# College and Career Readiness Standards Narrative Summary for Fourth Grade Mathematics

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1 College and Career Readiness Standards Narrative Summary for Fourth Grade Mathematics In Grade 4, instructional time should focus on three critical areas: (1) developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends; (2) developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; (3) understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry. 1. Students generalize their understanding of place value to 1,000,000, understanding the relative sizes of numbers in each place. They apply their understanding of models for multiplication (equal-sized groups, arrays, area models), place value, and properties of operations, in particular the distributive property. They develop fluency with efficient procedures for multiplying whole numbers; understand and explain why the procedures work based on place value and properties of operations; and use them to solve problems. Students apply their understanding of models for division, place value, properties of operations, and the relationship of division to multiplication as they develop, discuss, and use efficient, accurate, and generalizable procedures to find quotients involving multi-digit dividends. They select and accurately apply appropriate methods to estimate and mentally calculate quotients, and interpret remainders based upon the context. 2. Students develop understanding of fraction equivalence and operations with fractions. Students extend previous understandings about how fractions are built from unit fractions, composing fractions from unit fractions, decomposing fractions into unit fractions, and using the meaning of fractions and the meaning of multiplication to multiply a fraction by a whole number. 3. Students describe, analyze, compare, and classify two-dimensional shapes. Through building, drawing, and analyzing two-dimensional shapes, s deepen their understanding of properties of two-dimensional objects and the use of them to solve Operations and Algebraic Thinking Use the four operations with whole numbers to solve problems. Interpret a multiplication equation as a comparison Multiply or divide to solve word problems involving multiplicative comparison, Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Gain familiarity with factors and multiples. Generate and analyze patterns. Number and Operations in Base Ten Generalize place value understanding for multi-digit whole numbers. Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparison. Use place value understanding and properties of operations to perform multi-digit arithmetic. Page 1 of 61 Greenville Public Schools-

2 Number and Operations Fractions Extend understanding of fraction equivalence and ordering. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. Understand decimal notation for fractions, and compare decimal fractions. Measurement and Data Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. Represent and interpret data. Geometric measurement: understand concepts of angle and measure angles. Geometry Draw and identify lines and angles, and classify shapes by properties of their lines and angles. Page 2 of 61 Greenville Public Schools-

4 Topic: Operations and Algebraic Thinking The second ribbon is 4 times as long as the first ribbon because (16 = 4 x 4). *Have s complete a Venn diagram that is labeled multiples of - and multiples of. Have s complete the Venn by writing examples of the multiples in the sections of the Venn. Discuss how the numbers are that many times more than one factor and a different times more than the other. Venn Diagram Math Journal Questioning & Answering Sessions ( statements in your discussion. For example, say not only that the oil spill was large, but that 5 million gallons were spilled, or that the oil spill was 40 times larger than the next-worst oil spill. Provide illustrations to support your reasoning. *Use a hundreds chart and ask s to shade in the multiples of 2, 3, 4, etc. Discuss how the numbers in the patterns are 2, 3, 4, etc times more that the starting number. Hundreds Chart Paper Pencils Page 4 of 61 Greenville Public Schools-

6 Topic: Operations and Algebraic Thinking You have 10 pencils. Your partner has 3 times as many pencils. Students explain their results and record multiplication/division equations. Repeat using a scenario describing an additive comparison. For example: You have 10 pencils. Your partner has 13 pencils. Again, s explain and record their results, this time using addition/subtraction equations. Students share their results. Repeat with other scenarios. Chart Paper Higher Order Questioning ( *Put the following story problem up for s to see (chart paper, document camera). Then read it aloud. The bakery stocks 4 times as many donuts as muffins. If they stock 144 donuts, how many muffins are there? Distribute grid paper to s. Ask s to use the paper as a model to solve the problem. Give time for the s to solve the problem. Have the s come up to show their thinking while they explain their strategy. To help all s learn to think aloud and solve problems through mental math or estimation, have another repeat the strategy given in their own words. Repeat these steps using other word problems. Document Camera Grid paper Math Journal Higher Order Questioning Open-end Oral explanations Page 6 of 61 Greenville Public Schools-

7 Topic: Operations and Algebraic Thinking 1 st 4.OA3 Solve multistep word problems posed with whole numbers and having whole number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. How can patterns help me solve a problem? How can I identify appropriate operations to solve word problems? How do I interpret remainders in division word problems? Identify the differences among +, -,, and word problems. Perform +, -,, and with whole numbers. Interpret remainders and how they affect the whole number answer in a problem. Write equations using variables to represent the unknown for multi-step word problems. Evaluate the reasonableness of an answer by using estimation strategies or mental math strategies. Write an equation consisting of multiple operations to reflect the situation(s) in a word problem. Students work together in groups to create a problem-solving flow chart. Display the flow chart so that s can follow the steps as they work with math problems. Students explain how the flow chart can be used with 2-step or multi-step word problems. Allow s to provide feedback to each group as they apply the flow charts to a problemsolving situation. Help s translate word problems to pictures and symbols by drawing representational pictures of a problem. Use stick figures, tally marks, and cross-outs to show the action of a problem. Discuss the strategies used in the problemsolving process, including: -rounding or using compatible numbers to estimate the answer -drawing representational pictures or diagrams -using objects or manipulatives to create a model -using number lines -organizing data on a chart or table -determining reasonableness of answers. Chart paper Markers Computers Number lines Math Journal Stick figures Drawing paper Open-end Oral explanations Higher order ( ELA s: Determine the meaning of specific math words or phrases in the lesson. Create word problems using the specific vocabulary. Use precise math language to explain how to solve multistep word problems. Science s: The class has 144 rubber bands with which to make rubber band cars. If each car uses six rubber bands, how many cars can be made? If Page 7 of 61 Greenville Public Schools-

8 Topic: Operations and Algebraic Thinking Select a word problem that matches a specific equation. Solve addition and subtraction word problems that include the following situations: result unknown, total unknown, change unknown, difference unknown, bigger unknown, and smaller unknown. A perfect trio involves three whole numbers. Using three numbers, add the first two numbers together then divide the sum by the third number. Using the same three numbers, subtract the second from the first number and then multiple the difference by the third. The trio is perfect if the two outcomes are equal. Can you find three whole numbers that are perfect trios? Paper Pencils Open-end ( there are 28 s, at most how many rubber bands can each car have (if every car has the same number of rubbers)? Solve multiplication and division word problems that include the following situations: equal groups, arrays of objects, and comparison. Page 8 of 61 Greenville Public Schools-

9 Topic: Number and Operations in Base Ten2 1 st 4.NBT1 Recognize that in a multi digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that = 10 by applying concepts of place value and division. Why is it important to know what each place value represents in a multidigit number? How does our base- 10 number system work? How does understanding the base-10 number system help me add and subtract? Multiply and divide by multiples of 10. Show understanding of the relationship between place values by decomposing equations. Model place value relationships using base-ten blocks in the place value frame (ex. 10 x 50 represented as 5 tens each taken 10 times). Justify understanding by writing statements using times as many. Provide multiple opportunities in the classroom setting and use real-world context for s to read and write multi-digit whole numbers. Students need to have opportunities to compare numbers with the same number of digits, e.g., compare 453, 698 and 215; numbers that have the same number in the leading digit position, e.g., compare 45, 495 and 41,223; and numbers that have different numbers of digits and different leading digits, e.g., compare 312, 95, 5245 and 10,002. Students also need to create numbers that meet specific criteria. For example, provide s with cards numbered 0 through 9. Ask s to select 4 to 6 cards; then, using all the cards make the largest number possible with the cards, the smallest number possible and the closest number to 5000 that is greater than 5000 or less than Math Journal Chart paper Higher order Questioning Open-end Student Responses Exit Tickets ( ELA s: Explain the place value relationship between the thousands place and the hundreds place in your math journal. Students work with a partner and a set of base 10 blocks, including the thousand cube. Students form the number 1111 with base 10 blocks. Students line up or stack blocks to show that one thousand cubes is equivalent to 10 flats, that one flat is equivalent to 10 rods, and that one rod is equivalent to 10 units. Student partners discuss and describe a base ten block for 10,000 (a large Base 10 Blocks Higher order Questioning Open-end Student Responses Page 9 of 61 Greenville Public Schools-

10 Topic: Number and Operations in Base Ten2 rod formed from 10 of the thousand cubes) Students partners and also discuss and describe a base ten block for 100,000 (a flat that is formed from ten 10,000 rods or 100 of the thousands cubes). Discuss the pattern that the blocks illustrate (e.g., Each place value is ten times the value of the place to its right and one-tenth the value of the place to its left.). ( Place Value Triangle Deal 15 cards to each player facedown with one card in the top row, two cards in the second row, three in the third row, four in the fourth row, and five cards in the fifth row. Players turn over the top card. The player with the highest card completes the math talk sentence ( is greater/less than ) and wins a point. If there is a tie both players win a point. Players turn over the second row of cards which represents a 2-digit number. The player with the highest 2-digit number wins 2 points. Play continues with the remaining rows. The highest 3-digit number wins 3 points, the highest 4-digit number wins 4 points and the highest 5-digit number wind 5 points. Keep a record of your scores as you play. Shuffle the cards and play another round. Deck of Card Page 10 of 61 Greenville Public Schools- Paper Pencils Exit Tickets

11 Topic: Number and Operations in Base Ten2 The first player to reach 30 points wins the game. ( Page 11 of 61 Greenville Public Schools-

13 Topic: Number and Operations in Base Ten2 >, =, and < to record comparisons of two multi-digit numbers. different so that is where I would compare the two numbers. ( Place the following numbers on the board: 3, 2, 9, 6, 4. Ask s to use the digits to make the largest number possible (96,432). Then have s write that number in expanded form. Ask s to use all of the digits to make the smallest number possible (23,496). Then have s write that number in expanded form. Continue the activity with some of the following ideas: 1. A number larger than 30,000 that is divisible by A number between 25,000 and 60, An odd number smaller than 26,000. Chart Paper Higher order Questioning Page 13 of 61 Greenville Public Schools-

14 Topic: Number and Operations in Base Ten2 1 st 4.NBT 3 Use place value understanding to round multi digit whole numbers to any place. What information is needed in order to round whole numbers to any place value? How do I explain the rules for rounding? What strategies do I use to round multidigit whole numbers up to a million to any place value? Round multi-digit whole numbers up to the millions place. Use an open number line to show reasoning and understanding of rounding up to the millions place. Identify the largest and smallest number that rounds to a specified number. Create numbers that would round to a specified number (ex. List 2 numbers that would round to 100, ,789 would round to 100,00 and 104,999 would round to 100,000) and be able to explain the reasoning for your answer. Scenario I: The most popular boy band is coming to town for a concert. The concert tickets cost \$ Parking at the arena cost \$15. About how much will you pay to attend the concert? How do you know? Scenario II: Robert and his family traveled from Atlanta, Georgia to Washington D.C. to visit the Martin Luther King Monument. If they traveled the same route back to Georgia, about how many miles would they drive? Explain your answer? For the previous problem, determine the exact mileage for Robert s family s trip. Based on your estimation, is your answer reasonable? Explain. Pose the following problem to the class: What are two decimal numbers that can have an estimated sum of 9 and an estimated difference of 3? Permit time for the s to solve the problem. Discuss and evaluate the solutions with the class. Chart paper Computer games Chart paper Chart paper Chart paper Higher order Questioning Open-end Oral explanations Exit Tickets ( ELA s: Interpret math information presented visually, orally or quantitatively. (e.g., in charts, graphs, diagrams). Use your understanding of place value to explain rounding multi-digit numbers to a struggling classmate. Create rounding problems and have them to explain the process orally. Check for accuracy in their explanations. Page 14 of 61 Greenville Public Schools-

16 Topic: Number and Operations in Base Ten2 1 st 4.NBT 5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays and/or area models. Why is it important for me to understand place value and properties of operations to perform multi-digit arithmetic? What strategies can I use to illustrate and explain multiplication calculations through equations, rectangular arrays, and/or area models? Model multiplication by using base ten blocks, area model, and rectangular arrays. Find the product of up to a four-digit by a one-digit number. Explain how to find the product of up to a four digit by a onedigit and two-digit by a two-digit. Find the product of a two-digit by two digit number and explain the strategy that was used. Students use base ten blocks, area models, partitioning, compensation strategies, etc. when multiplying whole numbers and use words and diagrams to explain their thinking. They use the terms factor and product when communicating their reasoning. Multiple strategies enable s to develop fluency with multiplication and transfer that understanding to division. Use of the standard algorithm for multiplication and understanding why it works, is an expectation in the 5 th grade. This standard calls for s to multiply numbers using a variety of strategies. Base 10 blocks Chart paper Students use base ten materials Base 10 blocks and place value mats to model multiplication of multi-digit Place Value Mats numbers. For example, to show Formative 6 x 248, s build the factor 248 using 2 flats,4 rods, and 8 Unit cubes unit cubes. Students create six identical sets of 248 with base ten materials. Students then combine the sets, beginning with the unit cubes, making exchanges of unit cubes for rods as needed, then repeating with the tens, exchanging 10 rods for one flat, etc. Students explain their reasoning to a partner, using correct terminology, including compose, factor, and product. Students create simple Page 16 of 61 Greenville Public Schools- Higher order Questioning Open-end Formative s Summative s ( ELA s: Have s write what they learned in their math journals. Have them to provide sample problems using equations, rectangular arrays and/or area models to support their reasoning. Use formal English to explain the different strategies you learned when multiplying 2 two- digit numbers.

17 Topic: Number and Operations in Base Ten2 story problems to match their models. ( What s Your Problem? Ask s to create a multiplication problem that has a product between 500 and 600. Have s share their problems. This is a good opportunity for s to use their vocabulary knowledge of factor and product when sharing their problem. Chart paper Page 17 of 61 Greenville Public Schools-

18 Topic: Number and Operations in Base Ten2 1 st 4.NBT 6 Find whole number quotients and remainders with up to four digit dividends and one digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. How do I divide whole numbers using strategies based on place value, properties of operations, and the relationships between multiplication and division? Decompose numbers based on place value to find the quotient of a large-number divided by a one-digit number. Divide up to four-digit numbers that will result in whole numbers and remainders. Interpret remainders and how they affect the quotient. Model division by using the area model, rectangular arrays, and writing equations. Write an explanation describing how the quotient was found. Students make a rectangle and write 6 on one of its sides. They express their understanding that they need to think of the rectangle as representing a total of Students think, 6 times what number is a number close to 150? They recognize that 6 x 10 is 60 so they record 10 as a factor and partition the rectangle into 2 rectangles and label the area aligned to the factor of 10 with 60. They express that they have only used 60 of the 150 so they have 90 left. 2. Recognizing that there is another 60 in what is left they repeat the process above. They express that they have used 120 of the 150 so they have 30 left. 3. Knowing that 6 x 5 is 30. They write 30 in the bottom area of the rectangle and record 5 as a factor. Chart paper Story problems Math Journal Higher order Questioning Open-end Performancebased tasks Oral explanations Math Journal Writing Exit Tickets ( ELA s: Write about a time you took a short cut to complete a lengthy task in your math journal. Share your experience with a classmate. Page 18 of 61 Greenville Public Schools-

19 Topic: Number and Operations in Base Ten2 Students will work in small leveled groups to use base ten blocks or money to create equal groups to illustrate the division process and understand the concept of left over items. Base 10 Blocks ( Acting Out Division Pose the following problem to the class: All 24 people on the baseball team need to be driven to the game. Four players can fit in each car. How many cars are needed to transport all of the people? Use s to act out the problem in front of the remaining s. Discuss and record the division equation that reflects the solution to the problem (24 4 = 6). Story problems Story problems Math Journal Made Page 19 of 61 Greenville Public Schools-

20 Topic: Measurement and Data 2 nd 4.MD2 Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. How do I add and subtract fractions and mixed numbers? What strategies can I use to compute fractions and decimals? Convert larger units of measurement to smaller units of measurement in a single system. Recognize and identify units of measurements used to measure: length, capacity, weight, and time. Relate the size of a unit to a benchmark or mental image. Construct a number line diagram, marked in whole numbers and fractions or decimals, to represent a measurement scale. Read a measurement scale. Solve word problems related to measurement that include the following Have s complete this task: Chad has a school project that requires him to do some measuring. He has to build a structure that measures 5 feet tall using whatever materials he chooses. The materials he is looking at are cardboard, pipe cleaners, and wood. The pieces of cardboard each measure 18 inches and cost \$4.00. Each pipe cleaner is 12 inches long and cost \$1.50. The pieces of wood each measure 24 inches long and cost \$6.50. His mother only gave him twenty dollars to spend on the project. Which material(s) could Chad complete the project with, using the money his mom gave him? Which material would be his cheapest option? Most expensive? The teacher creates several story problems with holes in which numbers necessary for computation are missing. The teacher asks s for specified numbers (see example) to fill in the holes. After s supply the necessary numbers, s work in pairs or small groups to solve the problem, using the numbers supplied by the Story problem Chart paper Page 20 of 61 Greenville Public Schools- Math Journal Computers Story problems Chart paper Math Journal Higher order Questioning Open-end Performancebased tasks Oral explanations Formative s Formative ( ELA s: Explain about the procedures, ideas, or concepts you have learned in your math journal. Tell how you can use intervals of time, liquids, volumes, masses, and money in your everyday life. Science s: Estimate the mass of a large hailstone that damaged a car on a used-car lot. Measure the volume of water in liters collected during a rainstorm.

21 Topic: Measurement and Data situations: result unknown, total unknown, both addends unknown, change unknown, difference unknown, bigger unknown, smaller unknown, unknown product, group size unknown, and number of groups unknown. Determine how many times larger a specific unit is than another specific smaller unit. Calculate area and perimeter using a given unit. class. Example: Mr. Knight rides in bicycle races. Last month he rode in the Pedal Away in the 300 K over a 5- day period. On day 1, he rode (insert a 2- digit number < 80) kilometers, and on day 2 he rode an additional (insert a 2- digit number between 50-60) kilometers. On days 3 and 4 he rode a total of 128 kilometers. How many meters did Mr. Knight have left to ride on day 5? If he rode 20 kilometers farther on day 3 than he rode on day 4, how many kilometers did he ride on day 4? How many meters is this? Present the following problem to your s: Malachi rode his bike 268 meters to his friend s house. He then rode his bike half a kilometer to the park. How many total meters has Malachi ridden his bike? Represent your answer using a model drawing or other representation. Have s discuss the process they used to solve the problem. Chart paper Made ( A coastline reduces by an average of 4 feet per year. In an 18-month period, approximately how much of the coastline has been lost? Page 21 of 61 Greenville Public Schools-

23 Topic: Operations and Algebraic Thinking 2 nd 4.OA4 Find all factor pairs for a whole number in the range Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range is a multiple of a given one-digit number. Determine whether a given whole number in the range is prime or composite. What strategies can I use to determine if a whole number in the range is a prime or composite number? What strategies do I use to identify all the factor pairs for a whole number in a range 1-100? How do I determine if a whole number is a multiple of a given one digit number/ List factors for a given whole number. Classify numbers as prime or composite. List multiples of a given single digit number. Decide if a number is a multiple of a given one-digit number. In this lesson s will learn how to find all the factor pairs of a number by using area models. Click the link below to access the lesson. Click the link below to access the lesson find-all-the-factor-pairs-of-anumber-using-area-models * Pose this problem to your s. Randi bought some tickets to win a bicycle. The tickets are listed below. The winning ticket number was described as being a multiple of 2 and having the most factors. What was the winning ticket number? How many factors are there? Explain your reasoning. Winning tickets: Computers Paper Pencils Chart paper Higher order Questioning Open-end Oral explanations Exit tickets Student responses ( ELA s: Compare and contrast prime and composite numbers using a Venn Diagram. Share your findings with a classmate. *Definitions of prime and composite numbers should not be provided, but determined after many strategies have been used in finding all possible factors of a number. *Provide s with counters to find the factors of numbers. Have them find ways to separate the counters into equal subsets. Counters Post-it notes Math Journal Formative Page 23 of 61 Greenville Public Schools-

24 Topic: Operations and Algebraic Thinking Set up post-it notes with prime and composite numbers (maybe 5 of each or more if you want enough for the entire class). Using the post it notes, have s sort their numbers in two groups. Have them discuss their processes for sorting the numbers. ( Reverse Digits Have s work with a partner to find pairs of reverse digits that are both prime numbers. i.e.: (13, 31), (17, 71), 37, 73), (79, 97). Lead s into a discussion of why these are the only pairs less than 100 that fit this criteria. This leads into a great opportunity for you to also discuss divisibility rules. Chart Paper Made Student Responses Page 24 of 61 Greenville Public Schools-

26 Topic: Operations and Algebraic Thinking rule Add 3 or starting with the yellow hexagon, use two red trapezoids, three blue rhombuses, followed by six green triangles, followed by two yellow hexagons, etc. Student pairs present their patterns to the class while other groups of s confer and determine the rule for the mystery pattern. ( Pattern Block Functions Distribute pattern blocks to s or groups of s. Ask s to identify attributes of the shapes/blocks (corners, sides, etc.) Create a function table that shows 1 square has 4 corners. Ask s how many corners 2 squares would have. Record the information in the function table. Repeat this for different amounts of squares. Have s choose a pattern block shape. Using the number or corners of the number of sides, create a function table that represents this relationship. Have s share the function table that they created and the rule that explains the relationship with the class. Pattern blocks Function tables Formative Page 26 of 61 Greenville Public Schools-

27 Topic: Number and Operations - Fractions 2 nd 4.NF1 Explain why a fraction a/b is equivalent to a fraction (n a)/(n b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. What strategies can I use to calculate equivalent fractions? Plot, label, and identify fractions on a number line. Use a variety of visual fraction models (tape diagram, number line diagram, or area model). Write 1 and other whole numbers as a fraction. Partition a whole into smaller parts to model a fraction that is equivalent to the fraction that is already being represented. Create an equivalent fraction for a given fraction by partitioning a whole into smaller parts or by combining parts to make larger parts. Use writing to justify why two fractions are or are not equivalent. Begin instruction about equivalent fractions with concrete examples and activities. Students who are inexperienced with fractions or those who need additional help will benefit from the real life connections. Many foods can be cut into parts, then cut again to show the relationship between halves and fourths, thirds and sixths and so forth. Some foods, like dough, can be recombined to show equivalencies. Instruction should also include work with dividing groups of items such as candies into fractional sets and recombining them to create equivalent fractions. Be sure to make connections between the concrete fraction demonstration and the written representation. Each gets 2 square pieces of paper (8 by 8 ).( models first) (while s watch), followed by s following the teacher on each step). First divide both of the papers into four parts by folding it into a half and then into another half. Unfold the paper, which should now show 4 equal parts (fourths). Using only one of the papers, equally tear out four pieces (you should now have four equal pieces of one of the paper. Put 1 piece up and ask if it is equivalent to another piece (yes). Fraction examples Squared paper (8 by 8 ) Chart paper Page 27 of 61 Greenville Public Schools- Higher order Questioning Open-end Student Responses Formative s Exit tickets Formative s ( ELA s: Read math texts about plots, fractions and number lines. Write opinion pieces on math topics, supporting a point of view with reasons and information. Add visual displays of equivalent fractions to support your reasoning.

28 Topic: Number and Operations - Fractions Put 2 pieces up and ask if they are equal to one half of the other second paper (yes). See figure 1 below. Repeat with other shapes like circles divided into 4 parts. Write the fractions ½ and 2/4 and challenge s to come up with what symbol (> = <) goes in between the two fractions. Challenge s to show and prove their answers using any of the available materials. ( Have s use pattern blocks to discover equivalent fractions. For example, one trapezoid is ½ of the hexagon, and the triangle is 1/3 of the trapezoid. Pattern blocks Made Page 28 of 61 Greenville Public Schools-

29 Topic: Number and Operations - Fractions 2 nd 4.NF2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. In what ways can I use a visual fraction model to justify the comparisons between fractions? What strategies can I use to compare and order two fractions with unlike numerators and denominators? Make comparisons of fractions by using a variety of visual fraction models (tape diagram, number line diagram, or area model). Creating equivalent fractions by finding common denominators. Decompose fractions with the same denominator to justify comparisons. Draw a model to justify conclusions when comparing two fractions. Evaluate the reasonableness of a conclusion based on the benchmark fractions of 0, ½, and 1. Engagement activity: start with comparing whole numbers using the symbols >, =, <. (Use numbers that your s are comfortable with or depending on ability level have the first group do single digit # s, second group double digit numbers, and third group triple digit numbers. Start with equivalent fractions on the board and ask if they are equivalent. Next, ask s to close their eyes and imagine half a cake, half a pizza, half an apple, etc. followed by having them imagine the same objects but this time in quarters. Between each picture have them open their eyes, draw what they saw, along with the corresponding fraction. Fraction bars really helps conceptualize how a bigger denominator means a smaller fraction when the numerator stays the same. As a culminating activity have s color in and then compare the fractions. Give each pair of s a set of digit cards. Have s pull two digit cards and make a fraction with the cards. Have the s record and draw a picture of their fraction. Have s write a fraction that is greater than that the fraction they recorded, Have the s do the activity a second time writing a Fraction bars Page 29 of 61 Greenville Public Schools- Paper Pencils Set of digit cards Drawing paper Chart Paper Math Journal Higher order Questioning Open-end Performancebased tasks Student Responses Questioning and Answering Sessions Student Responses ( ELA s: Write explanatory pieces on how to compare two fractions supporting your point of view with reasons and information. Art s: Create a poster of equivalent fractions that represent common denominators.

30 Topic: Number and Operations - Fractions fraction that is less than the fraction they created. Bring the class together and share findings. ( Pull a Fraction, Make a Bigger Fraction Give each pair of s a set of digit cards. Have s pull two digit cards and make a fraction with the cards. Have the s record and draw a picture of their fraction. Have s write a fraction that is greater than the fraction they pulled. Have the s do the activity a second time writing a fraction that is less than the fraction they created this time. Bring the class together and share findings. Set of digit cards Drawing paper Chart Paper Math Journal Student Responses Formative Page 30 of 61 Greenville Public Schools-

32 Topic: Number and Operations - Fractions Write an equation that represents a specific fraction with its decomposed parts that equal that fraction. Determine if the sum of a set of fractions equal a given fraction. Use a variety of visual fraction models (tape diagram, number line diagram, or area model) to justify decompositions. c. Use a variety of visual fraction models (tape diagram, number line diagram, or area model). Create an equivalent fraction for a mixed number (write it as an improper fraction) in order to add or subtract. of the five sides by adding the two fractions together. b. Give each a card on which a fraction is written (all fractions have the same denominator). Ask s to find a partner. Partners combine the two fractions and then decompose the two fractions into a sum of different fractions. Partners continue pairing combining fractions to create new sums and decomposing fractions into addends until all s are in one large group. b. Write about how to compare two fractions with different numerators and different denominators using key vocabulary in simple and complex sentences. Write your reasoning in your math journal. b. Have s look at the calendar and write fractions for the number of weekend days and the number of weekdays for the current month. Have them explain why, no matter which month they use, the sum of the two factions will always equal 1. Index Cards Paper Pencils Math Journals Calendars Drawing paper Rulers Formative ( Art s: Use plain M & M s to create a poster that represents fractions of the different colors in a bag. Write the fraction that each color represents under the cluster of M & M s. Display your poster and explain your findings to a classmate. Represent the sum as a mixed number by joining a sufficient Page 32 of 61 Greenville Public Schools-

34 Topic: Number and Operations - Fractions Write an equation that represents a word problem. Model with mathematics. d. Create a poem to retell how to solve word problems involving addition and subtraction of fractions referring to the same whole that has like denominators. Share your poem with your classmates. Paper Pencils ( Make sense of problems and preserve in solving them. Look for and make use of structure. d. Write the following on the board: Ron has 12 flowers. He gives ¼ to Patrick and ½ to Michelle. How many flowers did Ron keep? How many did Ron give to Patrick and Michelle? Have s draw 12 flowers. Guide them to circle and count ½ of them. Then label the flowers circled with the label ½. Next, have s count the flowers and divide them in 4 equal groups. Have them label each group outside of the ½ they already circled with the label ¼. Ask s to write an equation showing the problem. Chart paper Story problems Circles Page 34 of 61 Greenville Public Schools-

36 Topic: Number and Operations - Fractions 2 nd 4.NF6 Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram. What strategies can I use to convert fractions with a denominator of 10 to an equivalent fraction with a denominator of 100? What does a decimal represent? How do I locate decimals on a number line? Write a decimal number as a fraction with a denominator of 10 or 100. Write a fraction with the denominator as 10 or 100 for decimal number. Create an equivalent fraction with 100 on the bottom for a fraction that has 10 on the bottom. Locate and label a decimal number on a number line. Look for and make use of structure. Give each or pair of s a dry erase board, marker, and play money. A dollar bill, 100 pennies, and 10 dimes. Set the value of the dollar as one whole. Students line up the 10 dimes and write words and a fraction to represent the value of one tenth. On the dry erase board. Display a place value chart explaining that the value of the dime can also be represented as a decimal number. Students draw a chart on the dry erase board and say one tenth. Continue by showing the values for two dimes, three dimes, and so on. Use a similar process to show the fraction and decimal equivalents for one penny (one hundredth), 15 pennies, and so on. Display a large number line from 0 to 1 marked in tenths. Provide each with a different fraction (denominator of 10 or100 and located between 0 and 1) written on a sticky note. Students determine the location of the fraction on the number line by changing the fraction to a decimal equivalent. Students affix the fraction to the number line and explain the placement. Dry Eraser Boards Play Money Place Value Charts Chart Paper Number lines Sticky notes made assessments Student responses Higher order observation Exit tickets Formative assessment ( ELA s: Create a funny story using decimals as characters. Draw pictures of your characters to add visual displays to your story. Share your story with your classmates. Page 36 of 61 Greenville Public Schools-

37 Topic: Number and Operations - Fractions 2 nd 4.NF7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. What strategies can I use to compare decimals using greater than, less than, and equal to symbols? How do compare and order decimals to hundredths? Represent a decimal with a visual model (number line or base ten blocks). Make comparisons of decimals by using a variety of visual models (number line or base ten blocks). Justify comparisons with a visual model. Give each a set of coins and a sheet of 1 centimeter grid paper divided into 10 x 10 squares. Students determine the value of their coins, such as thirty two cents. Students then shade the 10 x 10 grid to represent the decimal value of the coins. Student find a partner and compare their decimals using >, < or =. Repeat; until s have five decimal comparisons. Students work with a partner. Provide pairs with number sentences that compare two decimals. Students determine whether the number sentence is true or false and explain their reasoning using decimal place value and/or visual models. Set of Coins Grid Paper Squares (10 x 10) Chart Paper Number sentences Higher order Questioning Open-end Student Responses ( ELA s: Write what you learned about comparing two decimals to hundredths using their size as a measuring tool. Support your point of view with reasons and information from the lesson. Make a three column chart with the headings: Near 0, About ½, and Close to 1. Write each decimal below under the correct heading on your chart: 0.80, 0.9, 0.49, 0.96, 0.12, 0.58, 0.02, 1.06, 0.35, 3. Write three more decimals in each column. Choose one decimal from each column and explain the reasoning you used placing it on the chart. Three column chart Formative Page 37 of 61 Greenville Public Schools-

38 Topic: Measurement and Data 3 rd 4.MD 1 Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two column table. For example, know that 1 ft is 12 times as long as1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36),... How do I convert larger units of measurement to smaller units of measurement within a given system? How do I construct larger units of measurement into smaller units of measurements within a given system? Recognize and identify units of measurements used to measure, length, capacity, weight, and time. Relate the size of a unit to a benchmark or mental image. Convert larger units of measurement to smaller units of measurements in a single system. Create a two-column table of measurement equivalents. Write an equation to represent a multiplication comparison of two different units. Look for and make use of structure. Look for and express regularity in repeated reasoning. Have s complete the following tasks: Johnathan and Sarrah got to be a part of their school's field day in June. Their class got to rotate around the field and play games. 1. The first game they took part in was a frisbee throw. The team that threw the frisbee the furthest won. Johnathan's team threw the frisbee 10 meters. Sarrah's team threw the frisbee 550 centimeters. Which team won? Explain your answer. 2. The second event was a relay race where teams had to race across the field with a cup filled with water. They had to dump whatever they didn t spill into a bucket. At the end of the race, Johnathan's team filled their bucket with 4 liters of water. Sarrah's team filled their jar with 3,000 milliliters of water. Sarrah claimed that her team won. Is she right? How do you know? 3. The last event was a running race. Each team had to run one mile. The team that ran the mile in the shortest time won. Johnathan was the timekeeper when Sarrah's team ran, and Sarrah was the timekeeper when Johnathan's team ran. The times were written on the chart below. Problems Page 38 of 61 Greenville Public Schools- Paper Pencils Formative assessments observations Student Responses Open-ended Questioning Summative assessments Exit tickets made assessments ( ELA s: Engage in collaborative discussions about how the metric measuring system is different from the U.S. standard measuring system, use the information learned in the lesson to support your point of view. Science s: Estimate, then measure, the masses of two objects being used in an investigation of the effect of forces; observe that the change of motion due to an unbalanced force is larger for the smaller

39 Topic: Measurement and Data Team Total Time Johnathan 8 minutes Sarrah 540 seconds Their friend Jamie s team finished in between Johnathan and Sarrah. What could be a time that it could have taken for Jamie s team to run the race. ( mass. Have s work with a partner. Select five small classroom objects to weigh on the balance scales. Create a three column table with the headings: Object, Estimated Weight (g), and Actual Weight. Estimate the weight of each object in grams. Record each estimate on your chart. Weigh each object using gram weights. Record the actual weight on your chart. Write three statements about your data. Examples: My closest estimate was. A variety of objects for weighing Balance Scales Paper bags Formative The heaviest/lightest object I weighted was.. The stapler weighted three times more than I estimated. The eraser weighted 10 grams less than my estimate. Page 39 of 61 Greenville Public Schools-

40 Topic: Measurement and Data Making a Pound Work with a partner, or small group. Without using a scale fill a bag with dried beans until you predict it weighs about 1 pound. Weigh the bag on a scale to determine if it is more than, less than, or exactly 1 pound. Continue to add to, or remove weight from your bag until you reach 1 pound. Record each attempt in a table with the headings: ATTEMPT; ACTUAL WEIGHT; MORE THAN, LESS THAN, OR EQUALTO 1 lb? What did you learn about 1 pound from completing this task? Scale Paper bags Dried beans Recording sheets Chart paper Student Responses Made ( Page 40 of 61 Greenville Public Schools-

41 Topic: Measurement and Data 3 rd 4.MD3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor. How do I apply perimeter and area formulas? What strategies can I use to calculate the area and perimeter for rectangles in word problems? How can I apply what I have learned about measurement? What are customary units of length? How can I solve measurement problems? How can I measure perimeter? How can I find area? What are metric units of length? Find the area and perimeter of rectangular figures in real world situations. Find the unknown length of a rectangular figure when one side length and the area of the rectangle are known or when one side length and the perimeter are known. Find the length and width of a rectangle that has a specific perimeter or a specific area or a specific perimeter and area together. Write a situation equation that can be used to find the missing length of a rectangle when the length of one side and area of the rectangle is known or the length of one side and perimeter of the rectangle is known. (The Progression Document states that Given a perimeter, area, and one dimension (e.g., perimeter = 30 units, area = 50 square units, and length = 10 units), s work in small groups to find and record a rectangle that meets the given specifications. Students use the information to show how the formulas for area and perimeter can then create a real-world scenario that could be applied to the parameters given. Groups exchange problems and solve, using the appropriate formulas. Put s in pairs. Give them rulers and or yardsticks. Ask them to find the perimeter of objects in the room such s their desks, the perimeter of the chalkboard, their math textbooks. Give them a sheet of paper and have them record the objects and the perimeters. Discuss how they found the perimeter. Did they measure all sides and add them up? Or if the object was a rectangle, did they measure the length and width and double it? Remind them to record the unit of measure. Elicit why this is important. Recording Sheets Rectangles varying sizes Measuring tape Page 41 of 61 Greenville Public Schools- Rulers Yardsticks Paper Chart paper Higher order Questioning Open-end Student responses made assessments ( Art s: Provide s with construction paper, rulers, and makers. Have them draw five rectangles of different sizes, and find the area and perimeter of each triangle. ELA s: Write about their experience in their math journals. Science s: In Hawaii, some houses are raised on stilts to reduce the impact of a tsunami. The force of the tsunami on an object is greater if the object presents greater area to the

42 Topic: Measurement and Data a situation equation refers to the idea that the constructs an equation as a representation of a situation rather that identifying the situation as an example of a familiar equation). Give s rulers and paper. Ask s to design a polygon with the following parameters: 1. Polygons should have no more than 6 sides. 2. The perimeter should total between 15 and 25 inches. 3. Write the perimeter inside the shape. Rulers Paper Formative ( incoming wave. Based on a diagram of a stilt house, determine how much area the stilts present to an incoming wave. How much area would the house present to an incoming wave if it were not on stilts? Use a protractor and ruler to create a composite figure using the given specifications, and determine the area of parts of the figure. The bedroom in Samantha s dollhouse is a rectangle 26 centimeters long and 15 centimeters wide. It has a rectangular bed that is 9 centimeters long and 6 centimeters wide. The two dressers in the room are each 2 centimeters wide. One measures 7 centimeters long and the other measures 4 centimeters long. Create a floor plan of the bedroom containing the bed and dressers using your ruler and protractor. Find the area of the open floor space. In the bedroom after the furniture is in place. Paper Pencils Ruler Protractor Made Page 42 of 61 Greenville Public Schools-

44 Topic: Measurement and Data cs.org/contentstandards/tasks/1039 Partner Plots Have s collect data as a class. For example, have them work with a partner to count the number of jumping jacks a completes in one minute. Have them take turns counting for each other as the teacher calls out the times. Have the s record the data on a post-itnote and place it on the board creating a class line plot. Have the s work in pairs to write 3 or 4 questions that can be answered by the line plot. Have partner groups exchange questions. Partner groups answer the questions. Bring class together and have class share questions/answers and decide if they agree with the answers provided. Recording sheets Post-it notes Formative ( parent. Make a similar line plot for plants grown with insufficient water. Page 44 of 61 Greenville Public Schools-

45 Topic: Measurement and Data 3 rd 4.MD5 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement: a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by that turns through 1/360 of a circle is called a one degree angle, and can be used to measure angles. b. An angle that turns through n one degree angles is said to have an angle measure of n degrees. How do I define an angle measure as the fraction of the circular arc between two rays with a common endpoint? How do I calculate n one-degree angles as having a measurement of n degrees? a. Identify and angle. Identify benchmark angles (90, 180, 270, 360 ). Recognize that angles are measured within degrees of a circle. Write an angle s measurement as a fraction. Explain that an angle measurement is a fraction of a circle. Categorize angles based on their measurement (acute, obtuse, right, straight, reflex). Construct examples of an angle with a specific measurement using a protractor. Measure a given angle using a protractor. a. Students can understand this concept by using two rulers held together near the ends. The rulers can represent the rays of an angle. As one ruler is rotated, the size of the angle is seen to get larger. Ask questions about the types of angles created. Responses may be in terms of the relationship to right angles. Introduce angles as acute (less than the measure of a right angle) and obtuse (greater than the measure of a right angle). Have s draw representations of each type of angle. They also need to be able to identify angles in two-dimensional figures. a. Students can compare angles to determine whether an angle is acute or obtuse. This will allow them to have a benchmark reference for what an angle measure should be when using a tool such as a protractor or an angle ruler. a. Prepare a set of regular polygons for each group of s (a regular octagon, hexagon, pentagon, and a square.) Instruct s to measure the angles in the polygons and discuss what they notice. Debrief as a group and note that all the angles in a regular polygon are Page 45 of 61 Greenville Public Schools- Rulers Drawing paper Straw Index cards Typing paper Regular octagon Hexagon Pentagon Square Higher order Questioning Open-end Student responses made assessments Formative assessment Student Responses ( ELA s: Using geometric shapes have s create 3-D images of a car, house sun, etc Use technology to reproduce images. Encourage s to use the information they learned from the lesson to create each image. Write about their experiences in their math journals.

46 Topic: Measurement and Data b. identify an angle. equal. Paper ( Recognize that angles are measured within degrees of a circle. Recognize benchmark angles (90, 180, 270, 360 ). Explain that an angle measurement is a fraction of a circle. b. Prepare a set of angles cards for each group of 3 s. Have s work in groups to categorize the given angles (acute, right, and obtuse). Elicit a description for each type of angle with regard to their measures: Acute- less than 90 Pencils Set of angle cards Math Journal Categorize angles based on their measurement (acute, obtuse, right, straight, reflex). Right - 90 Obtuse greater than 90 and less than 180 Construct examples of an angle with a specific measurement using a protractor. Measure a given angle using a protractor. b. Two s were discussing the angles formed by the clock on the wall. They both agreed it was 3:00. Leslie argued that the angle formed was ¼ of the circle and 90 while Katie disagreed and said 3:00 showed ¾ of the circle and was 270º. Use what you know about angles to decide who was correct. Be prepared to explain why. Clocks Paper Pencils Student Responses Page 46 of 61 Greenville Public Schools-

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