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More Patterns with Products

Number and Operations
Location: Unknown

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.

To assess prior knowledge, ask students to look at the multiplication charts they began filling out in the previous lesson. Encourage them to add any facts they feel sure that they know by heart. Then call out a product and ask students who have entered that product on the chart to name one pair of factors. Repeat with several products, including 0. A sample chart is shown:

1274 tiles2 

To begin the lesson, have students open the Sieve of Eratosthenes.

appicon Sieve of Eratosthenes

The Sieve is usually used to locate primes, but in this case it will be used to look at patterns in products.] Select the "Show Multiples" option, then enter a number. If necessary, remind the students of the meanings of "factor," product," and "multiple." When the products of that number are displayed, ask the students to name as many things as they can about the pattern they see. Repeat with other numbers from 2 through 8. Then ask what the students think the pattern would be if one factor were 0. [The answers will all be 0.]

Repeat this question, naming the factor as 1. [The products will be the other factor.] Now choose the multiples of 9 to display, and ask the students whether they see any patterns. [For example, the tens digits increase by 1, the units digits decrease by 1, the sum of the digits is 9, or the multiples are on a diagonal.]

Now assign the students to groups of three or four students each, and distribute the Rules for Card Games. This time students will play a new card game, Good Times. After they have played for several minutes, ask them to return to their seats and take out their My Multiplication Chart activity sheets, which they began in the previous lesson.

pdficon  Rules for Card Games 

pdficon My Multiplication Chart

Ask them to add any multiplication facts they are now sure of to the chart. Then have pairs of students exchange charts and ask each other the facts that are on the chart. If a student misses a fact, ask the partner to make a small mark by the fact to indicate that they need to practice it further. [Marking missed problems with a highlighter is a strategy that may benefit some students.]

Assessment Options

  1. At this stage of the unit, it is important for students to know how to:
    • Describe patterns in multiplication products
    • Predict and explain the results of multiplying by 0 and by 1
  2. As you ask the guiding questions, you may wish to encourage the students to raise other questions that they may have.
  3. Use the Class Notes recording sheet to document progress.
1. Students who need additional practice may use the Times Table tool.
2. Move on to the last lesson, Keeping It All Together.

Questions for Students 

1. What products did you see when the computer showed multiples of 2? Of 7? Of 9? Of 10? How can knowing this help you learn the multiplication facts?

2. What happens when one factor is 1? When one factor is 0? How can knowing this help you learn the multiplication facts?

[When one factor is 1, the product is equal to the other factor; When one factor is 0, the product is 0; This makes it easier to remember any of the multiplication facts where 0 or 1 are a factor.]

3. What is alike about 6 × 5 and 5 × 6? What is different?

[The product is 30 in each case; Where you begin skip counting is different.]

4. Write the products you say when you skip count by nines to 100. What do you notice about the sum of the digits in the products? What do you notice about the number in the tens place? How does knowing how to find products when one factor is 9 help with other multiplication facts?

[The sum of the digits is 9; The number in the tens place increases by ten each time; Student responses may vary.]

Teacher Reflection 

  • Which students have learned only a few multiplication facts? What activities should I plan for them?
  • What extension activities are appropriate for students who have learned all or almost all their multiplication facts?
  • What adjustments will I make the next time that I teach this lesson?
Number and Operations

Looking for Patterns

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

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:

  • Describe patterns in multiplication.
  • Explain the results of multiplying by 0 and by 1.

NCTM Standards and Expectations

  • Understand various meanings of multiplication and division.
  • Understand the effects of multiplying and dividing whole numbers.
  • Develop fluency in adding, subtracting, multiplying, and dividing whole numbers.

Common Core State Standards – Mathematics

Grade 3, Algebraic Thinking

  • CCSS.Math.Content.3.OA.C.7
    Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

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.

Common Core State Standards – Practice

  • CCSS.Math.Practice.MP6
    Attend to precision.