1. Introduction to Eureka Math “A Story of Units”
Note that I’ve purposely put the “instructional” things last. These are things that you’ll work together to figure out better than I, but I want you to get an idea how to use the models/manipulatives since this will likely be a new way of thinking of these concepts.
Tricia Bevans, University of Oregon

2. 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.

3. What do these have in common?
= 96
5/9 +1/3 = 8/9
= 80
225°+135°=360°
12.7cm +7.62cm=20.32cm

4. What do these have in common?
= dozen + 3 dozen = 8 dozen
5/9 +1/3 = 8/ ninths + 3 ninths = 8 ninths
= 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
All of these are 5+3=8 if we use appropriate units.

5. The Importance of the Unit
The example in the last slide highlights why the Eureka/EngageNY K-5 curriculum is called “A Story of Units”

6. The Importance of the Unit
Units connect seemingly unrelated concepts in Elementary Math

7. The unit: a single object
Counting: 1 apple, 2 apples, 3 apples,…. Adding and Subtracting: 5 apples+3 apples = 8 apples
Here we’re building basic number sense about how to put together and break apart things of the same unit.

8. The unit: a ten
Counting: 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
We can use the same skills of putting together and breaking apart things of the same unit with tens

9. The units: tens and ones
Counting: 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 ones or = 86 Exchanging units: 1 ten = 10 ones
This is the basis for place value and standard algorithms

10. 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
Skip counting, of course builds the foundation for multiplication. Adding and subtracting and exchanging units provide a framework for memorizing multiplication facts (the examples here are: “If I know 5 x2, then I can get 6 x 2 by adding one more two” and “Fours are 2 twos so (5x2)x2= 5 x 4 and for provide powerful strategies for working through more difficult problems. This also lays a foundation for proportional reasoning and algebra in middle and high school.

11. 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
Now we’ve got an intuitive basis for adding and subtracting fractions as well as the equivalent fractions needed to get the “common denominator” (which is just another way of saying to make the fractions have the same units!) Combining these with the skip counting from before….

12. 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.
We can use units of bottles, buckets, and glasses along with a visual model (wait and see how!) to answer various questions about this situation.

13. “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”

14. A framework for effective mathematics learning
Meaning Method Mastery Be

15. 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)

16. Meaning Example - 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.

17. Method
Processes to solve clearly delineated mathematics problems.

18. Method Example: Methods for multiplication include
Procedures and algorithms, such as the multi-digit multiplication algorithm
Strategies such as reasoning that
11x6 = 10x6 +1x6=60+6

19. Mastery The ability to use a mathematical concept:
In (age-)appropriate mathematical reasoning.
In authentic real-world applications.
In the development of more advanced concepts.

20. Mastery Example: 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

21. Meaning Method Mastery
Students 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

22. 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?”)
Because application problems can serve both meaning and mastery objectives, and they are an ideal time to include attention to the Standards for Mathematical Practice, Eureka math pays special attention to them. A visual model bridges the context to the more abstract mathematical process needed to solve it.

23. Activity: Introduction to Concrete and Pictorial Models in Eureka Math
Form 7 groups: Each group will become “expert” in a type of model/manipulative used in Eureka. Use “How to Implement a Story of Units” as a reference.
Group 1: Number Towers, Number Paths, Number Line
Group 2: Number Bonds
Group 3: Bundles and Base Ten Blocks
Group 4: Rectangular Areas, Rectangular Arrays
Group 5: Tape Diagrams
Group 6: Money, Number Discs, Place value chart
Group 7: Rekenrek, Ten Frame
Each group makes a poster to represent the key features of their model, possibly including (what it is, how to use, strengths, an example of when to use it, connections between models if they have more than one to present)
After this, each group will present their poster to the whole group.

24. Activity: Coherence of Models
Form groups (or use existing groups) of participants (3-6 in a group). Give each group a set of cards with the name of each of the models including some extra blank cards and Four (larger?) cards labeled LINEAR MODEL, AREA MODEL, SET MODEL, OTHER
Have participants sort into these four groups, then
Further sort by the order in which they would introduced these models for each group.
Share out: Any observations about coherence? Did they notice any models that worked in more than one place?

25. Activity: Using Various Models
Break?
Grade Band teams or grade level teams. Give each a different problem/concept and have them think about and/or research how to use as many models as possible to represent the solution to the problem.
K: KNBTA1 Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (such as 18 = ); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.
1: 1OA.C.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., = = = 14); decomposing a number leading to a ten (e.g., = = = 9); using the relationship between addition and subtraction (e.g., knowing that = 12, one knows = 4); and creating equivalent but easier or known sums (e.g., adding by creating the known equivalent = = 13).
2: 2NBT.B5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
3: 3OA.D8 Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. (EX. Masha had 120 stamps. First, she gave her sister half of the stamps and then she used three to mail letters. How many stamps does Masha have left? )
4: 4NBT.B5 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.
5: 5NF.A1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/ /12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)
6: 6.NS.B3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
If possible, have groups share out.

26. Adapting Instructional Elements of a Lesson

27. General Resources Eureka Math and Berkeley Pacing Guides
Achieve the Core Focus by Grade Document
Teacher resources
5th: ccss5.com
3rd:
K-8: =

28. Fluency This should be brief Builds student confidence
Reinforces conceptual understanding and automaticity
Reviews foundational skills in a fun way for the day’s concept

29. 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

30. Application Problem
Allows 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

31. 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

32. Concept Development Key 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

33. 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)

34. Problem Sets Often 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.

35. 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.

36. Student Debrief
Allows 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

37. 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

38. Meaning, Method, Mastery
Talk 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?

39. Assessments in Eureka
The 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

40. End of Module Assessment
An 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)

41. Eureka Resources Google Drive Folder
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Bethel's Experience with Eureka