6th Grade Mathematics - Orange Public Schools / Overvie · 2016-05-04 · Finding unknown angle measures using several angle properties Essential Questions ... How do you use angle

5th Grade Mathematics Unit 4 Curriculum Map:

ORANGE PUBLIC SCHOOLS OFFICE OF CURRICULUM AND INSTRUCTION

OFFICE OF MATHEMATICS

Unit 4: Marking Period 4: April 9 – June 21

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Table of Contents

I. Unit Overview p. 2

II. MIF Lesson Structure p. 7

III. Pacing Guide p. 10

IV. Pacing Calendar p. 14

V. Unit 4 Math Background p. 17

VI. PARCC Assessment Evidence/Clarification Statements

p. 18

VII. Connections to the Mathematical Practices p. 19

VIII. Visual Vocabulary p. 21

IX. Potential Student Misconceptions p. 25

X. Teaching Multiple Representations p. 27

XI. Assessment Framework p. 31

XII. Performance Tasks p. 33

XIII. Additional Assessment Resources p. 43

XIV. Extensions and Sources p. 44


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Unit Overview

Unit 4: Chapters 6,7,10,12

In this Unit Students will be:

Recognizing that base and height are measurements of a triangle that are used to find its

area

Comparing numbers by division and expressing this comparison as a ratio

Understanding the relationships between percents, fractions and decimals

Using percents to solve real world problems

Finding unknown angle measures using several angle properties

Essential Questions

What information is necessary to find the area of a triangle?

What is the relationship between area of a rectangle and area of a triangle?

How is a ratio used to compare two quantities or values?

What is the connection between a ratio and a fraction? What is the difference?

How can I model and represent ratios?

What is the relationship between a decimal, fraction, and percentage and how do we

change from one form to another?

How do you use angle and line properties to solve for an unknown angle measure?

Enduring Understandings

Chapter 6: Area

Base and Height of a Triangle

Area of a Triangle

Area of a Rectangle with Fractional Side Lengths

Chapter 7: Ratio

Comparing Numbers of Quantities

Forms of a Ratio

Equivalent Ratios

Solve Real-World Problems

Chapter 10: Percent

Percent, Fraction and Decimal

Percent of a Number

Solve Real-World Problems

Chapter 12: Angles

Angles of a Line (find unknown angle measures)

Angles at a Point (find unknown angle measures)

Vertical Angles (find unknown angle measures)


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Common Core State Standards

5.NF.4b

Find the area of a rectangle with fractional side

lengths by tiling it with unit squares of the

appropriate unit fraction side lengths, and show that

the area is the same as would be found by

multiplying the side lengths. Multiply fractional

side lengths to find areas of rectangles, and

represent fraction products as rectangular areas.

This standard extends students’ work with area. In third grade students determine the area of

rectangles and composite rectangles. In fourth grade students continue this work. The fifth grade

standard calls students to continue the process of covering (with tiles). Grids (see picture) below

can be used to support this work.

Example: The home builder needs to cover a small storage room floor with carpet. The storage

room is 4 meters long and half of a meter wide. How much carpet do you need to cover the floor

of the storage room? Use a grid to show your work and explain your answer. In the grid below I

shaded the top half of 4 boxes. When I added them together, I added ½ four times, which equals

2. I could also think about this with multiplication ½ x 4 is equal to 4/2 which is equal to 2.

Example:

In solving the problem ⅔ x ⅘, students use an area model to visualize it as a 2 by 4 array of

small rectangles each of which has side lengths ⅓ and ⅕. They reason that ⅓ x ⅕= 1/(3 x 5) by

counting squares in the entire rectangle, so the area of the shaded area is (2 x 4) x 1/(3 x 5) =

They can explain that the product is less than ⅘ because they are finding ⅔ of ⅘. They can

further estimate that the answer must be between ⅖ and ⅘ because ⅔ of ⅘ is more than ½ of ⅘

and less than one group of ⅘.

The area model and the line segments show that the area is the same quantity as the product of

the side lengths.


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5.NF.5a

Explain patterns in the number of zeros of the

product when multiplying a number by powers of

10, and explain patterns in the placement of the

decimal point when a decimal is multiplied or

divided by a power of 10. Use whole-number

exponents to denote powers of 10.

This standard calls for students to examine the magnitude of products in terms of the relationship

between two types of problems. This extends the work with 5.OA.1.

Example 1:

Mrs. Jones teaches in a room that is 60 feet

wide and 40 feet long. Mr. Thomas teaches in

a room that is half as wide, but has the same

length. How do the dimensions and area of

Mr. Thomas‟ classroom compare to Mrs.

Jones‟ room? Draw a picture to prove your

answer.

Example 2:

How does the product of 225 x 60 compare to

the product of 225 x 30? How do you know?

Since 30 is half of 60, the product of 22 5x 60

will be double or twice as large as the product

of 225 x 30.

Example:

¾ x 7 is less than 7 because 7 is multiplied by a factor less than 1 so the product must be less

than 7.

5.G.3

Understand that attributes belonging to a category of two

dimensional figures also belong to all subcategories of that

category. For example, all rectangles have four right angles

and squares are rectangles, so all squares have four right

angles.

This standard calls for students to reason about the attributes (properties) of shapes. Student

should have experiences discussing the property of shapes and reasoning.

Example: Examine whether all quadrilaterals have right angles. Give examples and non-

examples. Example: If the opposite sides on a parallelogram are parallel and congruent, then

rectangles are parallelograms

A sample of questions that might be posed to students include: A parallelogram has 4 sides with

both sets of opposite sides parallel. What types of quadrilaterals are parallelograms? Regular


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polygons have all of their sides and angles congruent. Name or draw some regular polygons. All

rectangles have 4 right angles. Squares have 4 right angles so they are also rectangles. True or

False? A trapezoid has 2 sides parallel so it must be a parallelogram. True or False?

The notion of congruence (“same size and same shape”) may be part of classroom conversation

but the concepts of congruence and similarity do not appear until middle school.

TEACHER NOTE: In the U.S., the term “trapezoid” may have two different meanings.

Research identifies these as inclusive and exclusive definitions. The inclusive definition states:

A trapezoid is a quadrilateral with at least one pair of parallel sides. The exclusive definition

states: A trapezoid is a quadrilateral with exactly one pair of parallel sides. With this definition,

a parallelogram is not a trapezoid. North Carolina has adopted the exclusive definition.

(Progressions for the CCSSM: Geometry, The Common Core Standards Writing Team, June

2012.)

5.G.4 Classify two-dimensional figures in a hierarchy based on

properties.

This standard builds on what was done in 4th grade. Figures from previous grades: polygon,

rhombus/rhombi, rectangle, square, triangle, quadrilateral, pentagon, hexagon, cube,

trapezoid, half/quarter circle, circle, kite

A kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that

are beside (adjacent to) each other.

Student should be able to reason about the attributes of shapes by examining: What are ways to

classify triangles? Why can‟t trapezoids and kites be classified as parallelograms? Which

quadrilaterals have opposite angles congruent and why is this true of certain quadrilaterals?, and

How many lines of symmetry does a regular polygon have?

TEACHER NOTE: In the U.S., the term “trapezoid” may have two different meanings.

Research identifies these as inclusive and exclusive definitions. The inclusive definition states:

A trapezoid is a quadrilateral with at least one pair of parallel sides. The exclusive definition

states: A trapezoid is a quadrilateral with exactly one pair of parallel sides. With this definition,

a parallelogram is not a trapezoid. North Carolina has adopted the exclusive definition.

(Progressions for the CCSSM: Geometry, The Common Core Standards Writing Team, June

2012.)


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Example: Create a Hierarchy Diagram using the following terms:

M : Major Content S: Supporting Content A : Additional Content

polygons – a closed plane figure

formed from line segments that meet

only at their endpoints.

quadrilaterals - a four-sided polygon.

rectangles - a quadrilateral with two

pairs of congruent parallel sides and

four right angles.

rhombi – a parallelogram with all

four sides equal in length.

square – a parallelogram with four

congruent sides and four right angles


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MIF Lesson Structure LESSON STRUCTURE RESOURCES COMMENTS

Chapter Opener

Assessing Prior Knowledge

The Pre Test serves as a

diagnostic test of readiness

of the upcoming chapter

Teacher Materials

Quick Check

PreTest (Assessm‟t

Bk)

Recall Prior

Knowledge

Student Materials

Student Book (Quick

Check); Copy of the

Pre Test; Recall prior

Knowledge

Recall Prior Knowledge (RPK) can take place

just before the pre-tests are given and can take

1-2 days to front load prerequisite

understanding

Quick Check can be done in concert with the

RPK and used to repair student

misunderstandings and vocabulary prior to the

pre-test ; Students write Quick Check answers

on a separate sheet of paper

Quick Check and the Pre Test can be done in

the same block (See Anecdotal Checklist; Transition

Guide)

Recall Prior Knowledge – Quick Check – Pre

Test

Direct

Involvement/Engagement

Teach/Learn

Students are directly

involved in making sense,

themselves, of the concepts –

by interacting the tools,

manipulatives, each other,

and the questions

Teacher Edition

5-minute warm up

Teach; Anchor Task

Technology

Digi

Other

Fluency Practice

The Warm Up activates prior knowledge

for each new lesson

Student Books are CLOSED; Big Book is

used in Gr. K

Teacher led; Whole group

Students use concrete manipulatives to

explore concepts

A few select parts of the task are

explicitly shown, but the majority is

addressed through the hands-on,

constructivist approach and questioning

Teacher facilitates; Students find the

solution

Guided Learning and

Practice

Guided Learning

Teacher Edition

Learn

Technology

Digi

Student Book

Guided Learning Pages

Hands-on Activity

Students-already in pairs /small, homogenous

ability groups; Teacher circulates between

groups; Teacher, anecdotally, captures student

thinking

Small Group w/Teacher circulating among

groups

Revisit Concrete and Model Drawing;

Reteach

Teacher spends majority of time with

struggling learners; some time with on level,

and less time with advanced groups

Games and Activities can be done at this time

DIR

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AG

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PR

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GU

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Independent Practice

A formal formative

assessment

Teacher Edition

Let‟s Practice

Student Book

Let‟s Practice

Differentiation

Options

All: Workbook

Extra Support:

Reteach

On Level: Extra

Practice

Advanced: Enrichment

Let’s Practice determines readiness for

Workbook and small group work and is used

as formative assessment; Students not ready

for the Workbook will use Reteach. The

Workbook is continued as Independent

Practice.

Manipulatives CAN be used as a

communications tool as needed.

Completely Independent

On level/advance learners should finish all

workbook pages.

Extending the Lesson Math Journal

Problem of the Lesson

Interactivities

Games

Lesson Wrap Up Problem of the Lesson

Homework (Workbook

, Reteach, or Extra

Practice)

Workbook or Extra Practice Homework is

only assigned when students fully understand

the concepts (as additional practice)

Reteach Homework (issued to struggling

learners) should be checked the next day

End of Chapter Wrap Up

and Post Test

Teacher Edition

Chapter Review/Test

Put on Your Thinking

Cap

Student Workbook

Put on Your Thinking

Cap

Assessment Book

Test Prep

Use Chapter Review/Test as “review” for the

End of Chapter Test Prep. Put on your

Thinking Cap prepares students for novel

questions on the Test Prep; Test Prep is

graded/scored.

The Chapter Review/Test can be completed

Individually (e.g. for homework) then

reviewed in class

As a „mock test‟ done in class and doesn‟t

count

As a formal, in class review where teacher

walks students through the questions

Test Prep is completely independent;

scored/graded

Put on Your Thinking Cap (green border)

serve as a capstone problem and are done just

before the Test Prep and should be treated as

Direct Engagement. By February, students

should be doing the Put on Your Thinking

Cap problems on their own.

IND

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PR

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PR

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TRANSITION LESSON STRUCTURE (No more than 2 days)

Driven by Pre-test results, Transition Guide

Looks different from the typical daily lesson

Transition Lesson – Day 1

Objective:

CPA Strategy/Materials Ability Groupings/Pairs (by Name)

Task(s)/Text Resources

Activity/Description


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Pacing Guide

Activity Common Core Standards Estimated Time (# of block)

Lesson Notes

Mini Assessment 5.NBT.5-7

5.NBT.5, 5.NBT.6, 5.NBT.7 ½ block

Pre-Test 6 K.G.5, 2.G.1, 3.MD.5, 3.MD.5.a, 3.MD.5.b, 4.MD.3, 4.OA.3, 5.NF.4.b

½ block

Chapter Opener 6/Recall Prior Knowledge 1

3.MD.5.b, 4.MD.3, 4.MD.5, 4.G.1

1 block

Finding the Area of a Rectangle with Fractional Side Lengths 6.1

5.NF.4.b 1 block

Guide students to see that they will get the same area by counting the small squares inside a rectangle as by multiplying the side lengths.

Base and Height of a Triangle 6.2

N/A 1 block

Reinforce the concept that the height of a triangle is always perpendicular to the base.

Finding the Area of a Triangle 6.3

5.G.3, 6.G.1 1 block

Some students may have difficulty visualizing the Hands On Activity. Consider modeling it with the whole class before they try it on their own.

Chapter 6 Wrap Up/Review

1 block Reinforce and consolidate chapter skills and concepts

Chapter 6 Test-Review no TP

3.MD.8, 4.G.2, 5.NF.4.b, 6.G.1

1 block

Mini Assessment 5.G.1-2

5.G.1-2 ½ block

Review 2 blocks Review/Reteach

concepts that need to be readdressed

Pre-Test 7 1.OA.6, 3.NF.1, 3.OA.1, 3.OA.3, 4.NF.1, 4.OA.3

1/2 block

Chapter Opener 1.OA.6, 3.NF.1, 4.NF.1 1 block


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7/Recall Prior Knowledge 7

Finding Ratio 7.1 6.RP.1 1 block Emphasize that order is important when writing a ratio.

Equivalent Ratios 7.2 5.NF.5, 5.NF.5.a, 6.RP.1 1 block

Review how to simplify fractions using the greatest common factor before teaching the concepts in the Learn Box on page 299.

Real-World Problems: Ratios 7.3

6.RP.3 2 blocks

When reviewing the Math Journal problem on page 309, accept any answer that is logical based on the model.

Ratios in Fraction Form 7.4

6.RP.1 1 block

Have students act out the activities in the Learn sections of this lesson using models or snap cubes.

Comparing Three Quantities 7.5

5.NF.5.a, 6.RP.1 1 block

Throughout this lesson, ask students to work in pairs/groups and ask each other questions about each procedure they learn.

Real-World Problems: More Ratios 7.6

5.NF.5.a, 6.RP.3 2 blocks

In this lesson, work through and discuss each Guided Practice exercise together with the class. Then have students complete the Let’s Practice exercises on their own.


1 block Reinforce and

consolidate chapter skills and concepts

Chapter 7 Test-Review no TP

3.MD.8, 6.RP.1, 6.RP.3 1/2 block

Mini Assessment 5.MD.1

5.MD.1 ½ block

Review 2 blocks Review/Reteach


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concepts that need to be readdressed

Pre-Test 10 3.NF.1, 4.NF.6, 5.MD.1, 5.NBT.7

1/2 block


4.NF.1, 4.NF.6 1 block

Percent 10.1 N/A 2 blocks

To help students remember how to express a percent as a fraction, put the % sign and 100 on the board. Point out that both the percent symbol and 100 are a line segment with 2 zeros. This can help them remember that a percent is written as a fraction with 100 as the denominator.

Expressing Fractions as Percents 10.2

4.NF.1 1 block

After going through the Learn Activity and the Guided Practice with the students, it may be a good idea to discuss with them which method they prefer and why.

Percent of a Number 10.3

6.RP.3.c 1 block

To make this lesson relevant and make realistic connections for your students, have them solve problems related to their daily lives whenever possible.

Real World Problems: Percent 10.4

6.RP.3.c 1 block

Work through and discuss each Guided Learning exercise with the students before they work on them on their own.




Chapter 10 OMIT


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Test-Review w/TP

Pre-Test 12 4.G.1, 4.G.2, 4.MD.7 1/2 block


4.MD.5, 4.MD.6, 4.G.1 1 block

Angles on a Line 12.1 4.MD.7 1 block

For the Hands On Activity, you may choose to have students use a protractor to draw the angles to prove that they do in fact form a line.

Angles at a Point 12.2

4.MD.7 1 block

You may want to point out that angles at a point can also be central angles of a circle.

Vertical Angles 12.3 7.G.5 1 block

Have students color code the lines so that the vertical angles are apparent.




Chapter 12 Test-Review w/TP

OMIT


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Pacing Calendar

APRIL Sunday Monday Tuesday Wednesday Thursday Friday Saturday

1 2

3 4 5 6 7 8 9

10 11 District Closed

12 District Closed

13 District Closed

14 District Closed

15 District Closed

16

17 18 PARCC Testing*

19 PARCC Testing

20 PARCC Testing

21 PARCC Testing

22 PARCC Testing

23

24 25 Mini Assessment 5.NBT.5-7

26 Unit 3 Catch Up

27 28 29 Chapter 6 Area Pre Test

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PARCC Testing for grade 5 will fall anywhere between April 4

th and May 13

th.


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MAY Sunday Monday Tuesday Wednesday Thursday Friday Saturday

1 2 3

4

5 Test Prep 6

6 Mini Assessment 5.G.1-2

7

8 9 10 Chapter 7 Ratio Pre Test

11 12 13 14

15 16 17 18 19

20

21

22 23

24 Test Prep 7

25 Full Day Staff Only

26 Mini Assessment 5.MD.1

27 Chapter 10 Percent Pre Test

28

29 30 District Closed

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JUNE Sunday Monday Tuesday Wednesday Thursday Friday Saturday

1 2 3

4

5 6

7 8 Test Prep 10

9 Chapter 12 Angles Pre Test

10

11

12 13

14

15 Test Prep 12

16 Unit 4 Catch Up and Review

17 28

19 20

21 12:30 Dismissal Students Only



24

25

26 27 28 29 30


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Unit 4 Math Background Chapter 6: Area

In Grades 3 and 4, students learned about angles. In Grade 4, students also learned two methods for finding the areas of squares and rectangles. Chapter 7: Ratio

In Grade 1, students learned to compare numbers using subtraction. In Grade 3, students learned about fractions and how to express them in simplest form. In Grade 4, students were introduced to the unitary method for solving problems using a model. Chapter 10: Percent In Grade 3, students learned to find equivalent fractions and simplify fractions. In Grade 4, students learned to use models to relate fractions and decimals.

Chapter 12: Angles

In Grade 3, students were introduced to the terms point, line, line segment, ray, and angle. They also learned to identify right angles and perpendicular lines. In Grade 4, students learned to name and measure angles, and find an unknown angle measure for a pair of angles that make up a right angle. Transition Guide References: Chapter 10: Percent

Transition Topic: Money and Decimals

Grade 5 Chapter 10

Pre Test Items

Grade 5 Chapter 10

Pre-Test Item

Objective

Additional Support for the

Objective: Grade 4 Reteach

Additional Support for the

Objective: Grade 4

Extra Practice

Grade 4 Teacher Edition Support

Item 2

Divide up to a four-digit number by a one-digit number with regrouping, and with or without remainders.

Support for this objective is included in Chapter 2.

4A Chapter 3 Lesson 4


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PARCC Assessment Evidence/Clarification Statements

CCSS Evidence Statement Clarification Math Practices

5.NF.4b-1 Apply and extend previous

understandings of multiplication to

multiply a fraction or whole number

by a fraction. b. Multiply fractional

side lengths to find areas of

rectangles, and represent fraction

products as rectangular areas

i) 50% of the tasks present students with the rectangle dimensions and ask students to find the area; 50% of the tasks give the factions and the product and ask students to show a rectangle to model the problem

MP.2, MP.5

5.NF.5a Interpret multiplication as scaling

(resizing), by: a. Comparing the size

of a product to the size of one factor

on the basis of the size of the other

factor, without performing the

indicated multiplication.

i) Insofar as possible, tasks are designed to be completed without performing the indicated multiplication. ii) Products involve at least one factor that is a fraction or mixed number.

MP.7, MP.8

5.G.3 Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.

i) A trapezoid is defined as “A quadrilateral with at least one pair of parallel sides.”

MP.5, MP.7

5.G.4 Classify two-dimensional figures in a hierarchy based on properties.

i) A trapezoid is defined as “A quadrilateral with at least one pair of parallel sides.”

MP.5, MP.7


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Connections to the Mathematical Practices

1

Make sense of problems and persevere in solving them

Mathematically proficient students in fifth grade should solve problems by applying their understanding of operations with whole numbers, decimals, and fractions including mixed numbers. They solve problems related to volume and measurement conversions. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, “What is the most efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a different way?”.

2

Reason abstractly and quantitatively In fifth grade, students should recognize that a number represents a specific quantity. They connect quantities to written symbols and create a logical representation of the problem at hand, considering both the appropriate units involved and the meaning of quantities. They extend this understanding from whole numbers to their work with fractions and decimals. Students write simple expressions that record calculations with numbers and represent or round numbers using place value concepts.

3

Construct viable arguments and critique the reasoning of others

In fifth grade, mathematically proficient students may construct arguments using concrete referents, such as objects, pictures, and drawings. They explain calculations based upon models and properties of operations and rules that generate patterns. They demonstrate and explain the relationship between volume and multiplication. They refine their mathematical communication skills as they participate in mathematical discussions involving questions like “How did you get that?” and “Why is that true?” They explain their thinking to others and respond to others’ thinking.

4

Model with mathematics

In fifth grade, students experiment with representing problem situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, making a chart, list, or graph, creating equations, etc. Students need opportunities to connect the different representations and explain the connections. They should be able to use all of these representations as needed. Fifth graders should evaluate their results in the context of the situation and whether the results make sense. They also evaluate the utility of models to determine which models are most useful and efficient to solve problems.

5

Use appropriate tools strategically

Mathematically proficient fifth graders consider the available tools (including estimation) when solving a mathematical problem and decide when certain tools might be helpful. For instance, they may use unit cubes to fill a rectangular prism and then use a ruler to measure the dimensions. They use graph paper to


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accurately create graphs and solve problems or make predictions from real world data.

6

Attend to precision

Fifth graders should continue to refine their mathematical communication skills by using clear and precise language in their discussions with others and in their own reasoning. Students use appropriate terminology when referring to expressions, fractions, geometric figures, and coordinate grids. They are careful about specifying units of measure and state the meaning of the symbols they choose. For instance, when figuring out the volume of a rectangular prism they record their answers in cubic units.

7

Look for and make use of structure

Mathematically proficient fifth grade students look closely to discover a pattern or structure. For instance, students use properties of operations as strategies to add, subtract, multiply and divide with whole numbers, fractions, and decimals. They examine numerical patterns and relate them to a rule or a graphical representation.

8

Look for and express regularity in repeated reasoning

Fifth graders should use repeated reasoning to understand algorithms and make generalizations about patterns. Students connect place value and their prior work with operations to understand algorithms to fluently multiply multi-digit numbers and perform all operations with decimals to hundredths. Students explore operations with fractions with visual models and begin to formulate generalizations.


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Visual Vocabulary Visual Definition The terms below are for teacher reference only and are not to be memorized by students. Teachers should first present these concepts to students with models and real life

examples. Students should understand the concepts involved and be able to recognize and/or use them with words, models, pictures, or numbers.

CHAPTER 6


22

CHAPTER 7


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CHAPTER 10

CHAPTER 12


24


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Potential Student Misconceptions

- Chapter 6:

Lesson 6.2- Students tend to name the bottom side of the triangle as its base. Remind them that the base can be any side of the triangle. Suggest that they locate right angle symbol to help find the base.

Lesson 6.3- When using the formula for the area of a triangle, many students multiply the base by the height, but forget to multiply the product by ½. Suggest that students write the formula with the ½ in a different color.

- Chapter 7:

Lesson 7.1-One of the most common errors students make when writing ratios is writing the quantities in the incorrect order. Have students who still make this error label the quantities in their ratio as they labeled in the Learn examples on pages 289 and 290.

Lesson 7.2-Some students might have difficulty finding the missing terms in Exercises 10 to 17. Review the methods for finding the missing terms on page 300. Have students choose their preferred method to complete the exercises.

Lesson 7.3-Some students may have difficulty solving real world problems involving ratios when the different types are presented together. For those students, have then review the corresponding example in the Learn box that showed that particular type of problem.

Lesson 7.4-Some students may not understand what the number sentences for Exercises 2 and 3 require since they did not see this format in the Learn activity. Point out that these exercises are asking them to find out how many times one unit or quantity is than the other. Refer students to page 312 to remind them how to do this.

Lesson 7.5- Some students may forget whether they need to multiply or divide in order to find the missing terms in each ratio. Remind students that when they are finding missing terms for the set of ratios whose terms are greater, they multiply by the multiplying factor. However, when the missing terms are in the set of ratios whose terms are lesser, then they divide by the greatest common factor.

Lesson 7.6-Exercise 7 may be difficult for some students to solve if they cannot find the correct answers for b and c. Have students draw models to show each girl’s age.

- Chapter 10:

Lesson 10.1- Some students express a decimal as a percent by simply dropping the decimal point and writing a percent sign. This will result in incorrect answers when changing a tenths decimal to a percent, as in Exercises 17 and 18. Have students always append a zero to decimals that have only tenths before writing them as percents.

Lesson 10.2-Some students have difficulty solving multi-step problems such as Exercises 14,17 and 18. Be sure they understand what they need to do before changing the fractions to percents in part “a” of each problem.

Lesson 10.3- Some students may subtract one given percent from another in


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Exercises 9 and 12to find the percent of the remainder. Remind students that they need to subtract the given percents from 100% in order to find the percent of the remainder.

- Chapter 12:

Lesson 12.1- Guide students to reproduce any additional information, not depicted in the angle diagram, but required for finding the unknown angle measure.

Lesson12.2- Students may have difficulty with Exercise 7 because the figure includes an angle whose measure is greater than a straight angle. If necessary, show students how to extend one of the rays to create a straight angle so they can see that the measure of angle g is greater than 180 degrees.

Lesson 12.3- Students may have difficulty identifying vertical angles, especially when there are more than 2 intersecting lines. Suggest that students trace the figures and color code the lines so that the vertical angles are visually apparent.


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Teaching Multiple Representations

Concrete and Pictorial Representations

Area Model

x

x

Triangle (Base and Height)

Multiple Representations Framework


28

Area of a Triangle


29

Ratios

3 to 4 3:4

Bar Model 4:8 2:4 4:8 and 2:4 are equivalent ratios


30

Percent (“per hundred”)

Array

Hundreds Grid 25% shaded Number Line

Bar Model 75% shaded


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Assessment Framework

Unit 3 Assessment / Authentic Assessment Framework

Assessment

CCSS

Estimated Time

Format

Mini Assessment 5.NBT.5-7

5.NBT.5, 5.NBT.6, 5.NBT.7 25 minutes Individual

Authentic Assessment 13 5.NBT.3 25 minutes Individual

Pre Test 6 K.G.5, 2.G.1, 3.MD.5, 3.MD.5.a, 3.MD.5.b,

4.MD.3, 4.OA.3, 5.NF.4.b 40 minutes Individual

Chapter Test/Review 6 3.MD.8, 4.G.2, 5.NF.4.b,

5.NBT.7, 6.G.1 40 minutes Individual

Test Prep 6 5.NF.4.b, 6.G.1 40 minutes Individual

Authentic Assessment 14 5.NF.4.b, 5.NF.6 25 minutes Individual

Mini Assessment 5.G.1-2 5.G.1-2 15 minutes Individual

Pre Test 7 1.OA.6, 3.NF.1, 3.OA.1, 3.OA.3, 4.NF.1, 4.OA.3

40 minutes Individual

Chapter Test/Review 7 3.MD.8, 6.RP.1, 6.RP.3 40 minutes Individual

Test Prep 7 5.NF.4.a, 6.RP.1, 6.RP.3.a 40 minutes Individual

Authentic Assessment 15 5.MD.5.c 25 minutes Individual

Mini Assessment 5.MD.1 5.MD.1 10 minutes Individual

Pre Test 10 3.NF.1, 4.NF.6, 5.MD.1,

5.NBT.7 40 minutes Individual

Chapter Test/Review 10 6.RP.3.c 40 minutes Individual

Test Prep 10 6.RP.3.c 40 minutes Individual

Authentic Assessment 16 (optional)

5.MD.5 25 minutes Individual

Pre Test 12 4.G.1, 4.G.2, 4.MD.7 40 minutes Individual

Chapter Test/Review 12 4.MD.7, 7.G.5 40 minutes Individual

Test Prep 12 4.MD.7 40 minutes Individual


32

PLD Genesis Conversion

Rubric

Scoring PLD 5 100

PLD 4 89

PLD 3 79

PLD 2 69

PLD 1 59


33

Performance Tasks – Authentic Assessments Name:_______________________________________ Comparing Decimals on the Number Line

a. Which is greater, 0.1 or 0.01? Show the comparison on the number line and explain.

b. Which is greater, 0.2 or 0.03? Show the comparison on the number line and explain.

c. Which is greater, 0.12 or 0.21? Show the comparison on the number line and explain.

d. Which is greater, 0.13 or 0.031? Show the comparison on the number line and explain.

5th Grade Authentic Assessment #13 Comparing Decimals on the Number Line


34

Performance Task Scoring Rubric: Comparing Decimals on a Number Line 5.NBT.3: Read, write, and compare decimals to thousandths.

Mathematical Practices: 2 and 6

SOLUTION:

a. 0.1>0.01 because 0.1 is to the right of 0.01 on the number line.

b. 0.2>0.03because 0.2 is to the right of 0.03 on the number line.

c. 0.21>0.12because 0.21 is to the right of 0.12 on the number line.

d. 0.13>0.031because 0.13 is to the right of 0.031 on the number line.

Level 5:

Distinguished

Command

Level 4: Strong

Command Level 3: Moderate

Command Level 2: Partial

Command Level 1: No

Command

Student gives all 4 correct answers. Clearly constructs and communicates a complete response based on explanations/reasoning using the:

placement on

Student gives all 4 correct answers with a minor error in an explanation. Clearly constructs and communicates a complete response based on explanations/reasonin

Student gives 3 correct answers. Constructs and communicates a complete response based on explanations/reasoning using the:

placement on

Student gives 2 correct answers. Constructs and communicates an incomplete response based on explanations/reasoning using the:

placement on

Student gives less than 2 correct answers. The student shows no

work or

justification.


35

the number line

meaning of the < and > symbols

value of each digit

Response includes an efficient and logical progression of steps.

g using the:

placement on the number line


value of each digit

Response includes a logical progression of steps

the number line


value of each digit

Response includes a logical but incomplete progression of steps. Minor calculation errors.

the number line


value of each digit

Response includes an incomplete or Illogical progression of steps.


36

Performance Tasks – Authentic Assessments Name:_______________________________________

Solve and show all work.

Part 1:

There are two design proposals for a new rectangular park in town.

• In design one,

of the area of the park is going to be a rectangular grass area and

of the grass

area will be a rectangular soccer field.

• In design two, only

of the park is going to be a rectangular grass area and

of the grass area will

be a rectangular soccer field.

Which design (one or two) will have a bigger soccer field? Explain your answer. Draw a diagram that

can be used to compare the size of the soccer field in the two designs. Label the values

and

on the

diagram.

Part 2:

Presley and Julia are cutting 1 ft. square poster board to make a sign for the new park. Presley cut her

poster so that the length of the top and bottom are

ft and the length of the sides are

ft. Julia cut her

poster so that the lengths of the top and bottom are

ft and the length of the sides are

ft.

Draw a diagram of each poster board. Label the values on the diagram.

How are their poster boards similar and different? Justify your reasoning.

5th Grade Authentic Assessment #14 –New Park


37

Performance Task Scoring Rubric: New Park

5.NF.4.b: Apply and extend previous understandings of multiplication to multiply a fraction or whole

number by a fraction. 5.NF.6: Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by

using visual fraction models or equations to represent the problem.

Mathematical Practices: 1, 4, and 5

SOLUTION:

Part 1:

The two designs have the same size area. Students may use a variety of representations to model the

soccer fields, one of which might include a diagram like the one below.

Design 1:

Color represents the grass and blue represents the soccer field.

of

is

.

Design 2:

Color represents the grass and blue represents the soccer field.

of

is

.

Students may use equations to represent and solve the problem. Such equations would include

x

=

or

x =

They may identify the commutative property as a way to justify the soccer fields being the same size.

Part 2:

Note that the only difference between the two rectangles is that one is a rotation of the other.

In particular, they have the exact same area. Note that the area also corresponds to the same areas that

we saw in Part 1 by finding

of

and

of

.


38

Level 5:

Distinguished

Command

Level 4: Strong



Command Level 1: No

Command

Student gives all correct answers. Clearly constructs and communicates a complete response based on explanations/reasoning using the following:

Area of a rectangle

Visual fraction models

Equations Response includes an efficient and logical progression of steps.

Student gives all correct answers with a minor error in the explanation. Clearly constructs and communicates a complete response based on explanations/reasoning using the following:

Area of a rectangle


Equations Response includes a logical progression of steps

Student gives correct answers but one of the diagrams or explanations is incomplete. Constructs and communicates a complete response based on explanations/reasoning using the following:

Area of a rectangle


Equations Response includes a logical but incomplete progression of steps. Minor calculation errors.

Student gives correct answers to one part only. Constructs and communicates an incomplete response based on explanations/reasoning using the following:

Area of a rectangle


Equations Response includes an incomplete or Illogical progression of steps.

Student gives no correct answers. The student

shows no

work or

justification.


39


Use two different colored pencils or crayons to show your work.

John was finding the volume of this figure. He decided to break it apart into two separate rectangular

prisms. John found the volume of the solid below using this expression: (4 x 4 x 1) + (2 x 4 x 2).

Decompose the figure into two rectangular prisms and shade them in different colors to show one way

John might have thought about it. Briefly explain your reasoning.

Phillis also broke this solid into two rectangular prisms, but she did it differently than John. She found

the volume of the solid below using this expression: (2 x 4 x 3) + (2 x 4 x 1).

Decompose the figure into two rectangular prisms and shade them in different colors to show one way

Phillis might have thought about it. Briefly explain your reasoning.

5th Grade Authentic Assessment #15 – Breaking Apart Composite Solids


40

Performance Task Scoring Rubric: Breaking Apart Composite Solids 5.MD.5.c: Recognize volume as additive. Find volumes of solid figures composed of two non-

overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this

technique to solve real world problems.

Mathematical Practices: 7

SOLUTION:

John‟s Picture Phillis‟ Picture

Level 5:

Distinguished

Command

Level 4: Strong



Command Level 1: No

Command

All parts correct Clearly constructs and communicates a complete response based on explanations/reasoning using :

Operations of multiplication and addition and relating it to volume

The reasoning that volume is additive

Response includes an efficient and logical progression of steps.

All parts correct but explanation contains a minor error Clearly constructs and communicates a complete response based on explanations/reasoning using:



Response includes a logical progression of steps

One part is incomplete but shows logical reasoning Constructs and communicates a complete response based on explanations/reasoning using:




One part correct Constructs and communicates an incomplete response based on explanations/reasoning using:



Response includes an incomplete or Illogical progression of steps.

No parts correct The student

shows no

work or justification.


41


Solve and show all work.

A box 2 centimeters high, 3 centimeters wide, and 5 centimeters long can hold 40 grams

of clay. A second box has twice the height, three times the width, and the same length as

the first box. How many grams of clay can it hold?

5th Grade Authentic Assessment #16 – Box of Clay


42

Performance Task Scoring Rubric: Box of Clay 5.MD.5: Understand concepts of volume and relate volume to multiplication and to addition.

Mathematical Practices: 4

SOLUTION 1: Geometric Visualization

The second box has 3 times the width and the same length as the first, smaller box. So we can fit three of

the smaller boxes inside the second box to make one layer which will be 2 cm high. The second box is 2

times as high as the smaller one so we can add one more layer of three smaller boxes to fill the second

box.

This means that it takes 6 small boxes to fill the large box so the large box holds six times as much as

the small box. Since the small box holds 40 grams of clay, the large box holds

6×40=240

grams of clay.

SOLUTION 2: Arithmetic comparison of volumes

The first box is 2 centimeters high, 3 centimeters wide, and 5 centimeters long so it has volume

2cm×3cm×5cm=30 cubic centimeters

and it holds 40 grams of clay. The second box is 4 centimeters high, 9 centimeters wide, and 5

centimeters long so its volume is

4cm×9cm×5cm=180 cubic centimeters.

Since the volume of the second box is 180÷30=6 times bigger, it can hold 6 times as much clay. So the

second box can hold 6×40=240 grams of clay.

Level 5:

Distinguished

Command

Level 4: Strong



Command Level 1: No

Command

Clearly constructs and communicates a complete response based on explanations/reasoning using the operations of multiplication and addition and relating it to volume. Response includes an efficient and logical progression of steps.

Clearly constructs and communicates a complete response based on explanations/reasoning using the operations of multiplication and addition and relating it to volume. Response includes a logical progression of steps

Clearly constructs and communicates a complete response based on explanations/reasoning using the operations of multiplication and addition and relating it to volume.


Clearly constructs and communicates a complete response based on explanations/reasoning using the operations of multiplication and addition and relating it to volume. Response includes an incomplete or Illogical progression of steps.

The student shows no work or justification.


43

Additional Assessment Resources Illustrative Math: http://illustrativemathematics.org/

PARCC: http://www.parcconline.org/samples/item-task-prototypes

NJDOE: http://www.state.nj.us/education/modelcurriculum/math/ (username: model; password: curriculum) DANA: http://www.ccsstoolbox.com/parcc/PARCCPrototype_main.html New York: http://www.p12.nysed.gov/assessment/common-core-sample-questions/ Delaware: http://www.doe.k12.de.us/assessment/CCSS-comparison-docs.shtml

http://illustrativemathematics.org/

http://www.parcconline.org/samples/item-task-prototypes

http://www.state.nj.us/education/modelcurriculum/math/

http://www.ccsstoolbox.com/parcc/PARCCPrototype_main.html

http://www.p12.nysed.gov/assessment/common-core-sample-questions/

http://www.doe.k12.de.us/assessment/CCSS-comparison-docs.shtml


44

Extensions and Sources Think Central https://www-k6.thinkcentral.com/ePC/start.do Common Core Tools

http://commoncoretools.me/ http://www.ccsstoolbox.com/ http://www.achievethecore.org/steal-these-tools Achieve the Core

http://achievethecore.org/dashboard/300/search/6/1/0/1/2/3/4/5/6/7/8/9/10/11/12 Manipulatives

http://nlvm.usu.edu/en/nav/vlibrary.html http://www.explorelearning.com/index.cfm?method=cResource.dspBrowseCorrelations&v=s&id=USA-000 Problem Solving Resources Illustrative Math Project

http://illustrativemathematics.org/standards/k8 The site contains sets of tasks that illustrate the expectations of various CCSS in grades K–8 grade and high school. More tasks will be appearing over the coming weeks. Eventually the sets of tasks will include elaborated teaching tasks with detailed information about using them for instructional purposes, rubrics, and student work. Inside Mathematics http://www.insidemathematics.org/index.php/tools-for-teachers Inside Mathematics showcases multiple ways for educators to begin to transform their teaching practices. On this site, educators can find materials and tasks developed by grade level and content area. Engage NY

http://www.engageny.org/video-library?f[0]=im_field_subject%3A19

IXL http://www.ixl.com/ Georgia Department of Education

https://www.georgiastandards.org/Common-Core/Pages/Math-K-5.aspx Georgia State Educator have created common core aligned units of study to support schools as they implement the Common Core State Standards. 5th Grade: http://ccgpsmathematicsk-5.wikispaces.com/5th+Grade Formative Assessment : http://ccgpsmathematicsk-5.wikispaces.com/K-5+Formative+Assessment+Lessons+%28FALs%29 Number Talks and Multi-grade Resources: http://ccgpsmathematicsk-5.wikispaces.com/Number+Talks+and+other+Multi+Grade+Resources Newark http://www.nps.k12.nj.us/IRC/Page/8237.html

https://www-k6.thinkcentral.com/ePC/start.do

http://commoncoretools.me/

http://www.ccsstoolbox.com/

http://www.achievethecore.org/steal-these-tools

http://achievethecore.org/dashboard/300/search/6/1/0/1/2/3/4/5/6/7/8/9/10/11/12

http://nlvm.usu.edu/en/nav/vlibrary.html

http://www.explorelearning.com/index.cfm?method=cResource.dspBrowseCorrelations&v=s&id=USA-000

http://www.explorelearning.com/index.cfm?method=cResource.dspBrowseCorrelations&v=s&id=USA-000

http://commoncoretools.wordpress.com/2011/11/08/what-weve-been-working-on/

http://illustrativemathematics.org/standards/k8

http://www.insidemathematics.org/index.php/tools-for-teachers

http://www.engageny.org/video-library?f%5b0%5d=im_field_subject%3A19

http://www.ixl.com/

https://www.georgiastandards.org/Common-Core/Pages/Math-K-5.aspx

http://ccgpsmathematicsk-5.wikispaces.com/5th+Grade

http://ccgpsmathematicsk-5.wikispaces.com/K-5+Formative+Assessment+Lessons+%28FALs%29

http://ccgpsmathematicsk-5.wikispaces.com/K-5+Formative+Assessment+Lessons+%28FALs%29

http://ccgpsmathematicsk-5.wikispaces.com/Number+Talks+and+other+Multi+Grade+Resources

http://ccgpsmathematicsk-5.wikispaces.com/Number+Talks+and+other+Multi+Grade+Resources


45

NY SAMPLE QUESTIONS Grade 5: https://docs.google.com/file/d/0Byj6JhSTYWXwcjd5emNDNzF3ekE/preview Howard County 5

th Grade: https://grade5commoncoremath.wikispaces.hcpss.org/home

OHIO

http://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-

Standards/Mathematics/Grade_5_Math_Model_Curriculum_March2015.pdf.aspx

Gates Foundations Tasks http://www.gatesfoundation.org/college-ready-education/Documents/supporting-instruction-cards-math.pdf Minnesota STEM Teachers’ Center

http://www.scimathmn.org/stemtc/frameworks Massachusetts Comprehensive Assessment System

www.doe.mass.edu/mcas/search Performance Assessment Links in Math (PALM) PALM is currently being developed as an on-line, standards-based, resource bank of mathematics performance assessment tasks indexed via the National Council of Teachers of Mathematics (NCTM). http://palm.sri.com/ Mathematics Vision Project

http://www.mathematicsvisionproject.org/ NCTM http://illuminations.nctm.org/ http://www.thinkingblocks.com/

https://docs.google.com/file/d/0Byj6JhSTYWXwcjd5emNDNzF3ekE/preview

https://grade5commoncoremath.wikispaces.hcpss.org/home

http://www.gatesfoundation.org/college-ready-education/Documents/supporting-instruction-cards-math.pdf

http://www.gatesfoundation.org/college-ready-education/Documents/supporting-instruction-cards-math.pdf

http://www.doe.mass.edu/mcas/search

http://palm.sri.com/

http://www.mathematicsvisionproject.org/

http://illuminations.nctm.org/

http://www.thinkingblocks.com/