# The Wheel Shop

In the problem The Wheel Shop, students use algebraic thinking to solve problems involving unknowns, equations, and simultaneous constraints. The mathematical topics that underlie this problem are variables, inverse operations, equations, equalities, inequalities, and simultaneous systems. 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.

PRE-K
In this task, students will count the number of wheels on a tricycle. They will use counters to represent the wheels and solve for how many wheels would be on 3 and 10 tricycles.

LEVEL A
In this level, students are presented with the task of considering a tricycle shop with 18 wheels and determining how many tricycles there are. Their task involves finding an unknown and “undoing” the straightforward question of how many wheels 6 tricycles have in all.

This level supports Common Core standards 3.OA.A.1 and 3.OA.A.2. Students can approach solving this problem using either multiplication with an unknown number of groups or division to find the number of tricycles there are if the shop has 18 wheels.

LEVEL B
In this level, students are presented with a new situation that involves components of bicycles and go-carts.

Students at an elementary level can use skills and strategies related to Common Core standard 4.OA.A.3 to solve this problem. They will use their understanding of the four operations to work through the problem in multiple steps—an important precursor to equations work in middle school. At a secondary level, this problem addresses standards 8.EE.C.8b and 8.EE.C.8c, where students write and solve two linear equations with two variables.

LEVEL C
In this level, students are presented with a situation that involves components of bicycles, adult tricycles, and tandem bicycles. The situation can be translated into systems of linear equations with three unknowns.

This level supports Common Core standard A-REI.C.6. Students write and solve a system of three linear equations with three unknowns.

LEVEL D
In this level, students are given a situation that can be translated into a system of three equations with four unknowns. Students are asked to define the relationship between two unknowns.

This level extends the work of Common Core standard A-REI.C.6. Students build on algebraic and symbolic methods for solving systems of equations, including substitution and balance. They reason to find the relationship between two of the unknowns. Students will look for and make use of the structure (MP.7) within the equations to help them find the relationship between the two unknowns.

LEVEL E
In this level, students are presented with a logic situation that involves using rational numbers, inequalities, and a set of constraints. Students are asked to find the number of bikes in the shop and the range of repairs that need to be made.

This problem supports Common Core standard S-CP.A.1 as students must interpret the number and types of repairs represented in the problem as a sample space. They must also think about the complement of what is known. In the last part of the problem, students must solve a system consisting of an inequality and an equation. Students can use algebraic techniques or graphing, as described in Common Core standard A-REI.D.12. Students must reason abstractly and quantitatively (MP.2) as they make sense of the problem and potential solutions. Students may choose to organize their thinking using a diagram or table.

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