3-5 Standards and Support Training Office of Standards and Learning Mary L. Ruzga mruzga@ed.sc.gov...

3-5 Standards and Support Training

Office of Standards and Learning

Mary L. Ruzgamruzga@ed.sc.gov Summer 2015

Introductions• Introduce yourself to your table mates

• Show of hands – How many• 3rd, 4th, 5th grade teachers• Principal/Assistant, District Leader, other?• have participated in Numeracy Leader

training?• have read the new standards?• are comfortable with the new standards?

What will we do today:

1. Quick overview of format of the standards

2. Review the standards by engaging in activities which explain and demonstrate the need for possible new pedagogical techniques.

3. Review and explore the 3-5 Phase I Mathematics Support Document

Accessing the Standards and Resourceswww.ed.sc.gov

BASIC COURTESIES/REMINDERS

Please turn off phones (emergency vibrate)

Please refrain from texting, emailing, etc.

No formal break – take care of personal needs

Logistics (lunch, restrooms)

Participate and respect those around you

Content Standards Process Standards

Portrait of SC Graduate

Process Standards

Mathematical Process Standards

1. Make sense of problems and persevere in solving them.

2. Reason both contextually and abstractly.3. Use critical thinking skills to justify

mathematical reasoning and critique the reasoning of others.

4. Connect mathematical ideas and real-world situations through modeling.

Source: South Carolina College- and Career-Ready Standards for Mathematics 2015

Mathematical Process Standards

5. Use a variety of mathematical tools effectively and strategically.

6. Communicate mathematically and approach mathematical situations with precision.

7. Identify and utilize structure and patterns.

Content Standards

Format of Mathematics Content Standards

K – 8 Grade Level Content Standards

In grades K – 8:GradeLevel.KeyConcept.Standard

Number (e.g., 5.NSBT.1) or, if applicable,

GradeLevel.KeyConcept.Standard NumberStandardLetter (e.g., 3.NSF.1a)

Format of Math Content Standards

Termsincluding - references content that must be

mastered (See 5.NSF.1)

e.g. - references possible illustrative examples. (See 3.G.1)

i.e. - references the only examples or terms that should be used. (4.NSF.1)

TermsFluently and fluency describe a student’s ability

to compute with accuracy, flexibility, and efficiency (Kilpatrick, Swafford, & Findell, 2001).

Real-world refers to authentic contexts through which students engage in mathematics and should serve as a stepping-stone for thinking about important mathematical concepts.

K 1st Grade 2nd Grade 3rd Grade 4th Grade 5th GradeNumber Sense

Number Sense and Base Ten

Number Sense - Fractions

Number Sense and Operations – Fractions Decimals

Algebraic Thinking and Operations

Geometry Geometry Geometry Geometry Geometry Geometry

Measurement and Data Analysis

Progression of Key Concepts

Digging Into the Standards

Geometry Key Concept

Classification RequirementsPrevious Grades

Recognize 4-sided Shapes

Rectangles

Squares Rhombi

Classification Requirements3rd Grade - Read 3.G.1

Recognize 4-sided Shapes by Properties asQuadrilaterals

Rectangles

Squares Rhombi

Quadrilaterals

Classification Requirements4th Grade – Read 4.G.1 and 4.G.2

Recognize Impact of Parallel and Perpendicular Lines

Classification Requirements5th Grade - Read 5.G.3 and 5.G.4

Recognize Shape Classification by Properties

1st Grade 2nd Grade 3rd Grade 4th Grade 5th Grade

Number Sense and Base Ten

Number Sense – Fractions

Number Sense and Operations – FractionsDecimals

Number Sense and Operations – Fractions Decimals

Algebraic Thinking and Operations

Geometry Geometry Geometry Geometry Geometry

Measurement and Data Analysis

Progression of Key Concepts

Digging Into the 3-5 Operations

Pick a grade 3-5 and quickly read through the standards. As you do so, mark or highlight the standards that mention multiplication and/or division – whole numbers, fractions, and decimals.

3rd through 5th Grade

Multiplication and Division of

Whole Numbers and Decimals and

Multiplication of Fractions

Digging Into the Standards

Scaffolding for Multiplication

“Instructional scaffolding is a learning process designed to promote a deeper level of learning. Scaffolding is the support given during the learning process which is tailored to the needs of the student with the intention of helping the student achieve his/her learning goals” (Sawyer, 2006).

Pre-multiplication Knowledge – Multiplication Whole Numbers

Read 2.ATO.4

Step 1:Informal pre-assessment to check for understanding of repeated addition.

Step 2: Link repeated addition and multiplication and introduce the multiplication sign as more efficient than repeated addition.

Turn and tell your partner why these six representations on the screen show the same amount. is twelve• slices• Two plus two plus two plus two plus two plus

two equals 12

• 2 plus 2 plus 2 plus 2 plus 2 plus 2 equals 12

• 2 + 2 + 2 + 2 + 2 + 2 = 12

• 6 twos = 12

• 6 groups of two equal 12

Pre-multiplication Knowledge – Multiplication Whole Numbers 2 + 2 + 2 + 2 + 2 + 2 = 12

How many groups of two? 6 groups

6 x ________ = 12

What is the size of each group? two in each group

6 x 2 = 12

Two used six times 6 times 2

Turn and tell your partner why the representations on the screen all equal twelve. is twelve• slices

• 2 + 2 + 2 + 2 + 2 + 2 = 12

• 6 twos = 12

• 6 x 2 = 12

Step 2 - review Link repeated addition and multiplication and introduce the multiplication sign as more efficient than repeated addition.

Built on 2.ATO.4

Step 3: Continue to link repeated addition and multiplication, use the multiplication sign and introduce the importance of equal groups.

Re-read 2.ATO.4Note the words “equal addends (groups)”

These are equal groups, turn to your partner and tell why they are equal.

Write a repeated addition and a multiplication sentence for this picture.

Check my work by writing an addition sentence and counting to find the total number of objects.

Use your addition sentence as you talk to your partner about why you agree or disagree with my work.

3 x 4 = 12

• 2 + 2 + 2 + 2 + 2 + 2 = 12

• 6 twos = 12

• 6 x 2 = 12

3 x 4 = 12

4 + 4 + 3 = 11

Step 3 - reviewContinue to link repeated addition and multiplication, use the multiplication sign and introduce the importance of equal groups.

Complete Lesson 1, Problem Set 3.1

Sample Debrief Questions Might Include:• On the first page, what did you notice about the

answers to your problems? • Discuss the relationship between repeated

addition and the unit form 2 groups of three or 3 groups of two, depending on the drawing.

• Discuss the relationship between repeated addition, unit form, and the multiplication sentence 3 × 2 = 6.

• Review the new vocabulary presented in the lesson: equal groups, multiplication, and multiply.

Step 4: Introduce array as a more efficient method to see that groups are equal; identify rows and columns, make arrays and link to multiplication.

While the term array was introduced in 2nd grade (2.ATO.4) it was with repeated addition and must now be treated as new vocabulary because we are linking it to multiplication.

We can count and add, OR if the size of each group is the same (equal) we can use repeated addition or multiplication to find the total number of objects in groups like this.

• What is the symbol for multiplication?• Write a repeated addition and a multiplication

equation for this picture. • Share with your partner.

Turn to your partner and talk about how these two groups are alike and how they are different.

• Write a repeated addition and a multiplication expression for each.

This organized picture is called an array – like you used with repeated addition in 2nd grade.

Arrays are made of rows and columns. Rows tell how many in a group, columns tell how many groups. Use the 3 x 10 dot arrangement to trace and show rows/columns.

How many groups? How many in each group? 3 x 4

This drawing shows 3 groups of 5. Re-draw the picture as an array with 3 rows of five.

Write a multiplication expression to describe your array.

Informal assessment with dots and colored paper.

Step 4: – review Introduce array as a more efficient method to see that groups are equal; identify rows and columns, make arrays and link to multiplication.

Work Lesson 2, Problem Set 3.1

Debrief/discuss the task.

Review new vocabulary: row, array, number of groups, and size of groups.

Prompt students to notice arrays around the room and possibly think of arrays in real world situations

Now that students have some understanding of the meaning of multiplication and the link to repeated addition, multiplication can be introduced in story problems, as a way to learn – not just facts to be memorized.

Multiplication - Whole Numbers

By introducing multiplication in this manner, students not only know the meaning of multiplication but can use repeated addition as a strategy to solve single digit by single digit problems, if they don’t know the multiplication fact. Time should be spent ensuring that students have a firm understanding of the 1, 2, 3, 4, 5 facts before moving to other facts. - why?

Read: 3.ATO.5

Pre-multiplication Knowledge – Multiplication Whole Numbers (3.ATO.5)

Examples of the Distributive property

Don’t know 3 x 9 Know 3 x 5 and 3 x 4; then 3 x 9 = 3 groups of 5 + 3 groups of 4 3(5 + 4)

Don’t know 3 x 7 Know 3 x 5 and 3 x 2; then 3 x 7 = 3 groups of 5 + 3 groups of 2 3(5 + 2)

Multiplication, Perimeter, and Area – Whole Numbers (3.MDA.5c)

What is the length? What is the height/width? 3 x 4 Area is 12; the factors are the dimensions.

Division as an unknown Factor Problem – 3.ATO.4, 3.ATO.6, 4.NSBT.6)

• If 12 objects are shared equally into 3 groups, how many objects will be in each group? (Group size unknown) 3 x B = 12

• If 12 objects are to be shared/divided equally into groups of 4, how many groups will there be? (Number of groups unknown) B x 4 = 12

Multiplication - Whole Numbers

Standards explicitly addressed so far:

• 3. ATO.1(relationship between factors)

• 3.ATO.3 (arrays part)

• 3.ATO.4 (relationship among product/dividend, factors/quotient/divisor

• 3.ATO.5 (properties)

• 3.ATO.6 (unknown factor)

• 3.ATO.7 (multiplication and division fluency)

• 3.MDA.5c (area related to multiplication)

• 4.ATO.2 (number of groups/size of groups)

Multiplication – Multi-digit Whole Numbers and Decimals

Read • 4.NSBT.5• 4.NSBT.6• 5.NSBT.6

How does what we have done so far relate to those standards?

Multiplication – Multi-digit Whole Numbers and Decimals (4.NSBT.5)

3 x 32 = ___

Use Base ten Blocks to Build a model/array

10 units 10 units 10 units 2 units

3 x 10 = 30 3 x 10 = 30 3 x 10 = 30 3 x 2 = 6

3 (10 + 10 + 10 + 2)

Relate Multiplication and Division (4.NSBT.6)

If 96 objects are shared equally into 3 groups, how many objects are in each group? (Group Size Unknown)

If 96 objects are to be divided equally into groups of 32 each, how many groups will there be? (Number of groups Unknown)

( shared in 3 to a bag, how many bags are needed?there are a total of 96 objects divided into 3 groups, how many are in each array) with 3 groups of 32. Label the model/array.

3 x 32 = 963 groups of 32 equals 96

32 units

Multiplication – Multi-digit Whole Numbers and Decimals (4.NSBT.5)

Use Base Ten Blocks to Build a Model for 14 x 23

14 (10 + 10 + 3) = 322

10 x 10 = 100

10x3=30

4 x 10 = 40 4 x 10 = 40

4x3 = 12

10 10 3230

Save your Base Ten Model

Multiplication – Multi-digit Whole Numbers and Decimals SAVE YOUR Base Ten Model (4.NSBT.5)

Use the grid paper to draw the model (array)

that shows 14 groups of 23. Label the dimensions and area.

14 x 23 = 322

14 groups of 23 = 322

23 Units – Group Size

Relating Multiplication and Division SAVE YOUR MODEL(4.NSBT.6)

If 322 objects are shared equally into 14 groups, how many objects are in each group? (Group Size Unknown)

If 322 objects are to be divided equally into groups of 23 each, how many groups will there be? (Number of groups Unknown)

14 groups of 23 = 322

23 Units – Group Size

Multiplication – Multi-digit Whole Numbers and

Decimals - SAVE YOUR MODEL

Read 5.NSBT.7

Look at your model, What is 14 x 23?

What is tenths x tenths?

So, 1.4 x 2.3 = 3.22

Think about and then talk with your neighbor -How does this relate to your base ten block model?

Multiplication – Multi-digit Whole Numbers and Decimals - SAVE YOUR Base Ten MODEL

5NSBT.7

• Use grid paper to again draw the same whole number model/array you used for 14 x 23. Label it this time with the decimal factors 1.4 x 2.3. Clearly label the whole number and decimal portions of the factors. (It is cm grid paper so

you can lay your model on grid paper and trace around it. Be sure to trace/outline the wholes, ten strips and hundreds cubes.)

Then compare with your neighbor and discuss how the whole number and the decimal models relate. Save the base ten block model.

2 3 tenths

4 tenths

1.4 x 2.3 = 3.22

Save the base ten block model)

Relating decimal multiplication and decimal division. . . (5.NSBT.7) (Related to 3.ATO.6)

• What is 322 ÷ 23 = ______?• What is hundreds divided by tens? • What is hundredths divided by

tenths?So. . . 3.22 ÷ 2.3 = 1.4

Multiplication – Multi-digit Whole Numbers and Decimals (4.NSBT.5, 5.NSBT.5, 5.NSBT.7)

• Use Base Ten Blocks to build an array for 16 groups of 12. Save the model.

• Use the grid paper to draw your model. Label the dimensions and area.

• Solve 16 x 12 using an algorithm. Compare the product to your model/array.

• Compare all of the above with your neighbor.

• SAVE THE BASE TEN MODEL

Multi-digit multiplication – whole numbers and decimals. . . (4.NSBT.5, 5.NSBT.5, 5.NSBT.7)

What is 16 x 12? What is tenths x tenths?What is 1.6 x 1.2 =?

• Use grid paper to draw the same model (array) you used for whole number 16 x 12. Label it this time with the decimal factors 1.6 x 1.2

• Clearly label the whole number and decimal portions of the factors. Then compare with your neighbor. How do the whole number and decimal models compare?

1 2 tenths

1.2x1.61.92

1 2 tenths

Relating decimal multiplication and division.

What is 192 ÷ 16 ?What is hundreds divided by tens? What is hundredths divided by tenths?So. . . 1.92 ÷ 1.2 = 1.6

Multi-digit multiplication – whole numbers and decimals. . .

Strategies for making decimal arrays:• Think whole numbers• Think size of group and number of groupso Think size of group and form one group.o Think how many groups and make that

many groups of the size needed.o Arrange the groups in an array and trade to

use the fewest number of pieces.

The tendency in the US is to have children solve a lot of problems in a class period and focus on the correct answer rather than focusing in-depth on one or two well-chosen problems at a time, the relationship between the structures of the problems and the variety of approaches to the solutions. John A. Van De Walle

4.ATO.1

3.ATO.4, 4.ATO.2

3.ATO.3 3.ATO.3 3.ATO.3

3.MDA.5-6

4.ATO.1 4.ATO.1

3.MDA.5-6 3.MDA.5-6

3.ATO.7

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Joining our Edmodo Groups

SCDE Mathematics Team: https://edmo.do/j/snmcix

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Joining our Edmodo Groups

Support Document Overview

Mathematics Support Document

Review Process: Work by grade levels – YOUR DECISONo Move into Grade Alike Groupso Open browser

Explain Phase I and View “Overview” “Overview”

o Edmodoo http://ed.sc.gov – search “Math Support Document”

o Direct URL

http://ed.sc.gov/agency/ccr/Standards-Learning/Mathematics.cfm See yellow sheet

Mathematics Support Document

Review Process: http://maryruzga.weebly.com Grade Levels

Review/explore Share out one activity/lesson/resource

Evaluation and Certificate Link

My Contact InformationMary L. Ruzga

mruzga@ed.sc.govThank you for spending time with me today.

3-5 Standards and Support Training Office of Standards and Learning Mary L. Ruzga mruzga@ed.sc.gov...

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