V-patterns, Beams and Hair Growth

How are the Process standards illustrated throughout the lesson?
Problem Solving:

• In the v-pattern problem, students must develop a means of predicting how many birds will be in the v-pattern if the v-number is known. This gets the students looking for patterns within the numbers and thinking algebraically without specifically asking them to generate a formula. It is not enough to come up with the correct answer as the process is part of the learning objectives. 
• The students working to find a relationship between the length of a beam and the number of rods required to create it spend an extensive amount of time establishing a process to come up with the answer. During this time they create tables and diagrams. Ultimately they come up with a formula, but the process of developing it is as important as the solution itself. 
• Students were given the challenge to come up with a formula to predict hair length if provided with the original hair length and the growth rate. Students were encouraged to create a table to assist in seeing the relationship between hair length and time. By doing so they were able to generate a more abstract form of representation of this relationship, a formula. 

Reasoning and Proof:

• Through their examination of V-patterns, students must identify patterns within the numbers and prove that they are not random occurrences, but predictable and always true. When they observe that the number of birds in a v-pattern is always an odd number, they must justify their understanding with reference to the observed patterns. 
• As the students present their differing formulas for finding the number of rods for any given length of beam, they are asked to share how they came to the formula and demonstrate how they know it is true. 
• Students trying to generate a formula to describe the relationship between hair length, growth rate, and time were consistently asked to justify their understanding using the information provided in the question. 

Communication:

• In the V-pattern lesson, students are able express themselves orally to give the rationale behind their process and solution. They also are encouraged to model their thinking with tables and manipulatives. 
• During the rod and beam lesson, students spend time working in pairs on a solution and then must communicate that solution to the rest of the group. They use both formal and informal means, including diagrams, tables and formulas to illustrate their understanding. 
• Students working with hair growth rates communicated their understanding orally and through the use of tables and graphs. 

Connections:

• In the v-pattern lesson, students are able to connect repeated addition and multiplication to their process of predicting the number of birds at any given v-number. The real world connection was to the pattern in which geese fly as they migrate.
• When examining the relationship between the number of beams and bars, students were able to connect the project to real word the real world application of building construction. They used multiplication, addition and subtraction in this process, making a connection between new math learning and the math they have already mastered.
• This lesson examining the relationship between growth rate and hair length was something all students had experienced. The teacher connected it to specific student in the classroom, in addition to fun facts about the person holding the world record for the longest hair. As for connecting it to past math lessons, the teacher reminded one student that a formula was simply a means of describing the relationship between numbers without actually using numbers themselves. 

Representation:

• In the v-pattern lesson, students use manipulatives and tables in the process of determining a pattern between the different v-numbers.
• Students drew diagrams to assist in visualizing the relationship between the length of a beam and the number of rods needed to construct it. They also used more formal representation in the form of tables and, ultimately, a formula. 
• In the hair growth lesson, students created graphs to study the relationship between hair length and time. They were encouraged to examine each other's graphs for similarities and differences and even encouraged to critique their own work. 

How do you see the Standards of Mathematical Practice being portrayed?

1. Make Sense of problems and persevere in solving them. In all three problems, the teacher acted as a facilitator, encouraging students to persist in finding and communicating a solution. The teachers used guiding questions to get students to express their thinking. 
2. Reason abstractly and quantitatively. The students working with the V-pattern problem were provided with manipulatives that eased them from the realm of real-world context to that of decontextualized numeric relationships. This allowed them gain an informal algebraic understanding without overburdening them with new vocabulary. The students with the beam problem were able to draw diagrams to assist in their understanding. The oldest students, those working with hair growth rates, moved into more abstract reasoning with minimal tangible support. Their more formal understanding of algebraic relationships allowed them to work with numbers and formulas rather than manipulatives.
3. Construct viable arguments and critique the reasoning of others.  In all three lessons, students communicated their understanding visually and with words and were challenged to explain their process and reasoning. 
4. Model with mathematics.  In all three instances, the students used mathematics to model real-life phenomena. They developed these models in small groups and, at times, needed to correct the representation when they discovered an error in their reasoning. 
5. Use appropriate tools strategically. In all three lessons, the students were provided with the tools to represent their work. All three sets of student pages had tables complete with labels and partially completed. If students were already familiar with using and creating data tables, they might have left this to the students to create. 
6. Attend to precision. In all three lessons, students needed to communicate precisely about what the letters in their formulas meant. In the earlier lessons, they used terms such as V-number and Length. In the last lesson they used current and next. 
7. Look for and make use of structure.  The purpose of all of these lessons is for to help students develop an increasingly better understanding of algebraic relationships. This means looking at a series of numbers and detecting a relationship between them that is predictable in some fashion. 
8. Looking for and expressing regularity in repeated reasoning. The students were able to see repeated addition or multiplication as the relationship between numbers in the provided patterns. In the hair growth lesson, they were able to determine that such regularity results in a linear relationship that can be described graphically and with a measure of slope.