6th Grade Mathematics - Orange Board of Education · Web view6th Grade Unit 1: Ratios and Proportions September 9th – October 9th 3 Author Kelsey Marlow Created Date 09/01/2014

6th Grade MathematicsRatios and Proportions Unit 1 Curriculum Map: September 9th – October 9th

ORANGE PUBLIC SCHOOLS

OFFICE OF CURRICULUM AND INSTRUCTION

OFFICE OF MATHEMATICS

6th Grade Unit 1: Ratios and Proportions September 9th – October 9th

Table of Contents

I. Unit Overview p. 2

II. CMP Pacing Guide p. 3

III. Pacing Calendar p. 4-5

IV. Math Background p. 6

V. PARCC Assessment Evidence Statement p. 7-9

VI. Connections to Mathematical Practices p. 10

VII. Vocabulary p. 11

VIII. Potential Student Misconceptions p. 12

IX. Teaching to Multiple Representations p. 13-15

X. Unit Assessment Framework p.16

XI. Performance Tasks p.17-21

XII. Assessment Check p. 22

XIII. Summative Task p. 23

XIV. Extensions and Sources p. 24

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2


Unit OverviewIn this unit students will …

- Strengthen sense of and understanding of proportional reasoning

- Develop and use multiplicative thinking

- Develop the understanding that a ratio is a comparison of two numbers or quantities

- Find percents using the same processes for solving rates and proportions

- Solve real-life problems involving measurement units that need to be converted

Enduring Understandings

- A ratio is a number that relates two quantities or measures within a given situation in a multiplicative relationship (in contrast to a difference or additive relationship).

- Ratios can express comparisons of a part to whole, (a/b with b ≠ 0)

- Fractions are part-whole ratios, meaning fractions are also ratios. Percentages are ratios and are sometimes used to express ratios.

- Both part-to-whole and part-to-part ratios compare two measures of the same type of thing. A ratio can also be a rate.

- A rate is a comparison of the measures of two different things or quantities; the measuring unit is different for each value.

- Ratios use division to represent relations between two quantities.

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CMP Pacing Guide

Activity Common Core Standards Estimated TimeUnit Readiness Assessment (CMP3)

5.NBT.A.1, 4.NBT.A.2, 5.NF.B.3, 5.NF.B.7, 5.NBT.A.3b

1 Block

Comparing Bits and Pieces(CMP3) Investigation 1

6.RP.A.1, 6.RP.A.3, 6.RP.A.3a, 6.NS.B.4

4 Blocks

Assessment: Partner Quiz (CMP3)

6.RP.A.1, 6.RP.A.3, 6.RP.A.3a, 6.NS.B.4

½ Block

Comparing Bits and Pieces (CMP3) Investigation 2

6.RP.A.1, 6.RP.A.2, 6.RP.A.3, 6.RP.A.3b, 6.NS.B.4

2 Blocks

Assessment: Check Up 1 (CMP3)


½ Block

Performance Task 1 6.RP.A.2 ½ Block


6.NS.C.6a, 6.NS.C.6c, 6.NS.C.7b, 6.NS.C.7c

4 Blocks



½ Block


6.RP.A.1, 6.RP.A.3, 6.RP.A.3b, 6.RP.A.3c, 6.NS.B.2

2 Blocks



½ Block

Decimal Ops(CMP3) Investigation 4

6.RP.A.1, 6.RP.A.2, 6.RP.A.3c, 6.NS.B.2, 6.NS.B.3

2½ Blocks

Unit 1 Assessment 6.RP.A.1, 6.RP.A.2, 6.RP.A.3a, 6.RP.A.3b, 6.RP.A.3c

1 Block

Performance Task 2 6.RP.A.3c ½ Block

Total Time 19 ½ Blocks

4


Pacing Calendar

SEPTEMBERSunday Monday Tuesday Wednesday Thursday Friday Saturday

1Labor Day

2OPENING DAY

SUP. FORUMPD DAY

3

PD DAY

4

PD DAY

5

PD DAY

6

7 8PD DAY?

91st Day for students

10Unit 1:Ratios & ProportionsReadiness Assessment

11 12 13

14 15 16 17Assessment:Partner Quiz

18 19Performance Task 1 Due

20

21 22Assessment: Check Up 1

23 2412:30 pmDismissal for students

25 26Assessment: Check Up 2

27

28 29 30

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OCTOBERSunday Monday Tuesday Wednesday Thursday Friday Saturday

1Assessment: Check Up 3

2 3 4

5 6Assessment:Unit 1 Assessment

7 8 9Unit 1 Complete

Performance Task 2 Due

10 11

12 13 14 15 16 17 18

19 20 21 22 2312:30 pm Dismissal for students

24 25

26 27 28 29 3012:30 pm Dismissal for students

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Unit 1 Math Background6


Rational numbers are a focal point for middle school students. The goal of this unit is to help students deepen their understanding of equivalent fractions and develop this understanding as they explore ratios. Throughout the unit students will learn to compare with ratios for specific cases. This will assist them in improving their multiplicative thinking and prepare them for proportional reasoning.

During their elementary mathematics education, students were exposed to the area model for fractions. In this unit the students work with more linear models in order to extend the manner in which they reason about rational numbers, understand equivalence, as well as perform operations on rational numbers which is explored further in a later unit. These models include fraction strips, percent bars, and number lines.

Throughout the unit, students use rate tables as a way to express equivalent ratios and compute unit rates. For most of this unit, ratios are not written as fractions. The intent is to keep the notation for part–whole fractions and rational numbers apart from the notation for ratio comparisons to help develop understanding. When the word fraction appears, it is used to represent part of a whole. The learning here will help lay the foundation for the work on ratios and unit rates that will come later in the year as well as the following year. In grade 7, students will use fraction notation to express ratios and will explore ratios in more detail.

PARCC Assessment Evidence StatementsCCSS Evidence Statement Clarification Math

PracticeCalculator?

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s6.RP.1 Understand the concept of a

ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

i) Expectations for ratios in this grade are limited to ratios of non-complex fractions. (See footnote, CCSS p 42.) The initial numerator and denominator should be whole numbers.

2 No

6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b≠0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”

i i) Expectations for unit rates in this grade are limited to non-complex fractions. (See footnote, CCSS p 42.)The initial numerator and denominator should be whole numbers.

2 No

6.RP.3a Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

a. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane.

Use tables to compare ratios.

The testing interface can provide students with a calculation aid of the specified kind for these tasks.

i) Expectations for ratios in this grade are limited to ratios of non-complex fractions. (See footnote, CCSS p 42.) The initial numerator and denominator should be whole numbers.

2, 4, 5, 7, 8

Yes

6.RP.3b Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to

i) See ITN Appendix F, Table F.c, “Minimizing or avoiding common drawbacks of selected response,” specifically, Illustration 1 (in contrast to the problem

“A bird flew 20 miles in 100 minutes. At that speed, how long would it take the bird to fly 6 miles?”)

2, 8, 5 Yes

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mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

ii) The testing interface can provide students with a calculation aid of the specified kind for these tasks.

iii) Expectations for unit rates in this grade are limited to non-complex fractions. (See footnote, CCSS p 42)

iii) The initial numerator and denominator should be whole numbers.

6.RP.3c 1

Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity).

i) The testing interface can provide students with a calculation aid of the specified kind for these tasks.

ii) Pool should contain tasks with and without contexts

iii) Expectations for ratios in this grade are limited to ratios of non-complex fractions. (See footnote, CCSS

p 42.) The initial numerator and denominator should be whole numbers.

2, 7, 5, 8 Yes

6.RP.3c-2

Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

c. Solve problems involving finding the whole, given a part and the percent.

i) The testing interface can provide students with a calculation aid of the specified kind for these tasks.

ii) Expectations for ratios in this grade are limited to ratios of non-complex fractions. (See footnote, CCSS

p 42.) The initial numerator and denominator should be whole numbers.

2, 7, 5, 8 Yes

6.RP.3d Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

i) Pool should contain tasks with and without contexts

ii) Tasks require students to multiply and/or divide dimensioned quantities

iii) 50% of tasks require

2, 6, 7, 5, 8

Yes

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d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

students to correctly express the units of the result.

The testing interface can provide students with a calculation aid of the specified kind for these tasks.

iv) Expectations for ratios in this grade are limited to ratios of non-complex fractions. (See footnote, CCSS p 42.) The initial numerator and denominator should be whole numbers.

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Connections to the Mathematical Practices

1

Make sense of problems and persevere in solving them- Make sense of real-world rate and proportion problem situations by representing the

context in tactile and/or virtual manipulatives, visual, or algebraic models- Understand the problem context in order to translate them into ratios/rates

2

Reason abstractly and quantitatively- Understand the relationship between two quantities in order to express them

mathematically- Use ratio and rate notation as well as visual models and contexts to demonstrate

reasoning

3

Construct viable arguments and critique the reasoning of others- Construct and critique arguments regarding the proportion of a whole as

represented in the context of real-world situations- Construct and critique arguments regarding appropriateness of representations

given ratio and rate contexts, EX: does a tape diagram adequately represent a given ratio scenario

4Model with mathematics

- Model a problem situation symbolically (tables, expressions, or equations), visually (graphs or diagrams) and contextually to form real-world connections

5Use appropriate tools strategically

- Choose appropriate models for a given situation, including tables, expressions or equations, tape diagrams, number line models, etc.

6

Attend to precision- Use and interpret mathematical language to make sense of ratios and rates- Attend to the language of problems to determine appropriate representations and

operations for solving real-world problems.- Attend to the precision of correct decimal placement used in real-world problems

7

Look for and make use of structure- Use knowledge of problem solving structures to make sense of real world problems- Recognize patterns that exist in ratio tables, including both the additive and

multiplicative properties- Use knowledge of the structures of word problems to make sense of real-world

problems

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Look for and express regularity in repeated reasoning- Utilize repeated reasoning by applying their knowledge of ratio, rate and problem

solving structures to new contexts- Generalize the relationship between representations, understanding that all formats

represent the same ratio or rate- Demonstrate repeated reasoning when dividing fractions by fractions and connect

the inverse relationship to multiplication- Use repeated reasoning when solving real-world problems using rational numbers

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VocabularyTerm DefinitionAbsolute Value The absolute value of a number is its distance from 0 on a number line. Numbers

that are the same distance from 0 have the same absolute value. For example, −3 and 3 both have an absolute value of 3.

Equivalent Fractions

Fractions that are equal in value, but may have different numerators and

denominators. For example, 23 and

1421 are equivalent fractions. The shaded part of

this rectangle represents both 23 and

1421 .

Mixed Number A number that is written with both a whole number and a fraction. A mixed number

is the sum of the whole number and the fraction. The number 2 12 represents 2

wholes and a 12 and can be thought of as 2 +

12

Opposite Two numbers whose sum is 0. For example, −3 and 3 are opposites. On a number line, opposites are the same distance from 0 but in different directions from 0. The number 0 is its own opposite.

Percent A fraction or ratio in which the denominator is 100; a number compared to 100Proportion An equation which states that two ratios are equalRate A comparison of two quantities that have different units of measureRate Table A table that shows the value of a single item in terms of another item. It is used to

show equivalent ratios of the two items.

Ratio Compares quantities that share a fixed, multiplicative relationshipRational Number A number that can be written as a/b where a and b are integers, but b is not equal

to 0Tape Diagram A thinking tool used to visually represent a mathematical problem and transform the

words into an appropriate numerical operation. Tape diagrams are drawings that look like a segment of tape, used to illustrate number relationships. Also known as Singapore Strips, strip diagrams, bar models or graphs, fraction strips, or length models.

Unit Rate A unit rate is a rate in which the second number (usually written as the denominator) is 1, or 1 of a quantity. For example, 1.9 children per family, 32 miles

per gallon, and 3 flavors of ice cream1banana split are unit rates. Unit rates are often found by

scaling other rates.Unit Ratio Ratios written as some number to 1

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Potential Student Misconceptions

- Often there is a misunderstanding that a percent is always a natural number less than or equal to 100. Provide examples of percent amounts that are greater than 100%, and percent amounts that are less than 1%.

- Students may not distinguish between proportional situations and additive situations. Students may not realize that although they may have added to find equivalent ratios, they did not add the same amount on both sides.

- Students may still not understand the need to keep the same rate when thinking proportionally.

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Teaching Multiple Representations

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Assessment Framework

Unit 1 Assessment FrameworkAssessment CCSS Estimated

TimeFormat Graded

?Unit Readiness Assessment

(Beginning of Unit)CMP3

5.NBT.A.1, 4.NBT.A.2, 5.NF.B.3, 5.NF.B.7, 5.NBT.A.3b

1 Block Individual No

Assessment: Partner Quiz(After Investigation 1)

CMP3

6.RP.A.1, 6.RP.A.3, 6.RP.A.3a, 6.NS.B.4

½ Block Group Yes

Assessment: Check Up 1(After Investigation 2)

CMP3


½ Block Individual Yes

Assessment: Check Up 2 (After Investigation 3)

CMP3


½ Block Individual Yes

Assessment: Check Up 3(After Investigation 4)

CMP3


½ Block Individual or Group

Yes

Unit 1 Assessment(Conclusion of Unit)Model Curriculum

6.RP.A.1, 6.RP.A.2, 6.RP.A.3a, 6.RP.A.3b, 6.RP.A.3c

1 Block Individual Yes

Unit 1 Performance Assessment FrameworkAssessment CCSS Estimated

TimeFormat Graded

?Performance Task 1

(Mid-September)Mangos for Sale

6.RP.A.2 ½ Block Group Yes; Rubric

Performance Task 2(Early October)Gianna’s Job

6.RP.A.3, 6.RP.A.3a ½ Block Individual w/

Interview Opportunity

Yes: rubric

Assessment Check 1 (optional)

6.RP.A.1, 6.RP.A.3a, 6.RP.A.3b, 6.RP.A.3c,6.NS.C.6c

Teacher Discretion

Teacher Discretion

Yes, if administered

Summative Tasks(optional)

6.RP.A.1, 6.RP.A.2, 6.RP.A.3

Teacher Discretion

Teacher Discretion

Yes, if administered

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Performance Tasks

Performance Task 1:

Mangos for Sale (6.RP.A.2)

A store was selling 8 mangos for $10 at the farmers market.

Keisha said,

“That means we can write the ratio 10 : 8, or $1.25 per mango.”

Luis said,

“I thought we had to write the ratio the other way, 8 : 10, or 0.8 mangos per dollar."

Can we write different ratios for this situation? Explain why or why not.

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Solution:

Yes, this context can be modeled by both of these ratios and their associated unit rates. The context itself doesn’t determine the order of the quantities in the ratio; we choose the order depending on what we want to know.

Performance Task Scoring Rubric:

3-Point Response The response shows complete understanding of the problem’s essential mathematical concepts. The student executes procedures completely and gives relevant responses to all parts of the task. The response contains few minor errors, if any. The response contains a clear, effective explanation detailing how the problem was solved so that the reader does not need to infer how and why decisions were made.

2-Point Response The response shows nearly complete understanding of the problem’s essential mathematical concepts. The student executes nearly all procedures and gives relevant responses to most parts of the task. The response may have minor errors. The explanation detailing how the problem was solved may not be clear, causing the reader to make some inferences.

1-Point Response The response shows limited understanding of the problem’s essential mathematical concepts. The response and procedures may be incomplete and/or may contain major errors. An incomplete explanation of how the problem was solved may contribute to questions as to how and why decisions were made.

0-Point Response The response shows insufficient understanding of the problem’s essential mathematical concepts. The procedures, if any, contain major errors. There may be no explanation of the solution or the reader may not be able to understand the explanation. The reader may not be able to understand how and why decisions were made.

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Performance Task 2:

Gianna’s Job (6.RP.A.3, 6.RP.A.3a)

Gianna is paid $90 for 5 hours of work.

a. At this rate, how much would Gianna make for 8 hours of work?

b. At this rate, how long would Gianna have to work to make $60?

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Solutions:

Solution: Making a table

a. This method uses a ratio table:

Time Worked (hours) Gianna's Earnings (dollars)

5 90

10 180

20 360

40 720

8 144

b. The first row is the given information and to get to the second row we multiply both entries of the first row by 2. To get from the second to the third row of the table we multiply by 2 again. From the third to the fourth for we multiply by 2 for a third time. Now 40 hours can be divided by 5 to give 8 hours so this is the last step. There are many other possible ways to arrive at the answer with a table. For example, since

8=(85 )×5we could move from the first row to the last in one step, multiplying the first row by

85 .

c. We again make a table and this time the goal is to get $60 in the earnings column and find out how many hours it takes for Gianna to earn this amount of money. We see that 60 is not a factor of 90 so we can’t get to 60 directly by dividing by a whole number. But 60 is a factor of 180 which is 2 × 90 so we use this:

Time Worked (hours) Gianna's Earnings (dollars)

5 90

10 180

103 60

d. It takes Gianna 103 hours or 3 hours and 20 minutes to make $60.

Solution: Making a double number line21


a. We are given that Gianna makes $90 in 5 hours. We can plot this information on a double number line, with money plotted on one line and time on the other:

The goal is to use the information given to work out what dollar amount will go along with 8 hours. One way to do this would be to work out the hourly wage and then multiply by 8. This is shown below with the first step drawn in purple and the second step in blue:

To find the hourly wage we have to divide the number of given hours by 5 and so we also divide the wages by 5. Next, to find the wages for 8 hours we multiply the hourly wage by 8. There are many other alternatives. The quickest method would be to multiply the given values of money and time

by 85 .

b. To find how long Gianna has to work to make $60 notice that $60 is 23 of $90. So we can first

take one third of the given values (in purple below) and then double these new values (in blue):

It takes Gianna 103 hours or 3 and a third hours to earn $60.

Solution: Using a unit rate

a. In order to find out how much Gianna makes in 8 hours, we can first find her hourly rate and then multiply by 8. Since Gianna makes $90 in 5 hours she will make $90 ÷ 5 in 1 hour. This means that Gianna makes $18 per hour. So in 8 hours she will make 8 × $18 = $144.

b. To find out how long it takes Gianna to make $60 we can find out how long it take her to make $1 and then multiply by 60. Since Gianna makes $90 in 5 hours she will make $1 in

5 ÷ 90 hours. This is 118 of an hour. Since Gianna makes $1 in

118 of an hour she will make $60

in 6018 hours. This is three and a third hours.

Although the solutions to (a) and (b) are conceptually similar, (a) feels more natural because we use the units of dollars per hour frequently when thinking of wages. For part (b), we use the units of hours per dollar which feel less familiar

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Performance Task Scoring Rubric:

3-Point Response The response shows complete understanding of the problem’s essential mathematical concepts. The student executes procedures completely and gives relevant responses to all parts of the task. The response contains few minor errors, if any. The response contains a clear, effective explanation detailing how the problem was solved so that the reader does not need to infer how and why decisions were made.

2-Point Response The response shows nearly complete understanding of the problem’s essential mathematical concepts. The student executes nearly all procedures and gives relevant responses to most parts of the task. The response may have minor errors. The explanation detailing how the problem was solved may not be clear, causing the reader to make some inferences.

1-Point Response The response shows limited understanding of the problem’s essential mathematical concepts. The response and procedures may be incomplete and/or may contain major errors. An incomplete explanation of how the problem was solved may contribute to questions as to how and why decisions were made.

0-Point Response The response shows insufficient understanding of the problem’s essential mathematical concepts. The procedures, if any, contain major errors. There may be no explanation of the solution or the reader may not be able to understand the explanation. The reader may not be able to understand how and why decisions were made.

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Assessment Check

Assessment Check 1

1. The table below shows the 2009 population of Tennessee represented by different age groups.

Based on this information, which ratio represents the percent of the total population who were over 65 and over age group to the percent of the total population who were in the 0 to 18 age group in Tennessee in 2009?

a. 1:8 b. 1:5 c. 13: 24 d. 13:17

2. Fill in the chart comparing slices per pizza.

3. A farmer was selling corn to the market. He sold it by the ton (2000 lbs.)a. If he sold 12 tons for $3600, then how much did one pound of corn cost the market?b. After purchasing the corn, the market found that one ton of corn equaled 6000 ears

of corn averages. How many ears per pound does that compute to?

4. Jack ran 4 miles in 45 minutes. Jill ran 7 miles in 64.5 minutes. a. How many miles/hour did each person run?b. Who ran faster? Explain how you know.

5. Kendall bought a vase that was priced at $450. In addition, she had to pay 3% sales tax. How much did she pay for the vase?

6. A submarine was situated 800 feet below sea level. If it ascends 250 feet, what is the new position?

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7. One day in July, the temperature at ground level at the airport was 90°. A pilot reported the temperature at 10,000 feet was 50°. How much did the temperature drop per 1000 feet?

Summative Tasks

Summative Task 6.RP.A.1, 6.RP.A.2, 6.RP.A.3

1. John, Marie, and Will all ran for 6th grade class president. Of the 36 students, 16 voted for John, 12 for Marie, and 8 for Will. What was the ratio of votes for John to votes for Will? What was the ratio of votes for Marie to votes for Will? What was the ratio of votes for Marie to votes for John?

2. Because no one got half the votes, they had to have a run-off election. Marie dropped out and convinced all her voters to vote for Will. What is the new ratio of Will’s votes to John’s?

3. John and Will also ran for Middle School Council President. There are 90 students voting in middle school. If the ratio of Will’s votes to John’s votes remains the same as it was in part (b), how many more votes will Will get than John?

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Extensions and Sources

Online Resources

http://www.illustrativemathematics.org/standards/k8 - Performance tasks, scoring guides

http://www.ixl.com/math/grade-6 - Interactive, visually appealing fluency practice site that is objective descriptive

https://www.khanacademy.org/math/arithmetic/fractions - Interactive, tracks student points, objective descriptive videos, allows for hints

https://www.khanacademy.org/math/arithmetic/rates-and-ratios

- Interactive, tracks student points, objective descriptive videos, allows for hints

http://www.doe.k12.de.us/assessment/files/Math_Grade_6.pdf - Common Core aligned assessment questions, including Next Generation Assessment Prototypes

https://www.georgiastandards.org/Common-Core/Pages/Math-6-8.aspx- Common core assessments and tasks designed for students with special needs

http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsGRADE8_Nov2012V3_FINAL.pdf- PARCC Model Content Frameworks Grade 8

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Documents

6th Grade Mathematics - Orange Board of Education · Web view6th Grade Unit 1: Ratios and Proportions September 9th – October 9th 3 Author Kelsey Marlow Created Date 09/01/2014