##
* *Reflection: Diverse Entry Points
Multiplication: x7 - Section 5: Independent Practice

The task of "finding patterns" provides students with a range of entry points. Some students will find simple patterns whereas others will find complex patterns.

During the lesson, a student came up to the board and showed me and odd and even pattern: Odd Even Pattern. To further develop his reasoning skills, I asked: *Does this always work? *He didn't know, so I encouraged him to continue the with another student to test his theory.

After a while, I followed up with these students: Does this Pattern Always Work?. It was great to see these students construct and check a theory.

Also, understanding and using patterns is key component of mathematical reasoning. I've noticed that the patterns students notice tend to begin at a simpler level (such as noticing colors or repeating numbers). Then, as students develop their thinking, the patterns become more complex (such as creating rule or equations for the pattern).

Also, independent work time is also a perfect time for students to share other patterns they saw or helpful strategies. At times, I had up to four students in line waiting to show me a strategy on the board! They were just so excited to share!

Here's a video of the same student explaining how he remembers which numbers are odd and even. The fact that he has remembered this from previous years shows how powerful modeling is: Even & Odd Numbers.

I can't wait to see this student's reasoning skills develop this year!

*Diverse Entry Points: Diverse Entry Points*

# Multiplication: x7

Lesson 7 of 7

## Objective: SWBAT multiply a multi-digit number by 7.

The goal of this seven-day Multiplication Kick-Off is to review multiplication facts and to build a deep understanding of why we multiply! These seven lessons provide a gradual learning progression that slowly increases with complexity. You could teach these lessons in the middle of a unit or at the beginning of a Multiplication Unit. I taught these lessons within my Measurement Unit at the beginning of the year. Here's why: I didn't want to wait until my multiplication unit to review multiplication facts and to teach students how to solve a simple algorithm. After teaching these lessons, I could then implement daily fact and algorithm homework practice. Here’s the order in which I taught these lessons:

The goal of this activity was to help make multiplication understandable, fun, and memorable! I wanted to give students a context to discuss multiplication in the upcoming lessons. Not only that, but students loved creating monster paper plates so student engagement was high! For each of the following lessons, student had their monster paper plates on their desks as a reference and visual aid. This worked! Students continually went back to this monster problem to reason with multiplication.

Day 2: Multiplication: x0, x1, x2

1. I started by teaching x0, x1, and x2 as these are the easiest multiplication facts. Many of my students were still mixing up 5 x 0 and 5 x 1. They didn’t truly understand the meaning behind x0 and x1.

2. Students used both a number line on paper and unix cubes to show how to multiply by 0, 1, and 2. The goal was for students to make the connection between repeated addition (something students are very familiar with) and multiplication.

3. To further build number sense and a deeper understanding of multiplication, students analyzed patterns they noticed between counting by ones and counting by twos.

4. Finally, we applied new learning to a simple algorithm. Students grasped this concept quickly and were very successful.

1. Next, we moved onto x4 facts so that we could build upon previous learning of x2 facts. It’s easier for students to learn their x4 facts when they understand x2 facts. They quickly catch on that 4 x 6 is when you “just take two jumps of 6 and then double it.“

2. Students used both a number line on paper and unfix cubes to show how to multiply by 4. The goal was for students to make the connection between repeated addition (something students are very familiar with) and multiplication.

3. To further build number sense and a deeper understanding of multiplication, students analyzed patterns they noticed between counting by fours and counting by twos.

4. Finally, we applied new learning to a simple algorithm. Again, students grasped this concept quickly and were very successful.

1. I decided to teach x3 and x6 next as students can use the x3 facts to get to x6 facts. To solve 5 x 6, you can first take five jumps of three (5x3) and then double it to get 5 x 6. For this reason, it’s easier for students to learn x6 facts right alongside x3 facts.

2. Students used both a number line on paper and unix cubes to show how to multiply by 3 and how to multiply by 6. The goal was for students to make the connection between repeated addition (something students are very familiar with) and multiplication.

3. To further build number sense and a deeper understanding of multiplication, students analyzed patterns they noticed between counting by threes and counting by fours. Finally, we applied new learning to a simple algorithm.

4. Again, students grasped this concept quickly and were very successful.

Day 5: Multiplication: x10, x5, x9

1. We moved onto x10, x5, and x9. Students discover how to use 10 to better understand x5 and x9 facts. “Times five” is just “half of x 10.” For example, to find 7 x 5, you can “take seven jumps of ten and then split the product in half.” Students also learn that 6 x 9 is the same as “six jumps of ten – six.” For this reason, it’s easier for students to learn x9 and x5 facts alongside x10 facts.

2. Students used both a number line on paper and unix cubes to show how to multiply by 3 and how to multiply by 6. The goal was for students to make the connection between repeated addition (something students are very familiar with) and multiplication.

3. To further build number sense and a deeper understanding of multiplication, students analyzed patterns they noticed between counting by fives and counting by tens as well as counting by nines and counting by tens.

4. Finally, we applied new learning to a simple algorithm. Again, students grasped this concept quickly and were very successful.

1. Next, students focused on x8 facts. Students discover how to use x4 facts to better understand x8 facts. For example, to find 8 x 5, you can “take five jumps of four and double the prouct.” For this reason, it’s easier for students to learn x8 alongside previously covered x4 facts.

2. Students used both a number line on paper and unix cubes to show how to multiply by 8 and how to multiply by 4. The goal was for students to make the connection between repeated addition (something students are very familiar with) and multiplication.

3. To further build number sense and a deeper understanding of multiplication, students analyzed patterns they noticed between counting by eights and counting by fours.

4. Finally, we applied new learning to a simple algorithm. Again, students grasped this concept quickly and were very successful.

The final facts that we covered were x7 facts. This is because x7 is the most difficult to connect with other facts. For this reason, it’s easiest if taught last!

2. Students used both a number line on paper and unix cubes to show how to multiply by 7. The goal was for students to make the connection between repeated addition (something students are very familiar with) and multiplication.

3. To further build number sense and a deeper understanding of multiplication, students analyzed patterns they noticed when counting by sevens.

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#### Opening Activity

*15 min*

I began by reviewing the Multiplication Vocabulary Poster by making the same hand motions as before. Teacher: *Multiplication! *Students: *Multiplication!* Altogether: *A fast way *(running motion with fists up, elbows bent, and arms moving back and forth)* to add the same number over... and... over *(Counting on fingers).

I hung up the student-created rhyme posters that were multiples of seven, including posters for a 7 x 4 Rhyme (already introduced), 7 x 6 Rhyme (already introduced), 7 x 7 Rhyme, 7 x 8 Rhyme (already introduced), and 7 x 9 Rhyme (already introduced).

I always teach rhymes to help students remember the more difficult multiplication facts. The trick is to practice them often. Otherwise students get words and numbers mixed up, which is counterproductive!

One by one, we practiced the rhymes. At first, I would say begin the rhyme: *7 x 4 is... *Then students would join in *...so great! 7 x 4 is 28!* Then we would say it altogether*. *I would prompt students: *Louder! 7 x 4 is* so great! 7 x 4 is 28! Next, I ask students to say it with a partner.

After we followed this process with all the rhymes, I had students quiz each other, "What's the rhyme for 7 x 7 ?

Later on, when students are solving multi-digit multiplication algorithms, they will reflect upon the rhymes to help them remember their facts.

*expand content*

**Common Core Connection**

Often times, students are expected to simply memorize multiplication facts without truly understanding the meaning behind the facts. This lesson engaged students in Math Practice 2: Reason abstractly and quantitatively. I wanted students to "make sense of quantities" using their monster plates, hundreds lines, unifix cubes in order to contextualize abstract equations.

**Number Line Model**

I passed out the Hundred Number Line inside the page protectors to each student. A number line is one of the best ways to relate multiplication to counting and build number sense. I asked students to get out their white board markers (thin works best) and erasers. I projected the Hundred Number Line so I could very explicitly provide directions.

**Monsters Problem**

I also asked students to spread out their Monsters on their desks. This was important as the monster problem provided students with a context for multiplication. Throughout today's lesson, we'll refer to the plates and students will use them once in a while to show their thinking. I started off by reviewing Lucy's Problem *Lucy is having a Monster Bash! She wants each guest to get ____ cookies. If she invites ____ friends, how many cookies will she need in all? *

*Let's say that Lucy wants to give away 7 cookies to each her guests. How many cookies would she need if 0 monsters came? "Zero!" Then I demonstrated how to take 0 jumps of 7 on the number line and marking where I landed... 0.*

*How many cookies would Lucy need if she is giving one cookie away and 1 monster comes to the party? *"Seven!" Then I demonstrated how to take 1 jump of 7 on the number line, marking where I landed... 7. Students continued making jumps of 7 on their own.

**Unifix Cubes**

To provide students with one more hands-on method to model their multiplication facts and observe patterns, I asked students to model how to count by 7s using Unifix cubes. This was probably the student's favorite model as they were able to participate in hands-on learning! Also, by intermixing the number line and Unifix cube models, students began to see the connections between the two models!

Once students had modeled counting by 7s up to 70, I asked to use their unifix cubes to find how many cookies would be needed if seven monsters come to Lucy's party. The goal was to make sure students were connecting the cubes to multiplication within a specific context. Here's what one group came up with: Counting by 7s. I loved watching them count by sevens and made sense of a problem using the number line model.

To encourage students to dig deeper, I asked them to begin to use their row of Unifix cubes to look for patters when multiplying by sevens. I gave students plenty of time to analyze multiples of seven.

Then I invited students to sit on the front carpet and I projected the number line on the board for students to refer to as they explained their patterns. I was amazed at what happened next! First, two students volunteered to show their thinking. They out that you can double multiples to get to other multiples. For example, double 14 equals 28: Part 1 - Doubling Multiples. I asked: *Does this always work?* Then, students figured out that double 28 equals 56. At this point, they thought it would "take forever" to get the double of 56 by counting by seven on the number line.

This was when another student wanted to explain how to find double 56 without a number line: Part 2- Doubling 56. I pushed studnet thinking a bit more by asking: *What is 112? *I wanted students to see if 2 sevens = 14, 4 sevens = 28, 8 sevens = 56, then 16 sevens = 112.

**In and Out Box Model**

To help students visualize this doubling process, I drew an In & Out Box Model. I decided to model how to double 10s before modeling how to double 7s. I wanted students to see: If you double the number of tens, you double the product. On "in" side, I wrote 1 and on the "out" side, I wrote 10. I explained: *If we have 1 monster coming to the party, and Lucy is giving away 10 cookies to each monster, then how many cookies will Lucy need? *10! *What if there's 2 monsters coming? *20! *What if there's four monsters coming? *At first, students said, "30!" but with time and discussion, realized the they just needed to double the 20 to get to 40. We continued this all the way up 1,280 monsters!

I drew another In & Out Box on the board, only this time, we would use the chart to count by sevens! I asked: *Can you use this in and out box to figure out how many sevens are in 112?* The students could hardly wait: Part 3- In & Out Box.

**The Distributive Property**

I excitedly said to students: *Wow! Your patterns are amazing. I love how you're thinking about numbers! Can I push your thinking a step further? What if I wanted to figure out the result of seven jumps of seven? Could I take three jumps of seven and then take four jumps of seven and add them together? *As I spoke, I wrote this equation (distributive property) on the board: (7 x 7) = (3 x 7) and (4 x 7). Students said, "Oh.... yeah!" Then we multiplied and added 21 + 28 = 49.

A student shared, "I noticed if you take three jumps of seven twice, it equals six jumps of seven." I wrote her equation on the board: *So if you take three jumps of seven twice *(7 x 3 x 2), *it equals six jumps of seven *(6 x 7)? In this video, you'll see that I try to further develop the student's thinking by encouraging her to see the relationships between numbers on both sides of the equation: Part 4 - Distributive Property.

At this point, I was thinking that I may have "taken it too far" by exposing students to the distributive property. But just then, a student came up and shared another way to distribute the 7. Instead of distributing the 7 across 3 and 4, he distributed the 7 across 2 and 5: Part 5 - Breaking Down Equations. I think the number line model really helped him to conceptualize this process.

Before moving on, we added some of the patterns discussed to our Patterns Poster.

**Listing x7 Facts**

Finally, we made a list of equations when multiplying by seven, starting with 7 x 0. Students created this list in their journals while a couple students completed the Seven Cookies Per Monster Poster. These lists will be helpful later on when students are solving multi-digit multiplication problems using the algorithm.

We were then able to line up all of our posters: Equation Posters x1-x5 andEquation Posters x6-x10 along the board.

##### Resources (15)

#### Resources

*expand content*

#### Guided Practice

*30 min*

At this point, I began teaching students students the multiplication algorithm. I like to teach the algorithm right alongside multiplication fact review. However, I start off very simple.

Using the grid side of student white boards (to help line up digits), students followed along. Here's the string of problems that we completed together. At first I modeled, then students completed them with me, and then students solved problems independently. After modeling first few problems, most students caught on quite quickly.

- 7 x 12
- 7 x 222
- 7 x 135
- 7 x 602
- 7 x 657
- 7 x 981

**Modeling the Algorithm**

Explicitly teaching the alogrithm followed by a gradual release of responsibility helps students understand and apply the algorithm successfully. When modeling the algorithm, I used the same words over and over.

For 7 x 135, I would say:

*7 x 5 is 35, write down the five, carry the three*

*7 x 3 is 21, plus the three is 24, write down the four, carry the two*

*7 x 1 is 7, plus two is 9, write down 9*

**Comma Placement**

If the product resulted in more than three digits, I would then encourage correct comma placement by underlining the first three digits while saying: *One, two, three*,* comma!*

**Release of Responsibility**

As students gradually become more and more independent, I left out words and expected them to fill in the blanks:

I would say:

*7 x ____ is _____, write down the _____, carry the _____*

*7 x _____ is _____, plus the _____ is _____, write down the _____*

*7 x _____ is _____, write down the _____*

* One, _____, _____ *(underlining the first three digits),

*_____*!*expand content*

#### Independent Practice

*20 min*

Before practicing independently, we reviewed the multiplication rhyme posters for 7 once more: 7 x 4, 7 x 6, 7 x 7, 7 x 8, and 7 x 9.

I passed out Algorithm x7 to each student.

During this time, I rotated around the room and asked students to explain their thinking. If I saw a student make a mistake, I tried to give the student time to catch the mistake themselves, or I asked guiding questions:

When students finished, they checked and revised their answers: Completed Student Work. Most students were able to successfully complete this page on their own! I believe the conceptualizing multiplication facts by using a number line, Unifix cubes, and a problem solving context helped students complete the more abstract algorithm process.

*expand content*

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- UNIT 1: Measuring Mass and Weight
- UNIT 2: Measuring Capacity
- UNIT 3: Rounding Numbers
- UNIT 4: Place Value
- UNIT 5: Adding & Subtracting Large Numbers
- UNIT 6: Factors & Multiples
- UNIT 7: Multi-Digit Division
- UNIT 8: Geometry
- UNIT 9: Decimals
- UNIT 10: Fractions
- UNIT 11: Multiplication: Single-Digit x Multi-Digit
- UNIT 12: Multiplication: Double-Digit x Double-Digit
- UNIT 13: Multiplication Kick Off
- UNIT 14: Area & Perimeter