Introduction to Eureka Math “A Story

of Units”

Tricia Bevans, University of Oregon

Session Objectives:Use Units to recognize connections between elementary

math concepts

Use concrete and pictorial models to understand connections between concepts in “A Story of Units”

Use basic concrete and pictorial models to represent problems

Understand the structure of “A Story of Units”

Understand basic principles for adapting modules and lessons from “A Story of Units” for your classroom.

What do these have in common?

60 + 36 = 96

5/9 +1/3 = 8/9

50 +30 = 80

225°+135°=360°

12.7cm +7.62cm=20.32cm

What do these have in common?

60 + 36 = 96 5 dozen + 3 dozen = 8 dozen

5/9 +1/3 = 8/9 5 ninths + 3 ninths = 8 ninths

50 +30 = 80 5 tens + 3 tens = 8 tens

225°+135°=360° 5(π/4) + 3(π/4) = 8(π/4)

12.7cm +7.62cm=20.32cm

5 inches + 3 inches = 8 inches

The Importance of the Unit

The Importance of the Unit

Units connect seemingly unrelated concepts in Elementary Math

The unit: a single object

Counting:

1 apple, 2 apples, 3 apples,….

Adding and Subtracting:

5 apples+3 apples = 8 apples

The unit: a tenCounting:

unit form—1 ten, 2 tens, 3 tens,…

or the regular way—10, 20, 30,….

Adding and Subtracting:

5 tens+3 tens = 8 tens

or 50+30=80

The units: tens and onesCounting:

unit form—1 ten, 1ten 1, 1ten 2, 1ten 3,…or the regular way—10, 11, 12, 13….

Adding and Subtracting: 5 tens and 2 ones +3 tens and 4 ones = 8 tens 6

onesor 52+34 = 86

Exchanging units: 1 ten = 10 ones

The units: twos, threes, fours…

Counting:

(skip counting) 2, 4, 6,…

Adding and Subtracting:

5 two’s and 1 two = 6 twos

Exchanging units:

2 twos = 1 four

So 10 twos = 5 fours

The units: halves, thirds, fourths…

Counting :

(skip counting) 1/2, 2/2, 3/2,…

Adding and Subtracting:

1 fourth+2 fourths=3 fourths

or 1/4 +2/4 = 3/4

Exchanging units:

2 fourths = 1 half

The units: bottles, buckets, or glasses

Each bottle holds 900 ml of water. A bucket holds 6 times as much water as a bottle. A glass holds 1/5 as much water as a bottle.

“A Story of Units”Characters—the units

The plot—what happens when they are added, subtracted, multiplied, divided,…

The setting—all of the units live harmoniously on the number line

The sequel—What happens when pairs of these get together: “A Story of Ratios”

A framework for effective mathematics learning

Meaning

Method

Mastery

Meaning

using and translating between equivalent definitions and models of a concept. (internal connections)

Connecting a concept to previous concepts, including relevant applications. (external connections)

MeaningExample - the meaning of multiplication

Encompasses the definitions and models:

Repeated addition.

Rectangular arrays.

Area of rectangles.

Scaling on the line.

Connects to addition and real world contexts with quantities accumulated at a constant rate.

Method

Processes to solve clearly delineated mathematics problems.

MethodExample: Methods for multiplication include

Procedures and algorithms, such as the multi-digit multiplication algorithm

Strategies such as reasoning that

11x6 = 10x6 +1x6=60+6

MasteryThe ability to use a mathematical concept:

In (age-)appropriate mathematical reasoning.

In authentic real-world applications.

In the development of more advanced concepts.

MasteryExample: Mastery of Multiplication could include

Explaining a strategy for multiplying

Using it effectively in a long division problem

Using it to solve an application problem

Meaning Method MasteryStudents will only achieve mastery when it is

based on both understanding the meaning of a concept and appropriate use of methods related to that concept.

Real-world applications serve both meaning and mastery goals

Read-Draw-Write Read. (“What do I see?”)

Draw and label. (“Can I draw something?”)

Write a number sentence (equation).

Write a word sentence (statement).

(“What conclusions can I make from my drawing?”)

Activity:Introduction to Concrete and Pictorial Models in

Eureka Math

Activity: Coherence of Models

Activity:Using Various Models

Adapting Instructional Elements of a Lesson

General ResourcesEureka Math and Berkeley Pacing Guides

Achieve the Core Focus by Grade Document

Teacher resources

5th: ccss5.com

3rd: http://blogs.bethel.k12.or.us/mjenson/

K-8:

http://www.oakdale.k12.ca.us/cms/page_view?d=x&piid=&vpid=1391596408603

FluencyThis should be brief

Builds student confidence

Reinforces conceptual understanding and automaticity

Reviews foundational skills in a fun way for the day’s concept

Strategic Fluency Strategies

Do same type fluency strategies in successive days

Refer to script in each module overview

Choose one review and one current content practice daily

Do not collect Sprints. Send home with students

Utilize whiteboards

Students might have three Sprints in a single lesson. Pick and choose here!

*** Question doing fluency activities that require lots of teacher prep

Application ProblemAllows opportunity to apply what they have

recently learned in concept development: (without direct instruction and new context/real-life situation)

Allows for the development of single and multi-step word problems

Students discuss solutions and learn from one another

Strategic Application Problem Strategies

Consider doing after problem set is complete (keep students busy)

Consider assigning the application problem during a short open time in the schedule (10 minute opening between subjects/activities)

Periodically do Fluency Practice and Application Problems for entire math lesson or homework

Make choices about which Application Problems to assign

Don’t spend large amounts of time teaching the application problem. Issue the problem, allocate the recommended time and move on to Concept Development

Concept DevelopmentKey section that introduces NEW concepts

Moves through concrete to pictorial to abstract (not always in this order)

This is where the manipulatives will fit in the lesson and hopefully develop the conceptual understanding as students see the meaning and method

Scripted and may or may not be followed verbatim

Strategic Concept Development Strategies

Pre-read through the script and teach this section with a teaching style that works for you and your students

Don’t skip the script sections that require student engagement

Keep it focused on the lesson’s objective

Modeling and guided practice are critical

Look at the Problem Sets and make sure students are set up to be successful

(the concept development supports the independent work sections)

Problem SetsOften contain three different levels of problems.

Level 1: computational and conceptual

Level 2: students demonstrate reasoning & justify their arguments

Level 3: problems ask students to model real-world situations and use more advanced math

skills

Think of this as being similar to Bloom’s Taxonomy.

Strategic Problem Set Strategies

Level 1 questions for each lesson are “must dos”

Keeping the lesson’s objective in mind, decide which problems are essential for students to complete.

Use your judgment and differentiate what gets assigned to your students based on their skills.

Student DebriefAllows students to clarify thinking and make

connections by talking and listening to their peers

Students are exposed to different ways of thinking and reasoning

Teacher can be a “Neutral Observer” of student ideas and their mathematical reasoning

This is where students review and discuss the work done in the Problem Set

Strategic Student Debrief Strategies

Select only one question to debrief that addresses the lesson objective

Pair students or create small groups

Consider having students share their debrief with the whole class

Use the application problem if it pertains to the lesson

Students write the debrief rather than discussing orally

Meaning, Method, MasteryTalk at your table how you see meaning, method

and mastery fitting in these first two sections of the lesson.

Is this similar or different from the program you have been teaching? What makes it similar or different?

Assessments in EurekaThe end goal is for students to use their

mathematical skills to solve real-world problems.

Traditionally students aren’t strong story problem solvers

Students will be asked to do multi-step story problems and explain their thinking

Assessment informs instruction and must be used on a daily basis

End of Module AssessmentAn assessment of the entire module’s content

where students must apply the learning to real-world problems

Look at the last few pages in the section behind the first brown sheet

Quantum Leap………students must apply their skills and explain their reasoning. These assessments are extremely rigorous (level 3)

Eureka Resources Google Drive Folder

https://drive.google.com/?tab=mo&authuser=0#folders/0B-El36w1l5BHZEFaRUluWGZpWUU

Bethel's Experience with Eureka