Overview

Session Summary

In this session you will learn how starter problems and cluster problems help students develop flexible strategies for whole number computation. You will look for evidence of the CCSS (Content and Mathematical Practices) as you look at student work, watch students at work and read classroom cases.

Session Focus

In this session you will:

• Examine the connection between students strategies and the U.S. Standard Algorithm
• Experience Cluster Problems to understand how they help students develop flexible strategies
• Examine and sort student work according to the CCSS Math Content Standards
• Make connections between the development of computational fluency in the elementary grades and the CCSS Content Standards and the Standards for Mathematical Practice

Getting Started: Connecting to the U.S. Standard Algorithm

Watch these 4^th grade students’ explain the connection between a notation showing partial sums and one showing the U.S. Standard Algorithm for addition.

Read

About the U.S. Standard Algorithm

These Teacher Notes focus on the U.S. Standard Algorithms (addition, subtraction and multiplication).

Notebook

Reflect on the following questions in your notebook.

• What do the students in the video and readings understand about the place value?
• What do the students understand about the operations?
• What prior work might the students have done that so assure their understanding of how the algorithm works? (Think about the math you have experienced, as well as the math you observed or read about students doing (K-5) during this course.)

Final Thoughts…

Activity 1: Related and Cluster Problems

Cluster problems provide students different starting points to solve a problem. Students break up the problem using related problems. Students are given a few problems (a cluster). The problems in the cluster often are ones that students can mentally solve. Many involve landmarks such as multiples of ten or solving half of the problem and doubling that product.

Introducing the Cluster Problem 46 x 25

Solve the problem using one or more of the starter problems.

Record your steps.

46 x 25
4 x 25 40 X 25	10 x 25 50 X 25

Listen to the teacher, as she introduces the cluster problem (shown below) to her students.

Nashaya’s Strategy

Watch Nashaya share her strategy.

Notebook

After viewing the video, reflect on the following questions in your notebook.

• What does Nashaya understand about the base-ten number system?
• What property did Nashaya use when she solved the problem?

Stephen

Stephen successfully solved this problem one way. He realized his second way was incorrect. He chose 4 x 25 as his first step and continued using the following related problems: 4 x 25, 10 x 25, 6 x 25.

As you watch Stephen explain his thinking, jot down his steps.

Why is Stephen incorrect? What did Stephen learn from his error?

Stephen shares his incorrect steps and how listening to Nashaya helped him see his error.

Notebook

In your notebook, record your reflections on the following questions:

• Where do you see evidence of students looking for and making use of structure such as place value and the properties of operations? (Math Practice 7)
• Where do you see evidence of students constructing viable arguments and critiquing the reasoning of others, that is, explain the reasoning underlying a strategy or solution? (Math Practice 3)

Practicing Cluster Problems

Use one or more of the following related problems to solve 42 x 19.
42 x 10 = 42 x 5 =	42 x 2 = 42 x 20 =

Use one or more of the following related problems to solve 450 ÷ 16.
160 ÷ 16 =   80 ÷ 16 =	320 ÷ 16 =    16 x 10 =

Read

Developing Computational Strategies That Make Sense.

Notebook

In your notebook, record your reflections on the following question:

• In what ways do cluster problems support students’ development of computational fluency?

Activity 2

You will examine samples of student work and determine where they fall according to CCSS Math Content Standards. The purpose of this activity is to understand how related content standards develop across grades.

Looking at Student Work

Look at the student work across the grades and use the CCSS Math Content Standards to think about the critical phases in the progression for number and operations across the grades.

Notebook

Use the student samples and descriptions of the CCSS grade level standards to craft your own description of how the ideas progress across the grades.

Cite specific examples. You may want to refer back to video clips or readings from previous sessions.

Activity 3

In this activity you will delve more deeply into the Mathematical Practices (MP). You will examine video and/or classroom cases for evidence of MP.

Bringing Math Practices to Life in the Classroom…
(adapted from http://investigations.terc.edu/curriculum_clrm/mathpracticestandards.cfm)

The teacher, the student, and the curriculum each have a role to play in making a classroom, that embodies the Standards for Mathematical Practice come to life.

The curriculum presents a sequence of tasks designed to focus on particular content standards and to foster the use of the Math Practices. When looking at a particular task, questions to think about include:

• What kind of task is it?
  (e.g., a decontextualized computation problem, a problem set in a context, a problem with more than one answer)
• What math does the task focus on?
• Does the task suggest the use of one or more MPs? In what ways do the MPs support students’ mathematical thinking and understanding of the math content?

Teachers must be able to see and recognize the various Math Practices, and they must be “mathematically proficient.”

• use their knowledge of the math, their students, and the curriculum to implement the curriculum.
• facilitate the creation of an environment in which the Math Practices can flourish.
• sometimes they purposefully focus on one particular Math Practice; other times, they capitalize on an opportunity that arises in the classroom.

Students are the critical piece of the puzzle. Where they are with an idea (developmentally and mathematically) and how they approach a problem affects how they employ the Math Practices. It’s important for all students to:

• - making sense of quantities and relationships
• - create a representation of the problem
• - apply prior knowledge
• - check to see if the solution is accurate
• - look for patterns and structure
• - communicate and defend reasoning using contexts, math tools and/or drawings

The teacher, the student, and the task each have a role to play. Successful implementation of the Standards for Mathematical Practice relies on the interaction of these three factors. A task might be rich and complex, but if students aren’t ready for the ideas or it’s taught very procedurally, it won’t result in deep and varied use of the Practices. On the other hand, what can seem like a simple computation problem can blossom into a rich and interesting task.

Examining Classroom Vignettes

In this activity you will examine classroom vignettes, either (video and/or classroom cases) for evidence of teacher moves/practices, student learning, the purpose of the task and identifying instances of specific CCSS Mathematical Practices (indicated on the table below). You will share your findings on the Vignette Discussion Board.

Choose at least two artifacts (video and/or a case).

For each vignette:

1. Read the case or view the video several times, focusing each time on either the teacher, students, task(s) and/or MP.
2. Respond to the following questions in your notebook after you read the case or view the video.
   • What do you notice about what the teacher does (e.g., questions the teacher asks, the structure of the math experience and classroom environment)?
   • What are the students doing and saying? How are they thinking about the relationship between the numbers in the problems and a strategy they can easily and accurately use?
   • How did the experience engage students and help them understand the meaning of the operation(s)?
   • What evidence in the case or video is there of the specific MP’s listed on the table?
3. Go to the Vignette Discussion to share your overall findings for each of the Vignettes you examined.

Classroom Cases	MP
Helping Students Build on Each Other’s Ideas (K)	3,4,7
From Making Tens to Place Value (Gr 2)	1,6,7
The Case of Ellen: Deciding What to Nudge (Gr 3)	1,5
The Case of Kyle and Asiah: Multiplying by Multiples of 10 (Gr 4)	1,3,6,7
Videos	MP
Story Problems (Gr 1) Setting Expectations (Session 2 Subtraction, Overview) Discuss Strategies (Session 1 Addition, Overview)	4, 5, 7
Ryshawn and Nicholas 4 x 29 (Gr 3) (Session 4 Multiplication, Activity 3)	1, 7
Jemea 29 x 12 (Gr 5) (Session 4 Multiplication, Activity 3)	3, 6, 7
Building Flexibility with Cluster Problem (Gr 4/5) (Session 6 - Click on Activity 1 tab at the top of this page) Introduction Nashaya Stephen and Teacher Reflection	1, 3, 7, 8

Discussion

Choose one Mathematical Practice (MP). Describe how the MP contributes to students’ computational fluency. Give examples from you personal experience or experiences from this course.

Share a take-away about computational fluency and the Common Core that you will integrate into your practice.

Remember to post early and return to the discussion forum throughout the week to read and respond to others’ posts.

Go to forum.

Common Core Alignment

The content in this session is aligned to the following Common Core State Standards and Math Practices:

MATH PRACTICES

MP1: Make sense of problems and persevere in solving them.

MP2: Reason abstractly and quantitatively

MP3: Construct viable arguments and critique the reasoning of others

MP4: Model with mathematics

MP5: Use appropriate tools strategically

MP6: Attend to precision

MP7: Look for and make sense of Structure

MP8: Look for and express regularity in repeated reasoning.

You may also wish to review the full set of Common Core State Standards in Math and/or videos of the Math Practices enacted in Investigations.

This chart was created by Bill McCallum in an attempt to provide some higher order structure to the practice standards, just as the clusters and domains provide higher order structure to the content standards.

MATH STANDARDS

Grade	CCSS Standards for Addition and Subtraction
K	K.OA.1 Represent addition and subtraction with objects, fingers, mental images, drawings, acting out situations, verbal explanations, expressions, or equations. K.OA.2 Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. K.OA.3 Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation. K.OA.4 For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation. K.OA.5 Fluently add and subtract within 5
1	1.OA.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. 1.OA.3 Apply properties of operations as strategies to add and subtract. 1.OA.4 Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8. 1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten; decomposing a number leading to a ten using the relationship between addition and subtraction and creating equivalent but easier or known sums by creating the known equivalent.
2	2.OA.1 Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. 2.NBT.5 Fluently add and subtract within 100 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
3	3.NBT.2 Fluently add and subtract within 1,000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
4	4.NBT.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm.
5	Note: While the content standards do not include addition and subtraction at grade 5, students continue to practice and apply these operations.
Grade	CCSS Content Standards Multiplication and Division
2	2.OA.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.
3	3.OA.1 Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7. 3.OA.2 Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8. 3.OA.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1 3.OA.4 Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes 3.NBT.3 Multiply one-digit whole numbers by multiples of 10 in the range 10–90 using strategies based on place value and properties of operations.
4	4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. 4.NBT.6 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
5	5.NBT.5 Fluently multiply multi-digit whole numbers using the standard algorithm. 5.NBT.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Key Learning

• The foundational aspects of computational fluency – accuracy, efficiency and flexibility- develop from Kindergarten to grade 5. At each of those grade levels, what it means to be computationally fluent with one or more of the four operations in the base-ten number system is different (e.g., range of numbers, tools and representations, sophistication of strategies).
• The teacher’s role in supporting students’ development of computational fluency is critical. It’s not the activities that develop fluency, it’s how the teacher uses the opportunities they create to move the students in that direction.
• Starter problems and cluster problems help students develop flexible strategies for whole number computation.
• By making observations and posing well-timed questions, teachers can invite students into the CCSS Math Content Standards and Standards for Mathematical Practice.

Notebook

Fill out the Session 6 Notebook Page to reflect on your own learning and record ideas that you want to remember and/or try with students.

Select the following links to go to the Notebook.

• Session 6: Overview
• Session 6: Activity 1, Nashaya
• Session 6: Activity 1, Stephen
• Session 6: Activity 1
• Session 6: Activity 2
• Session 6: Activity 3