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Looking for Patterns

Number and Operations
Location: Unknown

Students skip count and examine multiplication patterns. They also explore the commutative property of multiplication.

To assess prior knowledge, ask students to skip count by 10's. Then ask a volunteer to name a number they said when they skip-counted. Identify this as a product. Ask students to name the numbers that they would multiply together to get that product. Identify these as factors. Repeat with other numbers in the counting sequence.

10, 20, 30, 40, 50, 60, 70, 80, 90, …

30 = 5 × 6, so 5 and 6 are factors of 30.

To begin the lesson, give each student a copy of the 0—99 Chart and a supply of counters or cubes.

pdficon 0—99 Chart

Ask students to cover the numbers that are called as you skip count by 2’s. Direct students to observe and report the pattern that results. Note that columns are formed when the numbers are covered in this counting sequence. Engage students in a discussion about this pattern and why it occurs.

Have students clear their 0‑99 chart and ask them to cover the numbers with counters or cubes as you call out the 5’s counting sequence [5, 10, 15, 20, 25, 30, …]. Ask students to describe the resulting pattern. Engage students in a discussion about the comparison between the column pattern created when counting by 2’s and the one created when counting by 5’s. Encourage students to explain their thinking.

Model for students how to count by 3’s and cover the numbers that are included in this sequence. Discuss how the resulting pattern compares with the two previous patterns created when counting by 2’s and 5’s. Invite students to explain how and why the counting pattern for 3’s is different from the others. Then ask students to clear their 0‑99 chart.

Give each student crayons and tell them that they will now color the 0‑99 chart that they used in the previous activity. (To shorten the activity, you may wish to use only the top half of the chart.) When the students are ready, ask them to skip count by 2's in unison, coloring each number that they say with a red crayon. It may be helpful for some students to say 1 softly, 2 loudly, 3 softly, 4 loudly, and so on.

Repeat this same process with counting by 3's and coloring each number with a yellow crayon. Then have students skip count by 5's, coloring each number with a blue crayon. (A portion of the completed chart is shown below.)

1267 tiles1 

Call students' attention to numbers that have been colored with more than one crayon. For instance, some numbers on their chart are orange (colored both red and yellow), and ask students if these numbers represent a skip-counting pattern. [They will show the pattern of skip counting by 6; each number is divisible by both 2 and 3.] Repeat with the numbers that are colored purple (i.e., colored with both red and blue crayons). [These are products of 10, having both 2 and 5 as factors.] Ask the students why the number 15 is colored green (i.e., colored with both yellow and blue). [It has both 3 and 5 as factors.]

Assessment Options

  1. At this stage of the unit, it is important for students to know how to:
    • Skip count by twos, threes, and fives
    • Find products by adding equal sets
    • Define and use the commutative property
  2. The guiding questions will help the students focus on the mathematics in this lesson. These questions will also aid you in assessing the students' level of knowledge and skill. Documenting information about students' understanding and skills throughout the unit on the Class Notes may help you plan appropriate extension and remediation activities. You may also find this information useful when discussing the students' progress toward learning targets with the students themselves and with parents or caregivers, administrators, and colleagues.


  1. Say to students, "Let's skip count by 3 by singing the answers to the tune of "Are You Sleeping?" Is that easier or harder for you than just saying the words? [Singing slowly will make it easier for most students.]

    You may wish to refer to Skip-Counting Songs for Multiplication for song ideas for various factors.

    pdficon Skip-Counting Songs for Multiplication

  2. Move on to the next lesson, Looking for Calculator Patterns.

Questions for Students 

1. What numbers did you say when you skip count by 2? By 5? By 3? By 10?

[2, 4, 6, 8, etc.; 5, 10, 15, 20, 25, etc.; 3, 6, 9, 12, 15, etc.; 10, 20, 30, 40, 50, etc.]

2. How can we show that 3 × 6 has the same product as 6 × 3?

[3 groups of 6 produces 18; 6 groups of 3 produces 18.]

3. What is similar about skip counting by 5 and adding 5 + 5 + 5? What is different?

[For both, you get multiples of 5; Counting by 5 can be quicker than adding 5 + 5 + 5.]

4. How does knowing how to skip count by 2 help you skip count by 4?

[Student answers may vary.]

Teacher Reflection 

  • Which students remembered the commutative property? What experiences are necessary for those who did not?
  • When students were unable to fluently count by 3’s, what strategies would facilitate their recall of this counting sequence?
  • Which students are able to skip count by 2, 3, 5, and 10 rapidly and correctly? What extension activities would be appropriate for those students?
  • What adjustments will I make the next time that I teach this lesson?
Number and Operations

Looking for Calculator Patterns

Students use a web-based calculator to create and compare counting patterns using the constant function feature of the calculator. Making connections between multiple representations of counting patterns reinforces students understanding of this important idea and helps them recall these patterns as multiplication facts.
Number and Operations

More Patterns with Products

After using an interactive Web site to find patterns in the multiplication tables, the students practice multiplication facts and record their current level of mastery of the multiplication facts on their personal multiplication chart.
Number and Operations

Keeping It All Together

By playing card games and using the The Product Game applet, students practice the multiplication facts. As students continue to master their facts, the teacher closely monitors their progress.

Learning Objectives

Students will:

  • Skip count by twos, threes, fives and tens.
  • Find products by adding equal sets.
  • Explore the commutative property of multiplication.

NCTM Standards and Expectations

  • Recognize equivalent representations for the same number and generate them by decomposing and composing numbers.
  • Understand various meanings of multiplication and division.
  • Identify and use relationships between operations, such as division as the inverse of multiplication, to solve problems.

Common Core State Standards – Mathematics

Grade 4, Algebraic Thinking

  • CCSS.Math.Content.4.OA.C.5
    Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule ''Add 3'' and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.