This Math Talk focuses on analyzing symbolic statements about square roots. It encourages students to think about the meaning of the square root symbol and to rely on what they know about the relationship between squares and square roots to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students plot solutions to equations of the form on the number line (MP7).
Launch
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
Give students quiet think time and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies and record and display their responses for all to see.
Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards. Supports accessibility for: Memory, Organization
Activity
None
Decide mentally whether or not each statement is true.
Student Response
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Building on Student Thinking
Activity Synthesis
To involve more students in the conversation, consider asking:
“Who can restate ’s reasoning in a different way?”
“Did anyone use the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to ’s strategy?”
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I because . . . .” or “I noticed , so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. Advances: Speaking, Representing
6.2
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 think about square roots in relation to the two whole number values they are closest to. Students are encouraged to use numerical approaches to explain their thinking (MP3), especially the fact that is a solution to the equation .
If needed, students can draw a number line to help them reason about the magnitude of the given square roots, but this is not required.
Launch
Arrange students in groups of 2. Since the purpose of this activity is to reason about the whole numbers that a square root is close to, do not provide access to calculators. Encourage students to reference the class display listing perfect squares.
Give students 2 minutes of quiet work time followed by a partner then whole-class discussion.
Representation: Internalize Comprehension. Provide students with a number line that includes rational numbers to record their square root values. Supports accessibility for: Visual-Spatial Processing, Organization
Activity
None
The value of a square root of a number lies between two consecutive whole numbers. Which are those consecutive whole numbers for the following? Be prepared to explain your reasoning.
Activity Synthesis
The goal of this discussion is to make sure students understand how to reason about the value of a square root by comparing it to square roots that are whole numbers. Ask students:
“What strategy did you use to figure out the two whole numbers?” (I made a list of perfect squares and then found which two the number was between.)
“Did anyone use inequality symbols when writing their answers?” (Yes, for the first problem, I wrote .)
When students are in agreement about which two whole numbers each square root lies between, ask students to think more deeply about the relationship by asking them to figure out which of the two whole numbers it is closer to. Give 1–2 minutes of quiet work time, then invite students to share their thinking.
If not brought up by students, highlight that if a number is exactly halfway between two perfect squares, it is not true that the square root of that number is also halfway between the square root of the perfect squares. For example, students may think that is halfway between 4 and 6 since 26 is halfway between 16 and 36. It’s close since , but it’s slightly more than “halfway.”
If time allows, remind students of the graph they made in an earlier lesson showing the relationship between the side length and area of a square. The graph showed a non-proportional relationship, so making proportional assumptions about relative sizes will not be accurate.
6.3
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.
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.
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 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.)
Student Lesson Summary
In general, 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.
If students reason about exponents with statements like “ is in between 2 and 4 because is 4, and is 8” or “ is in between 11 and 12 because is 22, and is 24,” consider:
Asking students to clarify the meaning of a number raised to the power of 2.
Prompting students to refer to the class display listing perfect squares.
Activity
None
The numbers , , and are positive, and , , and .
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.
