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Content Supports - Activities and rich problems
The Wheel Shop: Problem of the Month
Discipline
Mathematics
Grades
Kindergarten, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
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Professional Commentary

The Problems of the Month (POM) are designed to be used school wide to promote a problem-solving theme at your school. Each problem is divided into six levels of related problems, starting with the Primary Level and followed by Levels A through E. This allows access and scaffolding for the students at different grade levels. The goal is for all students to have the experience of attacking and solving non-routine problems and developing their mathematical reasoning skills as recommended in the mathematical practices of the Common Core Standards for Mathematics. Each sub-problem is aligned to the mathematical content and practice standards of the Common Core.

Primary Level:

This task challenges a student to think about groups of three, using counters to think about the
number of wheels for 1,2,3, . . . 10 tricycles.

Level A: This task challenges a student to use the relationship between multiplication and division to find the number of tricycles that have 18 wheels.
Level B:

This task challenges a student to reason about a situation with two variables, seats and wheels.
Students can use guess-and-check or set up simultaneous equations to solve for the number of
bicycles and go-carts.

Level C:

This task challenges a student to use systems of equations with 3 unknowns to find the number of bicycles, tandem bicycles, and tricycles in a shop.

Level D:

This task challenges a student to analyze relationships with four variables and three equations.

Level E:

This task challenges a student to use logic in a situation that involves using rational numbers,
inequalities, and a set of constraints. (author/jk)


Common Core State Standards for Mathematics
Kindergarten
Counting and Cardinality
Know number names and the count sequence.
K.CC.A.1 
Count to 100 by ones and by tens.

K.CC.A.2 
Count forward beginning from a given number within the known sequence (instead of having to begin at 1).
K.CC.A.3 
Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).
Count to tell the number of objects.
K.CC.B.4 
Understand the relationship between numbers and quantities; connect counting to cardinality.
  1. When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object.
  2. Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.
  3. Understand that each successive number name refers to a quantity that is one larger.
K.CC.B.5 
Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1–20, count out that many objects.
Operations and Algebraic Thinking
Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
K.OA.A.1 
Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
Grade 1
Operations and Algebraic Thinking
Represent and solve problems involving addition and subtraction.
1.OA.A.2 
Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
Grade 2
Operations and Algebraic Thinking
Work with equal groups of objects to gain foundations for multiplication.
2.OA.C.4 
Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.
Grade 3
Operations and Algebraic Thinking
Represent and solve problems involving multiplication and division.
3.OA.A.1 
Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.
3.OA.A.2 
Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.
3.OA.A.3 
Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
3.OA.A.4 
Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = ◊ ÷ 3, 6 × 6 = ?.
Understand properties of multiplication and the relationship between multiplication and division.
3.OA.B.5 
Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)
3.OA.B.6 
Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
3.OA.D.8 
Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
Grade 4
Operations and Algebraic Thinking
Use the four operations with whole numbers to solve problems.
4.OA.A.3 
Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
Grade 7
Expressions and Equations
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
7.EE.B.4 
Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
  1. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
  2. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.
Grade 8
Expressions and Equations
Analyze and solve linear equations and pairs of simultaneous linear equations.
8.EE.C.8 
Analyze and solve pairs of simultaneous linear equations.
  1. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
  2. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
  3. Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.
High School - Algebra
Creating Equations
Create equations that describe numbers or relationships
HSA-CED.A.3 
Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
Reasoning with Equations and Inequalities
Solve systems of equations
HSA-REI.C.6 
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
Standards for Mathematical Practice
CCSS.Math.Practice.MP1 
Make sense of problems and persevere in solving them.
CCSS.Math.Practice.MP4 
Model with mathematics.
OTM Mathematics, Statistics and Logics