##
* *Reflection: Pacing
Methods of One-Digit by Two-Digit Multiplication - Section 2: Concept Development

In this reflection video you can hear my reflection about how my students did making connections between the two multiplication methods.

A few days ago, I was really thinking that I should also show another multiplication method to students, which would then naturally lead some students to the shortcut or standard algorithm. I'm really glad I didn't. Pre-Common Core, I think I would have done just that. I would have had a feeling that I needed to rush students in order to cover so much material. The Common Core Standards are refreshing and freeing to me as teacher. Knowing that I have fewer concepts to teacher and can go deeper with those concepts. When looking at standard 4.NBT.5, the standard does not specify that a student will know the standard algorithm in fourth grade. This a fifth grade skill. To see the exact language of the fifth grade standard you can click here to be taken to the Common Core Standards website.

When I polled my class near the end of the lesson, I asked them if they preferred one method over another at this point. The result was about half and half. I liked seeing that some prefer one over the other and reaffirming with my students that we are all different and learn differently. This difference in method preference also gave me an opportunity to remind my students that there are often various efficient ways to solve a problem, and one way isn't necessarily better than another.

My students are very excited to experiment with double digit by double digit multiplication. They can definitely sense the progression leading up to this skill and have been very excited in math class. Click here to jump ahead to when I introduce double digit by double digit multiplication.

*Making Connections*

*Pacing: Making Connections*

# Methods of One-Digit by Two-Digit Multiplication

Lesson 8 of 22

## Objective: SWBAT relate the area model of multiplication to numeric methods of multiplication.

#### Warm Up

*5 min*

Each student is given a card when they walk in the room. Each card has either a multiplication number sentence with a multiple of ten, or a product. Students are asked to find their "match" as quickly as they can. They may whisper to each other to get this done, but I ask them to use low voices. I time the students and tell them how fast they were able to do this activity. I end the day with this activity as well and see if students can beat their first time.

This review activity serves two purposes. One, it is an engaging activity in which students can review, practice, and apply their skills from the previous lessons. One of the shifts in the CCSS is rigor. Rigor requires conceptual understanding, procedural skill and fluency, and application with intensity. This activity allows students to build fluency. The CCSS defines procedural skill and fluency as students are expected to have speed and accuracy in calculation. Teachers structure class time and/or homework time for students to practice core functions such as multiplication facts so that students are able to understand and manipulate more complex concepts. Multiplying by multiples of ten is a skill I want my students to be fluent in so they can be successful with double digit by double digit multiplication.

The second reason I chose this activity as a fluency practice is it serves a quick formative assessment. I can see and observe students that are struggling and make adjustments to instruction immediately. If students struggle with this activity it gives me a clue that these same students will most likely struggle in the next lessons as I present further multiplication concepts. I can build in scaffolds during those lessons and continue to reteach this concept for students that have not mastered this skill.

#### Resources

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#### Concept Development

*45 min*

*This lesson's focus is CCSS 4.NBT.5 as well as Math Practice Standard 7 - Look For and Make Use of Structure*

*Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.*

One way I develop students’ capacity to "look for and make use of structure" is to help them identify and evaluate efficient strategies for solutions. This lesson gives students an opportunity to compare methods of multiplication and evaluate the methods for efficiency. Students usually are able to discern that drawing area models for every multiplication problem becomes labor intensive and takes a longer amount of time than the expanded notation method.

I start this lesson by modeling the area model of multiplication and the expanded notation methods. Check out the video below for information on these two methods. I don't show this video to students, but the way I model the two methods is very similar to this video.

I model another number sentence on the smartboard, and show the area model **and **the expanded notation method for the same number sentence. I put the two methods side by side in a T-chart. After letting students observe and analyze the two methods silently for about one minute, I then, I explicitly ask questions like:

How are these methods similar?

How are these methods different?

What patterns do you notice or see in each method?

Which method is your favorite right now?

Which method feels easiest for you to do?

Which method will take a longer amount of time to solve?

If they both seem efficient, and take the same amount of time, do you think when we learn double digit- by double digit multiplication, one might be more efficient? Take less time? *Note: this is when I really focus on Math Practice Standard 7 and breifly discuss when one of these methods might take longer than the other. At this point, using only double digits by one digit, both methods take about the same amount of time and are both efficient ways to multiply. As students solve more complex problems, various methods become more and less efficient based on student knowledge and the size of numbers to multiply. *

For the remainder of the lesson, students practice the area method and the expanded notation method on lined paper. I let students **choose** which problems they want to work on for the assignment. They must choose at least 5 one-digit by two-digit multiplication problems to model the area model for and use the expanded notation method to solve. I ask them to choose a double digit larger than 21 simply because I want them getting practice with large numbers. By limiting the smallest number to 21, students who are struggling with basic facts using 7, 8, and 9, can choose smaller double digits and still feel successful and challenged. Based on prior experience, students that do not have their multiplication facts mastered at this point struggle most with facts having 7, 8, or 9 as a factor.

Giving students a choice for this assignment is beneficial for me and them. Students that have their basic math facts mastered often choose problems that use 7, 8, and 9 as digits in their numbers. Students who have not mastered their facts often choose lower digits like 2, 3, 4, and 5 when designing the problems they want to solve. Since I want students to master the concept of double digit by one digit multiplication, letting them choose the numbers they want to use in their problems allows them to use facts they are comfortable with while still practicing an important fourth grade skill.

*Students finish this assignment as homework if they don't finish in class. About half of my students finished in class and half needed to take it home to finish. *

In the following video, you can listen to a student talking about what method she likes best. I wanted to make sure I related my questioning to Math Practice Standard 7 and 8 by asking her about patterns and similarities between the two methods. As you can see in the video, she can identify similarities, but needs some more guidance in her math talk and using correct concept terms like ones and tens when noting the similarities between the place values and partial products in both methods.

Click here for a sample of student work in which I share my ideas about this student's progress.

Click here for another sample of student work and what this student discovered by doing the two different methods side by side.

*expand content*

#### Student Debrief

*10 min*

I end the lesson with the activity students started the lesson with. Each student is given a different card with either a multiplication number sentence with a multiple of ten, or a product. Students then find their "match" as quickly as they can. Once they find their match, they stand with their partner on the outside edge of the classroom. They may whisper to each other to get this done, but I ask them to use low voices. I time the students again and I encourage them to beat their time from earlier.

The students got much faster the second time through. This turned out to be a great formative assessment. The last four to six students standing in the center of the room had difficulties multiplying by a multiple of ten. This was a great formative assessment because I really thought ALL my students could do this skill quite easily. I was reminded that I still have a few students that need more support.

*expand content*

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- UNIT 1: Getting to Know You- First Days of School
- UNIT 2: Multiplication with Whole Numbers
- UNIT 3: Place Value
- UNIT 4: Understanding Division and Remainders
- UNIT 5: Operations with Fractions
- UNIT 6: Fraction Equivalents and Ordering Fractions
- UNIT 7: Division with Whole Numbers
- UNIT 8: Place value
- UNIT 9: Geometry
- UNIT 10: Measurment
- UNIT 11: Fractions and Decimals

- LESSON 1: Multiplicative Comparison Problems
- LESSON 2: Finding Factors and Prime Numbers
- LESSON 3: Multiplication arrays
- LESSON 4: Mental Math and Multiplication with Tens
- LESSON 5: One digit by two digit Multiplication
- LESSON 6: Multiplying multiples of ten - Not your Daily Grind
- LESSON 7: Multiplying one digit by two digits using the AREA MODEL
- LESSON 8: Methods of One-Digit by Two-Digit Multiplication
- LESSON 9: Compare methods of one digit by double digit multiplication
- LESSON 10: Practice Makes Perfect
- LESSON 11: Two-Digit by Two-Digit Multiplication
- LESSON 12: Looking at Different Multiplication Methods
- LESSON 13: Multplication Application with Food Service Staff
- LESSON 14: Multiplication Methods using COMPUTERS!
- LESSON 15: Multiplication and First Quarter Assessment
- LESSON 16: Using Games to practice multi-digit multiplication
- LESSON 17: Multiplication Bingo - Game Day 2
- LESSON 18: Estimate Products
- LESSON 19: Multiplication and Problem Solving to Make Bracelets Day 1
- LESSON 20: Multiplication and Problem Solving to Make Bracelets Day 2
- LESSON 21: Bracelet Wrap Up
- LESSON 22: Multiplication Card Game and Factorial Fun