Today's Warm-Up mimics the previous day's because many students continue to struggle with the manipulation of integers which gets in the way of their ability to think about relationships between tables, graphs and equations. I want them to get to the point where they question an output that does not make sense in relation to the graph or rule. For example, as they calculate outputs for the table and have errors, I want them to notice the pattern does not hold so they go back to check their work. By giving them several days of practice on this, I hope to support the growth of their number sense and reasoning ability.
In preparation for the students first attempt at proving proficiency of the square roots of the first 20 squares, I provide each partner pair with an envelope of flash cards and give them 8 minutes to practice. This not only will give students who studied independently additional practice, it will also give those students who did not study exposure to the content that they are expected to master by the end of the week.
After 8 minutes of practice, I distribute the 10-question square root quiz to each student face down. Because I have my students seated at tables of four, I intentionally create four versions of every assessment to discourage answer sharing. I set the timer for 3 minutes as I want students to be able to recall at the level of automaticity. Without this, students will struggle with estimating square roots of non-perfect squares in the upcoming lessons. Once the timer sounds, I ask students who have not yet finished to turn their papers over as I collect them.
To begin today's formal lesson, I want to see what students remember from the previous day's introduction to Pythagorean Theorem. I display the Theorem and tell students to talk with their shoulder partners for three minutes about what they remember about it. I circulate throughout the room making note on my clipboard which students are able to articulate well so that I can call on them to share when the discussion time is over.
Once the timer sounds, I call on those students to review the concept by recoding, when needed, to clarify. For example, one students begins by explaining: "You square the sides and then add them together, then take the square root of that number." I respond as I write an example, "So what I heard you say was that you take the length of each leg of the triangle and square it. Then, you add those two numbers together. Is that right? So what does that number tell you?" This exchange allows me to reinforce academic vocabulary as well as review the concept before we move to applying it.
I begin the work time session by showing the first triangle and asking the students if we are looking for the length of a leg or the hypotenuse. I remind students that the right angle symbol 'points' to the hypotenuse of the triangle every time (a tip that will help in the coming days when we begin to solve for legs instead). I label the triangle's sides a, b and c and ask if it matters how I label the sides by taking a poll: Give me a thumbs up if you think it matters what I label each side. Give me a thumbs down if you think it doesn't matter. This gives me quick feedback about student understanding of the formula. I choose a student who was in agreement to explain his/her thinking and then reinforce the response with the example.
Once we have completed the first example, I move to the next one and encourage students to try it on their own. I move about the class intervening with scaffolded support as needed. For my special needs or ELL students, this usually involves asking, "When we the last example together, what did we do first?" Typically, just a small nudge will kick-start their thinking.
As students finish, I select a "volunteer" at random for the class cup of names (on Popsicle sticks) to walk me through the work. If that student lacks confidence or is unsure, I encourage them to 'phone a friend'. That student will tell him/her in a whisper what to do so that the original student is still responsible for telling me. It is important for me to maintain a safe learning environment, but still hold all students accountable to the learning.
I continue these procedures with the last two problems. At the end of the work time, I ask students to scale their learning using our learning scale (5= I could teach someone this concept, 4= I understanding this concept fully, 3= I'm starting to understand this concept; 2= I understand this concept a little, and 1= I don't understand this concept). This quick self-assessment gives me the necessary feedback to plan the following day's lesson.
After responding to the learning scale, I ask students to look back through their journals at the lengths of the triangles we have been working with the past two days and ask them to find anything in common. If no one notices or volunteers the information, I ask, "Did any of our triangle side lengths ever have decimals or fractions?" I go on to explain that all the triangles we have been working with so far have been part of a special group because of their side lengths. I then share the 'Pythagorean Triples' page and explain that each of the triples listed represent the side lengths of triangles. I pull out a few and show how squaring the two legs, the shorter lengths, and adding them together will give me the hypotenuse squared. I explain this is another example of math practice 7, looking for and making use of structure. As students work through practice problems, I encourage them to watch for these common triples.
As closure for the day's lesson, students must correctly answer one of five square root flash cards they are shown (the less familiar 14, 16, 17, 18, and 19) in order to leave. If they don't know the answer, they must move to the end of the line, where they will have the opportunity to hear the answers from their peers. I keep this activity very light and inject as much humor (e.g., sound effects) to make it a fun end of the class instead of a stressful one for struggling students.