Invite listeners to ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing, such as “How do you know that is between 1.5 and 2?” and “How did you determine the area of the square that you drew?” Give students 3–5 minutes to revise their first draft based on the feedback they receive. Advances: Writing, Speaking, Listening
Activity
None
Diego says that .
Use the square to explain why 2.5 is not a very good approximation for .
Find a point on the number line that is closer to . Draw a new square on the coordinate plane and use it to explain how you know the point you plotted is a good approximation for .
Student Response
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Building on Student Thinking
Activity Synthesis
The goal of this discussion is to make sure students understand both a visual and an algebraic strategy for checking the value of square root approximations and to connect this thinking to the number line.
Display the diagram from the Task Statement for all to see. Invite 1–2 students to share their reasoning for why 2.5 is not a very good approximation for . Emphasize that squaring a point on the number line can be visualized as the area of a literal square sitting on the number line. This can help us estimate the value of a square root.
Ask students, “2.5 might be a good approximation for the square root of what number?”. After a brief quiet think time, invite students to share their values. If not mentioned by students, make sure these two strategies are brought up:
Estimate the area of the square by decomposing and rearranging squares and rectangles to get an area of 6.25 square units.
Find the exact value by squaring. Since , the square root of 6.25 is 2.5.
5.4
Activity
Standards Alignment
Building On
Addressing
8.EE.A.2
Use square root and cube root symbols to represent solutions to equations of the form and , where is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that is irrational.
The purpose of this activity is for students to use rational approximations of irrational numbers to place both rational and irrational numbers on a number line and to reinforce the definition of a square root as a solution to an equation of the form . This is also the first time that students are asked to consider negative square roots.
After arranging students in groups of 2, ask, “What two whole numbers does lie between? Be prepared to explain your reasoning.” After a brief quiet think time, invite students to share their reasoning. For example, students may make a list of perfect squares and then find which two perfect squares the number 7 was between (2 and 3). Once 2–3 strategies have been shared, ask students to determine which two whole numbers lies between (7 and 8). Then have students continue with the activity as indicated in the Launch.
To account for the extended Launch, increase the timing of this activity to 15 minutes.
Launch
Arrange students in groups of 2. Since the goal of this activity is for students to approximate the location of a square root on a number line, do not provide access to calculators. Give students 2 minutes of quiet work time followed by a partner then whole-class discussion.
MLR1 Stronger and Clearer Each Time. Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their first draft response to their reasoning for how they placed , , and on the number line in the first problem. Invite listeners to ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing. Give students 3–5 minutes to revise their first draft based on the feedback they receive. Advances: Writing, Speaking, Listening
Activity Synthesis
The purpose of this discussion is to reinforce the idea that irrational numbers are still numbers on the number line, though their location cannot be found by subdividing the unit interval into parts and taking of them in the way that rational numbers written as can be.
Display the number line from the activity for all to see. Select groups to share how they chose to place values, recording them on the number line as they share. After each placement, survey the class and ask if students used the same or different reasoning. Invite any groups that used different reasoning to share with the class.
Conclude the discussion by asking students to share how they placed . (I found the approximate location of and then placed the same distance from 0 to the left.)
Representation: Develop Language and Symbols. Use virtual or concrete manipulatives to connect symbols to concrete objects or values. For example, use a kinesthetic representation of the number line on a clothesline. Students can place and adjust numbers on folder paper or cardstock on the clothesline in a hands-on manner. Supports accessibility for: Visual-Spatial Processing, Conceptual Processing
Lesson Synthesis
The goal of this discussion is to make sure that students understand that a square root can be approximated by finding the whole numbers it lies between and then testing values between those two whole numbers to determine a more accurate approximation. Here are some questions for discussion:
“How can we find the whole numbers that a square root lies between?” (Look at whole numbers whose squares are greater than and less than the number inside the square root symbol.)
“How can we get a better approximation?” (We can test values between those two whole numbers.)
“What two whole numbers does lie between?” (8 and 9)
“Test some numbers between 8 and 9. What is a better approximation?” (8.25 is a good approximation because is only slightly greater than 68.)
If time allows, show students how to use each guess to refine their next guess when estimating the value of a square root. For example, suggest an order like this:
Student Lesson Summary
Here is a line segment on a grid. How can we determine the length of this line segment?
By drawing some circles, we can tell that it’s longer than 2 units, but shorter than 3 units.
Two circles that have the same center are drawn on a square grid with radii 2 and 3. A line segment slanted upward and to the left is drawn such that the bottom endpoint is the center of the two circles and is 1 unit down and 2 units right of the top endpoint of the line segment.
To find an exact value for the length of the segment, we can build a square on it, using the segment as one of the sides of the square.
The area of this square is 5 square units. That means the exact value of the length of its side is units.
Notice that 5 is greater than 4, but less than 9. That means that is greater than 2, but less than 3. This makes sense because we already saw that the length of the segment is between 2 and 3 units.
We can approximate the value of a square root by observing the whole numbers around it and remembering the relationship between square roots and squares. Here are some examples:
is a little more than 8 because is a little more than , and .
is a little less than 9 because is a little less than , and .
is between 8 and 9 (it’s 8 point something) because 75 is between 64 and 81.
is approximately 8.67 because .
A number line with the numbers 8 through 9, in increments of zero point 1, are indicated. The square root of 64 is indicated at 8. The square root of 65 is indicated between 8 and 8 point 1, where the square root of 65 is closer to 8 point 1. The square root of 75 is indicated between 8 point 6 and 8 point 7, the square root of 75 is closer to 8 point 7. The square root of 80 is indicated between 8 point 9 and 9, where the square root of 80 is closer to 8 point 9. The square root of 81 is indicated at 9.
If we want to find the square root of a number between two whole numbers, we can work in the other direction. For example, since and , then we know that (to pick one possibility) is between 22 and 23. Many calculators have a square root command, which makes it simple to find an approximate value of a square root.
Standards Alignment
Building On
Addressing
8.EE.A.2
Use square root and cube root symbols to represent solutions to equations of the form and , where is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that is irrational.
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., ). For example, by truncating the decimal expansion of , show that is between and , then between and , and explain how to continue on to get better approximations.
Plot , , and on the number line. Be prepared to share your reasoning with the class.
Plot on the number line.
Student Response
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Building on Student Thinking
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., ). For example, by truncating the decimal expansion of , show that is between and , then between and , and explain how to continue on to get better approximations.
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., ). For example, by truncating the decimal expansion of , show that is between and , then between and , and explain how to continue on to get better approximations.
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., ). For example, by truncating the decimal expansion of , show that is between and , then between and , and explain how to continue on to get better approximations.
