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

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





Grade 4


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.










Problem Solving



Reasoning and Proof














Number Sense and Operations












Statistics and Probability











Problem Solving Strand


Students will build new mathematical knowledge through problem solving.


4.PS.1               Explore, examine, and make observations about a social problem or mathematical situation



Using the information in the table below determine how much time it takes a bus to travel from Town A to Town B.

Bus Schedule

Leaves Town A




Arrives in Town B









4.PS.2               Understand that some ways of representing a problem are more helpful than others



Discuss how drawing a picture can be helpful in solving the problem below: 

Diane is studying a model of her house.  The front of her house is in the shape of a pentagon.  Each side of the roof is 10 cm long, each wall is 13 cm tall, and the bottom is 15 cm long.   Draw a picture of the model of her house.  What is the perimeter of the pentagon?  


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



Alexis wants a bowl of ice cream for dessert.  She can choose chocolate, vanilla or strawberry.  The toppings she can have on her ice cream are sprinkles or caramel.  List all possible combinations of ice cream and topping Alexis can have, if she can only choose one flavor of ice cream and one topping.


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


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



After reading Fraction Action by Loreen Leedy, Fraction Fun by David Adler or Hershey's Fractions by Jerry Pallotta, use objects to model the fractional amounts mentioned in one of the books.


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



Have students create a word problem using the following information.  Team A in fourth grade collected 1,547 toys to donate.   Team B in fourth grade collected 872 toys to donate.


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



Francine ate 2 pieces of her birthday cake as shown in the picture below.  If each piece is of equal size, what fractional part of the cake is left to share?  Write a number sentence to show your answer.







Francine's Birthday Cake




4.PS.7               Represent problem situations in oral, written, concrete, pictorial, and graphical forms



Divide students into small groups.  Ask each group to add up how many books they have read in the past month.  Create a chart to represent the number of books each group has read.  Have students create a pictograph showing the number of books read by each group.  


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



Discuss with the class the appropriateness of using a line graph rather than a bar graph to represent changes over a period of time.


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


4.PS.9               Use trial and error to solve problems



Zachary is thinking of two numbers whose sum is 9.  Their product is 20.  The difference of the two numbers is 1.  What are the two numbers?


4.PS.10             Use process of elimination to solve problems



Which is the better estimate of the length of Eric's foot, 20 cm or 20 m? 


4.PS.11             Make pictures/diagrams of problems



Draw a picture to help you solve the problem below: 
David is shorter than Mouro.  Jose is taller than Mouro.  Carl is shorter than David.  Who is the tallest?


4.PS.12             Use physical objects to model problems



Use fraction strips to find the missing numerators:








4.PS.13             Work in collaboration with others to solve problems



In groups of three, have students measure the mass of a variety of objects to find an object that weighs between 10 grams and 15 grams.


4.PS.14             Make organized lists to solve numerical problems



Luke makes his lunch each day.  He can use rye, white, or wheat bread.  He likes turkey or ham sandwiches.  How many different kinds of sandwiches can he make if he uses one kind of bread and one choice of meat? 


4.PS.15             Make charts to solve numerical problems



Make a chart to solve the problem below:

Maria needs to order pencils and pens for the school store. She wants to order the same number of pencils as pens. Pencils cost 10 cents each and pens cost 12 cents each. If she spends $1.00 on pencils, how much does she spend on pens?


4.PS.16             Analyze problems by identifying relationships



Sarah has drawn four different quadrilaterals.  Identify and name the quadrilaterals based upon the sides and angles.


4.PS.17             Analyze problems by identifying relevant versus irrelevant information



Derek goes to the store every Tuesday.  Derek spent $25.79 this week at the store.  Anna likes to go to the store on Fridays.  She walks one mile to get there.  She spent $25.75 this week at the store.  Write an inequality that compares the amounts that Derek and Anna spent at the store.  Who spent more money at the store this week?


4.PS.18             Analyze problems by observing patterns



Kathleen earns money babysitting each day. After she gets paid, she counts the money she has saved. The chart below shows her totals:






Total Saved





If Kathleen continues to save money at the same rate, how many days will it take Kathleen to save a total of $24.00?


4.PS.19             State a problem in their own words



Restate the problem below so that it can be solved:

Josephina's table is 15 feet long. If tablecloths are packaged in yards, which package should she buy, if she is only concerned with covering just the length of the table?


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


4.PS.20             Determine what information is needed to solve a problem



What additional information is needed to solve the following problem? Mrs. Morlang told the third grade class that she would buy them all ice cream once they reached their goal in reading. If the class has read 105 books so far, how many more books do they need to read in order to reach their goal?


4.PS.21             Discuss with peers to understand a problem situation



Provide each pair of students 11 paper clips and 3 cups to solve the problem below: 

Using all 11 paper clips, place only an odd number of paper clips in each of the three cups. Make an organized list of your results. 


4.PS.22             Discuss the efficiency of different representations of a problem



Carmine's race times for the last 5 races were as follows:

Race One             4 minutes 7 seconds
Race Two             4 minutes 8 seconds
Race Three          4 minutes 3 seconds
Race Four            3 minutes 58 seconds
Race Five             3 minutes 56 seconds

Describe Carmine's race times, comparing the races.  Then show a more efficient way to show and compare Carmine's race times, such as with a line graph or chart.  


4.PS.23             Verify results of a problem



Have students estimate the number of objects in an Estimation Jar.  Verify the number by counting the objects.


4.PS.24             Recognize invalid approaches



Joe collects baseball cards.  He has 6,230 cards in his collection.  There are 975 cards that are duplicates.  He is going to give his friend Sam the duplicates.  He wants to know how many cards will remain in his

collection.  Below are 2 attempts to find the number of cards in his collection.  Is either approach correct?  If not, explain and solve correctly. 

                  6230                               6230
                 - 975                               - 975
                  3520                               6745       


4.PS.25             Determine whether a solution is reasonable in the context of the original problem



Fred can walk at least 2 miles in one hour.  Is it reasonable to say that Fred can walk 8 miles in 4 hours?


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Reasoning and Proof Strand


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


4.RP.1   Use representations to support mathematical ideas



Using a hundreds grid, demonstrate how shading in 17 boxes is equivalent to .17 or 17 hundredths.


4.RP.2              Determine whether a mathematical statement is true or false and explain why



Using various examples, have students determine whether the following statement is true or false:   

If a factor (odd or even) is multiplied by an even number, the product of the two numbers will always be an even number. 


Students will make and investigate mathematical conjectures.


4.RP.3              Investigate the use of knowledgeable guessing by generalizing mathematical ideas.



A new movie was playing for three nights at a local theater.  Friday night 167 people bought tickets, Saturday night 206 people bought tickets, and on Sunday night 154 people bought tickets.  Approximately how many people bought tickets to the movie?


4.RP.4              Make conjectures from a variety of representations



Tell the class that you are thinking of a three-digit number.  The number is less than 500, but greater than 400.  It is less than the difference between 600 and 150, but greater than the sum of 270 and 170.   The number is a multiple of 5.  Have the class find the number and show how you arrived at the solution. 


Students will develop and evaluate mathematical arguments and proofs.


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



A square can be drawn with four line segments and four points.   Using a geoboard, create a square using four rubber bands, with each rubber band representing a line segment.  Continue using a geoboard to justify the following claims:

Any quadrilateral has 4 vertices and 4 line segments
A triangle has 3 vertices and 3 line segments
A pentagon has 5 vertices and 5 line segments
A hexagon has 6 vertices and 6 line segments
An octagon has 8 vertices and 8 line segments


4.RP.6              Develop and explain an argument using oral, written, concrete, pictorial, and/or graphical forms



Maria and her brother Michael are trying to determine whose bedroom has the greatest floor area.  Maria's room dimensions are 12 feet for length and 14 feet for width.  Michael's room is 15 feet long and 11 feet wide.  Draw a figure to represent each bedroom.  Determine which bedroom has the largest area. 


4.RP.7              Discuss, listen, and make comments that support or reject claims made by other students



Following a student presentation about a problem and solution, encourage classmates to share observations and ask questions about the solution that the student presented.


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


4.RP.8              Justify an argument by trying many cases



Have students try many cases to justify that when three numbers are multiplied, the product of the numbers stays the same when the grouping of the factors changes.  Label this argument the associative property of multiplication.  Does this always hold true?  Explain.  Ask students to support by examples or refute by providing counterexamples. 


4.RP.9              Disprove an argument by finding counterexamples


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Communication Strand


Students will organize and consolidate their mathematical thinking through communication.


4.CM.1             Understand and explain how to organize their thought process



Diego has earned $45.00.  He wants to use the money to buy a pet iguana from the pet store.  Diego made a list of his needs for an iguana.  Will Diego have enough money to buy an iguana and the supplies necessary to take care of the iguana if he needs to purchase an iguana, 2 bags of rocks, a glass aquarium, a dish of water, 2 boxes of iguana food and a heat lamp?  Explain your answer.

Iguana -  $10.00             Dish for water - $4.40
Bag of rocks - $3.29      Box of iguana food - $2.37
Aquarium - $14.99        Heat lamp - $5.83


4.CM.2             Verbally explain their rationale for strategy selection



Have students explain the strategy they used for solving the problem in 4.CM.1a.  Discuss with the students the appropriateness of using estimation to solve the problem. 


4.CM.3             Provide reasoning both in written and verbal form



Have students explain verbally whether Diego has enough money to buy the pet iguana and supplies in 4.CM.1a.  Have them then write their explanations.


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


4.CM.4             Organize and accurately label work



Emilio works in a local store for 10 dollars an hour.  On Monday he worked 2 hours, on Tuesday he worked 4 hours, on Wednesday he worked 3 hours, on Thursday he worked 5 hours, and on Friday he worked 2 hours.  Create a chart to display the number of hours that Emilio worked and the amount of money he earned for each day of the week.


4.CM.5             Share organized mathematical ideas through the manipulation of objects, drawings, pictures, charts, graphs, tables, diagrams, models, symbols, and expressions in written and verbal form



Emilio works in a local store for 10 dollars an hour.  On Monday he worked 2 hours, on Tuesday he worked 4 hours, on Wednesday he worked 3 hours, on Thursday he worked 5 hours and on Friday he worked 2 hours.  Create a graph to display the number of hours that Emilio worked and the amount of money he earned for each day of the week. 


4.CM.6             Answer clarifying questions from others



Once a student has presented the data in 4.CM.5a, encourage students from the class to ask questions about the graph.  The student presenting should answer the clarifying questions.


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


4.CM.7             Restate mathematical solutions shared by other students



Michaela used 2 blank hundreds grids to add 46 hundredths and 63 hundredths.  She found that she had completely filled in one grid and filled in nine hundredths on another grid.  She restated the problem using the following number sentence:

  .46 + .63 = 1.09 

Using hundreds grids, add 54 hundredths and 60 hundredths. Then restate the task using a number sentence. 


4.CM.8             Consider strategies used and solutions found in relation to their own work



Elisa uses the multiplication facts 20 x 3 = 60 and 6 x 3 = 18 to help her solve
26 x 3 = 78.  How can you use this same strategy to solve 45 x 6?


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


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



Give each student a picture of a two-dimensional shape.  Have students describe the attributes of the given shape, using the following terms: 

right angle

Have the class guess what shape the students are describing. 


4.CM.10           Describe objects, relationships, solutions, and rationale using appropriate vocabulary



Give each student a picture of a three-dimensional shape.  Have students describe the attributes of the given shape, using the following terms.  


Have the class guess the shape that the student is describing.


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



Solve the number sentences below:

936 - ___ = 863

567 - ___ = 10 x 55

5,698 + ___ = 7,436

23,090 + ___ = 43,703


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Connections Strand


Students will recognize and use connections among mathematical ideas.


4.CN.1              Recognize, understand, and make connections in their everyday experiences to mathematical ideas



Wayne needed to buy 12 packages of hot dogs for a family picnic.  Find the total cost of hot dogs if each package costs $2.20.  How much change will he receive if he uses 3 ten-dollar bills to pay for the hot dogs?


4.CN.2              Compare and contrast mathematical ideas



Interpret the remainder in the problems below: 

The soccer club collected 250 dollars to give to their 6 club leaders.  If each leader receives the same amount, how much money should each leader get?  Explain your answer. 

The baseball team has a game at another school.  There are 15 students on the team.  If a school van can only transport 6 players, how many vans are needed to transport to the other school?  Explain your answer. 


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




Show the class a one-liter bottle of soda and discuss how 1 liter is equal to 1000 milliliters.  Then have students estimate and record the capacity of the various containers, followed by measuring and recording the actual capacity of the containers.  Have them compare their estimates to the actual capacity of the containers. 


Students will understand how mathematical ideas interconnect and build on one another to

produce a coherent whole.


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



Using money (dimes and pennies), relate 23 cents to .23 and 23 hundredths. 


4.CN.5          Model situations with objects and representations and be able to make observations.



Using blocks or containers, have students model and then state the rule that is being used for the following input/output box:










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


4.CN.6              Recognize the presence of mathematics in their daily lives



Have students create a list of ways mathematics is used in a grocery store.


4.CN.7              Apply mathematics to solve problems that develop outside of mathematics



Anthony has 14 days to read 280 pages in his social studies textbook.  How many pages should he read each day if he reads the same number of pages every day?


4.CN.8              Recognize and apply mathematics to other disciplines



Adam needs to measure 150 milliliters of water for a science experiment.  What can he use to measure the water efficiently?


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Representation Strand


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


4.R.1                 Use verbal and written language, physical models, drawing charts, graphs, tables, symbols, and equations as representations.



Write the following decimals on the board:    .43   .73   .05   .29

Have students order and compare the decimals, using money and/or a hundreds grid to show their work.  Have students write an inequality and share it with the class.


4.R.2                 Share mental images of mathematical ideas and understandings 



Have students close their eyes and envision the number that consists of 1 thousand, 3 hundreds, 4 tens, and 5 ones.  Then have the students write the number and read it back.  Repeat this process using other amounts.



Have students close their eyes and envision a road sign.  Have students describe the road sign and name the shape.


4.R.3                 Recognize and use external mathematical representations



Identify intersecting, perpendicular, and parallel lines in the classroom.  Discuss how railroad tracks are an example of parallel lines; the roads that meet at an intersection are an example of intersecting lines; and the lines a flagpole and the ground make is an example of perpendicular lines.


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



Discuss with the students that the weight of a paper clip is approximately 1 gram and the weight of a large book is about 1 kilogram.  After sharing this information, ask the students which unit should be used to measure the weight of their pencil, their desk, and an eraser? 


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


4.R.5                 Understand similarities and differences in representations



Discuss with students how 56 hundredths can be represented using money.


4.R.6                 Connect mathematical representations with problem solving



Nina needs to cut two pieces of string, one that is ½ the length of the line segment below and one that is ¼ the length of the line segment below:

Using a ruler, measure the line segment in inches. Determine what is 1/2 and 1/4 of the measured length. Discuss how Nina can use this information to help her cut the two pieces of string.


4.R.7                 Construct effective representations to solve problems



If one square represents 1 square foot on a grid, solve the problem below:  Janice wants to put carpet on the floor of her bedroom.  She measured her room and found the length of her room to be 12 feet and the width of her room to be 10 feet.  Represent Janice’s room using the grid paper and determine how many square feet of carpet is needed to cover Janice’s floor. 


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


4.R.8                 Use mathematics to show and understand physical phenomena (e.g., estimate and represent the number of apples in a tree)


4.R.9                 Use mathematics to show and understand social phenomena (e.g., determine the number of buses required for a field trip)


4.R.10               Use mathematics to show and understand mathematical phenomena (e.g., use a multiplication grid to solve odd and even number problems)


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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


4.N.1                 Skip count by 1,000’s



What is the next number in the pattern?  1295, 2295, 3295, _____ 



Which number belongs in the blank:  _____, 2295, 3295, 4295 



What number is 1000 less than 5295?


4.N.2                 Read and write whole numbers to 10,000



Write a 5-digit number with a 1 in the ten thousand place and a 3 in the hundreds place.



Write the word name for 6,734.



Write the greatest number using the digits 4, 6, 2, 9.



Write the smallest number using the digits 4, 6, 2, 9. 


4.N.3                 Use place value to compare numbers to 10,000



Show a table with the populations of 5 different states (e.g., New York, New Jersey, Pennsylvania, Massachusetts and Vermont). Have students identify:

·        the state on the table that has the largest population

·        the state on the table that has the smallest population

·        the states in order from the smallest population to the largest

·        population

Compare the population of two of the states using >, < or =.  



Fill in the blank with >, <, or = .  356,568 ___ 365,589


4.N.4                 Identify and use: 10 ones = 1 ten, 10 tens = 1 hundred, 10 hundreds = 1 thousand, 10 thousands = 1 ten thousand



How many hundreds are in the number 1,800?  How many tens are in the number 1,800? How many ones are in the number 1,800? 



Matt has 1,800 baseball cards.  Explain how Matt could stack the baseball cards into stacks of 10, 100, or 1000.



What is the least number of base-ten blocks you need to represent 10,000?  What is the most number of base-ten blocks you need to represent 10,000?


4.N.5      Recognize equivalent representations for numbers up to four digits and generate them by decomposing and composing numbers



Decompose the number 2,400


4.N.6                 Understand, use, and explain the associative property of multiplication



Determine if the following equation is true or false.

(4 x 6) x 5 = 4 x (6 x 5) 

Explain your answer.



Fill in the missing number in the number sentence.  (3 x 5) x 2 = 3 x ( ___ x 2).


4.N.7                 Develop an understanding of fractions as locations on number lines and as divisions of whole numbers



Order fractions on a number line.



Identify fractions on a number line.



Four children are sharing 10 chocolate bars.  If each chocolate bar is the same size and each child must receive an equal amount, how much will each child receive?


4.N.8                 Recognize and generate equivalent fractions (halves, fourths, thirds, fifths, sixths, and tenths) using manipulatives, visual models, and illustrations



Use an area model (e.g., hexagon from a set of pattern blocks) and make several outlines of the hexagon on a piece of paper.  Ask students to use their pattern blocks to find as many single-fraction names for the region as possible.  Ask students to write about their discoveries.



Draw a picture to show an equivalent fraction for ¼ and explain the picture.



Find different ways (with manipulatives, pictures, etc.) to show ½. 


What fraction pairs name the same amount?  (e.g., 2/5 and 4/10, 5/8 and 2/3).


4.N.9                 Use concrete materials and visual models to compare and order unit fractions or fractions with the same denominator (with and without the use of a number line)



Monique ate 1/3 of an apple pie while Joe ate 1/6 of a lemon pie.  Who had a bigger piece of pie if both pies are the same size?  Use words or pictures to explain your reasoning.



Order from greatest to least:  1/5, 1/8, 1/3, and 1/10.  Explain the way you ordered the fractions using visual models and/or illustrations. 



Which fraction is greater?  1/5 or 1/9?  Which fraction is greater, 5/7 or 3/7

Explain your answer. 



Label ¾ on the number line below.


4.N.10               Develop an understanding of decimals as part of a whole



Show a circle divided into tenths with 2 sections shaded.  Have students identify which describes the part of the circle that is shaded:




Four friends counted their change: Sarah has 9 dimes, Brian has 2 quarters and 2 pennies, Alyssa has 1 quarter and 7 dimes and Laura has 15 nickels. Write the amount of change for each person, using decimals.   Who is closest to having a whole dollar? 


4.N.11               Read and write decimals to hundredths, using money as a context



Write 1/10 in two different ways.



Three friends counted their change: Sarah has 3 dimes, Brian has 10 nickels, Alyssa has 2 quarters and 6 nickels, and Laura has 3 quarters and 3 nickels.  Show how much money each friend has by writing a decimal; then tell which friend has the least amount of money. 


4.N.12               Use concrete materials and visual models to compare and order decimals (less than 1) to the hundredths place in the context of money



Circle the coins to represent each decimal listed below:

Order the decimals from least to greatest.  Order the decimals from greatest to least. 



Fill in the blank with <, >, or =  .
0.37___ 0.67, 0.3 ___ 30



Is .67 greater than or less than six tenths?  Explain your answer.



Provide Richter Scale Ratings for earthquakes in 5 countries.  Ask the following questions.  Which earthquake had a rating of _____?  Which earthquake was weaker than _____ but stronger than _____?  What was the strongest earthquake?  What was the weakest earthquake?  Order the strengths of the earthquakes from greatest to least. 


Number Theory                 


4.N.13               Develop an understanding of the properties of odd/even numbers as a result of multiplication



Give at least three examples and explain your answer for each question below.  If you multiply 2 even numbers will the product be odd or even?  If you multiply 2 odd numbers will the product be odd or even?  If you multiply an even and an odd number will the product be odd or even? 



If you multiply the numbers 3 and 4, will the product be odd or even?  Explain using words and pictures.


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




4.N.14               Use a variety of strategies to add and subtract numbers up to 10,000



Last year 38,399 people attended a concert.  This year 10,000 more people attended the concert.  What was the attendance for this year?  Explain how you solved this problem.


4.N.15               Select appropriate computational and operational methods to solve problems



Maria entered 75,039 into the calculator.  Then she pressed one of the operation signs and the number 4,482.  The calculator displayed 70,557.  What operation key did Maria press?  Explain your answer. 


Randy has 36 baseball cards he wants to give to 3 of his friends.  Each friend will receive the same amount of baseball cards.  What number expression will help him determine how many baseball cards he will give to each friend?


4.N.16               Understand various meanings of multiplication and division



Display the different ways to represent the number expression 6 x 5 and record in the chart below. Do the same for 5 x 6.  Discuss differences and similarities.

Repeated Addition


Skip Counting




Area Model


Circle and Stars


Real World Connection




Display the different ways to represent the number expression 15 ÷ 3 and record in the chart below. 





Repeated Subtraction


Real World Connection


Multiplication as an Inverse




Find the product of two numbers using a calculator without using the multiplication key (e.g., 6 x 4 can be found by pressing + 4 = = = = = = ).  Ask students to explain how this works.



Find at least two ways to determine 61 ÷ 14 without using the divide key.  Explain how this works. 


4.N.17               Use multiplication and division as inverse operations to solve problems



What multiplication number expression could you use to solve 56 ÷7?  Explain your answer.


4.N.18               Use a variety of strategies to multiply two-digit numbers by one-digit numbers (with and without regrouping)



Find the product of 23 and 2 using a variety of strategies.

4.N.19               Use a variety of strategies to multiply two-digit numbers by two-digit numbers (with and without regrouping) 



Use a variety of strategies to find the product of 23 and 24.


4.N.20               Develop fluency in multiplying and dividing multiples of 10 and 100 up to 1,000



How could you use 4 x 12 = 48 to solve 400 x 12?



Multiply mentally:  7 x 200. 


4.N.21               Use a variety of strategies to divide two-digit dividends by one-digit divisors (with and without remainders)



Using place value models, repeated subtraction or traditional algorithm, divide 56 by 4.


4.N.22               Interpret the meaning of remainders



Amelia has a rope that is 32 feet in length.  She  wants to make 7-foot jump ropes.  How many jump ropes can she make? How much rope remains?



If a school van can hold 9 students, how many school vans would you need to take 23 students on a field trip?  How many seats will be empty? Discuss the remainder.



You have 45 pieces of candy to share equally with 7 children.  How many pieces of candy will each child receive?  How many pieces will you have left over? 



There are 46 ounces of lemonade in a pitcher.  How many 12-ounce bottles can you fill?  How many ounces of lemonade will be left over?


4.N.23               Add and subtract proper fractions with common denominators



Use pattern blocks to represent the number sentence 1/6 + 2/6. 



Miss Smith ate 2/8 of a pizza pie.  Mr. Smith ate 3/8 of the pizza pie.  What fractional amount of the pizza pie remains?


4.N.24               Express decimals as an equivalent form of fractions to tenths and hundredths



Show a rectangle divided into tenths with some of it shaded.  What fractional amount of the figure is shaded?  Write a decimal that represents the amount shaded.



Jonathan states that he has more money than Nancy.  Jonathan has $0.25 compared to Nancy who has ¼ of a dollar.  Using a hundreds grid prove that Jonathan’s statement is incorrect.  


4.N.25               Add and subtract decimals to tenths and hundredths using a hundreds chart



Display a hundreds chart with .30 shaded in and .12 of those that were shaded crossed out. Write a number sentence to match the picture.



Have students select items from a menu, fill out an order form, and calculate total cost. 


Students will compute accurately and make reasonable estimates.




4.N.26              Round numbers less than 1,000 to the nearest tens and hundreds



The attendance at last night’s basketball game was 9,287 people. Round this number to show the approximate number of people at last night’s game.


4.N.27               Check reasonableness of an answer by using estimation



Anthony added the two numbers 749 and 54. His answer was 695. Did he make a mistake? Explain how estimating could have helped him prior to calculating the sum.


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Algebra Strand


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


Variables and Expressions


4.A.1                 Evaluate and express relationships using open sentences with one operation



Marc went scuba diving with his family.  He saw a total of 30 jellyfish, lobsters and horseshoe crabs.  There were 2 more lobsters than jellyfish.  There were 2 fewer horseshoe crabs than jellyfish.  How many of each sea animal did he see?


Students will perform algebraic procedures accurately.


Equations and Inequalities


4.A.2                 Use the symbols <, >, =, and ≠  (with and without the use of a number line) to compare whole numbers and unit fractions and decimals (up to hundredths)



Order the fractions 1/5, 1/8, 1/3 and 1/10 from least to greatest.


4.A.3                 Find the value or values that will make an open sentence true, if it contains < or



Fill in the blank with a number that will complete the following inequality:

3 < _____

5 > _____


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


Patterns, Relations, and Functions


4.A.4                 Describe, extend, and make generalizations about numeric () and geometric patterns



Maria wrote the number pattern: 1295, 2295, and 3295.  Write the next three numbers in Maria’s pattern.  Write a rule to describe Maria’s number pattern.



Which number belongs in the blank of the following number pattern:

  _____, 2295, 3295, 4295.


4.A.5                 Analyze a pattern or a whole-number function and state the rule, given a table or an input/output box



Using the information in the chart below, state the rule.











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Geometry Strand


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




4.G.1                 Identify and name polygons, recognizing that their names are related to the number of sides and angles (triangle, quadrilateral, pentagon, hexagon, and octagon)



Use The Greedy Triangle by Marilyn Burns to help students recognize the types of polygons based on the number of sides.  What is the next polygon shape in the pattern:  triangle, quadrilateral, _____?  Given a sheet with pictures of various polygons, ask students to circle the quadrilaterals and explain why they are quadrilaterals.



Present a shape riddle to students (e.g., I have no corners, I have no sides, what shape am I?).   Have students create a shape riddle.


4.G.2                 Identify points and line segments when drawing a plane figure



Use dot paper, geoboards, or toothpicks with mini marshmallows to make polygons.


4.G.3                 Find perimeter of polygons by adding sides



Show students a rectangular model of Miss Young’s backyard.  Ask students the following question: 

How could Miss Young figure out how much fencing she would need to buy to enclose her backyard?

Then have students figure how much fencing she would need if her backyard was twice the size.  



Measure the perimeter of different objects in the classroom that are polygons, such as the blackboard, bulletin board, or door.



Carol’s School Supply Store sells paper to be used as a border at 20 cents per foot.  Miss Ortiz’s square bulletin board is 6 feet long on one side.  How much would it cost for Miss Ortiz to put border paper all the way around her bulletin board? 



The perimeter of a square is 20 inches.  How long is each side?  Explain how you determined the length of each side.


4.G.4                 Find the area of a rectangle by counting the number of squares needed to cover the rectangle.



Use the area model of multiplication so students can see what happens to the area of a rectangle with a length of 5 cm when it is multiplied by a width that is greater than or less than 1 cm.



If the classroom floor has square tiles, use masking tape to section off areas.  Have students find the area of various sections.


4.G.5                 Define and identify vertices, faces, and edges of three-dimensional shapes



Give students patterns on a piece of paper that, when assembled, form a three-dimensional shape.  Have students predict what shape will be created when the pattern is cut and assembled.  Have students create a chart identifying the number of vertices, faces, and edges.



Present a shape riddle to students such as the riddle below:

I have 6 faces, 8 vertices, and all 6 faces are squares, what am I? 

Have students come up with their own shape riddles.



Categorize three-dimensional shapes according to vertices, faces, etc.


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


Geometric Relationships


4.G.6                 Draw and identify intersecting, perpendicular, and parallel lines.



Using a geoboard or dot paper, create a pair of intersecting lines.




Using a geoboard or dot paper, create a pair of perpendicular lines.



Using a geoboard or dot paper, create a pair of parallel lines. 



Identify real-world examples of intersecting, perpendicular and parallel lines.



Is the statement below true or false?  Explain your answer. 

A rectangle has two different pairs of parallel sides


4.G.7                 Identify points and rays when drawing angles



Draw angles of various sizes and identify points and rays.


4.G.8                 Classify angles as acute, obtuse, right, and straight



Classify the angles of various pattern blocks.



Show different times on an analog clock and ask students to identify the type of angle.  Then ask students to give a time on an analog clock that would represent an acute, right, obtuse, or straight angle.



Review the four types of angles by engaging the students in "Angle Simon Says" where students use their arms to represent an acute, right, obtuse, or straight angle.


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Measurement Strand


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


Units of Measurement


4.M.1                Select tools and units (customary and metric) appropriate for the length being measured



Name something in the classroom that you would measure in inches.



Name something in the classroom that you would measure in feet.



Name something in the classroom that you would measure in yards.



Name something in the classroom that you would measure in meters.


4.M.2                Use a ruler to measure to the nearest standard unit (whole, ½ and ¼ inches, whole feet, whole yards, whole centimeters, and whole meters)



Ask students to measure objects in the classroom to the nearest inch, ½ inch, ¼ inch, foot, yard, centimeter, or meter.


4.M.3                Know and understand equivalent standard units of length:

12 inches = 1 foot

3 feet = 1 yard 



Order the following students in order of their height from shortest to tallest:  Carla, 62 inches; Melissa, 5 feet; Kerry, 55 inches; Tom, 6 feet.


4.M.4                Select tools and units appropriate to the mass of the object being measured (grams and kilograms)



Name something in the classroom that you would measure in grams.



Name something in the classroom you that would measure in kilograms.



Name something in the classroom that you would measure in centimeters.


4.M.5                Measure mass, using grams



Ask students to weigh a variety of items on a balance scale using grams (e.g., 10 paperclips).


4.M.6                Select tools and units appropriate to the capacity being measured (milliliters and liters)



Name something in the classroom that you would measure using milliliters.



Name something in the classroom that you would measure using liters.


4.M.7     Measure capacity, using milliliters and liters



Fill a variety of containers with water.  Have students measure the capacity of each container in milliliters or liters.


Students will use units to give meaning to measurements.




4.M.8                Make change, using combined coins and dollar amounts



Cathy bought a pencil for $0.25, a notebook for $2.19, and an eraser for $0.67.  She gave the cashier $5.00 to pay for the three items.  How much change should she receive?



A cashier has to give a customer $0.55 change.  If she has no quarters or nickels, list 2 coin combinations she could use to give the customer the correct change.


4.M.9                Calculate elapsed time in hours and half hours, not crossing A.M./P.M.



It takes Jeff 2 ½ hours to travel from New York to Connecticut.  If he arrived in Connecticut at 7:00 p.m. at what time did he leave New York?



Andy wanted to find out what time he would be done with his homework if he started at 10:45 and if he spent 25 minutes working on it.   To figure out the time, Andy added 10:45 and 25 together to come up with the time of 10:70.  Did Andy use the correct procedure to find out his finish time? 


4.M.10              Calculate elapsed time in days and weeks, using a calendar



Veronica’s birthday is September 25.  Her brother’s birthday is 10 days later.    What is the date of her brother’s birthday?


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Statistics and Probability Strand


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


Collection of Data           


4.S.1                  Design investigations to address a question from given data



Show students a table or graph containing data from a newspaper or magazine. Pose questions to students based on the information provided.


4.S.2                  Collect data using observations, surveys, and experiments and record appropriately



Pose a question to the class such as "What is your favorite pet?"  Ask students to survey at least 10 people and share their results with the class the next day. 


Organization and Display of Data              


4.S.3                  Represent data using tables, bar graphs, and pictographs



Represent data collected from observations, surveys, or experiments using tables, bar graphs, and pictographs.


Analysis of Data


4.S.4                  Read and interpret line graphs



Show students a line graph from a newspaper or magazine and pose the following questions to the students:

What is the pattern? 

What is the highest/lowest number?

What does the graph show?


Students will make predictions that are based upon data analysis.


Predictions from Data


4.S.5                  Develop and make predictions that are based on data



Show students a chart of average rainfall data in a specific geographic area for the past several years.  Based on the data, ask students to predict the average rainfall next year for the same month in the same geographic area.


4.S.6      Formulate conclusions and make predictions from graphs



Given a graph, have students make predictions and draw conclusions.



Have students determine how much time they spend on certain activities each day, such as reading, sleeping, playing outside, watching television, and eating.  They should present their findings to the class and use a pictograph, a bar graph or manipulatives to display their results.  Based on the information they presented, ask them to make one statement about their day. 


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