Today's Warm Up provides students two problems with which they can practice applying the Pythagorean Theorem and estimating the side length. This practice will lead us directly into today's lesson focus: Using a double number line to estimate square roots to the nearest tenth.
Double number lines provide a great structure for students to use when estimating square roots of non-perfect squares. While the majority of students have not used double number lines before, it is a helpful strategy to learn and one that students may choose to use again during our unit on ratios and proportional reasoning.
To introduce the lesson, I begin by asking student to look at the first example and share with their shoulder partner (a like-abled student) between what two whole numbers will the square root of 96 fall? I chose this number because even my lowest level student will recognize that 96 is very close to 100 and has learned that square root. I then randomly select a student volunteer from my cup of names and ask for his/her answer. I then confirm with the class. I write 81 on the left side of the number line and begin counting as I draw tick marks along the line, writing an extra dark one when I reach 96, until I reach 100 on the right side of the line.
I ask how many numbers are between 81 and 100 and once someone responds with 19, I ask what half of nineteen would be. I explain that I am trying to locate the midpoint on this number line as a benchmark of 9.5, which I write below the number line.
I direct students to look at the number line we have created. I ask if the square root of 96 falls to the left or the right of the midpoint? Students usually see that it is to the right of the midpoint, so I ask if anyone has an idea about how we might label that point using a decimal. Sometimes, a student will strong number sense estimates the answer to be 9.75 because s/he reasons that the square root of 96 seems to fall half way between 9.5 and 10.
I then touch just below the number line where I have animated a green number line to appear. I explain that a double number line can be very helpful when wanting to see relationships between number. I go on to explain that I have divided the green number line into tenths and direct students to notice that 9.5 on the green number line corresponds directly to 9.5 on the black number line. I then write the values of the first few ticks on the green number (9.1., 9.2, 9.3) and then stop and say to the students: "Using this technique, give me a thumbs up if you can tell me the estimate of the square root of 96 to the nearest tenth."
I select a student at random to give his/her answer and then ask him/her to come to the board and explain.
Once the first example is done, I move to the second following similar procedures to the first, but scaffolding the steps so that students can follow along in their journals. By the third example, I want student to begin on their own, so I provide limited support until students have reached the point of drawing the second number line. For students who struggle with conceptual experience with number lines, I remind them that this is not an exact strategy, but rather, we are estimating our answers.
Once we have completed the three sample problems, I ask students to self-evaluate on our class learning scale (5= I could teach someone this; 4=I understand this concept; 3= I am starting to understand this concept; 2= I understand this concept a little; 1= I do not understand at all). If the vast majority of the class rates a 3 or higher, I move on to work time. If not, I add an additional example or two to move students along.
For work time, I have created a partner's work page with four problems. Student partners work on the problems, alternating which one holds the pencil. The other student verifies the partner's work. If both partners struggle to understand, they know to ask me for support. I set a timer for 16 minutes and explain that as student groups finish, I want them to reach consensus with their table mates. Once consensus is reached, I ask a representative to record their table answers in the table.
Because students are just learning to use the double number lines, consensus among all seven tables is rare. Typically, groups are with a tenth or two of each other. I ask individuals to share their work if the discrepancy among answers is too large. This step in the lesson gives focus to two important Math Practice Standards: 3. Constructing viable arguments and critiquing the reasoning of others, and 6. Attending to precision. I want students to be comfortable with mathematical discourse and I also want them to understand the importance of estimating with accuracy and appropriate precision. I explain that although calculators can estimate square roots to six decimal places or more, that level of precision is not necessary at this level when calculating on paper.
For our lesson closure today, I am interested in seeing the level at which individual students are able to graph the unit work to this point. I provide one question to which students must respond. I intentionally include a hypotenuse length that is 11.9 (instead of the correct answer of 11.4) because students who are still unsure of the use of the double number line will often simply count tick marks on the first number line and use that number as the decimal (since 130 is nine ticks away from 121). This is a common misconception that is usually cleared up in the following days with repeated exposure to the use of the double number line.