Pearson Assessments



Third Grade Curriculum

Multiplication/Division Section 1

Suggested Number of Days: 5

The suggested number of days includes instruction, practice, and mixed review time. Please review materials in advance to allocate days based on the resources provided.

|Topic | |Page |

|Multiplication: Multiple |Part I: Multiple Representations – concrete to pictorial |3 |

|Representations |Teacher Completed Multiple Representation Page |23 |

| |Equal Group Template |24 |

| |Number Line Template |25 |

| |T-chart Template |26 |

| |Part II:Vocabulary and Relationship b/w Digits |27 |

| |Number Sentence Starter Strips |30 |

| |Part III: Multiple Book and Multiple Representation Rotations |31 |

| |Part IV: Use What You Know to Compose and Decompose |34 |

| |Part V: Bridging 2nd grade Models to Unit Bar G.P 1 |39 |

| |Guided Practice 2 and 3 |44 |

| |Student Copy Guided Practice 2 and 3 |46 |

NOTE: **Target Questions** are included for use in conjunction with the Teacher Notes. In the Practice Problems, some are marked with an “*”. It is suggested that you include these problems in your unit. There is also a model window pane problem on some target problems to use as a Guided Practice. Additional problems are also included as needed.

Multiplication: Multiple Representations

TEKS:

3.4(D) determine the total number of objects when equally-sized groups of objects are combined or arranged in arrays up to 10 by 10;

3.4(E) represent multiplication facts by using a variety of approaches such as repeated addition, equal-sized groups, arrays, area models, equal jumps on a number line, and skip counting

3.4(F) recall facts to multiply up to 10 by 10 with automaticity and recall the corresponding division facts;

3.4(K) solve one-step and two-step problems involving multiplication and division within 100 using strategies based on objects; pictorial models, including arrays, area models, and equal groups; properties of operations; or recall of facts.

VOCABULARY: equal groups, number lines, array model, area model, relationship, skip counting, times, factor, product, multiple representations, multiple, multiply, multiplication, row, column, compose, decompose

Student Background: In second grade, students only focused on the action of putting together equal groups with concrete objects. Students have neither experienced the unit bar in relation to multiplication nor have they experienced arrays.

Teacher Background: This section focuses on using multiple representations to help build students understanding of multiplication. Examples of multiple representations could be equal groups, number lines, array models, area models and skip counting. Other representations include properties of multiplication (multiplicative identity of 1, zero property, commutative, associative and distributive). Please note that we are NOT expecting students to recall the specific names of each property. The goal is to help students understand how we can compose and decompose numbers to facilitate algorithms and mental math.

It is important to note why we write 5 groups of 4 or 5 x 4. The 4 is the multiplicative, which tells you how many you have in a group. The 5 is the scaler that tells you how many groups of 5 you have. This is why the order of the factors matters when you are representing multiplication. The product is still the same, however when the order is different, so is the representation because the multiplicative and scaler have been reversed.

Because students have not had experience with using the unit bar in relation to multiplication, we have explicit teaching on how to bridge to the unit bar.

Also note that students are now expected to recall facts to multiply up to 10 by 10 with automaticity. We will focus on recalling the corresponding division facts in later sections.

You will also notice the students’ possible answers to questions are in parenthesis. Some of these might be correct responses and others are not.

Materials:

• manila paper or construction paper

• scissors, glue or tape

• Equal group templates to be cut into 5 equal groups (pg. 24)

• 5 hula hoops or jump rope or yarn to make circles on the floor

• number line sheet provided in sheet protectors per partner (pg.24)

• number line sheet provided cut into number line strips (pg.25)

• t-chart template (pg. 26)

• inch grid paper provided cut into 5 by 4 (MATH_3_A_MULTIPLICATION DIVISION INCH GRID PAPER 2014_RES.DOC)

• 8 ½ inch x 11 inch pre-cut paper into 4 quadrants(window pane style)

• Number sentence starter strips (pg. 37)

• Multiple book(MATH_3_A_MULTIPLICATION BOOK 2014_RES.DOC)

• 20 colored counters in baggie per partner groups

• dry erase markers

• multiplication action poster

• 20 large die cut circles (red or yellow)

Part I: Multiple Representations (concrete to pictorial)

** Note: The experience before label activity and lesson are written for a class with 20 students. If you have more than 20 students, the extra students can be your “captains.” If you have less than 20 students, you can participate with your class.

Experience Before Label (steps 1-19)

1. “Arr matey! Today we are going on some pirate boats to go on an adventure to a secret island. However, there are only a certain number of boats available depending on what time we leave.”

2. Place 4 hula hoops (boats) on the floor around the room.

3. “If we leave now, there are 4 boats available and each boat can only hold 5 of us. So, we have 30 seconds to get in the boats…GO!” *Note: students can stand inside or hold the sides of the hula hoops.

4. Once students are in the “boats,” ask them, “What do you see or notice?” (there are the same number of pirates in each boat)

5. “How many of us are going to get to go to the secret island?” (20)

6. “How did you figure that out?” (counted everyone) “Tell me more about how you counted.” (one at a time or by 5’s) “Why did you count by 5’s?” (because every boat had five so I could count each group by 5’s)

7. “Awesome! Oh no, we can’t leave now because there is a storm a brewin’.”

8. Have students hand you the hula hoops and place 2 hula hoops (boats) on the floor around the room.

9. “The storm has passed, so let’s load up. Oh geez, they only have 2 boats available. They say that we need 10 of us on each boat. We have 30 seconds to get in the boats. Please notice the boats are small, so please no pushing or shoving of your fellow mates. Ready…GO!”

10. Once students are in the “boats”, ask them “What do you see or notice? (there are the same number of pirates in each boat)

11. “How many of us are going to get to go to the secret island now?” (20)

12. “How did you figure that out?” (counted everyone) “Tell me more about how you counted.” (one at a time or by 10’s) “Why did you count by 10’s?” (because we had 2 groups of 10 and it was faster to count by tens than ones)

13. “Amazing job! Oh no, we still can’t leave because one of the boat’s sails has torn.”

14. Have students hand you the hula hoops and place 5 hula hoops (boats) on the floor around the room.

15. “They have fixed the sail. Woohoo! We are finally going to get to head to the secret island. Because we waited for so long, they now have 5 boats ready to take us. They said we can have 4 pirates on a boat at a time. Let’s get on a boat. Ready…Go.”

16. Once students are in the “boats”, ask them “What do you see or notice? (there are the same number of pirates in each boat)

17. “How many of us are going to get to go to the secret island now?” (20)

18. “How did you figure that out?” (counted everyone) “Tell me more about how you counted.” (one at a time or by 4’s) “Why did you count by 4’s?” (because each group had 4 and it’s faster than counting by 1’s)

19. “Let’s get to our secret island. Row, row, row back to your seats and you have 2 minutes to name your secret island.” Have a special pencil or sticker for students for their hidden treasure.

20. “I’m so glad we got to take a trip together! What were some things you noticed while trying to leave on our trip?” (it took us forever to get to leave, there were the same number of us in each boat, we counted by 5’s or 10’s depending on how many of us were in each boat, there were always 20 of us going on the trip)

21. “Fantastic! Tell me more about the same number on each boat and always 20 of us going.” (we put together boats with the same number in each boat and always got 20 people)

22. “Anyone know what operation we just experienced?” (addition) “Tell me more.” (we added 10 + 10 or 5 + 5 + 5 + 5)

23. “Exactly. Because we are now sophisticated 3rd graders, we can relate this to the operation of multiplication. When we put equal groups together (boats) to get a total (20), we call that multiplication.”

24. Pass out 20 counters for each set of partners.

25. “Now, we’re going to draw out our last experience with our pirate boats. Let’s use the dry erase markers to draw 5 circles (draw on table or dry erase board) to represent our 5 boats. How many people did we have in each boat?” (4) “Let’s put 4 counters in each circle.”

26. “What do you see or notice?” (Each circle has 4 counters, they each have the same amount)

27. “How did we determine that we had 20 pirates riding in all of the boats?” (counted by 4’s)

28. “Super. Let’s write the number 4 under each circle and put addition signs to show us that we are counting by 4’s. Let’s also show that this equals the total number of pirates. What was our total?” (20)

29. Pass out the manila paper or construction paper to each student. Have students title their paper ‘Multiple Representations.’ Note: It is suggested that you create the ‘Multiple Representation’ paper along with your students.

30. Also pass out the equal group strips (pg. 24) and have students glue on the top left quadrant on your manila paper.

31. “Now, let’s draw the counters we have in each circle and write the addition number sentence, just as we have it on the table.”

32. “Let’s talk more about what this means. Did we put together groups that had the same amount or different amount?” (same) “Tell me more about that.” (each boat or circle had 4 pirates or counters)

33. “Excellent! So we put groups together that had the same amount. Talk with your neighbor to see if you can fill out the following.” Teacher writes the following on teacher demo.

34. “Ok, I’ll give you a hint.”

35. “What did you and your neighbor discuss?” (there are 5 groups of 4) “Amazing, let’s write this under our circles on our manila or construction paper.”

36. “What is our total when we have 5 groups of 4?” (20). “So, if I count by 4 five times, it equals 20?” (yes) “Let’s record our total on our manila paper.”

37. “Guess what? You’ve just seen one way to show a multiplication problem. Let’s write 5 x 4 = 20 underneath 5 groups of 4 = 20. Notice that we can represent ‘groups of’ with a multiplication symbol (x) and write 5 x 4 = 20.”

38. “From what we’ve experienced today, what do you think multiplication means?” (to put together groups that have the same amount)

39. “Fabulous! Today we are going to look at many ways to show or represent multiplication. Drawing groups as circles or boxes and putting an equal amount in each group is one way we can represent multiplication.”

40. “If you’re ready to see some more ways, say OH YEAH!” (OH YEAH!)

41. Pass out the number lines in sheet protectors. (pg.25)

42. “What do you notice?” (number lines, numbers 0-25)

43. “Excellent. Discuss with your neighbor how we might show 5 equal groups of 4 on the number line. You can use your dry erase marker and draw on the number line.”

44. “Wow, that’s great! Let’s share why you thought to do that.” (we circled groups of 4 like we did on our paper before) Discuss the different ways students may have shown equal groups on the number line.

45. “Super job! Let’s look at another way to show our equal groups instead of using circles. Remember when we used number lines for addition and subtraction? What did we use to show our movement?” (arrows) “Yes, we’re going to use arrows to show our equal groups.”

46. “As we are counting on the number line, showing our equal groups, I’m going to record the multiples of 4 on a t-chart. I’ll give you a chance to make one with your partner in a moment.”

47. “Keep what you’ve done on the first number line and let’s move to the number line below. What if we don’t have any groups of 4? Where would we start on the number line?” (0) “Why?” (because we don’t have any groups of 4 to represent)

48. “Exactly. If we don’t have any groups, how many counters do we have?” (0) “Why?” (because we don’t have any place to put the counters, there are no groups of 4)

49. “Excellent. Let’s look at a t-chart.”

50. “What do you notice about the t-chart?” (one side is the number of groups and the other side is the total number of items)

51. “How many groups did we just talk about having on the number line?” (0) “How many total items did we say we would have with 0 groups of 4?” (0) “Cool. Let’s show this.”

52. “Now, let’s think back to how many pirates we had on each boat.” (4) “Correct. So if we have 1 group of 4, discuss with your neighbor how we can use an arrow to show this on our number line.”

53. “Does your arrow look like mine? If not, go ahead and correct it”

54. “Awesome! Our arrow starts at 0 and ends at what?” (4) “Why?” (because we had 4 in each group) “Good. Help me show this on our t-chart. How many groups of 4 have we shown on our number line?” (1) “Then, how many total items does our number line show?” (4)

55. “Now we want to show 2 groups of 4 or 2 boats with 4 pirates on each boat. I challenge you to work with your neighbor to determine how far your next arrow goes on the number line. Remember, we’re starting at 4 on the number line. Why?” (because we have already determined 1 group of 4) “You have 1 minute to work with your partner.”

56. “Does your number line look like mine?” Have students share why their number line looks different to help analyze their thinking. Then have them correct their number line.

57. “Help me show this on our t-chart. How many groups of 4 have we shown on our number line?” (2) “Then, how many total items does our number line show now?” (8)

58. “Super. How many groups of 4 do we want to show now? (3 groups of 4 or our 3 boats with 4 pirates on each boat. I challenge you to work with your neighbor to determine how far your next arrow goes on the number line. Remember, we’re starting at 8 on the number line. Why?” (because we have already determined 2 groups of 4) “You have 1 minute to work with your partner.”

59. “Does your number line look like mine? Have students share why their number line looks different to help analyze their thinking. Then have them correct their number line.

60. “Help me show this on our t-chart. How many groups of 4 have we shown on our number line?” (3) “Then, how many total items does our number line show now?” (12)

61. “Super. How many groups do we want to show now? (4 groups of 4 or our 4 boats with 4 pirates on each boat) I challenge you to work with your neighbor to determine how far your next arrow goes on the number line. Remember, we’re starting at what number on the number line?” (12) “Why?” (because we have already determined 3 groups of 4) “You have 1 minute to work with your partner.”

62. “Does your number line look like mine? Have students share why their number line looks different to help analyze their thinking. Then have them correct their number line.

63. “Help me show this on our t-chart. How many groups of 4 have we shown on our number line now?” (4) “Then, how many total items does our number line show?” (16)

64. “Ok, does anyone remember how many boats we used to get to the secret island?” (5) “Yes. Now I want you and your partner to complete your number line. How many additional groups do we need for 5 equal groups of 4?” (1) “Once you and your partner have completed the number line with 5 groups and your arrow shows the total number of items, raise your hand so I can check.” (An example of a completed Multiple Representation paper is on pg.23 )

65. “After I check your work, I’m going to hand you a paper number line for you to glue onto your manila paper. On your number line, I want you to transfer what we did with the dry erase number lines.”

66. “After completing the number line on your manila paper, glue the t-chart (pg. 27) under the number line on your paper. Please fill in the t-chart as I did on your own t-chart. Then continue to show the total number of items with 5 groups.” (An example of a completed Multiple Representation paper is on pg.23 )

67. “So, let’s re-cap.

a. Glue number line onto manila paper.

b. Transfer dry erase number line info onto your paper number line.

c. Glue t-chart onto manila paper.

d. Fill in t-chart using teacher demo and number line.”

68. “Does yours look like mine?”

****Remind students that we were skip counting by 4’s to find the total number of items each time we created a new group of 4.

69. “Let’s take a big stretch. Wow, what different ways have we represented 5 times 4 = 20?” (drawing circles with the same number in each group, 5 groups of 4, 5 x 4, a number line and a t-chart) “If you’re ready to learn a few more ways to represent 5 times 4 say I’M READY.” (I’M READY)

70. “Here we go. I need 4 students to volunteer to come stand in 1 row.” Choose 4 students and have them stand next to each other. Note: students do not have to hold hands.

71. “Why did you stand like that?” (because you asked us to stand in a row….that’s like when we sit on the floor in a row side by side)

72. “Awesome. I need 4 more students to come stand in a 2nd row in front of my first row.” It should look similar to the following.

73. “Can anyone tell me how many rows you think we might be making based on what we have experienced today?” (5) “Why?” (because we had 5 boats, we drew 5 groups, we jumped 5 equal groups on our number line)

74. “Perfect. We now need to make 5 rows of how many people?” (4) “Here’s a challenge. You have 30 seconds to make the other 3 rows with 4 people in each row……Go.”

75. Students should look like the following.

76. “How many people do you think we have standing up here in our 5 rows with 4 people in each row?” (20) “How did you determine that?” (counted by 4’s 5 times, looked at number line, t-chart or equal groups to see a total of 20).

77. “Awesome. You have created an array to represent 5 groups of 4. An array is made up of …….? Hint, what did we stand in?” (rows) “An array is also made up of columns. Can anyone share an example of a column?” (there are some in the hallway or library or outside in the front or back of the school) “Tell me more about them.” (they go from the floor up to the ceiling or up and down)

78. “Super, what direction did our rows go?” (side to side) “Yes, horizontal and our columns go?” (up and down) “Yes, vertical.”

79. “Can you show or point to a column within our class array?”

Note: we are introducing the terminology of columns to students, but will refer more to a row and the number in each row. For example: 5 rows with 4 in each row.

80. Now, hand each student a large die cut circle. “Those of you that came to make the first row, please place your circle on the floor in your spot and walk back to your seats.” It should look like the following.

81. Continue to give students large die cut circles by rows and have them place on the floor so it looks like the following.

82. “Wow, what did we make?” (an array) “What is an array?” (rows with the same amount in each row, or rows and columns)

83. “Let’s take out our counters and make the same array we made as a class. How many counters were in the first row?” (4)

84. “Go ahead and make the first row. Thumbs up when done so I can come check. As I check, continue making rows of 4. How many rows are we making?” (5 rows)

85. “Now let’s label the rows and number of counters in each row of the array using a dry erase marker. As well as label 5 rows with 4 counters in each row is 20 and 5 x 4 = 20 under the array.”

86. “Let’s draw the array on our Multiple Representation manila paper.” (An example of a completed Multiple Representation paper is on pg.23 )

87. Pass out the pre-cut inch grid paper (MATH_3_A_ MULTIPLICATION DIVISION INCH GRID PAPER 2014_RES)to each student.

88. “Now, we are going to move our counters onto our grid paper to create another type of array using squares.” It should look similar to the following.

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89. “What do you notice?” (it looks the same) “Tell me more.” (We still have 5 rows with 4 in each row)

90. “Way to go. How is it different?” (it has lines) “Tell me more.” (it has lines that go side to side and up and down) “Yes, some are horizontal to show rows and some are vertical to show columns.”

91. “Let’s trace the lines around our model with a marker and then take the counters off. Go ahead and glue this to the right of the array we drew of our counters.” (An example of a completed Multiple Representation paper is on pg.23 )

92. “Tell me what you see?” (5 rows with 4 in each row) “Awesome. This is called an array model. Let’s label our array model.”

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93. “Are you ready to be blown away? If so say, ‘COWABUNGA DUDE!’” (COWABUNGA DUDE!)

94. Pass out the pre-cut quadrant copy paper (as noted in materials section) to each student.

95. “Take the paper I gave you and place it on top of the array model. Can you see your outline of your array model?”

96. “Great, trace the outline onto the blank paper I just handed you. Move it to the right of the array model.”

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97. “What do you notice?” (It doesn’t have any lines on the inside) “Do you think it represents the same amount of space as our array model?” (yes) “Why?” (because we traced it)

98. “Tell me more about what is the same?” (one side will still be 5 and one will still be 4) “Yes, tell me a little more about what you mean?” (there are still 5 rows and 4 in each row, we just can’t see them)

99. “Awesome. You are exactly correct. This is what we call an area model. Let’s write area model on the inside of our new representation.

100. “How would you describe an area model?” (it’s just like the array, but we don’t see the squares)

101. “Super, work with your partner to label the sides of your area model.”

102. “How did you label the sides of the area model?” (the left side is 5 and the top is 4) “Why?” (because we have 5 rows with 4 in each row)

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103. “Talk with your neighbor to discuss if you think the area model represents or covers the same amount of surface or space as the array model. You have 20 seconds…Go!”

104. “What did you decide?” (Yes, it does) “Why?” (because it still represents 5 rows with 4 in each row)

105. “When we have 5 rows with 4 in each row, how much does that represent?” (20) “Why?” (because 5 x 4 = 20 or we have 5 rows with 4 in each row)

106. “Great. Let’s place an equal sign between the two models and label 5 x 4 = 20 underneath our area model.”

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Part II: Vocabulary and Relationship between Digits

1. “Let’s look at our Multiple Representation page we created. You have 10 seconds to share with your partner what we represented...GO.”

2. “What did you and your partner discuss?” (5 groups of 4 equals 20) “Tell me more.” (we represented 5 groups of 4 in many different ways)

3. “Think about the following statement. We had 1 row of 4 and then we copied that row until we made a row of 4 five times.” You may have to show them the picture below to help students relate to their Multiple Representation page.

4. “Thumbs up if you agree with this statement or thumbs down if you do not agree. If you agree, share with me why?” (It’s like we made a stamp. Once we made the first row, we stamped it until we saw it 5 times.)

5. “Yes, so we can say 5 groups of 4 is the same as 5 copies of 4 or 5 x 4.” Pass out number sentence starter strips (pg. 37 ) and have students glue it on the top right corner of our Multiple Representation page. Then have them fill in the following.

6. “Let’s take a look at our numbers 5, 4 and 20 in our number sentence.”

7. Discuss how 5 and 4 are called factors and 20 is called a product.

8. Have students take one arm across the chest to make one part of the multiplication symbol and say “a factor.” Then have them take their other arm to complete the multiplication symbol and say “times a factor equals a product.”

9. “Now, let’s look back at our numbers. Share with your neighbor how we could write 5 x 4 = 20 in a different order with the 20 being first. I’ll get you started.”

10. “What did you come up with?” (20 = 4 x 5) “Good. Let’s see if that represents 5 groups of 4. With 4 being first, this means 4 groups of 5. Is this what we drew?” (No) Note: This is in reference to the important relationship between the multiplicative and the scalor noted in the teacher background.

11. “Ok, you’re almost there. 20 = ? Anyone else have a different number sentence?” (20 = 5 x 4) “Great. Let’s see if this represents 5 groups of 4. With 5 being first, this means 5 groups of 4. Awesome.”

12. “Now, let’s write 20 = 5 x 4 under 5 x 4 = 20 on our Multiple Representations paper.”

13. “Let’s talk about what this means by writing the product first. How might we say this outloud?” (20 equals 5 times 4 or 5 groups of 4)

14. “Awesome. Let’s talk about how much bigger 20 is than the 1 group of 4 we started with. How many times did we have a group of 4?” (5 times) “So, 20 is 5 times as much as 1 group of 4?” (yes, because we copied the group of 4 five times)

15. “Can we also say that 20 is 5 times bigger than 4?” (yes) “Why?” (because we counted 4 five times to get to 20 and 20 is bigger)

16. “Let’s write this next statement under 20 = 5 x 4 on our Multiple Representations paper.”

17. Note: This relationship will be discussed each day of completing the Multiple Book. We want students to eventually come up with the generalization that the product is ____ (scalor) times as much as _______(multiplicative).

Part III: Multiple Book and Multiple Representation Rotations

Materials: hundred chart, counters, colored pencils or crayons, link to interactive 100 chart below that can be displayed on the Smartboard



• Multiple book (MATH_3_A_MULTIPLICATION BOOK 2014_RES.DOC)

1. Have students stand in a circle and as a class count by 2’s until you see that most students understand the pattern.

2. Now, have students count by 5’s, but this time students can count one at a time by saying a multiple while going around the circle.

3. Share with students that they have just said the multiples of 2’s and 5’s. “What are some things that you think of when you hear the word ‘multiple’?” (more than 1 of an item, multiplying, counting by….)

4. “Today we are going to talk about what a multiple is. If we just said the multiples of 2’s and 5’s, what do you think a multiple is?” Allow students to share their thoughts about what a multiple is. Guide them to understand that a multiple is related to multiplication and is the product of the factors you are multiplying. For example, when we counted by 2’s, (2, 4, 6….) we said 6. 6 is a multiple of 2 because 3 groups of 2 = 6 or 3 x 2 = 6.

5. Have students look at their multiple book and the 100 chart on the smartboard. Have students help you find the multiples of 2 on the smartboard. You can choose a color and touch the numbers on the hundreds chart to shade in the multiples while students can record the multiples in their book using colored pencils or crayons.

6. It is suggested you play the two’s song from the Multiplication Motivation CD.

7. After all multiples are shaded, ask “What do you see? What do you notice?” (even numbers in the ones place, the ones place always has a 0, 2, 4, 6, or 8, or other patterns they see.)

8. “Now, let’s connect our multiples to multiplication. Share with your partner what we did yesterday?” (showed lots of ways to represent multiplication)

9. “Do you think we found some multiples yesterday?” (Yes) “Why?” (because we were skip counting by 4’s and found the total amount using equal groups)

10. “Today, let’s use the representations we’ve learned to show 5 groups of 2. What’s another way to say 5 groups of 2?” (5 times 2 or 5 x 2)

11. “We’re going to draw our representations on the left side of our booklet for 5 groups of 2. We’ll do the array model together.” Below shows a completed example of 5 groups of 2 = 10 or 5 x 2 = 10 in one section on the left side. When drawing equal groups, we will begin by drawing dots inside the groups. As we complete more multiples in our booklet, we want students to begin writing the actual number in the groups instead of dots.

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12. Now, students will rotate through 3 tables to work with table groups to complete the remaining sections of the left side in the Multiple book. Each table can have a card that tells the students which part they should fill out. Allow about 3-5 minutes at each station.

13. Note that tables 1,2 and 3 are doing the same rotations as 4, 5 and 6. Students will only rotate through 1 set of tables.

14. The suggested warm up for each day is to go to the next page in the multiple book. Once on the next page, students can shade in the multiples and show the multiple representations of the fact provided. Students will use what they have done to fill in their multiple book. They do not need to rotate each day.

Part IV: Use What You Know to Compose and Decompose

**Note: Students will record this information in the middle of their Multiple Book where there are 2 blank pages.

**It is suggested you use a cup or baggie to represent a group and some counters or beans to represent items in the group.

1. “Let’s use what we know to see if we can make learning our multiplication facts easier and come up with some generalizations about multiplication?”

2. “Here we have 1 cup. Do we have anything in the cup?” (no)

3. “So, what does my cup represent?” (a group)

4. “What digit do we use to represent that there is not anything in the cup?” (0)

5. “Talk with your neighbor to finish the following.” Teacher writes the following on board.

6. “What did you discuss?”

7. “Perfect. What if we write ______ x 0 = _______. How would you fill that in?” (1 x 0 = 0)

8. “Great. Now let’s look at 2 cups. Do we have anything in them?” (no) So, talk with your neighbor to fill in the following. ______ x 0 = _______”

9. “What did you discuss?” (2 x 0 = 0)

10. “Fantastic. What if we had 3 cups, 4 cups or even 100 cups with nothing in them? What would be our product?” (0) “Why?” (because if we have groups with nothing in them, we don’t have any items. That would be 0)

11. “Could we say then that anytime we multiply a number by 0 we get ____?” (0)

12. “What if we didn’t have any cups?” (then we wouldn’t have any groups)

13. “Could we say that 0 groups of 1 item is 0 too?” (Yes) “Why?” (because we don’t have any groups with items in them)

14. “Woohoo! Let’s write this in our multiple book.” (pg. 11)

15. “Let’s take a different look.”

16. “Now, what if we have 1 cup and have 2 items in the cup? What does our cup represent?” (a group)

17. “How many items are in the cup?” (2) “How do we write 1 group of 2?” (1 x 2)

18. “What is my product if we have 1 x 2?” (2)

19. “Talk with your partner about how to fill in the following.” Teacher writes the following on board.

20. “What did you come up with?” (1 x 2 = 2) “Why?” (because 1 group of 2 items equals 2)

21. “What if we have 1 cup, but have 3 items in the cup. What number sentence can I write?” (1 x 3 = 3)

22. “Super. Why?” (because 1 group of 3 equals 3)

23. “Now, what if we have 1 cup with 4 items. What would the number sentence look like?” (1 x 4 = 4) “Why?” (because 1 group of 4 equals 4)

24. “Here’s a challenge. What if we had 1 cup with 102 items in it. What would my number sentence be?” (1 x 102 = 102) “Why?” (because 1 group of 102 equals 102)

25. “Share with your partner something you are noticing.”

26. “What did you discuss?” (1 times a number equals that same number)

27. “Great. Yes, 1 x a = a”

28. “Now, let’s change it up a bit. Here I have 3 cups and each cup has 1 item. What do my cups represent?” (3 groups) “How many are in each group?” (1)

29. “What is the number sentence that would represent 3 groups of 1?” (3 x 1)

30. “What is our product?” (3)

31. “Wow, so 3 x 1 = 3”

32. “Let’s try one more. What if I have 25 cups and 1 item in each cup? What would that number sentence look like?” (25 x 1 =)

33. “What is the product of 25 x 1?” (25)

34. “Awesome. What are you noticing now?” (a number x 1 = the same number)

35. “Yes, a x 1 = a. Let’s write this in our Multiple Book.”

36. “Say ‘Woohoo’ if you’re ready to learn how to use what we know to make things easier?” (WOOHOO!)

37. “Here we go. Let’s look at 6 x 7 in our Multiple book on pg. 12.”

38. “Is 6 x 7 a fact that we can answer really quickly?” (some students will say yes and some will say no. For those that say yes, let them know that they will learn another way to look at solving more difficult facts for them)

39. “Let’s look at the array model for 6 x 7 on pg.12. Where do I label the 6 and the 7?”

| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

40. “Because 6 x 7 is a more difficult fact, we can look at the 6 or the 7 and break or decompose one of them up depending on which facts we know best. I know my 2’s and 5’s the best so let’s break or decompose one of the numbers up into a 2 and a 5. Which number would I break up into 2 and 5? The 6 or the 7?” (the 7) “Why?” (because 5 + 2 = 7)

41. “Can anyone see where we can split the 7 in our array into 5 and 2?”

| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

42. “Let’s show the decomposition of 7 on our paper.”

43. “By looking at the array model, how do you think breaking the 7 into 5 + 2 will help us solve 6 x 7?” Maybe we could multiply 6 x 5)

| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

44. “We could. What is 6 x 5?” (30) “How did you figure that out?” (counted by 5 six times)

45. “Is that our answer to 6 x 7?” (No) “What do you think we might do now?” (Maybe do 6 x 2) “Awesome. What is 6 x 2?” (12) “How did you figure that out?” (counted by 2’s 6 times)

| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

46. “Is that the answer to 6 x 7?” (No)

47. “Hmmm, so we figured out that

6 x 5 = 30

And 6 x 2 = 12

What do you think we have to do with the products of 30 and 12 to find the product of 6 x 7?” (add 30 and 12)

48. “What is 30 + 12?” (42)

49. “Do we agree that we can now add 30 and 12 to find the product of 6 x 7?” (yes)

50. “That’s awesome. We used the facts we knew of counting by 2’s and 5’s to help us solve 6 x 7.”

51. “Now, let’s see the fancy way to write this.”

52. “When we see this number sentence, what does it mean based on what we just did?” (We multiply 6 and 5, then multiply 6 and 2, then add the products together)

53. “Wow, here’s a fancy way to write that.”

54. Have students record the last two number sentences in their Multiple book.

55. Note: This is only one way this could be done. You could show students to break apart the 6 into 3 + 3 and do (3 x 7) and then double that. The goal is to help students use what they know to help them solve multiplication problems. It will be different per student based on their knowledge.

Part V

Bridging 2nd grade Models to Unit Bar for Multiplication

TEKS: 3.5b Represent and solve one- and two-step multiplication and division problems within 100 using arrays, strip diagrams and equations.

Vocabulary: factor, product, sum, total, repeated addition, equal groups, multiple

Concrete Materials: two-color counters or transparent chips or color tiles and student copies of story problems

Students can work in pairs or small groups for this activity.

NOTE: When teaching multiplication and division both operations have groups, both have a number in each group and both have a total. Using the vocabulary words (groups, each and total) is essential to help kids understand the concept. However, we do NOT want to label G.E.T. in the model drawing. Students have a tendency to over-generalize the use of the G.E.T. strategy by using it on problems that do not involve multiplication and division.

Guided Practice 1 – In this example we are going to be bridging the 2nd grade drawings to the 3rd grade unit bar.

1. The story problem below should be displayed. Teacher will model the 4-step process while students use manipulatives.

Nadyia gets 4 colored counters from her mom each time she does one chore. Last week she did 6 different chores. How many colored counters did Nadyia earn altogether?

2. Each pair of students will need 24 counters and a dry erase marker. Students may draw on table, small dry erase boards, or page protectors placed under manipulatives.

3. The teacher will model the “Four-Step Problem Solving” process by looking for the main idea first. (Be sure that all students really know the meaning of a “chore.”)

4. “Let’s look at the details. What is a group or what do you think of when you hear the word group?” Give them time to think of words for “group” or ways to describe group… ( group has items or members in it)

5. “What represents the groups in the story?” (chores). “Why do they think that the chores are the groups in the story?” (Because she earns 4 counters for each chore.)

6. “Let’s read the first sentence out loud.” (Nadyia gets 4 colored counters from her mom each time she does one chore.) “Let’s draw a picture for our details. “How many groups should we draw?” (1). Why? (because she gest 4 counters for 1 chore) Have students use dry erase to draw the group. (they can be circles or squares) “Share with your partner how many counters we should put in the group? (4) Why? (Because she got 4 colored counters for each chore.)

**As the students build Chore 1 concretely, the teacher can draw the picture of the model on the Smart Board in the detail/known box of the 4-step process.

7. “Now, read the next sentence.” (Last week she did 6 different chores.) “How many more groups do we need to draw?” (5) “Why?” (because we need 6 groups in all) “Why?” (because she did 6 chores) Have the students continue to build the groups until all 6 groups are labeled and 4 counters have been placed in each group representing the 6 chores.

**As the students continue to build Chore 2, 3, ……6 concretely, the teacher can draw the picture of the model in the details/known section of the window pane.

8. “Let’s read our question now.” (How many colored counters did Nadyia earn altogether?) “What does our question want us to find?” (total number of counters)

9. “Super. If we think back to last year, where did we put our question mark?” (We didn’t) “Do any of you remember what we did last year to show a total amount from our equal groups?”

10. “Look at this picture for a hint.”

11. “Does this help?” (Oh yeah, we did something called a total bar) “That’s correct. What did we do with all of our counters to find the total?” (put them all in the total bar) Students can draw the total bar and the arrows from the groups into the total bar as you model it.

12. “Does this total bar remind you of anything we have done this year? Hint: think of operations” (addition and subtraction) “Super, what do we draw to help us decide an operation?” (a unit bar) “Exactly. Today we are going transform the total bar into a unit bar.”

13. “Because we are doing a unit bar, what do we need to put in front of our now unit bar?” (who and what) “Awesome. Who is our problem about and what is it about?” (Nadyia and her counters)

14. “Now, instead of moving all of our counters done at once, let’s move the 4 counters from chore 1 down into the unit bar and draw a line after it to make 1 group. Be sure to label the equal groups under the unit bar.

15. Continue to do this with the remaining 5 groups and place the ? on the top of the unit bar.

16. “What does our unit bar represent?” (6 groups with 4 in each group)

17. In the strategy, write 6 groups of 4. Also, have the students write the number sentence 6 x 4. (In this example, the students can count the counters to see that 6 groups of 4 equals 24.) Talk with students about the answer to a multiplication problem is called the PRODUCT. Note: When solving multiplication problems, students may use any of the multiple representations illustrated on their manila ‘Multiple Representation’ page.

18. In the how, students can say they made 6 equal groups of 4 and put together the equal sets of 4. Or, I multiplied 6 and 4 to find the product.

Examples from multiple representation page

Guided Practice 2: (use unit bar with pictures inside each group)

2. Lauree has 4 empty pages in her photo album. She can place 3 pictures on each page. How many pictures can she put in her photo album?

Guided Practice 3 (unit bar using numbers)

Note: Let students know that they are now big 3rd graders and instead of drawing pictures in our unit bar, we are going to write the actual number.

Guided Practice

1. Nadyia gets 4 colored counters from her mom each time she does one chore. Last week she did 6 different chores. How many colored counters did Nadyia earn altogether?

2. Lauree has 4 empty pages in her photo album. She can place 3 pictures on each page. How many pictures can Lauree put in her photo album?

3.

-----------------------

Additional Resources:

MATH_3_A_MULTIPLICATION DIVISION INCH GRID PAPER 2014_RES

MATH_3_A_MULTIPLICATION BOOK 2014_RES

MATH_3_A_MULTIPLICATION DIVISION SECTION 1 MINI 1 2014_RES

MATH_3_A_MULTIPLICATION DIVISION SECTION 1 MULTIPLICATION PRACTICE 2014_RES

4 + 4 + 4 + 4 + 4 = 20

Multiple Representations

4 + 4 + 4 + 4 + 4 = 20

Multiple Representations

4 + 4 + 4 + 4 + 4 = 20

_______ groups of ________

_______ groups of ____4____

Multiple Representations

5 groups of 4

4 + 4 + 4 + 4 + 4 = 20

Multiple Representations

5 groups of 4 = 20

4 + 4 + 4 + 4 + 4 = 20

Multiple Representations

5 groups of 4 = 20

5 x 4 = 20

4 + 4 + 4 + 4 + 4 = 20

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

# of groups

Total # of items

# of groups

Total # of items

# of groups

Total # of items

# of groups

Total # of items

0 0

Reminder, teacher is doing a demo for students of the t-chart

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

# of groups

Total # of items

0. 0

1. 4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

# of groups

Total # of items

0. 0

1. 4

2. 8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Total # of items

# of groups

0. 0

1. 4

2. 8

3. 12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

# of groups

Total # of items

0. 0

1. 4

2. 8

3. 12

4. 16

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

# of groups

Total # of items

0. 0

1. 4

2. 8

3. 12

4. 16

5. 20

Column 1

Row 1

Column 1

Row 1

Column 1

Row 1

5 rows with 4 counters in each row is 20

5 x 4 = 20

1

Rows

2

3

4

5

Rows

Array Model

5 rows with 4 in each row is 20

5 x 4 = 20

Array Model

Rows

5 rows with 4 in each row is 20

5 x 4 = 20

Rows

Area Model

Array Model

Rows

5 rows with 4 in each row is 20

5 x 4 = 20

=

Rows

Array Model

Area Model

5 rows with 4 in each row is 20

5 x 4 = 20

5 x 4 = 20

=

**Note: This is what the students work will look like once computed.

It is suggested students glue down half of the number line and fold it if there is not enough room

4 + 4 + 4 + 4 + 4 = 20

5 groups of 4 = 20

5 x 4 = 20

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

X4

# of groups

Total # of items

0. 0

1. 4

2. 8

3. 12

4. 16

5 20

5 groups of 4 is 20

5 x 4 = 20

Area Model

1

Rows

2

Array Model

Rows

3

=

4

=

5

=

=

5 rows with 4 in each row is 20

5 x 4 = 20

5 x 4 = 20

5 rows with 4 counters in each row is 20

5 x 4 = 20

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0

1

2

3

4

5

# of groups

Total # of items

X

0

1

2

3

4

5

# of groups

Total # of items

X

0

1

2

3

4

5

# of groups

Total # of items

X

0

1

2

3

4

5

# of groups

Total # of items

X

0

1

2

3

4

5

# of groups

Total # of items

X

0

1

2

3

4

5

# of groups

Total # of items

X

5 copies of 4 = 20

5 groups of 4 = 20

5 x 4 = 20

5 x 4 = 20

5 x 4 = 20

=

20

5 copies of 4 = 20

5 groups of 4 = 20

5 x 4 = 20

20 = 5 x 4

5 copies of 4 = 20

5 groups of 4 = 20

5 x 4 = 20

20 = 5 x 4

20 is 5 times as much as 4

5 copies of 4 = 20

5 groups of 4 = 20

5 x 4 = 20

20 = 5 x 4

20 is 5 times as much as 4

20 is 5 times bigger than 4

____ groups of ____

____ X ____= ____

____ = ____ X ____

____ is ____ times bigger than ____

____ is ____ times as many as ____

____ groups of ____

____ X ____ = ____

____ = ____ X ____

____ is ____ times bigger than ____

____ is ____ times as many as ____

Number Sentence Starter Strips

____ groups of ____

____ X ____= ____

____ = ____ X ____

____ is ____ times bigger than ____

____ is ____ times as many as ____

____ groups of ____

____ X ____ = ____

____ = ____ X ____

____ is ____ times bigger than ____

____ is ____ times as many as ____

____ groups of ____

____ X ____= ____

____ = ____ X ____

____ is ____ times bigger than ____

____ is ____ times as many as ____

____ groups of ____

____ X ____ = ____

____ = ____ X ____

____ is ____ times bigger than ____

____ is ____ times as many as ____

____ groups of ____

____ X ____= ____

____ = ____ X ____

____ is ____ times bigger than ____

____ is ____ times as many as ____

____ groups of ____

____ X ____ = ____

____ = ____ X ____

____ is ____ times bigger than ____

____ is ____ times as many as ____

____ groups of ____

____ X ____= ____

____ = ____ X ____

____ is ____ times bigger than ____

____ is ____ times as many as ____

____ groups of ____

____ X ____ = ____

____ = ____ X ____

____ is ____ times bigger than ____

____ is ____ times as many as ____

13

14

19

20

23

22

24

17

16

15

21

18

25

0

1

6

7

10

9

11

4

3

2

8

5

12

2

Multiples of _______

0-0

1. 2

2. 4

3. 6

4. 8

5. 10

X

2 + 2 + 2 + 2 + 2 = 10

Equal Groups

Array Model

T-Chart

2

5

groups of

2

Area Model

5

5

____ groups of ____

____ X ____ = ____

____ = ____ X ____

____ is ____ times bigger than ____

____ is ____ times as many as ____

5 rows with 2 in each row

1

Equal Groups with repeated addition

Equal Groups with repeated addition

Table 1

Table 6

Number lines and t-chart

Number lines and t-chart

Table 2

Table 5

area model

area model

Table 4

Table 3

______ group of 0 = _________

___1__ group of 0 = ____0____

a x 0 = 0 a = a number

0 x a = 0

_____ x 2 = 2

1 x a = 0 a = a number

a x 1 = 0

6 x 7

5 + 2

6 x 5 = 30

6 x 5 = 30

6 x 2 = 12

6 x 7

6 x (5 + 2)

(6 x 5) + (6 x 2)

Chore 1

Chore 6

Chore 5

Chore 4

Chore 3

Chore 2

Chore 1

Chore 1

Chore 2

Chore 3

Chore 4

Chore 5

Chore 6

N. C

Chore 1

Chore 2

Chore 3

Chore 4

Chore 5

Chore 6

Chore 2

Chore 3

Chore 4

Chore 5

Chore 6

Chore 1

N.C

Ch 1

?

Ch 2

Ch 3

Ch 4

Ch 5

Ch 6

N. C

# colored counters N has

Done on table

4 + 4 + 4 + 4 + 4 + 4 = 24

6 groups of 4 = 24

6 x 4 = 24

**made 6 equal groups of 4

** put together 6 equal sets of 4

**multiplied 6 and 4

?

L. ph

# pictures in photo album

Pg 1 Pg 2 Pg 3 Pg 4

4 groups of 3

4 x 3 = 12

I put together 4 groups of 3.

I multiplied 4 times 3 to find the product.

Group 1 Group 2 Group 3

Putting Together Equal Sets

Multiplication

?

Multiplication

Putting Together Equal Sets

At the point, it is suggested that you connect the action poster of putting together equal sets to the unit bar through a teacher demo. By putting the original action poster in a sheet protector, you can use a dry erase marker to show the unit bar on top of the picture as shown below.

Jessica got a new address book to write all of her friends’ information. A page of her address book is shown below.

If Jessica filled out 5 pages of her address book, how many friends’ information does she have?

# friends’ information

J. f

8 8 8 8 8

Pg 1 Pg 2 Pg 3 Pg 4 Pg 5

?

5 groups of 8

5 x 8 = 40

X8

1 8

2 16

3 24

4 32

5 40

Multiply to find the product of 5 and 8

When drawing the unit bar, use the vocabulary of putting together equal sets to find the total.

Jessica got a new address book to write all of her friends’ information. A page of her address book is shown below.

If Jessica filled out 5 pages of her address book, how many friends’ information does she have?

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