The State Education Department / The University oF the
   State of New York / Albany, NY 12234

                                   Curriculum, Instruction, and Instructional Technology Team - Room 320 EB

                                   www.emsc.nysed.gov/ciai

    email: emscnysmath@mail.nysed.gov

 

 

 

Grade 6

 

Sample Tasks for PreK-8, developed by New York State teachers, are clarifications, further explaining the language and intent of the associated Performance Indicators. These tasks are not test items, nor are they meant for students' use.

 

 

 

 

 

Strands

Process

Content

 

Problem Solving

 

 

Reasoning and Proof

 

 

Communication

 

 

Connections

 

 

Representation

 

 

 

 

Number Sense and Operations

 

 

Algebra

 

 

Geometry

 

 

Measurement

 

 

Statistics and Probability

 

 

 

 

 

 

 

 

 

 

Problem Solving

 

Students will build new mathematical knowledge through problem solving.

 

6.PS.1               Know the difference between relevant and irrelevant information when solving problems

 

6.PS.1a

Using the diagram below: 

                        a.   Name the diameter of circle O.

                        b.   Name a radius of circle O.

                        c.   Name a central angle of circle O.

                        d.   Name a chord of circle O.

6.PS.1b

There are seven marbles in a bag. 1 marble is blue, 4 marbles are red and 2 marbles are green.  Kelsey draws one red marble from the bag. She does not put the red marble back into the bag.  Kelsey draws a second marble from the bag.  What is the probability the second marble will be red?

 

6.PS.2               Understand that some ways of representing a problem are more

efficient than others

 

6.PS.3               Interpret information correctly, identify the problem, and generate possible strategies and solutions

 

6.PS.3a

Adrianna is making lemonade for a picnic.  She uses 6 cups of water to one can of lemonade mix.  How many cups of water will Adrianna need for 5 cans of lemonade mix? 

 

6.PS.3b

In January, Selena was recording the low temperature of the day for Monday through Friday.  The temperatures were –11°, -25°, -19°, 0, -7°.  Have students write the temperatures in order from least to greatest. 

 

6.PS.3c

Deanna spent 3/4 of an hour reading and 1/3 of an hour playing her clarinet.  How long did Deanna spend reading and playing her clarinet? 

 

6.PS.3d

Naomi has 3/4 lb. of mixed nuts and she wants to make 1/8 lb. packages of mixed nuts.  How many 1/8 lb. packages of mixed nuts can Naomi make?

 

6.PS.3e

Jordan’s cat weighed 1 3/4 lb. when he brought her home.  If the cat now weighs 3 1/2  times that weight, how much does the cat weigh?

 

6.PS.3f

During a thunderstorm Yomara’s brother said that for every 1 mile between her and the lightening there is a 5 second gap between the lightening and the thunder. Yomara counts 30 seconds between seeing the lightening and hearing the thunder.  How far away is the lightening? 

 

6.PS.3g

Tony wants to know how much water is in the rectangular swimming pool at the park.  What tool(s) will Tony use to determine the amount of water in the park’s swimming pool?

 

6.PS.3h

Miguel is making a sundae, using one flavor of ice cream, one sauce and one topping. His choices of ice cream are chocolate, vanilla or strawberry.  His choices for sauce are chocolate or marshmallow sauce and for toppings he can choose walnuts, sprinkles, or coconut.  How many different sundaes can Miguel make?  What is the probability Miguel will choose a sundae with vanilla ice cream, chocolate sauce, and walnuts?

 

Students will solve problems that arise in mathematics and in other contexts.

 

6.PS.4            Act out or model with manipulatives activities involving mathematical content from literature

 

6.PS.5               Formulate problems and solutions from everyday situations

 

6.PS.5a

A chair in a furniture store costs $200.  It is on sale for 10% off the price.  If Ben buys the chair at the sale price, how much money will he save? 

 

6.PS.5b

Deanna spent 3/4 of an hour reading and 1/3 of an hour playing her clarinet. How long did Deanna spend reading and playing her clarinet? 

 

6.PS.5c

Naomi has 3/4 lb. of mixed nuts and she wants to make 1/8 lb. packages of mixed nuts.  How

many 1/8 lb. packages of mixed nuts can Naomi make?

 

6.PS.5d

Jordan’s cat weighed 1 3/4 lb. when he brought her home.  If the cat now weighs 3 1/2  times that weight, how much does the cat weigh?

 

6.PS.6               Translate from a picture/diagram to a numeric expression

 

6.PS.6a

Write a proportion using the information in the chart below: 

Bats

2

4

6

8

10

Balls

5

10

15

20

25

 

6.PS.7               Represent problem situations verbally, numerically, algebraically, and/or graphically

 

6.PS.7a

Monique can buy 6 oranges for 2 dollars.  How much will it cost Monique to buy 24 oranges for her class?

 

6.PS.7b

Write an equation showing 2 more than a number divided by 3 equals 5.  

(n+2) ÷ 3 = 5

 

6.PS.8               Select an appropriate representation of a problem

 

6.PS.8a

Jasmine’s age in years is three more than twice Heather’s age.  Write an algebraic expression to represent Jasmine’s age. 

 

6.PS.9               Understand the basic language of logic in mathematical situations (and, or, and not)

 

6.PS.9a

There are 20 students in Jolene’s class.  She noticed 8 students are wearing jeans, 7 students are wearing t-shirts with the school logo, and 3 students are wearing both jeans and the school t-shirt.  Construct a Venn diagram to represent the data.  How many students are wearing jeans and a t-shirt?  How many students are wearing pants other than jeans and a t-shirt without the school logo?

 

Students will apply and adapt a variety of appropriate strategies to solve problems.

 

6.PS.10             Work in collaboration with others to solve problems

 

6.PS.11             Translate from a picture/diagram to a number or symbolic expression

 

6.PS.12             Use trial and error and the process of elimination to solve problems

 

6.PS.13             Model problems with pictures/diagrams or physical objects

 

6.PS.10a

Give pairs of students several rectangular prisms.  One student measures and records the length, width, and height of each prism to the nearest centimeter.  The other student calculates and records the volume.   

 

6.PS.14 Analyze problems by observing patterns

 

6.PS.14a

Ms. Richards is training for the Community Walkathon by increasing the number of minutes she walks each day by 5 minutes.  On the first day she walks for 20 minutes.  On the second day she walks for 25 minutes and on the third day for 30 minutes.  How many minutes will Ms. Richards walk on the 7th day?  Show the work you did to decide the number of minutes Ms Richards walked on the 7th day.  

 

Students will monitor and reflect on the process of mathematical problem solving

 

6.PS.16             Discuss with peers to understand a problem situation

 

6.PS.17             Determine what information is needed to solve problem

 

6.PS.17a

The high temperatures for the first week in March were 36º, 37º, 38º, 32º, 35º, and 39º Fahrenheit.  What is the range of the temperatures? 

 

6.PS.18             Determine the efficiency of different representations of a problem

 

6.PS.19             Differentiate between valid and invalid approaches

 

6.PS.20             Understand valid counterexamples

 

6.PS.21             Explain the methods and reasoning behind the problem solving strategies used

 

6.PS.22             Discuss whether a solution is reasonable in the context of the original problem

 

6.PS.22a

A circle has a diameter of 7.5 centimeters and Julian estimates the circumference to be 22 square centimeters.  Is this a reasonable estimate?  Explain why. 

 

6.PS.23             Verify results of a problem

 

6.PS.23a

Determine whether or not each pair of ratios below is a proportion using the product of the means equals the product of the extremes. 

  a.   5/6 = 30/36

  b.   6/8 = 9/15

  c.   4/3 = 12/9

  d.   8/5 = 11/7

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Reasoning and Proof

 

Students will recognize reasoning and proof as fundamental aspects of mathematics.

 

6.RP.1              Recognize that mathematical ideas can be supported using a variety of strategies
 

6.RP.2              Understand that mathematical statements can be supported, using models, facts, and relationships to explain their thinking
 

Students will make and investigate mathematical conjectures.

 

6.RP.3               Investigate conjectures, using arguments and appropriate mathematical terms

 

6.RP.4              Make and evaluate conjectures, using a variety of strategies

 

6.RP.4a

Using the diagram below, find the area of the sector of the circle (shaded) and explain how you determined the area of the sector, 90/360  = 1/4  = .25

 

Students will develop and evaluate mathematical arguments and proofs.

 

6.RP.5              Justify general claims or conjectures, using manipulatives, models, expressions,    and mathematical relationships

 

6.RP.6              Develop and explain an argument verbally, numerically, algebraically, and/or graphically

 

6.RP.6a

Find the circumference and area of a circle with a radius of 2 ft.  Explain the difference between the circumference of a circle and the area of a circle. 

 

6.RP.7              Verify claims other students make, using examples and counterexamples when  appropriate
 

Students will select and use various types of reasoning and methods of proof.

 

6.RP.8                  Support an argument through examples/counterexamples and special cases

 

6.RP.9                  Devise ways to verify results

 

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Communication

 

Students will organize and consolidate their mathematical thinking through communication.

 

6.CM.1             Provide an organized thought process that is correct, complete, coherent, and clear

 

6.CM.1a

Simone was confused about a question on a math test that used the associative property of multiplication.  Explain the associative property of multiplication to Simone.  Be sure to include examples in your explanation.

 

6.CM.1b  

Is 3/4  the reciprocal of 4/8?  Explain your answer.

 

6.CM.1c

Ryan thinks 24 is the same as 4 x 4 .  Do you agree or disagree?  Explain your answer.

 

6.CM.1d

Explain how to evaluate the equation n² = (4² + 3²) if n = 5. 

 

6.CM.1e

Solve for n: 2n - 3 = 5.   Explain the steps that you used to solve the equation. 

 

6.CM.1f

Candy and Thomas are organizing the science club’s camping trip.  Ms. Moore, the club leader, says they will need2 gallons of milk.  Candy brought 6 quarts of milk.  Did Candy bring enough milk?  Explain your answer.

 

6.CM.1g

Estimate the volume of a rectangular prism that is 14.8cm long, 9.2 cm wide and 5.6 cm high.  Explain the process that you used to find your estimate. 

 

6.CM.1h

The Student Council at Pine Ridge Middle School wants to know what sixth grade students would like to do on sixth grade Activity Day.  They ask every 10th sixth grade student entering the school building if they would prefer a swimming party or a trip to an amusement park.  Does this represent a random sample?  Explain your thinking. 

 

6.CM.1i

Nijea has two spinners. Spinner A is divided into 3 equal parts and labeled 1, 2, 3.  Spinner B is divided into 4 equal parts and labeled 4, 5, 6, 7.  List all possible outcomes if you spin spinner A then spinner B. 

 

6.CM.2             Explain a rationale for strategy selection

 

6.CM.3             Organize and accurately label work

 

6.CM.3a

On a coordinate plane plot and label the following points:

A (-2,3)

B (3,3)

C (3,-4)

D (-2,-4)

Connect the points in the order A, B, C, D, A and find the area of ABCD? 

 

6.CM.3b

Divide the class into small groups and have them design a survey about favorites.  Each group must choose a category of favorites, such as favorite TV show, favorite lunch or favorite sport.  Each group must select five possible choices in their category and survey a group of 30 - 50 students.  Have students record the data collected in a frequency table.  The frequency table should contain three columns: choices, tallies, number telling the frequency of the choice.

 

Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

 

6.CM.4             Share organized mathematical ideas through the manipulation of objects, numerical tables, drawings, pictures, charts, graphs, tables, diagrams, models, and symbols in written and verbal form

 

6.CM.4a

Solve the percent problems below: 

What is 25% of 72?

____ is 3% of 156

 

6.CM.4b

Write each percent below as a decimal and fraction in lowest terms.

a. 23%

b. 98%

c. 7%

d. 45%

 

6.CM.4c

Evaluate each expression.

a. (4 + 5)2  
b. 23 = 8 - 3  
c. 32 + 2 x 7 + 3  

 

6.CM.4d

Divide the class into small groups and have them design a survey about favorites.   Each group must choose a category of favorites, such as favorite TV show, favorite lunch or favorite sport.  Each group must select five possible choices in their category and survey a group of 30 - 50 students.  Have students record the data collected in a frequency table.  The frequency table should contain three columns: choices, tallies, number telling the frequency of the choice.

 

6.CM.5             Answer clarifying questions from others

 

Students will analyze and evaluate the mathematical thinking and strategies of others.

 

6.CM.6             Understand mathematical solutions shared by other students

 

6.CM.7             Raise questions that elicit, extend, or challenge others' thinking

 

6.CM.8             Consider strategies used and solutions found by others in relation to their own work

 

6.CM.8a

Discuss the following question with the class: 

Without a measuring cup how can you measure a cup of water? 

List all possible ways that are shared and ask students which method would work more effectively.

 

Students will use the language of mathematics to express mathematical ideas precisely.

 

6.CM.9             Increase their use of mathematical vocabulary and language when communicating with others

 

6.CM.9a

Define the identity property of addition.  Write two equations representing the identity property of addition.  Use decimals and integers in the equations. 

 

6.CM.9b

Ashton says his car gets 22 miles to a gallon of gas.  Is Ashton discussing a ratio or a rate?  Explain your answer. 

 

6.CM.10           Use appropriate vocabulary when describing objects, relationships, mathematical solutions, and rationale

 

6.CM.10a

Rajad wrote the equation 2 (4 + 1/2) = 2(4) + 2(1/2) on the board.  Which property of multiplication does the equation demonstrate?  Explain your answer. 

 

6.CM.11              Decode and comprehend mathematical visuals and symbols to construct meaning

 

6.CM.11a

Simplify the expressions below. 

a.  (1/2)      

b.  (6.75)3    

 

6.CM.11b

At Washington Middle School 100 sixth grade students were surveyed about their favorite academic subject. The results are shown in the circle graph below.  How many students chose math as their favorite subject?

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Connections

 

Students will recognize and use connections among mathematical ideas.

 

6.CN.1              Understand and make connections and conjectures in their everyday experiences to mathematical ideas

 

6.CN.1a

If a dog dish can hold a liter of water, how many times do you need to fill and pour a 250 ml cup of water into the dog dish to fill it?  Show the process that you used to find your answer.

 

6.CN.2             Explore and explain the relationship between mathematical ideas

 

6.CN.2a

Clark’s bedroom is 12 ft. long and 9 ft. wide.  Clark’s mother is buying new carpet for his bedroom.  Using the formula A = l x w, determine how much carpet Clark’s mother will need to buy in order to carpet his entire floor. 

 

6.CN.2b

Provide pairs of students a worksheet containing circles with a diameter and a radius drawn on each circle.  Have students take turns measuring and recording the length of diameter and the radius of each circle. Have students write a statement telling what they notice about the relationship between the length of the diameter and the radius of each circle.  

 

6.CN.2c

Candy and Thomas are organizing the science club’s camping trip.  Ms. Moore, the club leader, says they will need 2 gallons of milk.  Candy brought 6 quarts of milk.  Did Candy bring enough milk?  Explain your answer.

 

6.CN.3              Connect and apply mathematical information to solve problems

 

6.CN.3a

James has 76 model cars in his collection.  He wants to give 10 % of his collection to his brother.  How many cars will James give to his brother? 

 

6.CN.3b

In the diagram below, ABC is similar to DEF.  Find the length of DF and EF in centimeters.

 

6.CN.3c

Which polygon has a greater area: a triangle with a height of 10 cm and a base of 5 cm or a square with a side of 5 cm?  Show all your work and explain your answer.  

 

6.CN.3d

Tim’s garden is 6 ft. long and 4 ft. wide.  Find the area of Tim’s garden. 

 

6.CN.3e

Find the circumference and area of a circle with a radius of 2 ft.  Explain the difference between the circumference of a circle and the area of a circle. 

 

6.CN.3f

Using the diagram below, find the area of the sector of the circle (shaded) and explain how you determined the area of the sector, 90/360  = 1/4  = .25

 

6.CN.3g

On a piece of grid paper with the x and y axes marked, have students plot and label the following points:

A (3,2)

B(-6,1)

C (9, -3)

D (-5,2)

E (-2,-4)

F ( 4,-2)    

 

6.CN.3h

Provide students the following lists of units of capacity:  cup, gallon, liter, pint, quart and milliliter.  Ask students to identify the units that are metric units and identify two items that can be measured using the metric units. 

 

6.CN.3i

Randi’s math quiz scores are 77, 95, 79, 90, 81, 90 and 83.  Find the median, mode and mean for Randi’s quiz scores.

 

6.CN.3j

The high temperatures for the first week in March were 36º, 37º, 38º, 32º, 35º, and 39º Fahrenheit.  What is the range of the temperatures? 

 

Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole.

 

6.CN.4              Understand multiple representations and how they are related

 

6.CN.4a

Change each fraction below to a decimal.  Identify if the decimal is a terminating decimal or a repeating decimal.

  a. 1/5

  b. 5/8

  c. 4/11

  d. 13/20

 

6.CN.4b

Provide students the following lists of units of capacity:  cup, gallon, liter, pint, quart and milliliter.  Ask students to identify the units that are metric units and identify two items that can be measured using the metric units. 

 

6.CN.5              Model situations with objects and representations and be able to draw conclusions

 

6.CN.5a

Divide the class into groups.  Provide each group of students with a piece of string and a ruler to measure the diameter and circumference of a variety of objects (e.g., soup can, coffee lid) to the nearest millimeter.  Have students record the results and divide each circumference by its diameter.  Have the students look for a pattern and draw conclusions.  Then discuss the meaning of pi

 

Students will recognize and apply mathematics in contexts outside of mathematics.

 

6.CN.6              Recognize and provide examples of the presence of mathematics in their daily lives

 

6.CN.6a

Provide a list of customary units of capacity.  Have the class identify items that can be measured in these units (cups, pints, quarts, gallons).

 

6.CN.7            Apply mathematics to problem situations that develop outside of mathematics

 

6.CN.8            Investigate the presence of mathematics in careers and areas of interest

 

6.CN.9            Recognize and apply mathematics to other disciplines and areas of interest

 

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Representation

 

Students will create and use representations to organize, record, and communicate mathematical ideas.

 

6.R.1                 Use physical objects, drawings, charts, tables, graphs, symbols, equations, or objects created using technology as representations

 

6.R.1a

Write two equations that represent the zero property of multiplication. 

 

6.R.1b

Draw a number line from -10 to +10.  Plot the following integers on the number line.

+2, -7, -3, +8, -4, +1, 0, +6.

 

6.R.1c

Have students divide into pairs.  One student uses wooden cubes to build a rectangular prism.  The other student counts the length, the width and the height to calculate the volume of the prism.  They record the information on a class chart. Have students change roles several times.  Bring the class together to review results and look for patterns.  Discuss the development of the formula   V = l x w h to calculate the volume of a rectangular prism. 

 

6.R.2               Explain, describe, and defend mathematical ideas using representations

 

6.R.3               Read, interpret, and extend external models

 

6.R.4               Use standard and nonstandard representations with accuracy and detail

 

6.R.4a

In 2004 the population of New York State was eighteen million, nine hundred seventy-six thousand, four hundred fifty-seven. Write this number in standard form.


6.R.4b

Find the absolute value of each integer below: 

 

6.R.4c

Write each expression below in exponential form. 

a. 6 x 6 x 6 =

b. 7 x 7 x 7 x 7 x 7 =

c. 5 x 5 =

 

Students will select, apply, and translate among mathematical representations to solve problems.

 

6.R.5                 Use representations to explore problem situations

 

6.R.5a

If a dog dish can hold a liter of water, how many times do you need to fill and pour a 250 ml cup of water into the dog dish to fill it?  Show the process that you used to find your answer. 

 

6.R.6                 Investigate relationships between different representations and their impact on a given problem

 

6.R.6a

Change each fraction below to a decimal.  Identify if the decimal is a terminating decimal or a repeating decimal.

  a. 1/5

  b. 5/8

  c. 4/11

  d. 13/20

 

6.R.6b

The chart below shows the number of bottles each 6th grade class has collected for the recycling drive.  What would be the best way to display the data? Show your choice and explain your answer.

Number of Bottles Collected in a Week

Class        # of bottles

6A                75

6C              103

6K                87

6L                38

6P               110

6T                63

 

Students will use representations to model and interpret physical, social, and mathematical phenomena.

 

6.R.7                 Use mathematics to show and understand physical phenomena (e.g., determine the perimeter of a bulletin board)

 

 

6.R.7a

Clark’s bedroom is 12 ft. long and 9 ft. wide.  Clark’s mother is buying new carpet for his bedroom.  Using the formula A = l x w, determine how much carpet Clark’s mother will need to buy in order to carpet his entire floor. 

 

6.R.8                 Use mathematics to show and understand social phenomena (e.g., construct tables to organize data showing book sales)

 

6.R.8a

Adrianna is making lemonade for a picnic.  She uses 6 cups of water to one can of lemonade mix.  How many cups of water will Adrianna need for 5 cans of lemonade mix? 

 

6.R.8b

Naomi has 3/4 lb. of mixed nuts and she wants to make 1/8 lb. packages of mixed nuts.  How many 1/8 lb. packages  of mixed nuts can Naomi make?

 

6.R.9                 Use mathematics to show and understand mathematical phenomena (e.g., Find the missing value: (3 + 4) + 5 = 3  + (4 + ___ )

 

6.R.9a

Define the identity property of addition.  Write two equations representing the identity property of addition.  Use decimals and integers in the equations.

 

6.R.9b

Determine whether or not each pair of ratios below is a proportion using the product of the means equals the product of the extremes. 

  a.   5/6 = 30/36

  b.   6/8 = 9/15

  c.   4/3 = 12/9

d.      8/5 = 11/7

 

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 Number Sense and Operations Strand

 

Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems.

 

Number Systems

 

6.N.1                 Read and write whole numbers to trillions

 

6.N.1a 

In 2004 the population of New York State was eighteen million, nine hundred seventy-six thousand, four hundred fifty-seven. Write this number in standard form.

 

6.N.2                 Define and identify the commutative and associative properties of   addition and multiplication

 

6.N.2a 

Simone was confused about a question on a math test that used the associative property of multiplication.  Explain the associative property of multiplication to Simone.  Be sure to include examples in your explanation. 

 

6.N.3                 Define and identify the distributive property of multiplication over addition

 

6.N.3a 

Rajad wrote the equation 2 (4 + 1/2) = 2(4) + 2(1/2) on the board.  Which property of multiplication does the equation demonstrate?  Explain your answer. 

 

6.N.4                 Define and identify the identity and inverse properties of addition and multiplication

 

6.N.4a 

Define the identity property of addition.  Write two equations representing the identity property of addition.  Use decimals and integers in the equations. 

 

6.N.5                 Define and identify the zero property of multiplication

 

6.N.5a 

 Write two equations that represent the zero property of multiplication. 

 

6.N.6                 Understand the concept of rate

 

6.N.6a 

Adrianna is making lemonade for a picnic.  She uses 6 cups of water to one can of lemonade mix.  How many cups of water will Adrianna need for 5 cans of lemonade mix? 

 

6.N.7                 Express equivalent ratios as a proportion

 

6.N.7a 

Write a proportion using the information in the chart below: 

Bats

2

4

6

8

10

Balls

5

10

15

20

25

 

6.N.8                 Distinguish the difference between rate and ratio

 

6.N.8a 

Ashton says his car gets 22 miles to a gallon of gas.  Is Ashton discussing a ratio or a rate?  Explain your answer. 

 

6.N.9                 Solve proportions using equivalent fractions

 

6.N.9a 

Monique can buy 6 oranges for 2 dollars.  How much will it cost Monique to buy 24 oranges for her class?

 

6.N.10               Verify the proportionality using the product of the means equals the product of the extremes

 

6.N.10a 

Determine whether or not each pair of ratios below is a proportion using the product of the means equals the product of the extremes. 

  a.   5/6 = 30/36

  b.   6/8 = 9/15

  c.   4/3 = 12/9

  d.   8/5 = 11/7

 

6.N.11               Read, write, and identify percents of a whole (0% to 100%)

 

6.N.11a 

Solve the percent problems below: 

____ is 3% of 156

 

6.N.12               Solve percent problems involving percent, rate, and base

 

6.N.12a 

A chair in a furniture store costs $200.  It is on sale for 10% off the price.  If Ben buys the chair at the sale price, how much money will he save? 

 

6.N.13   Define absolute value and determine the absolute value of rational numbers (including positive and negative)

 

6.N.13a 

Find the absolute value of each integer below: 

 

 

6.N.14               Locate rational numbers on a number line (including positive and negative)

 

6.N.14a 

Draw a number line from -10 to +10.  Plot the following integers on the number line.

+2, -7, -3, +8, -4, +1, 0, +6.

 

6.N.15               Order rational numbers (including positive and negative)

 

6.N.15a 

In January, Selena was recording the low temperature of the day for Monday through Friday.  The temperatures were –11°, -25°, -19°, 0, -7°.  Have students write the temperatures in order from least to greatest. 

 

Students will understand meanings of operations and procedures, and how they relate to one another.

 

Operations

 

6.N.16               Add and subtract fractions with unlike denominators

 

6.N.16a 

Deanna spent  3/4 of an hour reading and 1/3 of an hour playing her clarinet.  How long did Deanna spend reading and playing her clarinet? 

6.N.17               Multiply and divide fractions with unlike denominators

 

6.N.17a 

 Naomi has 3/4 lb. of mixed nuts and she wants to make 1/8 lb. packages of mixed nuts.  How many 1/8 lb. packages of mixed nuts can Naomi make?

 

6.N.18               Multiply and divide mixed numbers with unlike denominators

 

6.N.18a 

 Jordan’s cat weighed 1 3/4 lb. when he brought her home.  If the cat now weighs 3 1/2  times that weight, how much does the cat weigh?

 

6.N.19               Identify the multiplicative inverse (reciprocal) of a number

 

6.N.19a 

 Is 3/4  the reciprocal of 4/8?  Explain your answer.

 

6.N.20               Represent fractions as terminating or repeating decimals

 

6.N.20a 

Change each fraction below to a decimal. Identify if the decimal is a terminating decimal or a repeating decimal.

  a. 1/5

  b. 5/8

  c. 4/11

  d. 13/20

 

6.N.21               Find multiple representations of rational numbers (fractions, decimals, and percents 0 to 100)

 

6.N.21a 

Write each percent below as a decimal and fraction in lowest terms.

a. 23%

b. 98%

c. 7%

d. 45%

 

6.N.22               Evaluate numerical expressions using order of operations (may include exponents of two and three)

 

6.N.22a 

Evaluate each expression.

a. (4 + 5)2  
b. 23 = 8 - 3  
c. 32 + 2 x 7 + 3  

 

6.N.23               Represent repeated multiplication in exponential form

 

6.N.23a 

Write each expression below in exponential form. 

a. 6 x 6 x 6 =

b. 7 x 7 x 7 x 7 x 7 =

c. 5 x 5 =

 

6.N.24               Represent exponential form as repeated multiplication

 

6.N.24a 

Ryan thinks 24 is the same as 4 x 4 .  Do you agree or disagree?  Explain your answer.

 

6.N.25             Evaluate expressions having exponents where the power is an exponent of one, two, or 

                         three

 

6.N.25a 

Simplify the expressions below. 

a.  (1/2)      

b.  (6.75)3    

 

Students will compute accurately and make reasonable estimates.

 

Estimation

 

6.N.26               Estimate a percent of quantity (0% to 100%)

 

6.N.26a 

James has 76 model cars in his collection.  He wants to give 10 % of his collection to his brother.  How many cars will James give to his brother? 

 

6.N.27               Justify the reasonableness of answers using estimation (including rounding)

 

6.N.27a 

Jillian divided 325 by 5 and got a quotient of 65. How can you use rounding to verify this quotient? 

 

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Algebra Strand

 

Students will represent and analyze algebraically a wide variety of problem solving situations.

 

Variables and Expressions            

              

6.A.1                 Translate two-step verbal expressions into algebraic expressions

 

6.A.1a 

Jasmine’s age in years is three more than twice Heather’s age.  Write an algebraic expression to represent Jasmine’s age. 

 

Students will perform algebraic procedures accurately.

 

Variables and Expressions

 

6.A.2                 Use substitution to evaluate algebraic expressions (may include exponents of one, two and three)

 

6.A.2a 

Explain how to evaluate the equation n² = (4² + 3²) if n = 5. 

 

6.A.3                 Translate two-step verbal equations into algebraic equations

 

6.A.3a 

Write an equation showing 2 more than a number divided by 3 equals 5.   (n+2) ÷ 3 = 5

 

6.A.4      Solve and explain two-step equations involving whole numbers using inverse operations

 

6.A.4a 

Solve for n: 2n - 3 = 5.   Explain the steps that you used to solve the equation. 

 

6.A.5                 Solve simple proportions within context

 

6.A.5a 

During a thunderstorm Yomara’s brother said that for every 1 mile between her and the lightening there is a 5 second gap between the lightening and the thunder. Yomara counts 30 seconds between seeing the lightening and hearing the thunder.  How far away is the lightening? 

 

6.A.6                 Evaluate formulas for given input values (circumference, area, volume, distance, temperature, interest, etc.)

 

6.A.6a 

Clark’s bedroom is 12 ft. long and 9 ft. wide.  Clark’s mother is buying new carpet for his bedroom.  Using the formula A = l x w, determine how much carpet Clark’s mother will need to buy in order to carpet his entire floor. 

 

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Geometry Strand

 

Students will use visualization and spatial reasoning to analyze characteristics and properties

of geometric shapes.

 

Shapes

 

6.G.1                 Calculate the length of corresponding sides of similar triangles, using proportional reasoning

 

6.G.1a 

In the diagram below, ABC is similar to DEF.  Find the length of DF and EF in centimeters.

 

6.G.2                 Determine the area of triangles and quadrilaterals (squares, rectangles, rhombi, and trapezoids) and develop formulas

 

6.G.2a 

Which polygon has a greater area: a triangle with a height of 10 cm and a base of 5 cm or a square with a side of 5 cm?  Show all your work and explain your answer.  

 

6.G.3                 Use a variety of strategies to find the area of regular and irregular polygons

 

6.G.3a 

Tim’s garden is 6 ft. long and 4 ft. wide.  Find the area of Tim’s garden. 

6.G.4                 Determine the volume of rectangular prisms by counting cubes and develop the formula

 

6.G.4a 

Have students divide into pairs.  One student uses wooden cubes to build a rectangular prism.  The other student counts the length, the width and the height to calculate the volume of the prism.  They record the information on a class chart. Have students change roles several times.  Bring the class together to review results and look for patterns.  Discuss the development of the formula   V = l x w h to calculate the volume of a rectangular prism. 

 

6.G.5                 Identify radius, diameter, chords and central angles of a circle

 

6.G.5a 

Using the diagram below: 

a.   Name the diameter of circle O.

b.   Name a radius of circle O.

c.   Name a central angle of circle O.

d.   Name a chord of circle O.

 

6.G.6                 Understand the relationship between the diameter and radius of a circle

 

6.G.6a 

Provide pairs of students a worksheet containing circles with a diameter and a radius drawn on each circle.  Have students take turns measuring and recording the length of diameter and the radius of each circle. Have students write a statement telling what they notice about the relationship between the length of the diameter and the radius of each circle.  

 

6.G.7                 Determine the area and circumference of a circle, using the appropriate formula

 

6.G.7a 

Find the circumference and area of a circle with a radius of 2 ft.  Explain the difference between the circumference of a circle and the area of a circle. 

 

6.G.8                 Calculate the area of a sector of a circle, given the measure of a central angle and the radius of the circle

 

6.G.8a 

Using the diagram below, find the area of the sector of the circle (shaded) and explain how you determined the area of the sector, 90/360  = 1/4  = .25

6.G.9     Understand the relationship between the circumference and the diameter of a circle

 

6.G.9a 

Divide the class into groups.  Provide each group of students with a piece of string and a ruler to measure the diameter and circumference of a variety of objects (e.g., soup can, coffee lid) to the nearest millimeter.  Have students record the results and divide each circumference by its diameter.  Have the students look for a pattern and draw conclusions.  Then discuss the meaning of pi

 

Students will apply coordinate geometry to analyze problem-solving situations.

 

Coordinate Geometry

 

6.G.10              Identify and plot points in all four quadrants

 

6.G.10a 

On a piece of grid paper with the x and y axes marked, have students plot and label the following points:

A (3,2)

B(-6,1)

C (9, -3)

D (-5,2)

E (-2,-4)

F ( 4,-2)    

 

6.G.11              Calculate the area of basic polygons drawn on a coordinate plane (rectangles and shapes composed of rectangles having sides with integer lengths)

 

6.G.11a 

On a coordinate plane plot and label the following points:

A (-2,3)

B (3,3)

C (3,-4)

D (-2,-4)

Connect the points in the order A, B, C, D, A and find the area of ABCD

 

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Measurement Strand

 

Students will determine what can be measured and how, using appropriate methods and formulas.

 

Units of Measurement

 

6.M.1                Measure capacity and calculate volume of a rectangular prism

 

6.M.1a 

Give pairs of students several rectangular prisms.  One student measures and records the length, width, and height of each prism to the nearest centimeter.  The other student calculates and records the volume.   

 

6.M.2                Identify customary units of capacity (cups, pints, quarts, and gallons)

 

6.M.2a 

Provide a list of customary units of capacity.  Have the class identify items that can be measured in these units (cups, pints, quarts, gallons).

 

6.M.3     Identify equivalent customary units of capacity (cups to pints, pints to quarts, and quarts to gallons)

 

6.M.3a 

Candy and Thomas are organizing the science club’s camping trip.  Ms. Moore, the club leader, says they will need 2 gallons of milk.  Candy brought 6 quarts of milk.  Did Candy bring enough milk?  Explain your answer.

 

6.M.4                Identify metric units of capacity (liter and milliliter)

 

6.M.4a 

Provide students the following lists of units of capacity:  cup, gallon, liter, pint, quart and milliliter.  Ask students to identify the units that are metric units and identify two items that can be measured using the metric units. 

 

6.M.5     Identify equivalent metric units of capacity (milliliter to liter and liter to milliliter)

 

6.M.5a 

If a dog dish can hold a liter of water, how many times do you need to fill and pour a 250 ml cup of water into the dog dish to fill it?  Show the process that you used to find your answer. 

 

Tools and Methods

 

6.M.6     Determine the tool and technique to measure with an appropriate level of precision: capacity

 

6.M.6a 

Tony wants to know how much water is in the rectangular swimming pool at the park.  What tool(s) will Tony use to determine the amount of water in the park’s swimming pool?

 

Students will develop strategies for estimating measurements.

 

Estimation

 

6.M.7     Estimate volume, area, and circumference (see figures identified in geometry strand)

 

6.M.7a 

Estimate the volume of a rectangular prism that is 14.8cm long, 9.2 cm wide and 5.6 cm high.  Explain the process that you used to find your estimate. 

 

6.M.8                Justify the reasonableness of estimates

 

6.M.8a 

A circle has a diameter of 7.5 centimeters and Julian estimates the circumference to be 22 square centimeters.  Is this a reasonable estimate?  Explain why. 

 

6.M.9                Determine personal references for capacity

 

6.M.9a 

Discuss the following question with the class: 

Without a measuring cup how can you measure a cup of water? 

List all possible ways that are shared and ask students which method would work more effectively.

 

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 Statistics and Probability Strand

  

Students will collect, organize, display, and analyze data.

 

Collection of Data

 

6.S.1                  Develop the concept of sampling when collecting data from a population and decide the best method to collect data for a particular question

 

6.S.1a 

The Student Council at Pine Ridge Middle School wants to know what sixth grade students would like to do on sixth grade Activity Day.  They ask every 10th sixth grade student entering the school building if they would prefer a swimming party or a trip to an amusement park.  Does this represent a random sample?  Explain your thinking. 

 

Organization and Display of Data

 

6.S.2                  Record data in a frequency table

 

6.S.2a 

Divide the class into small groups and have them design a survey about favorites.   Each group must choose a category of favorites, such as favorite TV show, favorite lunch or favorite sport.  Each group must select five possible choices in their category and survey a group of 30 - 50 students.  Have students record the data collected in a frequency table.  The frequency table should contain three columns: choices, tallies, number telling the frequency of the choice.

 

6.S.3                  Construct Venn diagrams to sort data

 

6.S.3a 

There are 20 students in Jolene’s class.  She noticed 8 students are wearing jeans, 7 students are wearing t-shirts with the school logo, and 3 students are wearing both jeans and the school t-shirt.  Construct a Venn diagram to represent the data.  How many students are wearing jeans and a t-shirt?  How many students are wearing pants other than jeans and a t-shirt without the school logo?

 

6.S.4                  Determine and justify the most appropriate graph to display a given set of data (pictograph, bar graph, line graph, histogram, or circle graph)

 

6.S.4a 

The chart below shows the number of bottles each 6th grade class has collected for the recycling drive.  What would be the best way to display the data? Show your choice and explain your answer.

Number of Bottles Collected in a Week

Class        # of bottles

6A                75

6C              103

6K                87

6L                38

6P               110

6T                63

 

6.S.5                  Determine the mean, mode and median for a given set of data

 

6.S.5a 

Randi’s math quiz scores are 77, 95, 79, 90, 81, 90 and 83.  Find the median, mode and mean for Randi’s quiz scores. 

 

6.S.6                  Determine the range for a given set of data

 

6.S.6a 

The high temperatures for the first week in March were 36º, 37º, 38º, 32º, 35º, and 39º Fahrenheit.  What is the range of the temperatures? 

 

6.S.7                  Read and interpret graphs

 

6.S.7a 

At Washington Middle School 100 sixth grade students were surveyed about their favorite academic subject. The results are shown in the circle graph below.  How many students chose math as their favorite subject?

Students will make predictions that are based upon data analysis.

 

Predictions from Data

 

6.S.8                  Justify predictions made from data 

 

6.S.8a 

Ms. Richards is training for the Community Walkathon by increasing the number of minutes she walks each day by 5 minutes.  On the first day she walks for 20 minutes.  On the second day she walks for 25 minutes and on the third day for 30 minutes.  How many minutes will Ms. Richards walk on the 7th day?  Show the work you did to decide the number of minutes Ms Richards walked on the 7th day.  

 

Students will understand and apply concepts of probability.

 

Probability     

 

6.S.9                  List possible outcomes for compound events

 

6.S.9a 

Nijea has two spinners. Spinner A is divided into 3 equal parts and labeled 1, 2, 3.  Spinner B is divided into 4 equal parts and labeled 4, 5, 6, 7.  List all possible outcomes if you spin spinner A then spinner B. 

 

6.S.10               Determine the probability of dependent events

6.S.10a 

There are seven marbles in a bag. 1 marble is blue, 4 marbles are red and 2 marbles are green.  Kelsey draws one red marble from the bag. She does not put the red marble back into the bag.  Kelsey draws a second marble from the bag.  What is the probability the second marble will be red?

 

6.S.11               Determine the number of possible outcomes for a compound event by using the fundamental counting and use this to determine the probabilities of events when the outcomes have equal probability.

 

6.S.11a 

Miguel is making a sundae, using one flavor of ice cream, one sauce and one topping. His choices of ice cream are chocolate, vanilla or strawberry.  His choices for sauce are chocolate or marshmallow sauce and for toppings he can choose walnuts, sprinkles, or coconut.  How many different sundaes can Miguel make?  What is the probability Miguel will choose a sundae with vanilla ice cream, chocolate sauce, and walnuts?

 

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