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 5

 

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.

 

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

 

5.PS.2               Understand that some ways of representing a problem are more efficient than others

 

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

 

5.PS.3a

Luis has 471 baseball cards.  He wants to share them equally among two friends and himself.  How many baseball cards will each person receive?

 

5.PS.3b

Seth and Juan are going to see a movie.  The movie begins at 2:15 pm and ends at 3:45 pm.  How long is the movie? 

 

5.PS.3c

The Berns family is on vacation. On Monday they traveled 86 miles; on Tuesday 91 miles; on Wednesday 103 miles; on Thursday 84 miles; and on Friday 98 miles.  What is the mean of the number of miles the Berns family traveled in a day?

 

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

 

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

 

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

 

5.PS.5a

At the end of the baseball season the top three hitters were Ethan, Maggie and John.  Ethan's batting average was 0.282,  Maggie's batting average was 0.223, and John's batting average was 0.272.  Write the batting averages in order from the highest to the lowest.

 

5.PS.5b

Aaliyah and Katie went to the candy store. Aaliyah bought 3/8 lb. of gumdrops.  Katie bought 2/8 lb. of jellybeans.  What is the total weight of the candy the girls bought?

 

5.PS.5c

Andy is going to the gym to play basketball.  He  is carrying a pair of sneakers and a sweatsuit.  The sneakers weigh 3 1/4 lbs and the sweatsuit weighs 2 2/4 lbs.  How much weight is Andy carrying?

 

5.PS.5d

Jade went to the store.  She bought a gallon of milk for  $2.59, a box of cereal for $3.89 and 5 oranges for $1.59.  Estimate how much money Jade spent at the store. Explain your answer.

 

5.PS.5e

Caleb is putting a fence around a rectangular garden.  His garden is 12 ft. long and 6 ft. wide.  How many feet of fence does Caleb need to enclose his garden?

 

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

 

5.PS.6a

Provide students a worksheet with 8 angles of different sizes on it.  Have students use a protractor to measure the size of each angle.

 

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

 

5.PS.7a

Carol and Amber made banners.  Carol used 2/3 of a yard of felt.  Amber used 3/4 of a yard of felt.  Write an inequality comparing the amount of felt Carol used to the amount of felt Amber used. 

 

5.PS.7b

Shontel and Ellen were in a running race. Shontel finished the race in 2.45 minutes.  Ellen finished in 2.36 minutes.  Write an inequality comparing Shontel's time to Ellen's time.

 

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

 

5.PS.8a

Have students count and record the number of students in their class, divided into boys and girls.   Ask the students:  How many boys are in your class?  How many girls are in your class? What ratio compares the number of boys to girls?  What ratio compares the number of girls to the number of students in your class?  What ratio compares the number of boys to the number of students in your class?

 

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

 

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

 

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

 

5.PS.10a

Give each pair of students a basket with 8 triangles in it.  Each triangle is labeled with a letter.   Have students select one triangle from the basket and keep selecting triangles until they find one similar to the first triangle selected.  Have them keep a record of the pairs of similar triangles.  (Triangles should have the same shape but not necessarily the same size.  Corresponding sides are in proportion and corresponding angles are congruent).

 

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

 

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

 

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

 

5.PS.13a

Provide pairs of students three coins. 

Have students take turns flipping the coins and recording each outcome.  (e.g., 3 heads, 2 heads and 1 tail, 1 head and 2 tails,  3 tails, etc.).  Discuss the list of all possible outcomes with the class. 

 

5.PS.14             Analyze problems by observing patterns

 

5.PS.14a

Brittany wrote the number pattern  2, 10, 50, 250.  What are the next two terms in Brittany's pattern? Write an algebraic expression to describe Brittany's pattern. 

 

5.PS.15             Make organized lists or charts to solve numerical problems

 

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

 

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

 

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

 

5.PS.17a

Amanda has 100 coins in her coin collection.

10 coins are from Africa.
20 coins are from Europe.

15 coins are from Asia.
5 coins are from Australia.
20 coins are from South America.

 The rest of the coins are from North America. 

What percentage of the coins are from North America?

 

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

 

5.PS.18a

The number 3,405,179  was in a newspaper article. Write the number in expanded form.

 

5.PS.19             Differentiate between valid and invalid approaches

 

5.PS.20             Understand valid counterexamples

 

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

 

5.PS.21a

Harley says the factors of 12 are 1,2,3,5,6,12.  Do you agree with Harley?  Explain your answer.

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

 

5.PS.22a

Estimate the sum of 2 3/4  + 1 2/4  + 4 3/4 . Explain the process you used to find your estimate.

 

5.PS.22b

Stephanie estimates she will need 6 meters of border for a bulletin board that is 2.6 meters long and 1.5 meters high.  Is this a good estimate?  Explain your answer. 

 

5.PS.23             Verify results of a problem

 

5.PS.23a

Ask students to list three 7-digit numbers in no particular order, then exchange their list with another student and have them write the numbers in order from least to greatest. Give to another student to check for accuracy.

 

5.PS.23b

Give pairs of students 12 tiles with the numbers 2,3,4,5,6,7,8,9,10,11,12, and 25 written on them.  Each student chooses a tile and records the first 5 multiples of the number.  Students check each other for accuracy.  Students continue to choose tiles until no more tiles are left.

 

5.PS.23c

If n = 2, evaluate each expression below:

  (5 + n) + 4
  2n + 5 x 2
  (8 - n) x 9

 Compare your answer with another student and explain how you arrived at the answers.

 

5.PS.23d

Write an equation to determine how many baseballs Jerome has, and then solve the equation.

Jerome and Marcus have 13 baseballs.  Marcus has 5 baseballs.  How many baseballs does Jerome have?

 

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Reasoning and Proof

 

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

 

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

 

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

 

5.RP.2a

Have students fold a piece of rectangular paper to form triangles, then cut out a triangle from the paper and label the angles A,B and C. Then have students carefully tear the angles from the triangle and place the angles so all the vertices are together.  What kind of angle is formed?  What is the measurement of this angle? What conclusion can they make about the sum of the angles of a triangle? 

Students will make and investigate mathematical conjectures.

 

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

 

5.RP.3a

Provide students with a worksheet with large quadrilaterals on them.  Ask students to measure and label the

four interior angles (in degrees) of quadrilaterals.  Then have them add the measurements of the angles together and record the total in the center of each quadrilateral.  What conclusion can they make about the total number of degrees of the interior angles of a quadrilateral?

 

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

 

Students will develop and evaluate mathematical arguments and proofs.

 

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

 

5.RP.5a

Tyler started with the fraction 1/2.  He thinks 4/8 and 3/6 are equivalent to 1/2.  Do you agree with Tyler?  Explain why.

 

5.RP.5b

Give a pair of students an envelope with the numbers 2 to 30 written on paper squares.  Have students choose a number and write each number and its factors on a record sheet.  Have students compare their work for accuracy.  Have them circle the common factors and draw a square around the greatest common factor. (GCF).

 

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

 

5.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.

 

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

 

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Communication

 

Students will organize and consolidate their mathematical thinking through communication.

 

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

 

5.CM.1a

If n = 2, evaluate each expression below:

 (5 + n) + 4                                                       2n + 5 x 2
                (8 - n) x 9

Compare your answer with another student and explain how you arrived at the answers.

 

5.CM.1b

Have the class work together to answer the following question: 

If you don't have a ruler or a yardstick, how can you measure an inch, a foot, and a yard?

List suggestions and ask each student to decide which ideas work best for them and why.

 

5.CM.2             Explain a rationale for strategy selection

 

5.CM.2a

John estimated the product of 86 and 54 is 4,500.  Explain whether this is a reasonable answer.

 

5.CM.2b

Solve the equation below: 

n + 23 = 56.

Explain the process that you used to solve the equation. 

 

5.CM.2c

Explain why the nonstandard measurements below are good personal references for the length of a centimeter:

· the width of your hand  
· the length of your arm from elbow to wrist
· the width of your thumb

 

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

 

5.CM.3             Organize and accurately label work 

 

5.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

 

5.CM.4a

Provide a small group of students a copy of a newspaper.  Each group should have a different newspaper.  Have students scan the newspapers looking for examples of charts, surveys, graphs, etc.  Have each group make a display of examples collected and make a presentation to the class on the types of statistical data they found in the newspapers.

 

5.CM.5             Answer clarifying questions from others

 

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

 

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

 

5.CM.6a

Give each student a list of five objects in the classroom to measure.  Ask students to measure each object to the nearest centimeter and then verify measurements with a partner.

 

5.CM.7             Raise questions that elicit, extend, or challenge others’ thinking 

 

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

 

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

 

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

 

5.CM.9a

Name the constant(s) and variable(s) in the expression 2n + 6.

 

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

 

5.CM.10a

Provide a small group of students a copy of a newspaper.  Each group should have a different newspaper.  Have students scan the newspapers looking for examples of charts, surveys, graphs, etc.  Have each group make a display of examples collected and make a presentation to the class on the types of statistical data they found in the newspapers.

 

5.CM.10b

Provide each student a number cube with the sides labeled 1,2,3,4,5,6.  Ask students to answer the following questions: 

· If the cube is rolled once, list the sample space.

 

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

 

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Connections

 

Students will recognize and use connections among mathematical ideas.

 

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

 

5.CN.1a

Mrs. Rivera the librarian bought 6 shirts for the library helpers.  Each shirt cost $6.95.  She paid for the shirts using a fifty dollar bill.  How much change did Mrs. Rivera receive?

 

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

 

5.CN.2a

Sarah needs 2 1/4 cups of flour for her cookie recipe.  She has 11/4 cups of flour in her flour container.  Does Sarah have enough flour for her recipe? Explain your answer. 

 

5.CN.2b

Provide students with a worksheet containing pairs of similar triangles.  Have students find the ratio between the corresponding sides, based on their measurements for each pair of triangles.  The ratio should be the same for all three corresponding sides.

 

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

 

5.CN.3a

Have the class find all of the prime numbers less than 100, following the directions below.  Use a hundreds chart.

1. Cross out 1.  It has only one unique factor. 
2. Circle 2.  It is the smallest prime number. 
3. Cross out all numbers divisible by 2.
4. Repeat step 3 with the number 3, the next prime number.
5. Continue this process - circle the next prime number and cross out its multiples - until  you reach 100.
6.  List all the prime numbers.

(Steps 1 to 6 are called the Sieve Eratosthenes, a way to sift out the composite numbers and leave the prime numbers behind.)

 

5.CN.3b

If n = 2, evaluate each expression below:

  (5 + n) + 4
  2n + 5 x 2
  (8 - n) x 9

Compare your answer with another student and explain how you arrived at the answers.

 

5.CN.3c

Write an equation to determine how many baseballs Jerome has, and then solve the equation.

Jerome and Marcus have 13 baseballs.  Marcus has 5 baseballs.  How many baseballs does Jerome have?

 

5.CN.3d

Using the formula P = 2l + 2w to find the perimeter of a rectangle, calculate the perimeter of a rectangle that has a length of 12. 5 cm and a width of 4.8 cm.

 

5.CN.3e

Provide students a worksheet with 10 triangles on it.  Have students informally measure the length of the sides of each triangle using an index card and identify each triangle as an equilateral, isosceles, triangle, or a scalene triangle.

 

5.CN.3f

Paul knows the measurement of two angles of a triangle.  Angle A measures 65 degrees and angle B measures 35 degrees.  Find the measure of angle C in degrees. 

 

5.CN.3g

Plot and label the points below on grid paper.

A(3,2)
B(3,5)
C(5,5)
D(5,2)

Connect the points in the order that they were plotted. Label the length of each side of the polygon and find the perimeter of the shape.

 

5.CN.3h

Kim has a new puppy.  He wants to put fence around a yard for his puppy.  What mathematical tools do you recommend Kim use to decide how much fence he needs? 

 

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

 

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

 

5.CN.4a

In groups of three, ask each student to write a 7-digit number on an index card.  Have the students exchange the cards and write the number on the card in word form.  The three students check the cards for accuracy.

 

5.CN.4b

The number 3,405,179  was in a newspaper article. Write the number in expanded form.

 

5.CN.4c

Give each student 7 red, 8 blue, and 5 green counters.  Have them separate the counters by color and record the number of each color.   Have students write a ratio comparing red counters to green counters in three different ways  (e.g., 7 to 5, 7:5, 7/5).

 

 

5.CN.4d

Ask students to rename the following fractions in lowest terms.

  a.  10/12
  b.  15/20
  c.  3/18
  d.  3/7
  e.  12/27

 

5.CN.4e

Provide students with a worksheet containing five line segments with the end points of each line segment clearly marked. Have students measure each line segment to the nearest inch, 1/2  inch,  1/4  inch and 1/8  inch, keeping a record of their measurements.  Have students verify their measurements.

 

5.CN.4f

Provide pairs of students a ruler, yardstick, and a piece of string 36 inches long. Have students measure and record the length of the string in inches, feet and yards.   Have the students record their results on a class chart.  Discuss with the class the relationships among the three units of measurement and create a table showing these relationships.

 

5.CN.4g

Complete the equalities below:  

9 cm = ____ mm
2 km = ____ cm
20 cm = ____ dm

 

5.CN.4h

Have students measure the length of a bulletin board in inches.  Then ask students to change the measurement to feet, inches and feet, yards, or inches and yards.

 

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

 

5.CN.5a

Stephanie estimates she will need 6 meters of border for a bulletin board that is 2.6 meters long and 1.5 meters high.  Is this a good estimate?  Explain your answer. 

 

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

 

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

 

5.CN.6a

The number 3,405,179  was in a newspaper article. Write the number in expanded form.

 

5.CN.6b

Amanda has 100 coins in her coin collection.

10 coins are from Africa.
20 coins are from Europe.
15 coins are from Asia.
5 coins are from Australia.
20 coins are from South America.

 The rest of the coins are from North America. 

What percentage of the coins are from North America?

 

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

 

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

 

5.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.

 

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

 

5.R.1a

Provide students with an illustration of a measuring cup with marks on one side for 1/3, 2/3 and 1.  Provide another illustration with marks on the other side for 1/4, 2/4, 3/4 and 1. Correctly label 1/3, 2/3, 1/4, 1/2, 3/4, 1 on both cups.

 

5.R.1b

Provide students with a worksheet containing quadrilaterals.  Have the students cut out the quadrilaterals and group them into two piles:  parallelograms and non-parallelograms. Record results by gluing the quadrilaterals to a record sheet divided into two sections, one section for parallelograms and the other section for non-parallelograms. Have students verify work with a partner.

 

5.R.1c

Provide students with a worksheet containing pairs of congruent triangles.  Have students label the first set of corresponding sides with one line, the second pair of corresponding sides with two marks, and the third pair of corresponding sides with three marks.

 

5.R.1d

Provide students with grid paper. Have students draw the axes on the grid paper and plot and label the following points:  

A(3,2)
B(5,7)
C(10,6)
D(6,5)
E(7,1)

 

5.R.1e

Have students plot and label the points below on grid paper and then connect the points in the order they were plotted. 

A(4,2)
B(4,3)
C(5,4)
D(6,4)
E(7,4)
F(7,2)
G(6,1)
H(5,1)
A(4,2)

 What shape is formed?

 

5.R.1f

Construct a line graph from the information listed below: 
Normal Monthly Temperature in Fahrenheit for Albany, New York

January       22
February     25
March        36
April         47
May           58
June           66
July            71
August        69
September  61
October      48
November  39
December  28
(Source:  World Almanac, 2004) 
 

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

 

5.R.2a

Seth is 2 years older than his brother Ishmael.  Write an algebraic expression to represent Seth's age.

 

5.R.2b

Provide students with colored pencils and a worksheet containing 10 basic
geometric shapes.  Ask students to draw the line(s) of symmetry for each shape using a different colored pencil for each line of symmetry and recording the number of lines of symmetry for each shape.

 

5.R.2c

Stephanie estimates she will need 6 meters of border for a bulletin board that is 2.6 meters long and 1.5 meters high.  Is this a good estimate?  Explain your answer. 

 

5.R.2d

Title     Growth of a Tree
y axis   labeled Height in Feet
       number scale - interval of 1 
x axis    labeled  End of Year
      1st, 2nd, 3rd, 4th, 5th, 6th   
points plotted and connected by a line
(0,0)  (1st year, 1 1/2)  (2nd year, 3)  (3rd year, 4 1/2)  (4th year, 6)   (5th year, 7 1/2)

Have students describe the graph and explain how tall the tree will be at the end of the sixth year.

 

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

 

5.R.3a

Title     Growth of a Tree
y axis   labeled Height in Feet
       number scale - interval of 1 
x axis    labeled  End of Year
      1st, 2nd, 3rd, 4th, 5th, 6th   
points plotted and connected by a line
(0,0)  (1st year, 1 1/2)  (2nd year, 3)  (3rd year, 4 1/2)  (4th year, 6)   (5th year, 7 1/2)

Have students describe the graph and explain how tall the tree will be at the end of the sixth year.

 

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

 

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

 

5.R.5                 Use representations to explore problem situations

 

5.R.5a

Use a triangle and a square to create a five-part pattern.  Shade or rotate either one or both shapes.  Draw the pattern so that it repeats 3 times.

 

5.R.5b

Provide students a worksheet with 10  triangles on it and a sheet of tracing paper.  Each triangle is identified with a letter.  Have students trace the first triangle and place it over each of the other triangles to decide if the two triangles are congruent.  Remind them that the traced triangle can be rotated.  Have students keep a record of the pairs of congruent triangles and verify results with other students.

 

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

 

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

 

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

 

5.R.7a

Have students measure the length of a bulletin board in inches.  Then ask students to change the measurement to feet, inches and feet, yards, or inches and yards.

 

5.R.7b

Kim has a new puppy.  He wants to put fence around a yard for his puppy.  What mathematical tools do you recommend Kim use to decide how much fence he needs? 

 

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

 

5.R.8a

Provide each student a number cube with the sides labeled 1,2,3,4,5,6.  Ask students to answer the following questions: 

· If the cube is rolled once, list the sample space.
· If the cube is rolled once, what is the probability of rolling an even number?  
· If the cube is rolled once, what is the probability of rolling an odd number?  

 

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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

 

5.N.1                 Read and write whole numbers to millions.

 

5.N.1a 

In groups of three, ask each student to write a 7-digit number on an index card.  Have the students exchange the cards and write the number on the card in word form.  The three students check the cards for accuracy.

 

5.N.2                 Compare and order numbers to millions.

 

5.N.2a 

Ask students to list three 7-digit numbers in no particular order, then exchange their list with another student and have them write the numbers in order from least to greatest. Give to another student to check for accuracy.

 

5.N.3                 Understand the place value structure of the base ten number system.

  10 ones = 1 ten

  10 tens = 1 hundred

  10 hundreds  = 1 thousand

  10 thousands = 1 ten thousand

  10 ten thousands = 1 hundred thousand

  10 hundred thousands = 1 million

 

5.N.3a 

The number 3,405,179  was in a newspaper article. Write the number in expanded form.

 

5.N.4                 Create equivalent fractions, given a fraction.

 

5.N.4a 

Tyler started with the fraction 1/2.  He thinks 4/8 and 3/6 are equivalent to 1/2.  Do you agree with Tyler?  Explain why.

 

5.N.5                 Compare and order fractions including unlike denominators (with and without the use of a number line) Note: Commonly used fractions such as those that might be indicated on ruler, measuring cup, etc.

5.N.5a 

Provide students with an illustration of a measuring cup with marks on one side for 1/3, 2/3 and 1.  Provide another illustration with marks on the other side for 1/4, 2/4, 3/4 and 1. Correctly label 1/3, 2/3, 1/4, 1/2, 3/4, 1 on both cups.

 

5.N.6                 Understand the concept of ratio.

 

5.N.6a 

Have students count and record the number of students in their class, divided into boys and girls.   Ask the students:  How many boys are in your class?  How many girls are in your class? What ratio compares the number of boys to girls?  What ratio compares the number of girls to the number of students in your class?  What ratio compares the number of boys to the number of students in your class?

 

5.N.7                 Express ratios in different forms.

 

5.N.7a 

Give each student 7 red, 8 blue, and 5 green counters.  Have them separate
the counters by color and record the number of each color.   Have students write a ratio comparing red counters to green counters in three different ways  (e.g., 7 to 5, 7:5, 7/5).

 

5.N.8                 Read, write, and order decimals to thousandths.

 

5.N.8a 

At the end of the baseball season the top three hitters were Ethan, Maggie and John.  Ethan's batting average was 0.282,  Maggie's batting average was 0.223, and John's batting average was 0.272.  Write the batting averages in order from the highest to the lowest.

 

5.N.9                 Compare fractions using <, >, or =

 

5.N.9a 

Carol and Amber made banners.  Carol used 2/3 of a yard of felt.  Amber used 3/4 of a yard of felt.  Write an inequality comparing the amount of felt Carol used to the amount of felt Amber used. 

 

5.N.10               Compare decimals using <, >, or =

 

5.N.10a 

Shontel and Ellen were in a running race. Shontel finished the race in 2.45 minutes.  Ellen finished in 2.36 minutes.  Write an inequality comparing Shontel's time to Ellen's time.

 

5.N.11               Understand that percent means part of 100, and write percents as fractions and decimals.

 

5.N.11a 

Amanda has 100 coins in her coin collection.

10 coins are from Africa.
20 coins are from Europe.
15 coins are from Asia.
5 coins are from Australia.
20 coins are from South America.

 The rest of the coins are from North America. What percentage of the coins are from North America? 

 

Number Theory

 

5.N.12               Recognize that some numbers are only divisible by one and themselves (prime) and others have multiple divisors (composite)

 

5.N.12a 

Have the class find all of the prime numbers less than 100, following the directions below.  Use a hundreds chart.

1. Cross out 1.  It has only one unique factor. 
2. Circle 2.  It is the smallest prime number. 
3. Cross out all numbers divisible by 2.
4. Repeat step 3 with the number 3, the next prime number.
5. Continue this process - circle the next prime number and cross out its multiples - until  you reach 100.
6.  List all the prime numbers.

  (Steps 1 to 6 are called the Sieve Eratosthenes, a way to sift out the composite numbers and leave the prime numbers behind.)

 

5.N.13              Calculate multiples of a whole number and the least common multiple of two

                          numbers.

 

5.N.13a 

Give pairs of students 12 tiles with the numbers 2,3,4,5,6,7,8,9,10,11,12, and 25 written on them.  Each student chooses a tile and records the first 5 multiples of the number.  Students check each other for accuracy.  Students continue to choose tiles until no more tiles are left.

 

5.N.14               Identify the factors of a given number.

 

5.N.14a 

Harley says the factors of 12 are 1,2,3,5,6,12.  Do you agree with Harley?  Explain your answer.

 

5.N.15               Find the common factors and the greatest common factor of two numbers

 

5.N.15a 

Give a pair of students an envelope with the numbers 2 to 30 written on paper squares.  Have students choose a number and write each number and its factors on a record sheet.  Have students compare their work for accuracy.  Have them circle the common factors and draw a square around the greatest common factor. (GCF).

 

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

 

Operations

 

5.N.16               Use a variety of strategies to multiply three-digit by three-digit numbers  Note: Multiplication by anything greater than a three-digit multiplier/ multiplicand should be done using technology.

 

5.N.16a 

The Department of Environmental Conservation needs to plant 135 rows of trees with 236 trees in each row.  How many trees need to be planted?

 

5.N.17               Use a variety of strategies to divide three-digit numbers by one- and two-digit numbers Note: Division by anything greater than a two-digit divisor should be done using technology.

 

5.N.17a 

Luis has 471 baseball cards.  He wants to share them equally among two friends and himself.  How many baseball cards will each person receive?

 

5.N.18               Evaluate an arithmetic expression using order of operations including multiplication, division, addition, subtraction and parentheses.

 

5.N.18a 

Evaluate the expressions:  

   8 + 3 x 8
   (25  / 5) + 10
   10 x (12 - 4) + (2.7 ÷ 3)
   24 - 10 - 40 ÷ 10 ÷ 2
   (17 - 9) x (4 + 3)
 Verify your work with a partner.

 

5.N.19               Simplify fractions to lowest terms.

 

5.N.19a 

Ask students to rename the following fractions in lowest terms.     

  a.  10/12
  b.  15/20
  c.  3/18
  d.  3/7
  e.  12/27

 

5.N.20               Convert improper fractions to mixed numbers, and mixed numbers to improper  fractions.

 

5.N.20a 

Sarah needs 2 1/4 cups of flour for her cookie recipe.  She has 11/4 cups of flour in her flour container.  Does Sarah have enough flour for her recipe?   Explain your answer. 

 

5.N.21               Use a variety of strategies to add and subtract fractions with like denominators.

 

5.N.21a 

Aaliyah and Katie went to the candy store. Aaliyah bought 3/8 lb. of gumdrops.  Katie bought 2/8 lb. of jellybeans.  What is the total weight of the candy the girls bought?

 

5.N.22               Add and subtract mixed numbers with like denominators.

 

5.N.22a 

Andy is going to the gym to play basketball.  He  is carrying a pair of sneakers and a sweatsuit.  The sneakers weigh 3 1/4 lbs and the sweatsuit weighs 2 2/4 lbs.  How much weight is Andy carrying?

 

5.N.23               Use a variety of strategies to add, subtract, multiply, and divide decimals to thousandths.

 

5.N.23a 

Mrs. Rivera the librarian bought 6 shirts for the library helpers.  Each shirt cost $6.95.  She paid for the shirts using a fifty dollar bill.  How much change did Mrs. Rivera receive?

 

Students will compute accurately and make reasonable estimates.

 

 Estimation

 

5.N.24   Round numbers to the nearest hundredth and up to 10,000.

 

5.N.24a 

Round the number 156,784 to the nearest ten thousand.  Explain your answer.

 

5.N.25               Estimate sums and differences of fractions with like denominators.

 

5.N.25a 

Estimate the sum of 2 3/4  + 1 2/4  + 4 3/4 .  Explain the process you used to find your estimate.

 

5.N.26               Estimate sums, differences, products, and quotients of decimals.

 

5.N.26a 

Jade went to the store.  She bought a gallon of milk for  $2.59, a box of cereal for $3.89 and 5 oranges for $1.59.  Estimate how much money Jade spent at the store. Explain your answer.

 

5.N.27               Justify the reasonableness of answers using estimation

 

5.N.27a 

John estimated the product of 86 and 54 is 4,500.  Explain whether this is a reasonable answer.

 

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Algebra Strand

 

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

 

Variables and Expressions

 

 5.A.1                Define and use appropriate terminology when referring to constants, variables, and algebraic expressions.

 

5.A.1a 

Name the constant(s) and variable(s) in the expression 2n + 6.

 

5.A.2                 Translate simple verbal expressions into algebraic expressions.

 

5.A.2a 

Seth is 2 years older than his brother Ishmael.  Write an algebraic expression to represent Seth's age.

 

Students will perform algebraic procedures accurately.

 

Variables and Expressions 

5.A.3                 Substitute assigned values into variable expressions and evaluate using order of operations

 

5.A.3a 

If n = 2, evaluate each expression below:

  (5 + n) + 4
  2n + 5 x 2
  (8 - n) x 9

Compare your answer with another student and explain how you arrived at the answers.

 

Equations and Inequalities

 

5.A.4                 Solve simple one-step equations using basic whole-number facts

 

5.A.4a 

Write an equation to determine how many baseballs Jerome has, and then solve the equation. 

Jerome and Marcus have 13 baseballs.  Marcus has 5 baseballs.  How many baseballs does Jerome have?

 

5.A.5                 Solve and explain simple one-step equations using inverse operations involving whole numbers

 

5.A.5a 

Solve the equation below: 
n + 23 = 56.

Explain the process that you used to solve the equation. 

 

5.A.6                 Evaluate the perimeter formula for given input values.

5.A.6a 

Using the formula P = 2l + 2w to find the perimeter of a rectangle, calculate the perimeter of a rectangle that has a length of 12.5 cm and a width of 4.8 cm.

 

Students will recognize, use, and represent algebraically patterns, relations, and functions.

 

Patterns, Functions, and Relationships

 

5.A.7                 Create and explain patterns and algebraic relationships (i.e.,2,4,6,8...) algebraically: 2n (doubling)

 

5.A.7a 

Brittany wrote the number pattern  2, 10, 50, 250.  What are the next two terms in Brittany's pattern? Write an algebraic expression to describe Brittany's pattern. 

 

5.A.8                 Create algebraic or geometric patterns using concrete objects or visual drawings (e.g., rotate and shade geometric shapes).

 

5.A.8a 

Use a triangle and a square to create a five-part pattern.  Shade or rotate either one or both shapes.  Draw the pattern so that it repeats 3 times.

 

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Geometry Strand

 

Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes.

 

Shapes

 

5.G.1                 Calculate the perimeter of regular and irregular polygons

 

5.G.1a 

Caleb is putting a fence around a rectangular garden.  His garden is 12 ft. long and 6 ft. wide.  How many feet of fence does Caleb need to enclose his garden?

 

Students will identify and justify geometric relationships, formally and informally.

 

Geometric Relationships

 

5.G.2                 Identify pairs of similar triangles.

 

5.G.2a 

Give each pair of students a basket with 8 triangles in it.  Each triangle is labeled with a letter.   Have students select one triangle from the basket and keep selecting triangles until they find one similar to the first triangle selected.  Have them keep a record of the pairs of similar triangles.  (Triangles should have the same shape but not necessarily the same size.  Corresponding sides are in proportion and corresponding angles are congruent).

 

5.G.3                 Identify the ratio of corresponding sides of similar triangles.

 

5.G.3a 

Provide students with a worksheet containing pairs of similar triangles.  Have students find the ratio between the corresponding sides, based on their measurements for each pair of triangles.  The ratio should be the same for all three corresponding sides.

 

5.G.4                 Classify quadrilaterals by properties of their angles and sides.

 

5.G.4a 

Provide students with a worksheet containing quadrilaterals.  Have the students cut out the quadrilaterals and group them into two piles:  parallelograms and non-parallelograms. Record results by gluing the quadrilaterals to a record sheet divided into two sections, one section for parallelograms and the other section for non-parallelograms. Have students verify work with a partner.

 

5.G.5                 Know that the sum of the interior angles of a quadrilateral is 360 degrees.

 

5.G.5a 

Provide students with a worksheet with large quadrilaterals on them.  Ask students to measure and label the four interior angles (in degrees) of quadrilaterals.  Then have them add the measurements of the angles together and record the total in the center of each quadrilateral.  What conclusion can they make about the total number of degrees of the interior angles of a quadrilateral?

 

5.G.6     Classify triangles by properties of their angles and sides.

 

5.G.6a 

Provide students a worksheet with 10 triangles on it.  Have students informally measure the length of the sides of each triangle using an index card and identify each triangle as an equilateral, isosceles, triangle, or a scalene triangle.

 

5.G.7                 Know that the sum of the interior angles of a triangle is 180 degrees.

 

5.G.7a 

Have students fold a piece of rectangular paper to form triangles, then cut out a triangle from the paper and label the angles A,B and C.  Then have students carefully tear the angles from the triangle and place the angles so all the vertices are together.  What kind of angle is formed?  What is the measurement of this angle?  What conclusion can they make about the sum of the angles of a triangle? 

5.G.8                 Find a missing angle when given two angles of a triangle.

 

5.G.8a 

Paul knows the measurement of two angles of a triangle.  Angle A measures 65 degrees and angle B measures 35 degrees.  Find the measure of angle C in degrees. 

 

5.G.9                 Identify pairs of congruent triangles.

 

5.G.9a 

Provide students a worksheet with 10  triangles on it and a sheet of tracing paper.  Each triangle is identified with a letter.  Have students trace the first triangle and place it over each of the other triangles to decide if the two triangles are congruent.  Remind them that the traced triangle can be rotated.  Have students keep a record of the pairs of congruent triangles and verify results with other students.

 

5.G.10              Identify corresponding parts of congruent triangles.            

 

5.G.10a 

Provide students with a worksheet containing pairs of congruent triangles.  Have students label the first set of corresponding sides with one line, the second pair of corresponding sides with two marks, and the third pair of corresponding sides with three marks.

 

Students will apply transformations and symmetry to analyze problem solving situations.

 

Transformational Geometry

 

5.G.11              Identify and draw lines of symmetry of basic geometric shapes.

 

5.G.11a 

Provide students with colored pencils and a worksheet containing 10 basic
geometric shapes.  Ask students to draw the line(s) of symmetry for each shape using a different colored pencil for each line of symmetry and recording the number of lines of symmetry for each shape.

 

Students will apply coordinate geometry to analyze problem solving situations.

 

Coordinate

 

5.G.12              Identify and plot points in the first quadrant.

 

5.G.12a 

Provide students with grid paper. Have students draw the axes on the grid paper and plot and label the following points:  

A(3,2)
B(5,7)
C(10,6)
D(6,5)
E(7,1)

 

Geometry

 

5.G.13              Plot points to form basic geometric shapes (identify and classify).

 

5.G.13a 

Have students plot and label the points below on grid paper and then connect the points in the order they were plotted. 

A(4,2)
B(4,3)
C(5,4)
D(6,4)
E(7,4)
F(7,2)
G(6,1)
H(5,1)
A(4,2)   
                                      What shape is formed?

 

5.G.14              Calculate perimeter of basic geometric shapes drawn on a coordinate plane (rectangles and shapes composed of rectangles having sides with integer lengths and parallel to the axes)

 

5.G.14a 

Plot and label the points below on grid paper.

A(3,2)
B(3,5)
C(5,5)
D(5,2)

Connect the points in the order that they were plotted. Label the length of each side of the polygon and find the perimeter of the shape.

 

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Measurement Strand

 

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

 

Units of Measurement

5.M.1                Use a ruler to measure to the nearest inch, and  inch

 

5.M.1a 

Provide students with a worksheet containing five line segments with the end points of each line segment clearly marked. Have students measure each line segment to the nearest inch,  1/2  inch,  1/4  inch and 1/8  inch, keeping a record of their measurements.  Have students verify their measurements.

 

5.M.2                Identify customary equivalent units of length

 

5.M.2a 

Provide pairs of students a ruler, yardstick, and a piece of string 36 inches long. Have students measure and record the length of the string in inches, feet and yards.   Have the students record their results on a class chart.  Discuss with the class the relationships among the three units of measurement and create a table showing these relationships.

 

5.M.3                Measure to the nearest centimeter

 

5.M.3a 

Give each student a list of five objects in the classroom to measure.  Ask students to measure each object to the nearest centimeter and then verify measurements with a partner.

 

5.M.4                Identify equivalent metric units of length.

 

5.M.4a 

Complete the equalities below:  

9 cm = ____ mm
2 km = ____ cm
20 cm = ____ dm

 

5.M.5                Convert measurement within a given system.

 

5.M.5a 

Have students measure the length of a bulletin board in inches.  Then ask students to change the measurement to feet, inches and feet, yards, or inches and yards.

 

Tools and Methods

 

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

 

5.M.6a 

Kim has a new puppy.  He wants to put fence around a yard for his puppy.  What mathematical tools do you recommend Kim use to decide how much fence he needs? 

 

Students will use units to give meaning to measurements.

 

Units

 

5.M.7                Calculate elapsed time in hours and minutes.

 

5.M.7a 

Seth and Juan are going to see a movie.  The movie begins at 2:15 pm and ends at 3:45 pm.  How long is the movie? 

 

5.M.8                Measure and draw angles using a protractor.

 

5.M.8a 

Provide students a worksheet with 8 angles of different sizes on it.  Have students use a protractor to measure the size of each angle.

 

Students will develop strategies for estimating measurements.

 

Estimation

 

5.M.9                Determine personal references for customary units of length (e.g., your pace is approximately 3 feet, your height is approximately 5 feet, etc.)

 

5.M.9a 

Have the class work together to answer the following question: 

If you don't have a ruler or a yardstick, how can you measure an inch, a foot, and a yard?

List suggestions and ask each student to decide which ideas work best for them and why.

 

5.M.10              Determine personal references for metric units of length.

 

5.M.10a 

Explain why the nonstandard measurements below are good personal references for the length of a centimeter:

· the width of your hand  
· the length of your arm from elbow to wrist
· the width of your thumb

 

5.M.11              Justify the reasonableness of estimates.

 

5.M.11a 

Stephanie estimates she will need 6 meters of border for a bulletin board that is 2.6 meters long and 1.5 meters high.  Is this a good estimate?  Explain your answer. 

 

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Statistics and Probability Strand

 

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

 

Collection of Data

 

5.S.1                  Collect and record data from a variety of sources (e.g., newspapers, magazines, polls, charts, and surveys)

 

5.S.1a 

Provide a small group of students a copy of a newspaper.  Each group should have a different newspaper.  Have students scan the newspapers looking for examples of charts, surveys, graphs, etc.  Have each group make a display of examples collected and make a presentation to the class on the types of statistical data they found in the newspapers.

 

Organization and Display of Data

 

5.S.2                  Display data in a line graph to show an increase or decrease over time

 

5.S.2a 

Construct a line graph from the information listed below: 
Normal Monthly Temperature in Fahrenheit for Albany, New York

January       22
February     25
March        36
April         47
May           58
June           66
July            71
August        69
September  61
October      48
November  39
December  28
(Source:  World Almanac, 2004) 

 

Analysis of Data

 

5.S.3                  Calculate the mean for a given set of data and use to describe a set of data

 

5.S.3a 

The Berns family is on vacation. On Monday they traveled 86 miles; on Tuesday 91 miles; on Wednesday 103 miles; on Thursday 84 miles; and on Friday 98 miles.  What is the mean of the number of miles the Berns family traveled in a day?

 

Students will make predictions that are based upon data analysis.

 

Predictions from Data

 

5.S.4                  Formulate conclusions and make predictions from graphs.

 

5.S.4a 

Title     Growth of a Tree

y axis   labeled Height in Feet

 number scale - interval of 1 

x axis    labeled  End of Year

1st, 2nd, 3rd, 4th, 5th, 6th   

points plotted and connected by a line

(0,0)  (1st year, 1 1/2)  (2nd year, 3)  (3rd year, 4 1/2)  (4th year, 6)   (5th year, 7 1/2)


 

Have students describe the graph and explain how tall the tree will be at the end of the sixth year.

 

Students will understand and apply concepts of probability.

 

Probability

 

5.S.5                  List the possible outcomes for a single-event experiment.

 

5.S.5a 

Provide pairs of students three coins. 
Have students take turns flipping the coins and recording each outcome.  (e.g., 3 heads, 2 heads and 1 tail, 1 head and 2 tails,  3 tails, etc.).  Discuss the list of all possible outcomes with the class. 

 

5.S.6                  Record experiment results using fractions/ratios.

 

5.S.6a 

Provide each pair of students with a spinner divided into four equal sectors numbered 1, 2, 3, 4.   Ask students to spin the spinner 60 times and record their results in a tally chart.  Ask each pair for the experimental probability of spinning a number less than four from their data and record the information as a fraction on a class chart.  Discuss the results and compare the experimental probability to the theoretical probability. 

 

5.S.7                  Create a sample space and determine the probability of a single event, given a simple experiment (i.e., rolling a number cube)

 

5.S.7a 

Provide each student a number cube with the sides labeled 1,2,3,4,5,6.  Ask students to answer the following questions: 

· If the cube is rolled once, list the sample space.
· If the cube is rolled once, what is the probability of rolling an even number?  
· If the cube is rolled once, what is the probability of rolling an odd number?  

 

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