# Lesson Study Resource Kit - Grade 1 - Fluency and Flexibility with Numbers

### Overview

Addition of a two-digit number and a one-digit number: Using knowledge of composing-decomposing to understand the algorithm for addition with regrouping.

This mathematics resource kit uses video excerpts and written resources to support your reflection and learning about early elementary experiences that build fluency and flexibility with numbers. You can reflect alongside this lesson study team as they develop insight into helping students connect knowledge of single-digit numbers to addition of larger numbers.

### Quick Think/Write

Have you seen students in upper elementary or middle school use their fingers to calculate? What early elementary experiences would build fluency with mental calculation so students would not still rely on their fingers in later grades? Write down at least five experiences you know and then share your ideas with your colleagues.

### Some Examples of Grade 1 Student Experiences with Numbers

The following video excerpts are from first grade classrooms in Florida, California, and New York.  Students seen in these videos are using Japanese Elementary Mathematics.

The Kindergarten Japanese elementary mathematics curriculum devotes over 75% of total instructional time to numbers and operations. Games are a central feature of this instructional time and students build fluency composing and decomposing numbers to 5, then numbers to 10.   In first grade, students expand their knowledge of numbers and operations by sequencing numbers and quantities up to 100 and noticing the relative magnitude of numbers. Students make units of 10 and recognize that 10 can be thought of as 10 ones or as a unit of 10, and they learn to think of teen numbers as ten and some ones.

The excerpts from Pine Trail Elementary School in Florida are from Ms. Hajdin’s first grade classroom and span over a 3-month period, from January through March, leading up to the Research Lesson highlighted in this resource.  These first grade students have been using the Japanese Curriculum since Kindergarten.

The excerpts from Brookfield Elementary School in California and Harlem Village Academy in New York are taken from a series of lessons taught over several days by guest teachers Dr. Takahashi and Mr. Jackson, respectively.

1. View the following video clips and note your responses in the Table 1 provided here. Write down what you notice as you watch the clips.

Excerpt 1: Two Numbers Together

Students shake 5 counters in a box, half of which is covered by an opaque lid. They look at the visible counters and predict how many counters are hidden behind the opaque lid.

Excerpt 2: 100 Chart Game, Ten Frame Game, Different Ways to Compose 9

This video shows the excerpts of 3 games. In the 100 Chart Game, students take turns rolling one die (then two dice) and advance their game piece the same number of spaces as the number rolled. In the Ten-Frame Game, students look at a partially filled ten-frame, and identify the quantity needed to make 10. In Decompose This Number, students find different ways to make 9.

Reflect with your colleagues: How might these experiences build students' fluency and flexibility with numbers? What experiences do you use in your classroom to build students' fluency and flexibility with numbers?

After your discussion: Table 2 offers some examples of other teachers ideas.

2. Read the section ‘Development and Connection of Increasingly Abstract Representations of Mathematics in K and 1st grade’ from the Elementary Japanese Mathematics Teacher’s Editions (2012, Global Education Resources).  Share with your colleagues how you are building Math Practice Standards 2, 4, and 7 in your classroom. You may focus on one or more Standard. View Math Practice Standards.

3. In the next two video excerpts, you will see how students use what they know to develop their understanding of single-digit addition.  Write down what you notice in Table 1.

Except 3: Single-Digit Addition with Sums Less Than 10

Excerpt 4: Addition of Single-Digit Numbers with Sums More Than 10

Share with your colleagues what you noticed. After your discussion: Table 2 continued offers some examples of other teachers' ideas for excerpts 3 and 4.

Reflect with your colleagues about student learning: It may take time for students to connect their intuitive approaches to the approach of making ten that you watched in Excerpt 4. What are the key experiences that help students make the connection? For example, how might explaining/hearing other students explain solutions, board work, journals, manipulatives, etc. help students make a connection between decomposition and addition with regrouping?

4. If you would like to read more about this dilemma – honoring individual students’ methods while moving the whole class toward making ten – we suggest the following chapter from ‘Perspectives on Learning,' 66th Yearbook, NCTM:

Murata, A., Otani, N., Hattori, N. & Fuson, K.  (2004).  The NCTM Standards in a Japanese Primary School Classroom: Valuing Student’s Diverse Ideas and Learning Paths. Chapter 7 in ‘Perspectives on Learning, 66th Yearbook’ by NCTM.

### Planning

The lesson study team from Pine Trail Elementary School met weekly between January and March to investigate the sequence of experiences that would help all students see the power of making 10's.

1. As you watch the following clip, note down what the teachers do that supports their learning from each other and from the mathematics curriculum.

Excerpt 5:  Planning Highlights

Note:  If you would like to see more highlights of this group’s planning meetings, a longer version is available here.

3. Reflection: Choose one or more practices of this group that you would you like to cultivate with your planning team. For example, these might be practices such as: norm-setting, solving the student problem(s) first, anticipating students’ thinking, figuring out the learning trajectory across the unit, preparing board plans.  Write down one or more habits that you would like to nurture going forward.

### Research Lesson

1. Study the First Grade Lesson Plan “Addition of a Two-Digit Number to a One-Digit Number” and the Board Plan.  After looking at these plans, consider: If you were going to attend this lesson, what student data would you collect? Discuss your data collection focus with your colleagues.

2. Highlights of the research lesson follow in 3 excerpts.   Watch each excerpt and collect data (from what you can see of the lesson) using the focus you described to your colleagues.  Reflect with your colleagues about what you noticed about students' mathematical thinking.  Use the guiding prompts after each excerpt to discuss the evidence you collected.

Excerpt 6: Starts With 8+6 and Review
What do you notice about the way the student uses the blocks at the board to replicate how she solved the problem using the make a ten strategy? What are the similarities and differences between the two strategies?

Excerpt 7: 53+7
What do you notice about the different counting strategies students used during their work time at their desks?

Excerpt 8: Sharing Strategies 53+7 at the Board
What do you notice about the choice of student strategies presented and their order of presentation?

Excerpt 9: 42+8/94+6 and Relationship between 8+6 and 78+6
Notice how students react to the first demonstration of 78+6.  What elements of classroom routines and lesson design help students make sense of a challenging new mathematical idea?

### Post-Lesson Discussion

1. The instructor and team share observations on student thinking, followed by commentary by Dr. Tad Watanabe. Dr. Watanabe points out how important it is to think about the sequence of experiences - what kind of problem should come next and why. What next experience would you provide for these students? View highlights from post-lesson discussion:

Excerpt 10: Post-Lesson Discussion