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* *Reflection: Student Led Inquiry
Multiplication: x3, x6 - Section 5: Independent Practice

I loved, loved, loved this conversation with the student in the following video! Prior to taping, I was conferencing with the student as he was completing Algorithm Practice x6. He then turned to me and asked, "Is 6 x 7 the same as 7 x 6?"

I responded: *How could you find that out?* He pulled out his Hundred Number Line Page and went right to work: Communitive Property.MOV.

I loved how he explained that he wanted to test his "hypothesis!" I then guided him to write out an equation for each of his number line models.

I was soooo proud of him for seeking the answer to his own question (student-led inquiry).

Overall, this lesson went quite well. By using the same activities as yesterday, students easily built upon their understanding of multiplication facts from yesterday to make sense of facts today. Most students were able to meet the learning goal.

# Multiplication: x3, x6

Lesson 4 of 7

## Objective: SWBAT multiply a multi-digit number by 3 and 6.

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 six, including posters for a 6 x 6 Rhyme, 6 x7 Rhyme, 6 x 8 Rhyme, and 6 x 9 Rhyme. 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. Then, I had students quiz each other, "What's the rhyme for 6 x 7?

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

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**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 3 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 3 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? *"Three!" Then I demonstrated how to take 1 jump of 3 on the number line, marking where I landed... 3. Students followed along, making jumps on their own.

**Unifix Cubes**

To provide students with one more hands-on method to model multiplication, I asked students to model 3 x 0... 3 x 1... 3 x 2... 3 x 3... all the way up to 3 x 10. This was a great way for students to make a concrete model of the number line representation. Later on, students will use their number lines and Unifix cubes to list x3 facts.

I wanted to use their Unifix cube model to answer a word problem, so I asked students to use their Unix cubes to show me how many cookies Lucy would need for six guests. Here, a student explains How many cookies for 6 monsters? by taking a part 36 cubes (dividing into groups of 6). This was great as students were unknowingly making the connection between multiplication and division.

**Making Connections Between x3 and x6 Facts**

I encourage students to observe a pattern between multiplying by threes and multiplying by sixes, I asked students to build a Unifix cube number line counting by threes up to 24 and then a number line of Unix cubes counting by sixes up to 24. Students also did this on their Hundred Number Line Page. Students eagerly began building their number lines and were excited to look for patterns. This student is really Trying to See a Pattern!

As a class, we discussed the observed patterns. Students couldn't wait to come up to the front board. This student pointed out repeating numbers: Multiples of 3 are Multiples of 6. Another student saw the same pattern, but explained it in a different way: Two jumps of three equal one jump of six. This was exactly what I wanted students to notice. Eventually, students will discover the trick behind doubling and halving: If 4 x 3 = 12, then 2 x 6 = 12. We added discussed patterns to our Patterns Poster.

**Listing x3 and x6 Facts**

Before I could even ask, many students began making a list of equations for the x3 facts (just as we did yesterday). Next, they listed the equations for the x6 facts. I asked for student volunteers to complete the Three Cookies Per Monster Poster and Six Cookies Per Monster Poster on the board. Students were really beginning to make connections between the models and their math facts!

##### Resources (10)

#### Resources

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#### 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.

- 3 x 12
- 3 x 222
- 3 x 657
- 6 x 135
- 6 x 602
- 6 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 3 x 657, I would say:

*3 x 7 is 21, write down the one, carry the two*

*3 x 5 is 15, plus the two is 17, write down the seven, carry the one*

*3 x 6 is 18, plus one is 19, write down 19*

**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:

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

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

*6 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 6 once more: 6 x 4, 6 x 6, 6 x 7, 6 x 8, and 6 x 9.

I passed out Algorithm x3 and Algorithm x6 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: Algorithm Practice x6.

When students finished, they checked and revised their answers: Completed Algorithm Practice x3 and Completed Algorithm Practice x6.

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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