Let Me Count the Ways

Ages 6–10 · 8 minutes · Math

How many ways can 3 friends stand in a line? A simple question leads to some deep ideas.

Big ideas often come from simple questions. Students join in a discussion about ordering objects and play with ideas from a branch of math called combinatorics. There is even a special challenge at the end with a follow up lesson! No background knowledge is required.

Part of

Math Problem Solving

Standards alignment

Common Core State Standards

MP7 — Look for and make use of structure.
MP8 — Look for and express regularity in repeated reasoning.
3.MP1 — Make sense of problems and persevere in solving them.
3.MP6 — Attend to precision.

Alabama

MA19.K.5 — Count to answer "how many?" questions.

Alaska

1. — Make sense of problems and persevere in solving them.
7. — Look for and make use of structure.
6.SP.7 — Solve or identify solutions to problems involving possible combinations (e.g., if ice cream sundaes come in 3 flavors with 2 possible toppings, how many different sundaes can be made using only one flavor of ice cream with one topping?) L
8. — Look for and express regularity in repeated reasoning.

Arizona

MP1 — Make sense of problems and persevere in solving them.
MP7 — Look for and make use of structure.
MP8 — Look for and express regularity in repeated reasoning.

Arkansas

1.CAR.7 — Solve real-world problems involving addition of three whole numbers whose sum is less than or equal to 20.
K.NPV.3 — Identify the position of objects in a set using ordinal numbers (first, second, third, etc.).

California

MP1 — Make sense of problems and persevere in solving them.
MP7 — Look for and make use of structure.
MP8 — Look for and express regularity in repeated reasoning.

Colorado

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.

Connecticut

MP1 — Make sense of problems and persevere in solving them.
MP7 — Look for and make use of structure.
MP8 — Look for and express regularity in repeated reasoning.

Delaware

MP1 — Make sense of problems and persevere in solving them.
MP7 — Look for and make use of structure.
MP8 — Look for and express regularity in repeated reasoning.

Florida

MA.K12.MTR.1.1 — Actively participate in effortful learning both individually and collectively. Mathematicians who participate in effortful learning both individually and with others: • Analyze the problem in a way that makes sense given the task. • Ask questions that will help with solving the task. • Build perseverance by modifying methods as needed while solving a challenging task. • Stay engaged and maintain a positive mindset when working to solve tasks. • Help and support each other when attempting a new method or approach.
MA.K12.MTR.5.1 — Use patterns and structure to help understand and connect mathematical concepts. Mathematicians who use patterns and structure to help understand and connect mathematical concepts: • Focus on relevant details within a problem. • Create plans and procedures to logically order events, steps or ideas to solve problems. • Decompose a complex problem into manageable parts. • Relate previously learned concepts to new concepts. • Look for similarities among problems. • Connect solutions of problems to more complicated large-scale situations.
MA.K12.MTR.4.1 — Engage in discussions that reflect on the mathematical thinking of self and others. Mathematicians who engage in discussions that reflect on the mathematical thinking of self and others: • Communicate mathematical ideas, vocabulary and methods effectively. • Analyze the mathematical thinking of others. • Compare the efficiency of a method to those expressed by others. • Recognize errors and suggest how to correctly solve the task. • Justify results by explaining methods and processes. • Construct possible arguments based on evidence.
MA.K12.MTR.6.1 — Assess the reasonableness of solutions. Mathematicians who assess the reasonableness of solutions: • Estimate to discover possible solutions. • Use benchmark quantities to determine if a solution makes sense. • Check calculations when solving problems. • Verify possible solutions by explaining the methods used. • Evaluate results based on the given context.
MA.K12.MTR.2.1 — Demonstrate understanding by representing problems in multiple ways. Mathematicians who demonstrate understanding by representing problems in multiple ways: • Build understanding through modeling and using manipulatives. • Represent solutions to problems in multiple ways using objects, drawings, tables, graphs and equations. • Progress from modeling problems with objects and drawings to using algorithms and equations. • Express connections between concepts and representations. • Choose a representation based on the given context or purpose.

Georgia

3.MP.1 — Make sense of problems and persevere in solving them.
3.MP.6 — Attend to precision.
3.MP.7 — Look for and make use of structure.
3.MP.3 — Construct viable arguments and critique the reasoning of others.
3.MP.8 — Look for and express regularity in repeated reasoning.

Hawaii

MP1 — Make sense of problems and persevere in solving them.
MP7 — Look for and make use of structure.
MP8 — Look for and express regularity in repeated reasoning.

Idaho

3.MP.1 — Make sense of problems and persevere in solving them.
3.MP.4 — Model with mathematics.
3.MP.6 — Attend to precision.
4.MP.7 — Look for and make use of structure.
3.MP.3 — Construct viable arguments and critique the reasoning of others.

Illinois

MP.1 — Make sense of problems and persevere in solving them.
MP.7 — Look for and make use of structure.
MP.8 — Look for and express regularity in repeated reasoning.

Indiana

3-5.DI.1 — Understand and use the basic steps in algorithmic problem solving (e.g., problem statement and exploration, examination of sample instances, design, implementation, and testing).
3-5.DI.2 — Develop a simple understanding of an algorithm (e.g., search, sequence of events, or sorting) using computer-free exercises.
K-2.DI.2 — Understand how to arrange (sort) information into useful order, such as sorting students by birth date, without using a computer.

Iowa

SMP1 — Make sense of problems and persevere in solving them.
SMP7 — Look for and make use of structure.
SMP8 — Look for and express regularity in repeated reasoning.

Kansas

MP.1 — Make sense of problems and persevere in solving them.
MP.7 — Look for and make use of structure.
MP.8 — Look for and express regularity in repeated reasoning.

Kentucky

KY.K.CC.5.a — Count to answer “how many?” questions with as many as 20 things arranged in a line, a rectangular array, or a circle.

Louisiana

MP7 — Look for and make use of structure.
MP1 — Make sense of problems and persevere in solving them.
MP8 — Look for and express regularity in repeated reasoning.

Maine

Use counting and numbers to identify and describe patterns in the natural and designed world(s).
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.)

Maryland

MP1 — Make sense of problems and persevere in solving them.
MP7 — Look for and make use of structure.
MP8 — Look for and express regularity in repeated reasoning.

Massachusetts

K.CC.B.4.b — 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.

Michigan

MP7 — Look for and make use of structure.
K.CC.B.4.a — 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.

Minnesota

K.1.1.1 — Recognize that a number can be used to represent how many objects are in a set or to represent the position of an object in a sequence. For example: Count students standing in a circle and count the same students after they take their seats. Recognize that this rearrangement does not change the total number, but may change the order in which students are counted.
2.2.1.1 — Identify, create and describe simple number patterns involving repeated addition or subtraction, skip counting and arrays of objects such as counters or tiles. Use patterns to solve problems in various contexts. For example: Skip count by 5s beginning at 3 to create the pattern 3, 8, 13, 18, … . Another example: Collecting 7 empty milk cartons each day for 5 days will generate the pattern 7, 14, 21, 28, 35, resulting in a total of 35 milk cartons.

Mississippi

MP7 — Look for and make use of structure.
MP8 — Look for and express regularity in repeated reasoning.
1.MD.4 — Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Missouri

1.RA.A.2 — Solve problems that call for addition of three whole numbers whose sum is within 20.

Montana

MP1 — Make sense of problems and persevere in solving them.
MP7 — Look for and make use of structure.
MP8 — Look for and express regularity in repeated reasoning.

Nebraska

2.N.5 — Number and Algebraic Relationships: Students will create and solve problems involving addition and subtraction and work with equal groups of objects to gain foundations for multiplication.

Nevada

MP1 — Make sense of problems and persevere in solving them.
MP7 — Look for and make use of structure.
MP8 — Look for and express regularity in repeated reasoning.

New Hampshire

MP7 — Look for and make use of structure.
MP1 — Make sense of problems and persevere in solving them.
MP8 — Look for and express regularity in repeated reasoning.

New Jersey

1 — Make sense of problems and persevere in solving them.
7 — Look for and make use of structure.
8 — Look for and express regularity in repeated reasoning.

New Mexico

MP1 — Make sense of problems and persevere in solving them.
MP7 — Look for and make use of structure.
MP8 — Look for and express regularity in repeated reasoning.

New York

MP.1 — Make sense of problems and persevere in solving them.
MP.7 — Look for and make use of structure.
MP.8 — Look for and express regularity in repeated reasoning.

North Carolina

MP.8 — Look for and express regularity in repeated reasoning.
NC.1.OA.2 — Represent and solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, by using objects, drawings, and equations with a symbol for the unknown number.

North Dakota

3-5.MA.P — Learners can develop and carry out a logical plan to problem-solve situations, reflect on the reasonableness of solutions, and explore alternate strategies with guidance..
3-5.MA.R — Learners can reason logically based on experience and knowledge, citing evidence to support their reasoning and conclusions.
3-5.MA.C — Learners can make connections and summarize related ideas using supporting evidence.

Ohio

MP.7 — Look for and make use of structure.
MP.8 — Look for and express regularity in repeated reasoning.

Oklahoma

MAP4 — Develop Mathematical Reasoning
MAP5 — Develop a Productive Mathematical Disposition

Oregon

MP7 — Look for and make use of structure.
K.DR.A.1 — Generate questions to investigate situations within the classroom. Collect or consider data that can naturally answer questions by sorting and counting.

Pennsylvania

Look for and express regularity in repeated reasoning.
Reason abstractly and quantitatively.

Rhode Island

MP1 — Make sense of problems and persevere in solving them.
MP7 — Look for and make use of structure.
MP8 — Look for and express regularity in repeated reasoning.

South Carolina

2.PS.7b — Recognize mathematical repetition in order to make generalizations.
3.PS.7b — Recognize mathematical repetition in order to make generalizations.

South Dakota

MP1 — Make sense of problems and persevere in solving them.
MP7 — Look for and make use of structure.
MP8 — Look for and express regularity in repeated reasoning.

Utah

2.MP.7 — Look for and make use of structure.
K.MP.1 — Make sense of problems and persevere in solving them.
K.MP.8 — Look for and express regularity in repeated reasoning.
K.MP.3 — Construct viable arguments and critique the reasoning of others.
K.MP.6 — Attend to precision.

Vermont

MP1 — Make sense of problems and persevere in solving them.
MP7 — Look for and make use of structure.
MP8 — Look for and express regularity in repeated reasoning.

Virginia

4.PS.2b — Model and determine all possible outcomes of a given simple event where there are no more than 24 possible outcomes, using a variety of manipulatives (e.g., coins, two-sided counters, number cubes, spinners).

Washington

MP1 — Make sense of problems and persevere in solving them.
MP7 — Look for and make use of structure.
MP8 — Look for and express regularity in repeated reasoning.

Washington, D.C.

MP1 — Make sense of problems and persevere in solving them.
MP7 — Look for and make use of structure.
MP8 — Look for and express regularity in repeated reasoning.

West Virginia

MHM7 — Look for and make use of structure.

Wisconsin

M.1.MD.C.4 — Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.
M.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.

Wyoming

MP8 — Look for and express regularity in repeated reasoning.
1.MD.J.4 — Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.