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In the problem Tri-Triangles, students use algebraic thinking to solve problems involving patterns, sequences, generalizations, and linear and non-linear functions. The mathematical topics that underlie this problem are finding and extending patterns, creating generalizations, finding functions, developing inverse processes, exploring non-linear functions, and justifying solutions. In each level, students must make sense of the problem and persevere in solving it (MP.1). Each problem is divided into five levels of difficulty, Level A through Level E, to allow access and scaffolding for students into different aspects of the problem and to stretch students to go deeper into mathematical complexity.

In this task, students are shown a triangle pattern and use toothpicks to extend the pattern and identify the number of toothpicks that would be needed in subsequent patterns.

In this level, students view a pattern of triangles composed of toothpicks.  Their task is to determine the number of toothpicks that make up each pattern. 

In this level, students use multiplication and division within 100 to solve word problems in situations involving equal groups (3.OA.A.3).    
In this level, students examine a linear pattern that involves a constant. The task involves a set of triangular tables that are arranged adjacently in a row. The task asks students to determine the relationship between the number of tables and the number of people who can sit around the tables.  Students need to extend the pattern. They also find the inverse relationship, i.e., find the pattern number when given the total number of people seated. 

In this level, students are given a shape pattern and asked to extend the pattern and informally describe the rule of the pattern (4.OA.C.5).  

In this level, students determine how a pattern grows. Students need to see that the pattern grows by square numbers. They identify the relationship and then explain a valid process for finding these values. 

In this level, students  determine how a pattern of triangles grows. Students use their understanding of expressions and exponents (6.EE.A.1) to identify the relationship between the pattern and number of triangles and then explain a valid process for finding these values (6.EE.B.6, 6.EE.C.9).
In this level, students are asked to generalize a rule for finding a value in the triangular number sequence. They are also asked to explain the process for finding an inverse value for the triangular number sequence by finding the term when given the total.

Students explore a pattern of triangular numbers and write an explicit expression describing the pattern (F-BF.A.1a). Students use the expression to write a quadratic equation to find the pattern number given the triangular number (A-CED.A.1). The quadratic equation can be solved with reasoning, factoring, or the quadratic formula (A-REI.B.4b). Students explain their reasoning (SMP.3) and use visual models to explore the pattern (SMP.5).
In this level, students generate a closed expression for a sequence that grows as the sum of two exponential functions.  In addition, the students must justify their findings. This level builds on the work students do in level D. 

In this level, students build on their experience with triangular numbers from level D to generate an explicit expression to describe a pattern (F-BF.A.1a). As they build the explicit expression, they interpret parts of the expression independently and then combine the parts to arrive at the final expression (A-SSE.A.2). The rule that is developed may use exponential or polynomial functions. This level requires students to look for and make use of structure in the expressions they build (SMP.7).

Download the complete packet of Tri-Triangles Levels A-E here

You can learn more about how to implement these problems in a school-wide Problem of the Month initiative in “Jumpstarting a Schoolwide Culture of Mathematical Thinking: Problems of the Month,” a practitioner’s guide.  Download the guide as iBook with embedded videos or Download as PDF without embedded videos.

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