In the problem Growing Staircases, students use algebraic thinking to solve problems involving patterns, sequences, generalizations, and linear and non-linear functions. The mathematical topics that underline 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, the teacher starts by holding up a picture of 3 different staircases and asks the students to think about the number of steps and squares in the growing pattern. Then the students use square tiles to explore the number of tiles needed to build staircases with varying numbers of steps.
In this level, students view a three-step staircase composed of squares (six in total). Their task is to determine the number of squares that make up each step and the total needed for the staircase.
This level addresses Common Core standard K.CC.B.5 as it has students count the number of objects in an arrangement. This can also be used as an introduction to standard 1.OA.A.1 by having the students create an equation to add the number of blocks in each step to find the total.
In this level, students extend the pattern of blocks and determine the number of blocks needed for a given step. They also find the inverse relationship, i.e. the number of steps when given the total number of blocks.
This level addresses Common Core standard 4.OA.C.5 by having students identify a number or shape pattern that follows a rule and then use that rule to solve problems. Students will look for and express repeated reasoning (MP.8) in order to justify the number of blocks needed or the number of steps given a number of blocks.
In this level, students consider staircases made up of blocks, in which each step has one more block than the step immediately above it. They determine how many blocks are needed to make the nth step of such staircases, as well as the total number of blocks needed to make the entire staircase.
This problem can be modeled with an arithmetic sequence that can be represented recursively or with an explicit formula (F-LE.2, F-BF.2). Students might also fit a quadratic function to the data to find the general rule (S-ID.6a).
In this level, students consider staircases made up of blocks. The first staircase has 1 step and requires 1block, the second staircase has 2 steps and requires 6 blocks, and the third staircase has 3 steps and requires 18 blocks. They determine how many blocks are needed to make the top step of such staircases, as well as the total number of blocks needed for the base of the staircase. Additionally, they describe a process to determine the number of blocks needed to build a staircase with 30 steps.
This problem can be modeled with an arithmetic sequence that can be represented recursively or with an explicit formula (F-LE.2, F-BF.2).
In this level, students consider staircases made up of blocks.The first staircase has 1 step and requires 1 block, the second staircase has 2 steps and requires 6 blocks, and the third staircase has 3 steps and requires 18 blocks. They determine a formula for the number of blocks needed to build a staircase within steps.
This level addresses Common Core standards A-CED.2 and F-BF.1a by having students generalize and write expressions or equations that describe the relationships between the number of steps and the number of blocks. Students might derive the formula by modeling the staircases as portions of cubes or rectangular prisms (G-MG.1) or by recognizing that a cubic function models the data and then fitting that function to the data (S-ID.6a).
PROBLEM OF THE MONTH
Download the complete packet of Growing Staircases 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.
To request the Inside Problem Solving Solutions Guide, please get in touch with us via the feedback form.