Patterns In Multiplication (Days 1 - 5)

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Objective

SWBAT analyze patterns in multiplication products to explain patterns in multiplication.

Big Idea

As students develop fluency finding and explaining patterns with products helps them make sense of multiplication.

Warm-Up

10 minutes

To begin this 5 day lesson, I ask students to review patterns that include repetitive and growing patterns. Some types of repetitive patterns include AABBCC or ABCCBA.  An example of growing patterns includes AB, AABB, AABBCC, AABBCCDD.

Beginning with this type of review cues students to focus and analyze numbers patterns rather than thinking about a specific operation or solution. 

Each day I focus on creating a more complex pattern to challenge the students. This was also a way for me to engage gifted students with challenging patterns and sequences. Throughout the week I use both written patterns and patterns using of manipulatives. When students are writing the pattern, I use a new grid each day.  Centimeter grid paper works well for this. 

By the end of the week, the students choose to create their own patterns to challenge their partners. These patterns can be written on whiteboards, paper, or built with a model. 

Mini-Lesson: Multiplying by 2 and 5 (Day 1)

10 minutes

During the week, I present mini-lessons for multiples 2 and 5, 3 and 9, 6, 3, 6 and 9, 4 and 8 .  I chose this order, and combinations, of facts to build in complexity toward the end of the week.  To differentiate for gifted students, I chose to have them look at the patterns for 7 on the last day. A new, blank 100 chart is used each day. This lesson provides students with the opportunity to use Mathematical Practice 8, looking for and expressing regularity in repeated reasoning as they determine multiplies of a given number.    

The primary expectations throughout this week of pattern seeking are explained in the section following these mini-lessons.

I model with 2 and then 5 because these patterns are very familiar to my students from skip counting activities. They quickly identify the numbers for these patterns.  I have them fill out a 100 chart for each of these numbers, using two different colors and mark the squares in halves. I explained their challenge was to be able to write and explain the number patterns for the multiples of these two digits. Using sentence stems for these explanations such as:

Numbers that are multiplied by two have ones digits that follow the pattern of ______________. 

I provide a more complete sentence stem on the first day, and as the week progresses, the sentence stem can be modified or reduced so that the student is providing more of their own explanation.  This is something that can be differentiated for the students as needed.  

I ask the students what do they notice about the numbers multiplied by 2.  I want their focus to be on the pattern in the ones place of 2, 4, 6, 8, 0 and the increasing pattern in the tens of five ones, five twos, five threes, and so on. It is important for students to reason through and find this pattern themselves, rather than the teacher pointing it out. I facilitate only, and use visual supports (e.g. circling the numbers in the ones place with one color, or "re-presenting" this pattern separately but in close proximity to the multiples pattern) to assist as they begin to articulate what they notice. 

I then ask the students think about the numbers for multiples of 5.  This focus again is on the changing pattern in the tens digits.  This pattern includes 0 1, 1 2, 2 3, 3 4, 4 5, and so on. 

The screen cast below includes a demonstration and visual examples for clarification for replicating this lesson. 

Mini-Lesson: Multiplying by 3 and 9 (Day 2)

10 minutes

On the second day the students begin looking for similar patterns in the ones and tens places. Using the hundred chart,  I had them use two different colors and mark the squares in halves.  The ones digits for multiples of three include 3, 6, 9, 2, 5, 8, 1, 4, 7, 0.  The tens digits increase in a pattern of 0 0 0 0, 1 1 1, 2 2 2, 3 3 3 3.  It is important for students to realize that when a zero is in the ones place value there will be four multiples in those rows on the chart.  

Multiples of 9 are more obvious for students to see.  The ones place value counts backward -  0, 9, 8, 7, 6, 5, 4, 3, 2, 1, and the tens place value increases - 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 0.  The extra 9 includes 90 and 99.

It is important for students to reason through and find this pattern themselves, rather than the teacher pointing it out. I facilitate only, and use visual supports (e.g. circling the numbers in the ones place with one color, or "re-presenting" this pattern separately but in close proximity to the multiples pattern) to assist as they begin to articulate what they notice.  

The lesson for today introduces students to adding digits together to find another type of pattern. Adding digits together for multiples of 3 results in a number that is another multiple of 3.  For example adding the digits of the number twelve results in 1 + 2 = 3, and adding the digits for 36 results in 3 + 6 = 9.  Also, adding the digits for twenty-four results in 2 + 4 = 6.  I also present numbers such as three hundred twenty-seven with digits that add 3 + 2 + 7 = 12. 

Next, I ask the students to do the same with digits for 9 and to see what patterns they see.  The factors to 90 will have digits that add to a sum of 9.  The number ninety-nine is the first exception they will encounter and requires two addition steps 9 + 9 = 18, and 1 + 8 = 9.   

Mini Lesson: Multiplying by 6 (Day 3)

10 minutes

On the third day the students begin looking for similar ones and tens digit patterns. Using the hundred chart, the students mark the multiples of six.   The ones digits for multiples of six are 0, 6, 2, 8, 4, 0.  The tens digits increase in a pattern on the rows of the hundred chart of 0 0, 1  1, 2, 3 3, 4 4, 5, 6 6, 7 7, 8, 9  9 (two like factors, two like factors, one factor, two like factors, two like factors, one factor). 

Next, have the students look at adding the digits as they did with three and nine.  Factors of six do not follow the same pattern.  Numbers that are factors of six have differing sums.  This factor has two criteria which may be difficult for students to see, and may need some introduction for students.  The rules for six are that the number resulting from adding the two digits of a product are even and and a factor for 3. One of the numbers to model this with is the number 54.  It is even and the digits add to nine. Have the students check other numbers to verify this rule.

It is important for students to reason through and find this pattern themselves, rather than the teacher pointing it out. I facilitate only, and use visual supports (e.g. circling the numbers in the ones place with one color, or "re-presenting" this pattern separately but in close proximity to the multiples pattern) to assist as they begin to articulate what they notice.

Mini Lesson - Multiplying by 3, 6, and 9 Comparisons (Day 4)

10 minutes

Comparing the differences between the rules for 3 and 9 to the rule for factors of 6 will result in a chart requiring the use of three colors.  I  provide the students with the charts they had previously created, and I also create a master chart for the students to reference during the lesson. Have students mark the squares of the hundreds charts in narrow third strips rather than halves as they previously have done.  I model how to do this for the students for clarity.  

The ones digits for multiples of three include 3, 6, 9, 2, 5, 8, 1, 4, 7, 0. Again, I want to emphasize the critical importance of students "making these discoveries", rather than the teacher telling them.

Next, I have the students mark the numbers for factors of nine. The ones place value counts backward from 0, 9, 8, 7, 6, 5, 4, 3, 2, 1. Finally, they mark the factors for six.  The ones digits follow the pattern of 0, 6, 2, 8, 4, 0.

Using the rule for these factors to compare is demonstrated with the number 39.  When adding the digits in the product, say for example 39:  3 + 9 = 12 - again, a factor of 3.  However, 39 is an odd digit so it will only have two colors.  One of the numbers that will have three colors is 54.  The digits of 5 + 4 = 9 which is both a factor of 3 and 9, and the number is even.

It is important for students to reason through and find this pattern themselves, rather than the teacher pointing it out. I facilitate only, and use visual supports (e.g. circling the numbers in the ones place with one color, or "re-presenting" this pattern separately but in close proximity to the multiples pattern) to assist as they begin to articulate what they notice.  

Mini Lesson - Multiplying by 4 and 8 (Day 5)

10 minutes

On the last day the students use a new hundred chart and began marking the numbers for multiples of 4 and 8.  I have students use two different colors and mark the squares in halves. looking for similar ones and tens place digit patterns. The ones place for multiples of four include 0, 4, 8, 2, 6.  The tens place increase in a pattern of 0 0 0, 1 1, 2 2 2, 3 3, 4 4 4.  It is important for students to realize that the tens digits follow a pattern of three factors in when even and and two factors with odd digits in the tens place. 

Next, I have the students mark the chart for the multiples of 8.  This pattern for digits in the ones place is 0, 8, 6, 4, 2, 0.  The pattern for those in the tens place is 0, 1, 2, 3, 4, 4, 5, 6, 7, 8, 8, 9. I ask the students to first focus on understanding these patterns before trying our additional step of adding the digits in the ones and tens place.

It is important for students to reason through and find this pattern themselves, rather than the teacher pointing it out. I facilitate only, and use visual supports (e.g. circling the numbers in the ones place with one color, or "re-presenting" this pattern separately but in close proximity to the multiples pattern) to assist as they begin to articulate what they notice.  

Then, the students begin looking for patterns with adding digits.  This is familiar to them because our previous lesson focused on three and nine.  They will realize that adding digits for these numbers does not create the patterns we've been seeing.  8 x 2 = 16, when we add the two digits in the product the sum is seven,  8 x 4 = 32  --  3 + 2 = 5, and 8 x 8 = 64 -- 6 + 4 = 10.  The focus of this lesson is on the initial simple pattern we find in the ones and tens places.  As students move into fourth grade skills with division they will look at the number created with tens and ones for divisibility by four such as 316, 432, 824.  This is something that can be demonstrated for students during this lesson. 

Finally, the students analyze the numbers for multiples of eight. This is even more complex and requires students to look at the final three numbers because the ones digits pattern for four and eight are in different order.  In third grade the focus remains on the pattern for the separate tens and ones digits, but this can be differentiated here for gifted students to work through the same lesson with three digit numbers. 

Explain The Rules (Days 1-5)

10 minutes

Each day the students are responsible for explaining the rules and their thinking about the patterns.  This writing allows students to use their own words to describe patterns, describe how the ones digits and tens digits change.  

I chose to have the students write their thinking on the back of their charts each day so that they could be combined at the end of the week into one packet.  

Sentence stems can be provided to support students and keep their explanations focused on the objective of the lesson.  It also support differing academic and language levels for the students.