Content Strand: Number Sense and Base Ten
3rd Grade MathCurriculum GuideFirst Nine WeeksGrade 3Content Strand: Number Sense and Base Ten Standard: 3.NSBT.1 Use place value understanding to round whole numbers to the nearest 10 or 100.Related Standard: 3.NSBT.4Vocabulary: whole number, rounding, ones, tens, hundreds, place value, commaExample: Round the following number to the nearest 10: 893 (890)Round the following number to the nearest 100: 893 (900) Strategies/Activities:Rounding Rap:Find your number, Look right next door, 4 or less, just ignore.5 or more, add one more. Saying 5 or more, raise the score 4 or less, let it rest. Have students circle the place value they are rounding to and underline the digit to the right. Students will then decide if the circled digit should move up by one digit or remain the same.Initial activity: Give students index cards with various numbers rounded to the nearest 10 and 100 along with cups labeled 10 and 100. Students will then determine if the number is rounded to the nearest 10 or 100 by placing the card in the appropriate cup.Activity: Have students roll dice to determine a 3-digit number. Students will round their student-created numbers to the nearest 10 and 100 and record their answers in a chart.1. Tell the students that the rule for rounding to the next greatest ten is to see if the ones place is a 5, 6, 7. 8, or 9. If the ones digit is 0, 1, 2, 3, or 4 the tens digit will not change. Have students make number lines with numbers 1–50 and tell you which ones round to 0, 10, 20,30,40, or 50. 2. Call out numbers between 60–80 and have the students tell you which ones round to 60, to 70, or to 80. Extend the activity to round to hundreds and thousands (for early finishers). 3. Have students use scratch paper and spinners and work in groups of four. One of the students spins the spinner twice to get a two-digit number. The other students write the number down and round it to the nearest ten. They take turns spinning the spinner. Extend the activity to round to hundreds and thousands. You may also use dice for this activity. 4. Prepare number 20 cards with numbers from 11 to 99 on them. Place them face down in a stack. Have students draw a card from the stack and round it to the nearest ten. 5. Prepare number 20 cards with numbers from 101 to 999 on them. Place them face down in a stack. Have students draw a card from the stack and round it to the nearest ten or hundred. 6. Give students objects, such as confetti or candy to group into tens. Have them tell whether the group is to be rounded up or down to get to the nearest ten. 7. Have each student write the numbers 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 on cards. Call on students to write a three-digit number on the board. The other students then hold up the card that represents the nearest tens place. Extend the activity to round to hundreds and thousands (early finisher). 8. Use digit cards 0 to 9 and have students pick four cards and put them on a chart that has thousands, hundreds, tens, and ones. Then ask them to round the number to the nearest ten or hundred. Inquiry Skills (if applicable): Resources:Text: Envision Text Book- Lesson 2-5Exemplar Lesson: : APP: Sample Assessment Questions:Multiple Choice:Identify the value of 237 when rounded to the nearest 10.230240200300Open Ended: James has to find out how many animals are in the zoo. He knows there are 27 monkeys, 84 birds, 12 lions, and 32 elephants. James estimated there are about 150 animals in the zoo. Do you agree or disagree? Explain your answer.Grade 3Content Strand: Number Sense and Base Ten Standard: 3.NSBT.2- Add and subtract whole numbers fluently to 1,000 using knowledge of place value and properties of operations.Related Standard: 2.NSBT.5, 2.NSBT.6, 2. NSBT. 7, 3.NSBT.1, 3.NSBT.4Vocabulary: Addition, Subtraction, place valueExample: 576 + 321 = 897 432 + 27 = 27 + 432 716 – 319 = 397Strategies/Activities:This standard refers to fluently, which means accuracy, efficiency (using a reasonable amount of steps and time), and flexibility (using strategies such as the distributive property). Problems should include both vertical and horizontal forms, including opportunities for students to apply the commutative and associative properties. Students explain their thinking and show their work by using strategies and algorithms, and verify that their answer is reasonable. Do not rely only on the algorithm. The standard algorithm is introduced in 4th grade. Teach a variety of strategies. Allow students to use the strategy that best suits the problem and makes the most sense to them.More Examples:There are 178 fourth graders and 225 fifth graders on the playground. What is the total number of students on the playground?Student 1100 + 200 = 300 70 + 20 = 908 + 5 = 13300 + 90 + 13 = 403 studentsStudent 2I added 2 to 178 to get 180. I added 220 to get 400. I added the 3 left over to get 403.Student 3I know the 75 plus 25 equals 100. I then added 1 hundred from 178 and 2 hundreds from 275. I had a total of 4 hundreds and I had 3 more left to add. So I have 4 hundreds plus 3 more which is 403.SubtractionBy beginning with the expanded form strategy and then moving to the place value strategy, we are emphasizing the importance of place value. First, we break apart both numbers into expanded form, then, in the place value strategy, we break apart only the smaller number (subtrahend). This helps students to conceptualize the process of regrouping, rather than simply memorizing the steps without context. It is important that these strategies be taught with base ten blocks. The process of the subtracting with base ten blocks can be converted to written form, as the students progress. Once students master these strategies, they will have a conceptual understanding of the standard algorithm. (It is not necessary for students to be familiar with the term "subtrahend", but you may use it, if you choose)Expanded Form SubtractionShow students a sample three-digit math problem written vertically (468 - 457= ) on the board. Tell the students that today we are going to take what we know about place value, how to break numbers apart into expanded form, and addition and use that knowledge to solve this three-digit subtraction problem. Tell students that the strategy we will be using to solve this problem is called the expanded form of subtraction. Write a three-digit number "468" on the board and have the students demonstrate how to represent that number using base ten blocks. (4 hundreds, 6 tens, 8 ones) and in expanded form (400 + 60 + 8)Ask a student to represent the second number with base ten blocks. Again, have the students write the second addend in expanded form (300 + 50 + 7) and write it directly below 400+60+8. Emphasize the importance of lining up the place values, vertically.Tell the students that we will start with the ones place and work our way to the largest place in the problem. Ask students to attempt to solve each place value separately, writing the difference to each place out in expanded form (Difference = 100 + 10 + 1). Tell the students that just like with addition now we have to snap the problem back together to find the difference. (I have my students rewrite the problem vertically to make it easier for them to add.) Now we have found the difference to this problem. Start with a subtraction problem that does not require regrouping and have students work with those problems until they are comfortable with the strategy. After the students are comfortable with subtraction problems without regrouping, introduce a problem that requires them to regroup across a place value. For example, 633 - 528 = have students write the problem out in expanded form:Allow student to practice solving multiple problems with regrouping. Then show the students the expanded form strategy poster for them to refer to as they solve their problems. Place Value Strategy:Now that we have learned the expanded form strategy, we are going to use a similar but different strategy to solve the same problem. Write the problem from yesterday on the board vertically (468 - 457= ). Have students show the problem using base ten blocks. Tell students that today we are going to be learning a strategy called Place Value Strategy. In this strategy students will leave the first number like it is and then break the second number into expanded form.Now take each number from the problem written in expanded form and subtract it from the first number. Repeat this step for each place value in the second number. Repeat multiple problems using this strategy until the students are comfortable with it.Discuss with students that sometimes when we subtract we have to regroup. Show students a problem that requires regrouping in the tens place (428 - 186 =). Demonstrate regrouping this example, using base ten blocks and discuss why you have to regroup.Important Note: The term "regrouping" is more accurate than "borrowing". Borrowing implies that we will return the item borrowed. Instead, we are regrouping the values in the number. Write 186 in expanded form (100+80+6) and subtract the ones from 428 (428-6= 422). After subtracting the hundreds place, ask: "What is the value of this 2 in 422?" Answer: 20"What is the value of 8 in 186?" Answer: 80"Can I take 80 away, if I have 20?" No"Who can show me how I can regroup 422, so that I have enough tens to take away 80?"Using base ten blocks, have a student demonstrate regrouping 1 hundred from 422, so that there are 3 hundreds (flats), 12 tens (rods) and 2 ones.Have a student demonstrate how we represent this in the written problem. (In the hundreds place, cross out the 4 and place a 3. In the tens place, Cross out the 2 and place a 12.) Emphasize that 3 hundreds and 12 tens (200+120) is the same as 4 hundreds and 2 tens (300+20)Then have a student demonstrate subtracting 8 tens (rods) from the base ten block representation of 422 (3 hundreds, 12 tens, and 2 ones) Answer: 3 hundreds, 4 tens and 2 ones.Demonstrate how this is represented in the written problem. Complete the problem by having a student remove 6 ones from 348, in the base ten blocks representation, as well as in the written form. 428-186= 242Let students practice determining if they need to regroup or not until you feel that they grasp this concept.Then show the students the subtraction strategy posters (electronic version)for them to refer to as they solve their problems. Resources:Compass Learning:Mathematics 3MA3.02.01.03 Subtract 3- and 4-digit Numbers MA3.02.01.01 Add 2- and 3-Digit Numbers: Regroup MA3.02.01.02 Add 3- and 4-Digit NumbersMA3.02.01.03 Subtract 3- and 4-Digit NumbersMA3.02.01.04 Debug 3- Digit SubtractionMA3.02.01.05 Complete Computation PuzzlesMA3.02.01.06 Subtract 2- and 3-Digit Numbers: Regroup Text: Envision Text Book pages 32-41 and page 66. Exemplar Lesson: Websites: Assessment-Like Questions: Multiple Choice:Which addend was added to 341 to get the sum 774?A. 344B. 433C. 434D. 1,115Below is a table showing the daily attendance at a local water park. Daily Attendance DaysAttendanceMonday243Tuesday383Wednesday239Thursday427Friday438 How many more people came to the park on Friday than on MondayA. 195 B. 199 C. 11 D. 184Open Ended: Marshall solved the problem below:5 2 0 500 + 20 + 0 - 2 3 5 - 230 + 00 - 5 ________ ____________ 3 7 5What was his mistake?Find the error and explain what Marshall needs to do to make the problem correct. Grade 3Content Strand: Number Sense and Base Ten Standard: 3.NSBT.3 Multiply one-digit whole numbers by multiples of 10 in the range 10-90, using knowledge of place value and properties of operations.Related Standard: 3.ATO.7Vocabulary: Multiply, multipleExample: 30 X 4 = 120 5 x 20 = 100Strategies/Activities:This standard extends students’ work in multiplication by having them apply their understanding of place value. This standard expects that students go beyond tricks that hinder understanding such as “just adding zeros” and explain and reason about their products.For example, for the problem 50 x 4, students should think of this as 4 groups of 5 tens or 20 tens, and that twenty tens equals 200.The special role of 10 in the base-ten system is important in understanding multiplication of one-digit numbers with multiples of 10. For example, the product 3 x 50 can be represented as 3 groups of 5 tens, which is 15 tens, which is 150. This reasoning relies on the associative property of multiplication: 3 x 50 = 3 x (5 x 10) = (3 x 5) x 10 = 15 x 10 = 150. It is an example of how to explain an instance of a calculation pattern for these products: calculate the product of the non-zero digits, and then shift the product one place to the left to make the result ten times as large. (Progressions for the CCSSM; Number and Operation in Base Ten, CCSS Writing Team, April 2011, page 11) Begin by asking the students, “ If Ben runs 10 miles each week, how many miles will he run in 6 weeks?Also, students can draw an array that shows 6 groups of 10. Point to a row in the array and ask, “how many counters?” (10) What do the counters in the row represent? (How many miles Ben runs in a week.) “how many rows are there?” (6). Ask what they represent. ( 6 weeks). Ask, “ What multiplication facts does this array represent? (6 x 10). How many miles would Ben run in 6 weeks? ( 6 x 10 = 60).**Continue giving examples and build rigor as you go along by giving other multiples of 10. Resources:Text:Text: Envision Text Book 128, 129, 130. Exemplar Lesson: : Assessment-Like Questions:Kendra bought 7 tickets to the movies. Each ticket cost $10 each. How much did Kendra spend for all 7 tickets? A. $3 B. $17 C. $70 D. $50 Open Ended: Mr. Goodson invited 50 people to his house for a party. He bought 3 packs of hotdog buns. Each pack has 10 buns inside. Will he have enough buns for all of his guest? Explain your answer.Grade 3Content Strand: Number Sense and Base Ten Standard: 3.NSBT.4 Read and write numbers through 999,999 in standard form and equations in expanded form.Related Standard: N/aVocabulary: Standard Form, Expanded Form, Equations, word formExample:Write whole numbers such as 504,312 or 64,923. Have students read them aloud. Answer: The students would say, “five-hundred four thousand, three-hundred twelve” or sixty-four thousand, nine hundred twenty-three”. They will not use “and” while reading whole numbers.Display number words, such as five hundred seventy-nine thousand, two hundred fifteen, or fifty-eight thousand, four hundred eleven. Have students write them. Answer : 579,215 or 58,411.Write whole numbers in expanded form by composing and decomposing numbers using each place value of the number.Answer: 319,478 = 300,000 + 10, 000 + 9,000 + 400 + 70 + 8.Or 900,000 + 40,000 + 3,000 + 500 + 30 + 7 = 943,537 Strategies/Activities:To introduce standard form, (start with smaller numbers and then build up to 6-digit numbers) write the words "three thousand, five hundred six" on the board. Write just the words; not the numeral. Ask a volunteer to come to the board and write the words using digits. State that this number, 3,506, is written in standard form.Explain that standard form is simply the numerical form of a number. Write the words "three thousand, five-hundred six" on the board. Write just the words; not the numeral. Ask a volunteer to come to the board and write the words using digits. State that this number, 3,506, is written in standard form.Explain that standard form is simply the numerical form of a number. To get students to think about what this number means, have them write everything they know about the number in standard form. To focus students' thoughts, encourage them to make observations about each digit and what that digit represents in the number. Discuss their answers. Students may make these observations about 3,506:? It has four digits.? It includes ones, tens, hundreds, and thousands place.? It has a 3 in the thousands place.? It has a 0 in the tens place.? It has a 5 in the hundreds place.? It is a number written in standard form.? It is greater than 3,403 but less than 3,532.Another strategy to use is to allow students to use white boards to respond in word form, when given a whole number in standard form, and vice versa.Hand out base ten models to students. (If you are having students model numbers greater than thousands, you may want to use small laminated models). Teacher writes a number on the board, and allows students to build the number using the base ten models.Teacher writes: 2,364. Students will build the number.Ask “How many thousands (cubes) did I use? 2. What is the value of the cubes? 2,000. Teacher writes this on the board.How many hundreds (flats) do I have? 3. What is the value of this? 300How many tens (rods), etc.Teacher will eventually have 2,000 + 300 + 60 + 4. This will lead into expanded form.Expanded Form:Showing the value of each digit in a number individually.Students need to be engaged in activities in which they compose and decompose numbers using expanded and standard forms. Students often have difficulty when composing/decomposing numbers with zeros as digits. Offer several opportunities for practice with this. Group students into pairs and pass out the sets of place value cards, one set to each pair. Explain that pairs will play a game. The first student uses the place value cards to compose a number in expanded form, and shows it to the second student. The second student writes the standard form of the number on a slip of paper and shows it to the first student so he or she can confirm or reject. Have students reverse roles. : Envision Text Book- Lesson 1-2Exemplar Lesson: Assessment-Like Questions:Multiple Choice: Population of Local CountiesCountyPopulationDarlington68,681Florence138,326Lee18,347Mr. Smith lives in the county that has a populationof 60, 000 + 8,000 + 600 + 80 + 1. Which county does he live in? A. Darlington CountyB. Florence CountyC. Lee CountyD. None of the above Open Ended:Ms. Blue bought her son a new car for graduation and paid $21,802 for it. When she wrote the check, she wrote the amount like this: twenty-one thousand, eighty-two dollars. Did she write the correct amount? Why or why not. Explain your answer. Grade 3Content Strand: Number Sense and Base Ten Standard: 3.NSBT.5 Compare and order numbers through 999,999 and represent the comparison using the symbols >, =, or <.Related Standard: 2.NSBT.4,3.NSBT.1, 3.NSBT.4, Vocabulary: whole number, rounding, ones, tens, hundreds, place value, comma, compare, order, greatest, least, greater than, less than, equal to, symbolExample: 932,842 > 903,832 823 = 823 327,328 < 392,847Strategies/Activities: Make ten sets of digit cards for the numbers 0-9 and put them in ten bags. Working in pairs, each student draws six cards and makes the greatest number possible. Cards are replaced and game continues. The first one to win ten times, wins the game. Make twenty sets of digit cards for the numbers 0-9 and put them in twenty bags. Working in pairs, each student draws six cards and makes the least number possible. The one with the least number earns a point. Both students write the inequality in their math journal. The first one to win ten times, wins the game. Show students a systematic approach for comparing two numbers. (Ex. Compare digits one at a time starting on the left.) Have the students write the steps in a math journal. Prepare 20 different cards with numbers expressed in ten thousands (Ex. 65,890). Turn the cards over to where the numbers can’t be read. Have two students stand up and draw a card. The student with the least number places his/her card in a discard pile and sits down. The student with the greatest number continues to stand while another student tries to draw a larger card. Again, the one with the largest card stands. Continue until all the cards are used. Divide the class into pairs and have them prepare two sets of ten cards with digits of five or more. Each pair turns the cards face down and plays a game of MATCH using the cards. If they draw the same number they must compare them using the words “equal to”. If they draw different cards the next player must compare the cards using the words “less than” or “greater than”. Play “Higher/Lower” (Ex. A student guesses 60,000. Write the guess on the board. You say “No, it’s greater than 60,000.” Continue until the number has been guessed. Be sure to use the terminology: "greater than", "less than", or "equal to"). Have students create their own numbers by rolling dice. Using a chart to record the numbers, students will use symbols and words to compare the student-made numbers. Resources:Text: Envision Text Book- Lesson 1-6, 1-7Exemplar Lesson: : Assessment-Like Questions:Multiple Choice:Which of the following is correctly written?4,567 = 4,5006,740 = 6,4703,456 < 3,5468,765 > 9,675Open Ended: James has the numbers 8, 3, and 9 pulled out from a deck of cards. What are two numbers larger than 300 that can be created from the three digits? If he uses all three cards, could he make a number less than 300? Explain using a complete sentence.Grade 3Content Strand: Algebraic Thinking and Operations Standard: 3.ATO.1 Use concrete objects, drawings and symbols to represent multiplication facts of two single-digit whole numbers and explain the relationship between the factors (i.e., 0-10) and the product.Related Standard: 2.ATO.4, 3.ATO.3, 3.ATO.5, 3.ATO.7, 3.ATO.9Vocabulary: multiplication, whole numbers, factor, product. Strategies/Activities:Jim purchased 5 packages of muffins. Each package contained 3 muffins. How many muffins did Jim purchase? 5 groups of 3, 5 x 3 = 15. Describe another situation where there would be 5 groups of 3 or 5 x 3.Sonya earns $7 a week pulling weeds. After 5 weeks of work, how much has Sonya worked? Write an equation and find the answer. Describe another situation that would match 7x5. Warm up:· Skip count by twos, threes, and fours, etc. Use a multiplication chart.· Demonstrate multiplication using circles with groups of twos, threes, and fours, etc.1. Supply each student with objects, such as confetti. Have students use the objects to demonstrate various multiplication sentences (as in the example above). 2. Relate multiplication to addition. Use the laminated train boxcars and overhead markers to demonstrate multiplication number sentences. Call on students to make a drawing (hearts) on each boxcar to depict a number sentence that you supply, such as 7x4 (seven groups of four) or 4x7 (four seven times). They can also put only the numeral four on each of seven boxcars. 3. Use V patterns (by putting a v between two adjacent numbers in a pattern) to show thatyou add a number again and again in order to multiply. Give students various multiples to demonstrate using this type of patterning. Quadrant A4. Have students use hangers, string, and pictures to make mobiles that illustrate multiplication problems. Example: 2x4 (2 groups of 4) or 4x2 (four two times) could be shown as follows: 5. Circles and Stars Booklet: Give students several sheets of paper that has been cut into fourths. Staple the sheets together to make a small booklet. Give each pair of students a die. Student #1 rolls a die. This will represent the amount of circles the pair are to draw on the paper. Student # 2 rolls a die. This number will represent the amount of stars to put inside of each circle. They are then to write the multiplication sentence below the circles and stars. They may need to count the stars in all circles to get the product. This will give extra practice. Resources:Text: Envision Text Book pages 118-132Exemplar Lesson: Assessment-Like Questions:Multiple Choice:Which of the following represents this picture? Open Ended: Draw an array or picture model to show the product of 6 x 8.Grade 3Content Strand: Algebraic Thinking and OperationsStandard: 3.ATO.2- Use concrete objects, drawings and symbols to represent division without remainders and explain the relationship among the whole number quotient (i.e., 0-10), divisor (i.e., 0-10), and dividend.Related Standard: 3.ATO.4, 3.ATO.7Vocabulary: division, remainder, whole number, quotient, divisor, dividendExample:Strategies/Activities:Read the book :The Doorbell Rang, by Pat Hutchins to introduce the meaning of division. Teacher then can develop the lesson after the initial lesson using the terms dividend, divisor, and quotient. Strategies and Activites Cont.: 1. The teacher will tell the students that they are going to create some problems together before they work in their math groups.2. Use the docu-cam or smart board to show counters to help with the problems. Ask for a volunteer to suggest how many cookies were on grandmother’s cookie tray.3. Ask for a second volunteer to come up to the board and write a problem explaining how many people had to divide up the cookies. The class can decide for example that the grandmother and mother are also eating cookies.4. The teacher will model how to use a graphic organizer to record different ways to divide the cookies.5. After the students see the teacher model with several problems, talk about whether the answers seem reasonable.6. Next the teacher will distribute plastic counters to each of the students. The teacher will make more counters available for those students who want to experiment with large amounts of cookies.7. The teacher will tell the students to pretend the plastic counters are cookies. They are to explore ways to share the counters equally.8. The teacher will distribute graphic organizers to help the students’ do the problem associated with the book.9. The teacher will circulate around the classroom to answer questions and to challenge the students to think of different ways to divide the counters.10. The teacher will bring the students back to attention. 11. The teacher will ask the students with words and sign language to do a think, pair share about the different ways to divide the cookie counters. The students will be encouraged to use the overhead to demonstrate their thinking.12. The teacher will write a story problem with the class using the theme: The doorbell rang. She will ask for volunteers to provide the structure of the story.13. She/He will prompt the students until she/he gets information about the number of children and the number of cookies.14. The teacher will then instruct the students to write a story problem in their Math journal using the doorbell rang theme. They may work with a partner, or in a small group15. The students will share the story problem with their Math partner and answer each other’s problem.The teacher will define the words divisor, dividend and quotient. The teacher will ask a student to put the words divider, dividend and quotient on the word wall.Resources:Text: Envision Text Book pages 172-174Exemplar Lesson: Sample Assessment-Like Questions:Multiple Choice:Henry has 45 toy cars. His parking garage has 9 levels. If he wants to put an equal amount of toy cars on each level, how many cars will he put?5 cars12 cars36 cars54 carsOpen Ended: Darnell drew 28 pictures for his 4 grandparents. If he gave an equal amount of pictures to them, how many would each of them receive? Solve and use pictures, words, or symbols to justify your answer. Grade 3Content Strand: Algebraic Thinking and Operations Standard: 3.ATO.3 Solve real-world problems involving equal groups, area/array, and number line models using basic multiplication and related division facts. Represent the problem situation using an equation with a symbol for the unknown. Related Standard: 3.ATO.1Vocabulary: groups, area, array, number line, models, multiplication, division, facts, symbol, equation, variable, product, factorsExample: Strategies/Activities: 1. Supply grid paper to each student. Have students color the squares of the grid paper to represent various multiplication facts. Quadrant B2. Make strips of rectangular arrays using grid paper. Give each of the students one of the strips and have him/her write the two multiplication facts for the array. Then have each one share his/her findings with the class. Quadrant B3. Divide the class into groups of 4-5 students and give each group about one hundred 1” by 1” squares. Write a multiplication fact on the board and have them demonstrate it using the squares, dots, X marks, etc. in a rectangular array. Quadrant B4. Have students trace 1” by 1” squares to demonstrate multiplication of one- and two-digit numbers in rectangular arrays. Quadrant B5. Provide objects such as Cheerios, Cheez-its, or unifix cubes and have students demonstrate arrays of two-digit by one-digit multiplication. Quadrant BUse string to create a human number lineUse tape on the floor to create a large number line and use manipulative to solve various problems. Have students take buckets of objects and create multiplication problems by separating the objects into various groups.Resources:Text: Envision Text Book 4-1, 4-2Exemplar Lesson: Assessment-Like Questions:Multiple Choice:Open Ended: Lesley had a garden that had new flowers planted for the summer. She has 7 rows with 4 flowers in each row. How many flowers does she have in all? Draw an array or use a number line in order to show your work.3rd Grade MathCurriculum Guide2nd Nine WeeksGrade 3Content Strand: Algebraic Thinking and OperationsStandard: 3.ATO.4-Determine the unknown whole number in a multiplication or division equation relating three whole numbers when the unknown is a missing factor, product, dividend, divisor, or quotient.Related Standard: 3.ATO.5, 3.ATO.7Vocabulary: multiplication, division, equation, factor, product, dividend, divisor, quotientExample: The easiest problem structure includes Unknown Product (3 x 6 = ? or 18 ÷ 3 = 6). The more difficult problem structures include Group Size Unknown (3 x ? = 18 or 18 ÷ 3 = 6) or Number of Groups Unknown (? x 6 = 18, 18 ÷ 6 = 3). The focus of 3.OA.4 is to extend beyond the traditional notion of fact families, by having students explore the inverse relationship of multiplication and division.Students extend work from earlier grades with their understanding of the meaning of the equal sign as “the same amount as” to interpret an equation with an unknown. When given 4 x ? = 40, they might think:4 groups of some number is the same as 40 4 times some number is the same as 40 I know that 4 groups of 10 is 40 so the unknown number is 10 The missing factor is 10 because 4 times 10 equals 40.Equations in the form of a x b = c and c = a x b should be used interchangeably, with the unknown in different positions.Strategies/Activities:Write the following word problem on the board: Jose has 15 pieces of candy to give away. He would like to give each of his 3 friends an equal amount of candy. How many pieces of candy will Jose give each of his friends?Ask students for possible ways to approach the problem.Explain to students that we will use equal groups to determine the number of candy pieces that Jose can give to each of his 3 friends.Give each student a piece of blank paper. Ask students to draw equal groups to help Jose determine how many pieces of candy each of his friends will get. Circulate and take note of who is drawing 3 equal groups.Next ask a student to share the number of circles they drew. (3)Ask students to go back to the word problem to remind them how many pieces of candy Jose has to give out. (15)Ask students to start making a dot in the circles one at a time as they count to the total number of piece of candy. (15)Write the following equation on the board: 3xc=15Tell students that the 3 represents each of Jose's friends, the c represents the pieces of candy each friend will get and the 15 is the total number of pieces of candy that Jose has.Ask students to count the dots in their circles to find out the unknown factor in the problem (how many pieces of candy each friend would receive) …15Fact Family Triangles: Envision Text Book Pages 176-180Exemplar Lesson:: determine-unknown-whole-numbers-in- multiplication-or-division-equations Sample Assessment-Like Questions:Multiple Choice: Calculate the missing number in 4 x ? = 20.a. 4 b. 5 c. 6 d. 7Answer: B: Use trial-and-error to find a number that you can multiply by 4 to get 20. Since 4 1 means four groups of one object, 4 1 = 4. Try using other numbers for missing number.Open Ended: Ms. Nelson evenly distributes 32 crayons to 8 kindergartners. How many crayons does each student get? Explain your answer. a. 3 crayonsb. 4 crayonsc. 6 crayonsd. 8 crayonsAnswer: B: There are 32 crayons that are evenly divided among 8 students. Therefore, you can find number of crayons each student gets by dividing 32 by 8. The result is 4 crayons each.Grade 3Content Strand: Algebraic Thinking and Operations Standard: 3.ATO.5 Apply properties of operations (i.e., Commutative Property of Multiplication, Associative Property of Multiplication, Distributive Property) as strategies to multiply and divide and explain the reasoning.Related Standard: 2.NSBT.5, 3.ATO.7 Vocabulary: Commutative Property of Multiplication, Associative Property of Multiplication, Distributive Property, strategy, multiply, divide, Example:Commutative Property: 2 x 6 = 6 x 2Associative Property: 8x(8x9)=(8x8)x9Distributive Property: 7x8=(3x8)+(4x8)Strategies/Activities:Think of commutative property like a community- you all share the same thing so it doesn’t matter the order in which you get them (have students relate this to real life- getting in line, eating items on their plates, etc.)The associative property can be related to grouping and how even if things are grouped differently they can arrive at the same product (relate to grouping in the classroom- write an equation on the board, such as 3 groups of 2 in 2 sets in the room is (3x2)x2 or 3x(2x2)).The distributive property can be related to sharing or splitting. Relate this to the class and have students split themselves into groups. For example, if you have 24 students, they could split into (2x4)+(2x8) or (3x3)+(3x5). Have students use manipulatives to explore these properties, such as blocks, beans, or coins. Resources:Text:Envision Text BookCommutative Property Envision Text Book 4-3Distributive Property Envision Text Book 6-1Exemplar Lesson:: Assessment Questions:Multiple Choice:Which of the following would give you the same product at 8x9?9-872/99x89+8Draw pictures or write an explanation to show the commutative property.Which of the following represents the associative property of multiplication for (4x7)x9?4-(7x9)4x(7x9)(4+7)x94x(7+9)Explain the associative relationship (2x5)x7=2x(5x7) using words or a drawing.Sample Assessment Questions Continued:Which of the following represents the distributive property of multiplication for 9x7?(4x7)+(5x7)(5x7)+(5x7)(4x9)+(5x9)(4x7)+(3x7)Sara says that 8x3 can be written as (8x2)x(8x1). Is Sara correct? Explain your answer in a complete sentence. Grade 3Content Strand: Number Sense-Fractions Standard: 3.ATO.6- Understand division as a missing factor problem.Related Standard: 3.ATO.2, 3.ATO.7Vocabulary: division, factorExample: Find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8. Strategies/Activities: Since multiplication and division are inverse operations, students are expected to solve problems and explain their processes of solving division problems that can also be represented as unknown factor multiplication problems.Example:A student knows that 2 x 9 = 18. How can they use that fact to determine the answer to the following question: 18 people are divided into pairs in P.E. class? How many pairs are there? Write a division equation and explain your reasoning.Multiplication and division are inverse operations and that understanding can be used to find the unknown. Fact family triangles demonstrate the inverse operations of multiplication and division by showing the two factors and how those factors relate to the product and/or quotient. Resources:Text: Envision Text Book page 176Exemplar Lesson:: video: Sample Assessment-Like Questions:Multiple Choice:Open Ended: Tehya insists that 12 is the only number that will make this equation true.Kenneth insists that 3 is the only number that will make this equation true.Who is right? Why? Draw a picture to support your idea.Grade 3Content Strand: Algebraic Thinking and Operations Standard: 3.ATO.7 Demonstrate fluency with basic multiplication and related division facts of products and dividends through 100.Related Standard: 2.ATO.4, 3.ATO.4Vocabulary: multiplication, division, product, factor, dividend, divisorExample: This standard uses the word fluently, which means accuracy, efficiency (using a reasonable amount of steps and time), and flexibility (using strategies such as the distributive property). “Know from memory” should not focus only on timed tests and repetitive practice, but ample experiences working with manipulatives, pictures, arrays, word problems, and numbers to internalize the basic facts (up to 9 x 9).By studying patterns and relationships in multiplication facts and relating multiplication and division, students build a foundation for fluency with multiplication and division facts. Students demonstrate fluency with multiplication facts through 10 and the related division facts. Multiplying and dividing fluently refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently.Strategies/Activities:032321500Resources:Text: Envision Text Book pages 192-208Compass Learning:Mathematics 2 MA2.08.04.01 Explore: Divide Whole NumbersMA2.08.04.02 Practice: Divide Whole NumbersMA2.08.04.03 Check-up: Divide Whole NumbersMathematics 3MA3.04.01.01 Find Missing FactorsMA3.04.01.09 Divisors 1-9 Vertical FormMA3.04.01.10 Divisors 1-9 Horizontal FormExemplar Lesson: Assessment-Like Questions:Multiple Choice:LaChelle had 24 stamps. She placed an equal number of stamps on 8 letters. How many stamps did LaChelle put on each letter? A. 16 stampsB. 32 stampsC. 3 stampsD. 9 stampsJames and his friends are on vacation. He bought two different types of postcards for his friends to send out. How many animal and park postcards did he buy for each friend? Type of PostcardNumber of PostcardsNumber of Friends Animal 24 8 Park 24 6 A. James bought 3 Animal postcards and 4 Park Postcards.B. James bought 4 Animal postcards and 3 Park Postcards.C. James bought 32 Animal postcards and 30 Park Postcards.D. James bought 5 Animal postcards and 7 Park Postcards.Open Ended: Kelvin and Zayan each have 40 pennies. Kelvin put his pennies into 5 groups and Zayan put his pennies into 8 groups. Who will have more pennies in their group? Solve and Explain in the space below.Grade 3Content Strand: Algebraic Thinking and Operations Standard: 3.ATO.8- Solve two-step real-world problems using addition, subtraction, multiplication and division of whole numbers and having whole number answers. Represent these problems using equations with a letter for the unknown quantity.Related Standard: 2.ATO.1, 3.ATO.5Vocabulary:addition, subtraction multiplication, division, whole number, equation, two-step.Example:Students in third grade begin the step to formal algebraic language by using a letter for the unknown quantity in expressions or equations for one and two-step problems. But the symbols of arithmetic, x or . or * for multiplication and ÷ or / for division, continue to be used in Grades 3, 4, and 5.This standard refers to two-step word problems using the four operations. The size of the numbers should be limited to related 3rd grade standards. Adding and subtracting numbers should include numbers within 1,000, and multiplying and dividing numbers should include single-digit factors and products less than 100.This standard calls for students to represent problems using equations with a letter to represent unknown quantities. Example: Mike runs 2 miles a day. His goal is to run 25 miles. After 5 days, how many miles does Mike have left to run in order to meet his goal? Write an equation and find the solution (2 x 5 + m = 25).This standard refers to estimation strategies, including using compatible numbers (numbers that sum to 10, 50, or 100) or rounding. The focus in this standard is to have students use and discuss various strategies. Students should estimate during problem solving, and then revisit their estimate to check for reasonableness.Example:Here are some typical estimation strategies for the problem:On a vacation, your family travels 267 miles on the first day, 194 miles on the second day and 34 miles on the third day. How many total miles did they travel? 114300308610000035560000Strategies/Activities:Resources:Text: Envision Text Book pages 160 and 202Exemplar Lesson: Assessment-Like Questions:Multiple Choice:When I woke up, it was 2626 degrees. During the day, the temperature increased 1717degrees and then it began to decrease. When I went to bed, it was 2828 degrees.Which expression gives a reasonable estimate of how much the temperature t(t)decreased?Please choose from one of the following options.6858002165350Open Ended: Austin bought 5 fossils at the store. He paid $3 for each one. How much change will he get back from his $20 bill?Solve the problem and justify your answer. Grade 3Content Strand: Algebraic Thinking and Operations Standard: 3.ATO.9 Identify a rule for an arithmetic pattern (e.g., patterns in the addition table or multiplication table).Related Standard: 2.NSBT.8, 3.NSBT.2, 3.NSBT.3, 3.ATO.7, Vocabulary: rule, patternStrategies/Activities:This standard calls for students to examine arithmetic patterns involving both addition and multiplication. Arithmetic patterns are patterns that change by the same rate, such as adding the same number. For example, the series 2, 4, 6, 8, 10 is an arithmetic pattern that increases by 2 between each term.This standards also mentions identifying patterns related to the properties of operations. Examples: Even numbers are always divisible by 2. Even numbers can always be decomposed into 2 equal addends(14 = 7 + 7). Multiples of even numbers (2, 4, 6, and 8) are always even numbers. On a multiplication chart, the products in each row and column increase by the same amount (skip counting). On an addition chart, the sums in each row and column increase by the same amount.What do you notice about the numbers highlighted in pink in the multiplication table?Explain a pattern using properties of operations.When (commutative property) one changes the order of the factors they will still gets the same product, example 6 x 5 = 30 and 5 x 6 = 30. 1136655810250Teacher: What pattern do you notice when 2, 4, 6, 8, or 10 are multiplied by any number (even or odd)? Student: The product will always be an even number. Teacher: Why? Students need ample opportunities to observe and identify important numerical patterns related to operations. They should build on their previous experiences with properties related to addition and subtraction. Students investigate addition and multiplication tables in search of patterns and explain why these patterns make sense mathematically. Example: Any sum of two even numbers is even. Any sum of two odd numbers is even.Any sum of an even number and an odd number is odd. The multiples of 4, 6, 8, and 10 are all even because they can all be decomposed into two equal groups. The doubles (2 addends the same) in an addition table fall on a diagonal while the doubles (multiples of 2) in amultiplication table fall on horizontal and vertical lines. The multiples of any number fall on a horizontal and a vertical line due to the commutative property.All the multiples of 5 end in a 0 or 5 while all the multiples of 10 end with 0. Every other multiple of 5 is amultiple of 10.Students also investigate a hundreds chart in search of addition and subtraction patterns. They record and organize all the different possible sums of a number and explain why the pattern makes sense. 2667000Other Suggested Activities1. Laminate blank input/output cards and pass them and erasable markers out to the students. Give the students a rule (like in the above example) to follow and the input of the table. Have them supply the output. Quadrant B2. Supply each group with coins: quarters, dimes, nickels, and pennies. Have them make a pattern using value of each set of coins in the pattern. (Ex. three dimes, two quarters, six pennies, four nickels, three dimes, etc.). Then have them tell the pattern. Quadrant C3. Have students use V patterns to explain the pattern from one number to the next. 4. Write numbers from a predetermined pattern one at a time on the board. Make it a contest for the students to see who can come up with the rule of the pattern first. 5. Supply students with string, macaroni noodles, and fine-tipped permanent markers. Have themwrite numbers on “necklaces” to make numerical patterns for the other students to explain. Resources:Text: Envision Text Book page 126 (multiplication) Thinking Algebraically with Numbers and Shapes:- Patterns on a Hundreds Board- Coins in Your PocketNavigating Through Algebra:- Watch Them Grow- Calculator Patterns- Tiling a Patio- The Ups and Downs of Patterns Compass Learning:Mathematics 3 MA3.01.02.03 Sequence: Ascending Numbers 0-1000MA3.01.02.04 Sequence: Descending Numbers 0-1000Mathematics 4MA4.12.01.02 Apply PatternsMA4.08.01.03 Use Patterns Exemplar Lesson: Elementary Educators - Lessons_Plans/Mathematics/Grades_3-5/index.shtml Click on “Number Spiral Pattern”Sample Assessment-Like Questions:Multiple Choice:0374650Grade 3Content Strand: Number Sense-FractionsStandard: 3.NSF.1- Develop an understanding of fractions (i.e., denominators 2, 3, 4, 6, 8, 10) as numbers.3.NSF.1.a A fraction 1 (called a unit fraction) is the quantity formed by one part _______ when a whole is partitioned into equal parts;Related Standard:N/AVocabulary: fraction, numerator, denominator, wholeExample: This lesson is the introduction to fractions. Teachers can begin with an apple, and cut it into pieces and show that each “part” is a fraction. Teachers can use this lesson to allow students to create fraction strips or circles. Expectations in this domain are limited to fractions with denominators 2, 4, 6, 8, 10. Students develop an understanding of fractions, beginning with unit fractions. Students view fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole. Students understand that the size of a fractional part is relative to the size of the whole. Strategies/Activities:Using magnetic fraction circles, the teacher can tell the class a story about dinner last week when I was so hungry that he decided to order a whole pizza for himself. When his sister arrived to bring him something, she saw the pizza, and being hungry herself, asked to have some. He gave her one slice, but she said, “That’s not fair! I want the same amount that you’re eating!” So they decided to split the pizza in half and both got two equal pieces. Right as we were about to take the first bite, her husband called her, saying he was starving. She told him we were having pizza and he should come join us. The teacher tried to give him one slice, too, but he also complained that everyone should get the same amount, so we cut the pie into three equal pieces.You get the picture? This continued to fourths, fifths, sixths, and tenths. I made it funny by saying things like, “Just as I was pouring the garlic on my pizza, the bell rang again!” as well as, “At this point, I was wondering if I should just order another pizza,” and “I only had six seats at my table!” They loved the story and the visuals helped support their understanding that a fraction is an equal part of a whole.A great book for introducing fractions is “Full House, An invitation to Fractions” by Gail Dodd Resources:Text:A. Abrohms, Alison. 1001 Instant Math Manipulatives for Math B. Adler, David A. Fraction Fun Appendices A - HC. . Larson, Nancy. Saxon Math 1, An Incremental Development E. Mathews, Louise. Gator PieF. Murphy, Stuart J. Give Me Half ! The Mailbox: April/May 2000, pages 41-46 H. Wood, Don and Audrey. The Little Mouse, The Red Ripe Strawberry, and The Big Hungry BearExemplar Lesson:: Sample Assessment-Like Questions:Grade 3Content Strand: Number Sense-Fractions Standard: 3.NSF.1- Develop an understanding of fractions (i.e., denominators 2, 3, 4, 6, 8, 10) as numbers.3.NSF.1.b- A fraction */* is the quantity formed by parts of a size */*;Related Standard: N/AVocabulary: fraction, numerator, denominatorExample: This standard refers to the sharing of a whole being partitioned. Fraction models in third grade include area (parts of a whole) models (circles, rectangles, squares), number lines and Set models (parts of a group). In 3.NF.1b., students start with unit fractions (fractions with numerator 1), which are formed by partitioning a whole into equal parts and reasoning about one part of the whole, e.g., if a whole is partitioned into 4 equal parts then each part is 1?4 of the whole, and 4 copies of that part make the whole. Next, students build fractions from unit fractions, seeing the numerator 3 of 3?4 as saying that 3?4 is the quantity you get by putting 3 of the 1?4’s together. There is no need to introduce “improper fractions" initially. Some important concepts related to developing understanding of fractions include: Understand fractional parts must be equal-sized. The number of equal parts tell how many make a whole.As the number of equal pieces in the whole increases, the size of the fractional pieces decreases. The size of the fractional part is relative to the whole.One-half of a small pizza is relatively smaller than one-half of a large pizza. When a whole is cut into equal parts, the denominator represents the number of equal parts. The numerator of a fraction is the count of the number of equal parts.3/4 means that there are 3 one-fourthsStudents can count one fourth, two fourths, three fourths.Students express fractions as fair sharing or, parts of a whole. They use various contexts (candy bars, fruit, and cakes) and a variety of models (circles, squares, rectangles, fraction bars, and number lines) to develop understanding of fractions and represent fractions. Students need many opportunities to solve word problems that require them to create and reason about fair share.Initially, students can use an intuitive notion of “same size and same shape” (congruence) to explain why the parts are equal, e.g., when they divide a square into four equal squares or four equal rectangles. Students come to understand a more precise meaning for “equal parts” as “parts with equal measurements.” For example, when a ruler is partitioned into halves or quarters of an inch, they see that each subdivision has the same length. In area models they reason about the area of a shaded region to decide what fraction of the whole it represents. Strategies/Activities:Shade the figure below to represent the fraction Resources:Text: Envision Text Book pages 224-229Exemplar Lesson:: understand-a-fraction-1b-as-part-of-a-whole- partitioned-into-b-equal-parts understand-fractions Sample Assessment-Like Questions:Multiple Choice:Circle the shapes that shows equal fourths:Open Ended: Four friends want to share this pizza. Can each person have an equal share? Explain your answer. 5715001428750Grade 3Content Strand: Number Sense-Fractions Standard: 3.NSF.1- Develop an understanding of fractions (i.e., denominators 2, 3, 4, 6, 8, 10) as numbers.3.NSF.1.c- A fraction is a number that can be represented on a number line based on counts of a unit fraction;Related Standard: 3.NSF.1.a, 3.NSF.1.b, 3.NSF.1.dVocabulary: number line, interval, fraction, unit fraction, numerator, denominatorExample: Strategies/Activities:To have students label the number lines to find the equivalent fractions, you can have students count the number of spaces to identify the denominator. The tick marks indicate the numerator.Have students make a human number line. Students can hold number line cards and compare the fractions on the number line. Skills (if applicable): Resources:Text: Envisions Lesson 9-5 Pg. 230-231Exemplar Lesson:: Assessment-Like Questions:Multiple Choice:The number line below represents fractions on a number line. What fraction is represented at the letter B? ????Open Ended:01511300Grade 3Content Strand: Number Sense-FractionsStandard: 3.NSF.1- Develop an understanding of fractions (i.e., denominators 2, 3, 4, 6, 8, 10) as numbers.3.NSF.1.d-A fraction can be represented using set, area, and linear models.Related Standard: 3.NSF.1.a, 3.NSF.1.b, 3.NSF.1.cVocabulary: fraction, set, area models, linear models, numerator, denominator, representation, unit fractionExample: materials should not take the place of concrete materials. For student investment, you could have students bring in household items to use as manipulatives. Examples incude: rice, cotton balls, crayons, beans, blocks, etc.0000: Envisions Lesson 9-3 Pg. 226-227Exemplar Lesson:: Assessment-Like Questions:Multiple Choice:Using the model above, what fraction is not shaded?????Open Ended: Name the fraction represented by the set below that has just a chocolate drizzle. Explain how you found your answer.Grade 3Content Strand: Number Sense-FractionsStandard: 3.NSF.2- Explain fraction equivalence (i.e., denominators 2, 3, 4, 6, 8, 10) by demonstrating an understanding that; 3.NSF.2.a- Two fractions are equal if they are the same size, based on the same whole, or at the same point on a number line;Related Standard: 3.NSF.2.b, 3.NSF.2.c, 3.NSF.2.dVocabulary: fraction, numerator, denominator, number line, whole (shape)Example: All of the fractions below at the same point on the number line are equivalent because they are all related to the same whole (represented with the 0-1 number line at the top). For example, ? and 3/6.01143000Strategies/Activities:Use geoboards to explore equal sizes and then various ways to split that initial shape.Have students make a human number line. Students can hold number line cards and compare the fractions on the number line. Make sure that the first and last person in multiple number lines are the same distance apart so that when they are comparing they are equivalent wholes.To have students label the number lines to find the equivalent fractions, you can have students count the number of spaces to identify the denominator. The tick marks indicate the numerator.Fractions for Elementary TeachersResources:Text: Envisions Lessons 10-1 to 10-6 Pg. 246-259Exemplar Lesson: Assessment-Like Questions:Multiple Choice:Using the number lines provided, which two fractions are equivalent?? and ?? and ?? and 2/4 2/4 and 2/5Open Ended: Using the number lines to the left, identify two equivalent fractions and explain how you know these to be equivalent.-1904990Grade 3Content Strand: Number Sense-FractionsStandard: 3.NSF.2- Explain fraction equivalence (i.e., denominators 2, 3, 4, 6, 8, 10) by demonstrating an understanding that; 3.NSF.2-b-Fraction equivalence can be represented using set, area, and linear models;Related Standard: 3.NSF.2.a, 3.NSF.2.c, 3.NSF.2.dVocabulary: fraction, equivalent, set, area model, linear model, numerator, denominator, compareExample: Strategies/Activities:Use geoboards to explore equal sizes and then various ways to split that initial shape. Also, the geoboards are large enough to make two shapes the same size and then play with creating equal fractions with different denominators. This can be used to explore for an introduction or as a challenge by having students create models using the geoboards to compare fractions!Use objects, fraction pieces, and number lines to have students explore comparing fractions before asking them to complete problems on paper.Three groups could be used: Group A- sets, Group B- area models, Group C- linear modelsEmphasize the importance of having two models be the same size before comparing.Resources:Text: Envisions Lesson 10-1 Pg. 246-259Exemplar Lesson:: Assessment-Like Questions:Multiple Choice:Using the set above, which is greater: yellow chips and red chips or green chips and white chips?green chips and white chipsblack chips and green chipsyellow chips and red chipsthey are equivalent fractionsOpen Ended: Cheri says that the two models above are equivalent. Do you agree or disagree? Explain your answer using a complete sentence. ................
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