Let Me Count the Ways: Bonus Challenge

Let Me Count the Ways: Bonus Challenge

Ages 6–10 · 4 minutes · Math

Solve an extra challenge from the Let Me Count the Ways lesson with 4 friends!

In the lesson Let Me Count the Ways, students found how many ways 3 friends can stand in a line. But what about 4 friends in a line? We'll practice logical skills to solve the question by cleverly building on solutions to simpler problems. No background knowledge is required.

This lesson is best taken after Let Me Count the Ways.

Part of

Standards alignment

Common Core State Standards

  • 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.

Alabama

  • MA19.3.9 — Recognize and explain arithmetic patterns using properties of operations.
  • MA19.3.8 — Determine and justify solutions for two-step word problems using the four operations and write an equation with a letter standing for the unknown quantity. Determine reasonableness of answers using number sense, context, mental computation, and estimation strategies including rounding.

Alaska

  • 7. — Look for and make use of structure.
  • 1. — Make sense of problems and persevere in solving them.
  • 3.OA.3 — Use multiplication and division numbers up to 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).
  • 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

Arizona

  • MP7 — Look for and make use of structure.
  • MP8 — Look for and express regularity in repeated reasoning.
  • 3.OA.A.1 — Interpret products of whole numbers as the total number of objects in equal groups (e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each).
  • MP1 — Make sense of problems and persevere in solving them.

Arkansas

  • 3.CAR.6 — Solve real-world problems using multiplication and division within 100 involving equal groups, arrays, partitive and measurement division.
  • 2.CAR.7 — Solve one and two-step real-world problems involving addition and subtraction within 100 in situations of adding to, taking from, putting together, taking apart, and comparing unknowns in all positions.

California

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

Colorado

  • 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.
  • 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. (This evidence outcome is limited to problems posed with whole numbers and having whole-number answers; students should know how to perform operations in the conventional order of operations when there are no parentheses to specify a particular order.)

Connecticut

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

Delaware

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

Florida

  • MA.2.AR.3.2 — Use repeated addition to find the total number of objects in a collection of equal groups. Represent the total number of objects using rectangular arrays and equations.
  • 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.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.
  • MA.K12.MTR.3.1 — Complete tasks with mathematical fluency. Mathematicians who complete tasks with mathematical fluency: • Select efficient and appropriate methods for solving problems within the given context. • Maintain flexibility and accuracy while performing procedures and mental calculations. • Complete tasks accurately and with confidence. • Adapt procedures to apply them to a new context. • Use feedback to improve efficiency when performing calculations.

Georgia

  • 3.MP.7 — Look for and make use of structure.
  • 3.MP.8 — Look for and express regularity in repeated reasoning.
  • 3.PAR.3.6 — Solve practical, relevant problems involving multiplication and division within 100 using part-whole strategies, visual representations, and/or concrete models.
  • 2.NR.3 — Work with equal groups to gain foundations for multiplication through real-life, mathematical problems.

Hawaii

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

Idaho

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

Illinois

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

Indiana

  • PS.8 — Look for and express regularity in repeated reasoning.
  • PS.8 — Look for and express regularity in repeated reasoning.
  • 3.CA.7 — Solve real-world problems involving whole number multiplication and division within 100 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). (E)

Iowa

  • SMP7 — Look for and make use of structure.
  • SMP8 — Look for and express regularity in repeated reasoning.
  • 2.OA.C.4 — Use repeated addition to find the total number of objects arranged in equal groups and rectangular arrays; write an equation to express the total as a sum of equal addends.
  • SMP1 — Make sense of problems and persevere in solving them.

Kansas

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

Kentucky

  • MP8 — Look for and express regularity in repeated reasoning.
  • KY.3.OA.8 — Use various strategies to solve two-step word problems using the four operations (involving only whole numbers with whole number answers). 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.

Louisiana

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

Maine

  • 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.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.

Maryland

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

Massachusetts

  • MP.8 — Look for and express regularity in repeated reasoning.
  • 3.OA.D.8 — Solve two-step word problems using the four operations for problems posed with whole numbers and having whole number answers. 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.

Michigan

  • MP7 — Look for and make use of structure.

Minnesota

  • 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.
  • 3.1.2.3 — Represent multiplication facts by using a variety of approaches, such as repeated addition, equal-sized groups, arrays, area models, equal jumps on a number line and skip counting. Represent division facts by using a variety of approaches, such as repeated subtraction, equal sharing and forming equal groups. Recognize the relationship between multiplication and division.

Mississippi

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

Missouri

  • 3.RA.A.1 — Interpret products of whole numbers.
  • 3.RA.A.4 — Use multiplication and division within 100 to solve problems.
  • 3.RA.A.3 — Describe in words or drawings a problem that illustrates a multiplication or division situation.

Montana

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

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.
  • 2.N.5.c — Use repeated addition to find the total number of objects arranged in an array no larger than five rows and five columns and write an equation to express the total.

Nevada

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

New Hampshire

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

New Jersey

  • 7 — Look for and make use of structure.
  • 8 — Look for and express regularity in repeated reasoning.
  • 1 — Make sense of problems and persevere in solving them.
  • 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 and/or represent a context in which a total number of objects can be expressed as 5 × 7.

New Mexico

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

New York

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

North Carolina

  • MP.8 — Look for and express regularity in repeated reasoning.
  • NC.3.OA.1 — For products of whole numbers with two factors up to and including 10: • Interpret the factors as representing the number of equal groups and the number of objects in each group. • Illustrate and explain strategies including arrays, repeated addition, decomposing a factor, and applying the commutative and associative properties.

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.
  • 3.OA.1 — Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. (Note: These standards are written with the convention that a x b means a groups of b objects each; however, because of the commutative property, students may also interpret 5 x 7 as the total number of objects in 7 groups of 5 objects each).
  • MP.1 — Make sense of problems and persevere in solving them.

Oklahoma

  • MAP4 — Develop Mathematical Reasoning
  • 3.N.2.1 — Represent multiplication facts by modeling a variety of approaches (e.g., manipulatives, repeated addition, equal-sized groups, arrays, area models, equal jumps on a number line, skip counting).

Oregon

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

Pennsylvania

  • Make sense of problems and persevere in solving them.
  • Look for and express regularity in repeated reasoning.
  • CC.2.2.2.A.3 — Work with equal groups of objects to gain foundations for multiplication.
  • CC.2.2.3.A.1 — Represent and solve problems involving multiplication and division.

Rhode Island

  • MP7 — Look for and make use of structure.
  • MP8 — Look for and express regularity in repeated reasoning.
  • 3.OA.A.1 — Interpret products of whole numbers, e.g., interpret 5 x7 as the total number of objects in five groups of seven objects each. For example, describe a context in which a total number of objects can be expressed as 5 x 7.
  • MP1 — Make sense of problems and persevere in solving them.

South Carolina

  • 3.ATO.3 — Solve real-world problems involving equal groups, area/array, and number line models using basic multiplication and related division facts. Represent the problem situation using an equation with a symbol for the unknown.
  • 3.PS.3b — Compare and discuss the validity of various reasoning strategies.
  • 3.PS.7b — Recognize mathematical repetition in order to make generalizations.
  • 2.PS.7b — Recognize mathematical repetition in order to make generalizations.

South Dakota

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

Tennessee

  • 3.OA.A.1 — Interpret the factors and products in whole number multiplication equations (e.g., 4 x 7 is 4 groups of 7 objects with a total of 28 objects or 4 strings measuring 7 inches each with a total length of 28 inches).
  • 2.OA.C.4 — Use repeated 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. For example, a 3 by 4 array can be expressed as 3 + 3 + 3 + 3 = 12 or 4 + 4 + 4 = 12.
  • 3.OA.A.3 — Multiply and divide within 100 to solve contextual problems, with the unknown in any positions, in situations involving equal groups, arrays/area, and measurement quantities using strategies based on place value, the properties of operations, and the relationship between multiplication and division (e.g., contexts including computations such as 3 x ? = 24, 6 x 16 = ?, ? ÷ 8 = 3, or 96 ÷ 6 = ?).

Texas

  • 3.4D — determine the total number of objects when equally-sized groups of objects are combined or arranged in arrays up to 10 by 10;
  • 2.6A — model, create, and describe contextual multiplication situations in which equivalent sets of concrete objects are joined; and
  • 3.4E — represent multiplication facts by using a variety of approaches such as repeated addition, equal-sized groups, arrays, area models, equal jumps on a number line, and skip counting;
  • 3.4K — solve one-step and two-step problems involving multiplication and division within 100 using strategies based on objects; pictorial models, including arrays, area models, and equal groups; properties of operations; or recall of facts.

Utah

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

Vermont

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

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

  • MP7 — Look for and make use of structure.

Washington, D.C.

  • 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.

West Virginia

  • MHM7 — Look for and make use of structure.

Wisconsin

  • M.3.OA.A.1 — Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each.
  • M.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.
  • M.3.OA.D.7 — Solve two-step word problems, posed with whole numbers and having whole number answers, using the four operations. Represent these problems using one or two equations with a letter standing for the unknown quantity. If one equation is used, grouping symbols (i.e. parentheses) may be needed. Assess the reasonableness of answers using mental computation and estimation strategies.

Wyoming

  • 2.OA.C — Work with equal groups of objects to gain foundations for multiplication.
  • 4.MP7 — Look for and make use of structure.
  • MP8 — Look for and express regularity in repeated reasoning.
  • 4.MP1 — Make sense of problems and persevere in solving them.