# OA3-10 Patterns in Addition Tables

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8 Extensions 1. Look at boxes in the table a) How many boxes are there for the sum 0? b) How many boxes are there for the sum 1? c) How many boxes are there for the sum 2? d) How many boxes are there for the sum 3? e) How many boxes are there for the sum 4? f) How many boxes are there for the sum 5? g) How many boxes are there for the sum 6? Answers: a) 1, b) 2, c) 3, d) 4, e) 3, f) 2, g) 1 2. Use the table to answer the questions a) List all the addition sentences from the table for the sum 4. b) List the number pattern in the row for addend 3. c) List the number pattern in the row for addend 4. (MP.7) d) What is the same about the number patterns in parts b) and c)? (MP.7) e) What is different about the number patterns in parts b) and c)? f) List the number pattern in the column for addend 3. g) What do you notice about the number patterns in parts b) and f)? Answers: a) = 4, = 4, = 4, = 4, = 4; b) 3, 4, 5, 6, 7; c) 4, 5, 6, 7, 8; d) the gap is 1 for both number patterns; e) the number pattern in part b) starts at 3, while the number pattern in part d) starts at 4; f) 3, 4, 5, 6, 7; g) the number patterns are the same Teacher s Guide for AP Book 3.1 Unit 3 Operations and Algebraic Thinking D-21

16 (MP.7) 2. Add the first pair of numbers mentally. Then use your answer to add the next pair mentally. a) = = b) = = c) = = Answers: a) 45, 145; b) 52, 652; c) 65, 865 (MP.3) 3. Explain how you would add mentally. Answer: Add 2 to 48 to get 50. Since 7 = 2 + 5, add 5 to 50 to get 55. Teacher s Guide for AP Book 3.1 Unit 3 Operations and Algebraic Thinking D-29

17 OA3-12 Parts and Totals Pages Standards: 3.OA.D.8 Goals: Students will use tape diagrams to record the parts, totals, and differences in problems involving addition and subtraction. Prior Knowledge Required: Can add two numbers within 20 Can subtract two numbers within 20 Understands addition as how much/many altogether Understands subtraction as how much/many more Vocabulary: addend, addition sentence, difference, sum, tape diagram, total Materials BLM Blank Tape Diagrams (p. D-49) Before the lesson, print out enough copies of BLM Blank Tape Diagrams so that each student can have two. Using tape diagrams to find the total. Draw on the board: 6 green marbles 2 blue marbles green blue SAY: Let s say we have 6 green marbles and 2 blue marbles. This picture shows the number of green marbles and the number of blue marbles. Point to the top row of squares and ASK: How many squares are on top beside the word green? (6) SAY: Each square stands for one green marble. There are 6 squares because we have 6 green marbles. Pointing to the bottom row of squares, ASK: How many squares beside the word blue? (2) SAY: Each square stands for one blue marble. There are 2 squares because we have 2 blue marbles. SAY: This picture is called a tape diagram. A tape diagram uses squares or boxes in a row to stand for objects. A tape diagram can help you find the total number of two kinds of objects, such as green marbles and blue marbles. Let s use this tape diagram to find out how many D-30 Teacher s Guide for AP Book 3.1 Unit 3 Operations and Algebraic Thinking

18 marbles we have in total. Explain that the total is really the sum of the two parts, blue and green. Add a brace, the word total, and a blank line to the diagram on the board, as shown below: green blue total: ASK: How do we find the total number of marbles? (count the total number of boxes) SAY: There are 6 boxes on top, or 6 green marbles, plus 2 boxes on the bottom, or 2 blue marbles. ASK: What is 6 + 2? (8) SAY: There are 8 marbles in total. Write 8 in the blank beside total. Write = 8 below the tape diagram. Pointing to this addition sentence, explain that the number of green marbles (6) and the number of blue marbles (2) are the addends, and the total number of marbles (8) is the sum. Using tape diagrams to find the difference. ASK: How do we find the difference between the number of blue and green marbles? In other words, how do we find out how many more green marbles there are than blue marbles? (6 2 = 4, or count the extra squares in the row beside the word green ) Add to the diagram on the board: difference: green blue total: 8 Pointing to the first two boxes in the green row, ASK: If we pair up 2 green marbles with 2 blue marbles, how many green marbles are left over? (4) Pointing to the remaining 4 squares in the green row, SAY: These squares here show the difference. There are 4 more green marbles than blue marbles, so the difference is 4. Write 4 in the blank beside the word difference. Give each student one copy of BLM Blank Tape Diagrams for the following exercises. Exercises: Write green or blue in the blanks. Shade the boxes for green marbles and blue marbles. Then find the total and the difference. a) 8 green marbles, 3 blue marbles b) 2 green marbles, 5 blue marbles c) 5 green marbles, 9 blue marbles d) 10 green marbles, 7 blue marbles Answers: a) total: 11, difference: 5; b) total: 7, difference: 3; c) total: 14, difference: 4; d) total: 17, difference: 3 Problems where only one part is given. SAY: There are always two parts in these problems: the number of green marbles and the number of blue marbles. Sometimes you know only one of the parts and the difference or the total. For example, you might know the number of Teacher s Guide for AP Book 3.1 Unit 3 Operations and Algebraic Thinking D-31

20 Answers: a) total: 11, difference: 1; b) 9 green grapes, difference: 5, total: 13; c) 5 black cars, difference: 3, total: 13; d) 2 red apples, difference: 5, total: 9; e) 9 red apples, difference: 5, total: 13; f) 6 Science books, difference: 2, total: 10 (MP.1) 2. For an extra challenge, have students try to find both parts when only the difference and the total are given. At this stage, students should just guess and test their answers. They can use a tape diagram to help. Students should list the pairs of numbers that add to the given total, and test each pair until they find the pair with the given difference. a) 5 marbles in total, 1 more green marble than blue marble b) 5 marbles in total, 3 more blue marbles than green marbles c) 10 blue and green marbles in total, 2 more blue marbles than green marbles d) 10 marbles in total, 4 more green marbles than blue marbles e) 10 marbles in total, 8 more blue marbles than green marbles f) 10 marbles in total, 6 fewer blue marbles than green marbles Answers: a) 3 green marbles, 2 blue marbles; b) 1 green marble, 4 blue marbles; c) 4 green marbles, 6 blue marbles; d) 7 green marbles, 3 blue marbles; e) 1 green marble, 9 blue marbles; f) 8 green marbles, 2 blue marbles 3. Have students use sketches of tape diagrams (as shown below) to find the difference and total in problems that involve larger numbers. The scale is not important in the sketches, as long as the bar for the larger number is longer than the bar for the shorter number. Have students line up the digits to add or subtract by regrouping. a) 324 green marbles, 568 blue marbles difference: green 324 blue 568 total: b) 298 green marbles, 427 blue marbles difference: green 298 blue 427 Answers: a) difference: 244, total: 892; b) difference: 129, total: 725 total: Teacher s Guide for AP Book 3.1 Unit 3 Operations and Algebraic Thinking D-33

21 OA3-13 More Parts and Totals Pages Standards: 3.OA.D.8 Goals: Students will use tables to distinguish between the parts, the total, and the difference in problems involving addition and subtraction. Students will solve word problems involving addition and subtraction by distinguishing between the parts, the total, and the difference. Prior Knowledge Required: Can add and subtract 2-digit numbers Can distinguish between the total (or sum) of and the difference between two numbers Vocabulary: addend, difference, sum, table, tape diagram, total Materials: BLM Cards for Parts and Totals (p. D-50, see Extension 2) Using tables for parts and totals. Write on the board: 3 red apples, 5 green apples SAY: Now we will use a table to keep track of the red apples, the green apples, the total, and the difference. Draw on the board: Red Apples Green Apples Total Difference ASK: How many red apples do we have? (3) Write 3 in the Red Apples column. ASK: How many green apples? (5) Write 5 in the Green Apples column. ASK: How do we find the total number of apples? (add 3 and 5) What is 3 + 5? (8) Write 8 in the Total column. ASK: How do we find the difference? (subtract 3 from 5) PROMPT: How do we find how many more green apples than red apples? (5 3) What is 5 3? (2) Write 2 more green apples than red in the Difference column. SAY: Another way to help find the total and the difference is to draw a tape diagram, as we did in the previous lesson. Draw a tape diagram for this example, as in Lesson OA3-12. Write on the board: 6 red apples, 8 apples in total D-34 Teacher s Guide for AP Book 3.1 Unit 3 Operations and Algebraic Thinking

25 ASK: Which number in the table answers the question? ( 8 under Green) Go through each option and have students signal thumbs up or thumbs down. Circle the 8 in the Green column. Demonstrate the first two exercises below in the same manner before having students complete the rest. Point out that it is harder to make a tape diagram for large numbers, so a table is better. NOTE: A tape diagram sketch can be used for larger numbers, without showing the individual squares, as in Extension 3 of Lesson OA3-12. (MP.4) Exercises: Fill in the table. Circle the number in the table that answers the question. Red Green Total Difference a) Ed has 22 apples. 15 are green and the rest are red. How many red apples does he have? b) Randi has 23 red apples and 11 fewer red apples than green apples. How many apples does she have in total? c) Lynn has 17 green apples and 25 red apples. How many more red apples does she have than green? d) Jon has 14 red apples and 3 fewer green apples than red. How many apples does he have? e) Mona has 21 apples. 11 are red. The rest are green. How many more red apples does she have than green? f) Raj has 13 green apples and 4 more green apples than red. How many apples does he have altogether? Answers: a) red: 7 (circled), green: 15, total: 22, difference: 8; b) red: 23, green: 34, total: 57 (circled), difference: 11; c) red: 25, green: 17, total: 42, difference: 8 (circled); d) red: 14, green: 11, total: 25 (circled), difference: 3; e) red: 11, green: 10, total: 21, difference: 1 (circled); f) red: 9, green: 13, total: 22 (circled), difference: 4 SAY: You can make your own table to solve word problems. Write on the board: Nina has 7 pets. 5 are dogs and the rest are cats. How many cats does she have? Draw on the board: Total Difference ASK: What word should we write at the top of the first column? (Dogs) At the top of the second column? (Cats) Where does the number 7 belong? (Total column) Where does 5 belong? (under Dogs) How do we find the number of cats? (7 5 = 2) Write 2 under Cats. ASK: Which number answers the question? (the number of cats, which is 2) Do we need to find the difference? (no) Demonstrate the first exercise below before having students complete the rest. D-38 Teacher s Guide for AP Book 3.1 Unit 3 Operations and Algebraic Thinking

27 Answers: a) How many more red apples than green, b) Total number of apples, c) green apples, d) Total number of apples, e) How many more red apples than green, f) Number of red apples (MP.6) 3. Have students fill in + or in the circle, now in more challenging situations where the difference appears on the left side of =. Begin with examples that involve numbers (parts a) and b) below), and then move to examples without numbers. Remind students that the phrase how many more blue cars than red is not a question that they need to answer, but instead refers to the difference, i.e., the number of extra blue cars. a) 5 red cars 3 more blue cars than red = 8 blue cars b) 10 red cars 4 more red cars than blue = 6 blue cars c) red cars How many more blue cars than red = blue cars d) red cars How many more red cars than blue = blue cars e) blue cars How many more blue cars than red = red cars f) blue cars Answers: a) +, b), c) +, d), e), f) + How many more red cars than blue = red cars (MP.1) 4. For an extra challenge, have students try to find both parts when only the difference and the total are given, like in Extension 2 of the previous lesson. At this stage, students should just guess and test their answers. Students should list the pairs of numbers that add to the given total, and test each pair until they find the pair with the given difference. Red Apples Green Apples Total Difference a) 9 1 more green apple than red b) 7 3 fewer green apples than red c) 10 2 more red apples than green d) 10 6 fewer red apples than green D-40 Teacher s Guide for AP Book 3.1 Unit 3 Operations and Algebraic Thinking

28 Bonus: e) 21 1 fewer green apple than red f) 19 5 fewer green apples than red Answers: a) red: 4, green: 5; b) red: 5, green: 2; c) red: 6, green: 4; d) red: 2, green: 8; Bonus: e) red: 11, green: 10; f) red: 12, green: 7 Teacher s Guide for AP Book 3.1 Unit 3 Operations and Algebraic Thinking D-41

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