Lesson 5

The purpose of this warm-up is for students to analyze symbolic statements about square roots and decide if they are true or not based on the meaning of the square root symbol.

Display one problem at a time. Tell students to give a signal when they have an answer and a strategy. After each problem, give students 1 minute of quiet think time and follow with a whole-class discussion.

Poll students on their responses for each problem. Record and display their responses for all to see. If all students agree, ask 1 or 2 students to share their reasoning. If there is disagreement, ask students to share their reasoning until an agreement is reached.

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, especially the fact that \(\sqrt{a}\) is a solution to the equation \(x^2=a\), rather than less efficient geometric methods (which may not even work). Students can draw a number line if that helps them reason about the magnitude of the given square roots, but this is not required. However the reason, students must construct a viable argument (MP3).

Do not give students access to calculators. Students in groups of 2. 2 minutes of quiet work time followed by a partner then a whole-class discussion.

Discuss:

• “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 \(2<\sqrt5<3\).)

Once the class is satisfied with which two whole numbers the square roots lie between, ask students to think more deeply about their relationship. Give 1–2 minutes for students to pick one of the last two square roots and figure out which whole number the square root is closest to and to be ready to explain how they know. One possible misconception that could be covered here is that if a number is exactly halfway between two perfect squares, then the square root of that number is also halfway between the square root of the perfect squares. For example, students may think that \(\sqrt{26}\) is halfway between 4 and 6 since 26 is halfway between 16 and 36. It’s close, since \(\sqrt{26}\approx5.099\), but it’s slightly larger than “halfway.”

This is a good opportunity to remind students of the graph they made earlier 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.

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 the equation of the form \(x^2=a\). This is also the first time that students have thought about negative square roots. 

Do not provide students with access to calculators. Students in groups of 2. 2 minutes of quiet work time followed by a partner, then a whole-class discussion.

Display the number line from the activity for all to see. Select groups to share how they chose to place values onto the number line. Place the values on the displayed number line as groups share, and after each placement poll the class to ask if students used the same reasoning or different reasoning. If any students used different reasoning, invite them to share with the class.

Conclude the discussion by asking students to share how they placed -\(\sqrt{2}\) and why.