Modeling Math Ideas with Patterns and Algebraic Reasoning
Text Box 5.1 A Math Happening 5a: Building a Staircase
We are building a staircase.
Draw the blocks in the diagram to make the fourth step.
How many blocks in all are needed to make a staircase with five steps?
How many blocks does it take to build just the tenth step?
A staircase has 105 blocks. How many stairs does it have?
LESSON STUDY VIGNETTE-GROWING STAIRCASE PROBLEM
In this third-grade research lesson, students were shown a three-step staircase composed of squares and their task was to determine the number of squares that make up each step and the total needed to build a staircase for a deck. The mathematical topics that underlie this problem are extending patterns, creating generalizations, and justifying solutions. Students used manipulatives, pictures, numbers, and words to extend the pattern of the staircase. Many of them grabbed the cubes to build the staircase pattern, while some preferred to create the pattern on the graph paper. They immediately saw this as a growing pattern with each stage growing by one more cube (see Figure 5.1).
Supporting productive struggle in learning mathematics. The sample student work below is from a fifth-grade intervention classroom. The teachers reflected on their students’ work and shared that these students identified as low Tier II and Tier III and had previously failed their grade-level Math Standards of learning test. She states,
One of my most interesting observations was that one of my student’s that has an IEP and always states how hard everything is actually worked through the task quite easily which was very surprising to me. He drew his picture on the back of his paper and then worked. Students saw that they were able to use various ways to solve a problem and still arrive at the same answer. They gained confidence, because they stated that they normally struggle
Figure 5.1 Building with Lego manipulatives.
with these types of problems. They were able to justify their answers verbally and in words. Many realized that once they discovered what they were adding they didn’t have to build and count. They were able to take the number of steps previously and add 11, then 12, then 13 and then 14 to arrive at 105 blocks. Not all of my students finished the task, but they worked the whole class period. They all demonstrated perseverance in their work. I felt that this lesson was a valuable learning experience for my students. (Missy 5th grade Math Interventionist)
Selecting a task that is accessible to all is an important part of the teaching practices. All students can access the staircase problem. Students began to draw out the staircase and add up all the blocks it took to create the total number of steps they have to figure out (see Figure 5.2). We loved that students were able to engage in the sensemaking with manipulatives (unifix cubes, legos), but quickly realized that they could be more efficient with graph paper and numbers.
One of the teachers who revised the host lesson stated in her reflection that students started out building their staircase because “it seemed like the logical thing to do,” according to one member of this group. Once they reached eleven stair steps, they stopped because, “it was a waste of time. It works better with numbers.” Another student created a table to figure out the total number of blocks to find how many blocks created each step. This idea of efficiency in strategy will eventually lead the students into understanding that algebraic thinking will allow one to come to a general rule from the computation of all the strings of numbers.
Driscoll (1999) describes this as one of the habits of mind, which he calls “abstracting from computation.” This research lesson’s goal was not necessarily for students to come to a general rule, but rather, explore the growing pattern in the staircase using concrete, numbers, tables, and explanations to notice the efficiency in creating a table and how one can organize information. The graphical approach would be readily available after creating a table with x (stage) and y (the number of cubes) values.
Figure 5.2 Moving to more efficient strategies.
Establishing a Math Goal to Focus the Learning. Another related research lesson called the Growing Tower had a different mathematical goal. The host teacher stated that she was inspired to teach this lesson because, “I had an aha moment during the class that helped me have a full understanding of how a constant and a coefficient of a function relate to the graph as well as a concrete model of a function. I decided then to create a lesson to help my sixth-grade math students to see the same relationships.” The purpose of this lesson geared to grades 5-7 was for students to create algebraic equations to represent arithmetic and geometric patterns and determine the nth value. In addition, the lesson study team wanted to connect the rule that they come up with important mathematical vocabulary such as constant, coefficient, and the function table.
Students completed a Modeling Math Mat (see Figure 5.4) to demonstrate their thinking as they worked through the problem. The teachers urged students to show their mathematical proficiency with this pattern problem by using various representations to highlight their process of coming to a solution, these included the use of pictures, tables, graphs, explanations in words, and algebraic symbols. Math agenda for the lesson was for students to analyze the patterns in multiple ways and demonstrate these process standards.
- 1. Communication: Students will explain and communicate their thinking using a variety of formats including words, expressions, equations, and verbal discussion to represent their ideas.
- 2. Connections: Students will relate sequences and functions to concrete patterns.
- 3. Problem Solving: Students will recognize patterns from illustrations and will apply a variety of strategies to solve problems.
- 4. Reasoning: Students will use inductive and deductive reasoning to recognize patterns and apply rules for patterns.
5. Representation: Students will show their thinking via diagrams, tables, and/or words.
Teacher: “In one of the Modeling Math Mat, you will create a function table to represent the pattern you identify.”
Student 1: (speaking to himself) “A function table? What is a function table?” Student 2: (turning half way around toward student 1 behind him and whispering) “Dude, it’s an input output table.”
Student 1: “Huh? What is Input output?”
Pose purposeful questions. The host teacher incorporated the use of vocabulary throughout the lesson and made sure to connect students’ background knowledge and previous grade-level content (у-intercept, constants, coefficients, slope, rate of change, parallel lines, input/output, function table, table of values, verbal expression) into this lesson of extending patterns. The lesson study team also anticipated and prepared questions to pose to students to help students communicate and explain their thinking.
Do you notice a pattern? Can you use a picture or number sentence to help you extend the pattern? Determine what the next pattern will be? How can you organize the information you’re given? How do you relate the shapes to the next pattern?
Posing questions was especially important when checking a student’s general rule to their table of values for the staircase. For example, the host teacher asked students, “Does adding three (which works for у-values) work for going across from the x-values to the у-values?” They wanted to make sure that students could differentiate between the recursive rule and the function rule. They were asked to pay attention to visual cues that can be organized and translated to numeric sequences as they explored and interpreted the pattern task and generalized function rules.
-  How do students use words or diagrams to support their answer?
-  Do students see how the pattern is being extended?
-  Do the students see how to relate the stage number to the output of the function?
-  Do the students know how to build the formula for the arithmetic sequence?
-  Do more students look at the pattern horizontally or vertically or just numerically,(counting the blocks) is it decreasing, increasing, and how?
-  Should students start by talking through their reasoning/thinking before they connect numbers to the pattern? Should they write down their thinking verbally beforetranslating it numerically?
-  Do students know how to use a table to help organize their thinking (functionalthinking)?
-  What stays the same? What changes? Think about it! Examine the patterns for the growing towers below. Whatmathematical learning opportunities do the different patterns offer?