**Lesson #1: Real Numbers – ****Chalk Talk**

**Overview: **

‘Real Numbers’ is the first topic that we cover in Pre-Algebra. Students explore the real number system and discover the difference between rational and irrational numbers. From there, students further categorize rational numbers into integers, whole numbers, and natural numbers.

In order for students to build a deep understanding of the characteristics of rational numbers and irrational numbers, students must first understand what a ‘real number’ actually is. The purpose of this lesson is to explore students’ prior knowledge and interpretation of numbers.

**Objectives:**

- Students will be able to give the definition of a “Real Number”.
- Students will be able to categorize real numbers as Rational or Irrational.

**Standards:**

__NS.1__: Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

** ****Materials:**

- Teacher:
- Whiteboard/Markers

- Students:
- Pencil/Paper
- Laptop (for homework)

**Lesson Outline:**

__Introduction__: To start the lesson off, the teacher will write “What is a number?” on the whiteboard at the front of the room. This will be the first of two questions students will explore during the lesson.

__Explain Activity__: The teacher will then explain the process of a ‘chalk talk’. The teacher will tell students that they may freely come up to the board and write their thoughts/ideas, but they may not talk. They will be reminded that the point of a ‘chalk talk’ is to make learning visible. Students can write down questions and ideas, draw pictures, write numbers, etc. Nothing is off limits!

__Facilitate Chalk Talk #1__: Allow students a good amount of time to put their thoughts/ideas on the board. Walk around and make sure that students are participating without talking. As the teacher, feel free to write something on the board that may prompt a different thought or idea! Encourage students who are not participating by placing a marker on their desk. Ask students to elaborate on what they have already put on the board.

__Discussion #1__: Ask students to go back to their seats. Allow students to have some time to just look at the board as a whole. They can now talk to one another. Have them discuss any observations that they make. Have them jot down one or two ideas/thoughts that stood out to them. Come back as a class and discuss the question, “What is a number?” Allow all students to participate in the conversation.

__Facilitate Chalk Talk #2__: Erase the first ‘chalk talk’ and write a new question on the board: “How can we categorize numbers?” Facilitate this second ‘chalk talk’ the same way as the first.

__Discussion #2__: Facilitate a class discussion in the same manner as before around the question, “How can we categorize numbers?” Allow all students to participate in the conversation. This conversation should transition into the next step, ‘making the connection’.

__Making the Connection__: To end the lesson, introduce students to the terms ‘Rational’ and ‘Irrational’. Ask students to make connections between the way they were categorizing numbers and how these two terms now categorize numbers. This will also be a teacher-facilitated discussion. Make sure students leave with the correct definitions of these two types of real numbers.

**Assessment:**

__Homework__: Using a Prezi (prezi.com), students are to create some kind of graphic representation of ‘real numbers’. Students may categorize numbers in different ways, but must show examples of both irrational and rational numbers. Students will share their Prezi with the class. The teacher will use these assignments to facilitate a discussion in the near future.- Students will be given the following graphic organizer the next day:

**Social Media:**

__https://www.youtube.com/watch?v=cLP7INqs3JM&t=295s____https://www.youtube.com/watch?v=pYViAeeGdws&t=17s____https://twitter.com/EU_Precious/status/865707111346581505__

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**Lesson #2: Scatter Plots – Micro Lab Protocol (Revised)**

**Overview: **

‘Statistics’ is the last topic that we cover in Pre-Algebra. Students observe scatter plots and discuss relationships and patterns in data. Students are introduced to ‘lines of best fit’. In high school, students continue analyzing scatter plots and start the discussion of ‘correlation’.

This lesson just focuses on the patterns that students see in scatter plots. Students struggle with making meaning of these patterns in the context of problems. The Micro Lab Protocol is used in this lesson to get students used to interpreting data in front of their peers. The collaboration aspect of this lesson will allow students to engage in deep discussion on patterns in data.

**Objectives:**

- Students will be able to interpret a scatter plot.
- Students will be able to use context to make meaning of data represented in a scatter plot.

** ****Standards:**

__SP.1__: Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

** ****Materials:**

- Teacher:
- Scatter plots printed for groups

- Students:
- Pencil/Paper
- Notebook (for notes)

**Lesson Outline:**

__Introduction__: Put students in groups of 3 or 4. Pass out 3-4 different scatter plots that all look different in their correlation/pattern. Each student gets one scatter plot.

- Examples:

__Explain Activity__: Tell students that they will be given time to interpret the scatter plot, and tell their peers what they think the scatter plot is representing. Walk students through the entire process of the “Micro Lab”.

__Sharing__: Tell the first student in each group to start their interpretation of their scatter plot to the group. Remind the other students to be silent when their peer is presenting.

__Call for silence__: After the first student speaks, tell students to allow 20 seconds of ‘quiet time’. Tell students that this time should be used to take in and analyze what their peer just said. After they have spent some quiet time, students should write down notes on what their group member just said. This will be turned in to the teacher at the end of class.

__Do rounds 2 and 3__: Repeat with the other students in the groups.

__Commence Discussion__: Tell students that they may have 5 minutes to discuss (as a full group) what everyone said collectively. Encourage students to connect their ideas with one another. What were some similarities? What were some differences?

__Share the thinking__: As a whole class, we will discuss the ‘thinking’ process that students were observing. Make sure that the conversation is centered around the interpretation and analysis processes that were taking place.

__Reveal__: After the class discussion, reveal the context behind each scatter plot. Show the labels for the x- and y-axis for each graph. Give students the rest of class to discuss the similarities/differences between their own interpretations and the actual context.

**Assessment:**

__Notes__: Students should keep written notes of what each student said. These notes will be turned in to the teacher for a grade. The teacher should encourage students to listen first, and to take notes after the student in done talking (and after the ‘quiet time’). This will allow the teacher to visibly see the reflection process.

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Thanks for the post!

Bobby, How wonderful for me (and your readers) to see how you start and end your year with students. You wrote -“Students can write down questions and ideas, draw pictures, write numbers, etc. Nothing is off limits!” LOVE THIS! Yes, have the students engage in divergent thinking vs. the convergent thinking we often see in classes. The execution of your Chalk Talk is spot on and will be a terrific introduction to the concept of “numbers.” You found a “spot on” twitter link too… loved the rap that was included in the tweet.

Your micro lab will work well too with scatter plots- great choice. Your images, graphics and links really rounded out your post in the most engaging, thorough way.

Thank you very much! I appreciate your response!

Bobby, big fan of the Chalk Talk and I really like how you used it in the whole class setting. I also like the idea of the students not talking as they populate the whiteboard with the first round of ideas. Having them step back and take it all in, and then having them talk among themselves first is a great adaptation and should promote good discussion. Will you let them add to the board from their smaller discussions before opening up the whole class discussion? You indicated you would erase the board before question #2. Might be an opportunity to allow the students to take a picture of it, or maybe you take one and share so they have the visual to help them recall the discussion later.

The Micro Lab lesson is something I have never seen happen. I really like how it is set up to allow each student to have the floor. Expecting their fellow group members to take notes for future discussion ensures their attention. How much exposure to scatter-plots will they have had at the time of this lesson? Will they know some of the vocabulary usually involved with scatter plots, or will this be in their own words?

Hey Christ! Thanks for the reply. My students will know what a scatter plot is by the definition, and will be comfortable with plotting points. However, they haven’t had any exposure to interpreting data based on correlation. They also have graphed linear equations already, so they should be able to describe the general slope of the trend line (if they imagine one). We have not officially introduced line of best fit at this point.

Bobby,

I like how you have built in time during the Chalk Talk to allow students “look at the board as a whole”. I often tell my students the they have to let math marinate for a while sometimes before it makes sense. As in when new concepts are introduced, they can be overwhelming at first, but with time, become accessible. By allowing students to sit back and just see what everyone has written seems to go along this ideology. It is going to be intriguing to see students categorize numbers. I am guessing that some may use the category “big” and “small” and then have the joy of discussing that you can have large and small all in the same category. I don’t know, I think that this could be a fun conversation for students just learning about categorizing numbers. Great job!

Thanks Grady! The more I thought about it, the more my chalk talk seemed to connect to the ‘see-think-wonder’ routine. I agree, students should be given plenty of time to let the math marinate before jumping to conclusions or assumptions. Perhaps this could be extended to another day. The second day, students would come to class after having seen, thought, and wondered. Then, they could add more to the chalk talk. Thanks for the input!

I like this idea of Chalk talk because it is really challenging students in their writing and articulation of Math as well. What a great tool for you as a teacher to know what they are all thinking before you jump into the lesson. (Although I think I would have a hard time following the rule of no talking because I would want to ask questions of them haha 🙂 )Do you tell students ahead of time to write something different than their classmates or do you encourage them to just write whatever comes in their minds first, whether it is a repeat idea or not?

Thank you for the reply! I have done a chalk talk once before with my students. I just used the ‘no talking’ rule because that is how I had seen it done before. I also figured that this rule would give every student an equal opportunity to participate. Sometimes 2 or 3 students can direct a conversation and sway the thinking of their peers. I wanted the chalk talk to truly show me what students are thinking about a certain topic, without the help of their peers (at first).

It is interesting to think of the ways that you could go about modifying this activity. If students were aloud to talk during it, what would be the advantages/disadvantages? What if every student was only able to write one thing on the board? What if every comment had to be 1 single word? I think you could really change the dynamic of the chalk talk and explore different kinds of learning.

Bobby, I love the two questions that you chose for your chalk talks in your first lesson plan. Both encourage insightful conversations among students. Students may not always think about what numbers actually are and how they can be grouped, so asking these questions would lead students having a deeper understanding of number systems. It would also allow students to take ownership of their own learning, and would set them up to be “constructors” of their own knowledge (rather than waiting for a teacher to tell them what they need to know about numbers). Out of curiosity, how do you plan on refreshing students on categories of numbers other than irrational and rational during class discussion and when explaining the homework?

Thanks for the comment! I think the goal of the chalk talk is really to put students in the ‘driver’s seat’. They are responsible for continuing the conversation, and adding to it with their ideas/opinions. While this seems to be an ideal way of introducing a new topic (with students as constructors of knowledge), sometimes I worry about what the outcome could be. For example, what if the chalk talk leads to misconceptions about mathematics? Would it be ‘right’ for the teacher to step in and immediately correct students? Or would it be more effective for the teacher to let it play out and allow students to see the consequences of the particular misconception (such as a contradiction mathematically)? I haven’t had this happen during a chalk talk yet, but I do wonder about it.

My plan is to end class with the conversation on ‘what is rational?’ and ‘what is irrational?’. I think it is valuable for students to come up with their own ways of classifying numbers initially, but in the end they need to understand how real numbers are truly categorized in mathematics. The homework is meant to be more open-ended, so that we can continue the discussion on classifying numbers in class the next day.