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* *Reflection: Developing a Conceptual Understanding
Multiplication: x10, x5, x9 - Section 5: Independent Practice

Throughout these multiplication lessons, one student has continually come up to me and pointed out that any number times five falls between the shaded and non-shaded sections of the number line. I was reminded of how important it is to teach kids to make sense of numbers in relation to friendly numbers (5s, 10s, 20s...)

During this lesson, this student called me over to show me that she discovered the pattern again! Patterns with 5s.

Now that I look back at this, it reminds me how important it is for students to be provided with repeated opportunities to discover the same pattern. This really helps solidify concepts and patterns all the more.

Overall, this lesson went quite well. Students did a great job making the connection between their math facts and the algorithm. I think they were more successful at solving the multiplication algorithm because they were more familiar with the meaning behind each step, such as 5 x 6...

*Developing a Conceptual Understanding: Developing a Conceptual Understanding*

# Multiplication: x10, x5, x9

Lesson 5 of 7

## Objective: SWBAT multiply a multi-digit number by 5 and 9.

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 (1-digit x multi-digits). 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 nine, including posters for a 9 x 4 (already introduced) , 9 x 6 Rhyme (already introduced), 9 x 7 Rhyme, 9 x 8 Rhyme, and 9 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 9 x 6?

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 5 cookies to each her guests. How many cookies would she need if 0 monsters came? "Zero!" I demonstrated how to take 0 jumps of 5 on the number line and marking where I landed... 0.*

*How many cookies would Lucy need if she is giving one cookie away and 5 monster comes to the party? *"Five!" I demonstrated how to take 1 jump of 5 on the number line, marking where I landed... 5. Students followed along, making jumps on their own.

We repeated this same procedure with 10s as there are three number lines on the yellow Hundred Number Line Model.

**Unifix Cubes**

To provide students with one more hands-on method to model their multiplication facts and observe patterns, I asked students to build a concrete model of a number line by counting by 5s and another number line, counting by 10s. I was hopeful that students would begin to see doubling and halving connections: If 4 x 5 = 20, then 2 x 10 = 20.

I was proud to watch students students realize that some multiples of fives are multiples of tens and all multiples of tens are multiples of 5s: Patterns between 5s and 10s!

**Modeling x9 Facts**

This was a great opportunity of students to also model taking jumps of 9 on a number line and taking jumps of 10 on a number line. So, they created two rows of Unifix cubes, one counted by 9s and one counted by tens. Without my even asking, students immediately began observing patterns! I was incredibly impressed when the following conversation came up!!!! In this video, Patterns with 9s Part 1, a student points out, "There's one nine and then one is left over. There's another nine and then two are left over." I then said: *Here's what I hear you saying: When you take one jump of nine *(1 x 9),* it equals one less than ten *(1x10-1)*. *While I spoke, I wrote this equation: 1 x 9 = (1 x 10) -1. *And if you take two jumps of nine, it equals two jumps of ten minus two. *While I spoke, I wrote this equation: 2 x 9 = (2 x 10) -2.

Other students were then inspired! Here's what happened next! A student came up to the board and continued this pattern! Patterns with 9s Part 2. Then, unbelievably, this student came up and pointed out the repeating numbers in the equation: Patterns with 9s Part 3!!!

I just had to push my students' thinking one step further! *So what you're saying is that any number (n) x 9 = (that number x 10) - that number? *As I spoke, I wrote this equation on the board: n x 9 = (n x 10) - n. I asked if any students would like to come up and "test" our equation to see if it always works! Here's an example of a student who tested this conjecture: Patterns with 9s Part 4. I was so proud of my students!!!

This was a perfect opportunity to we add another pattern to our Patterns Poster!

**Listing x10, x5, x9 Facts**

To wrap up this activity, we made a list of equations when multiplying by ten (starting with 10 x 0), five (starting with 5 x 0), and nine (starting with 9 x 0). Students volunteered to complete the Nine Cookies Per Monster Poster, the Ten Cookies Per Monster Poster, and the Five Cookies Per Monster Poster. These lists will be helpful later on when students are solving multi-digit multiplication problems using the algorithm.

##### Resources (12)

#### 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.

- 5 x 12
- 5 x 222
- 5 x 657
- 9 x 135
- 9 x 602
- 9 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 5 x 657, I would say:

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

*5 x 5 is 25, plus the three is 28, write down the eight, carry the two*

*5 x 6 is 30, plus two is 32, write down 32*

**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:

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

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

*9 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 9 once more: 9 x 4, 9 x 6, 9 x7, 9 x 8, and 9 x 9.

I passed out Algorithm x5 and Algorithm x9 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. I was so proud to find this student using the number line model to help support his thinking: Student Using Number Line Model for Help.

When students finished, they checked and revised their answers: Completed x5 Algorithm and Algorithm x9.

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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