Pearson Assessments



|Topic | |Pages |

|Division using partial quotients and |Part I: Using SmartBoard Base Ten Blocks to Divide |1-5 |

|the area model | | |

| |Part II: Student uses Base Ten Block Manipulatives and Teacher Records the Area |6-22 |

| |Model | |

| |Part III: Student uses Base Ten Block Manipulatives while Teacher and Student |23-26 |

| |both Record the Area Model | |

| |Part IV: No Base Ten blocks are used while Student and Teacher Record the Area |27-28 |

| |Model | |

| |Part V: Only Student Records |29-30 |

| |Part VI: Looking at Other Area Models |31-32 |

|Connecting the Algorithm to the Base |Part VI: Connecting the Agorithm to the Base Ten blocks |33-37 |

|Ten blocks | | |

Fourth Grade Curriculum

Division Using Partial Quotients and the Area Model

Up to a Four-Digit Quotient with a One-Digit Divisor

Division using Partial Quotients and Area Models

4.4E- Supporting – represent the quotient of up to a four-digit whole number divided by a one-digit whole number using arrays, area models, or equations

4.4F – Supporting – use strategies and algorithms, including the standard algorithm, to divide up to a four-digit dividend by a one-digit divisor

4.4H – Readiness – solve with fluency one- and two-step problems involving multiplication and division, including interpreting remainders

Student Background – Students in 3rd Grade were given pictures in their story problems to divide. Students have also been taught to use number bonds (fact families) to find the quotient to basic facts up to 10 times 10. Because the focus is now on the process of dividing up to a four-digit dividend by a one-digit divisor, we might want to allow students to use a multiplication chart as an aid.

Math Vocabulary – dividend, divisor, quotient, remainder, array, area model, partial quotient

Materials – Base ten blocks, dry-erase boards and markers, SmartBoard, and SmartBoard file (MATH_4_A_DIVISION WITH PARTIAL QUOTIENTS SMARTBOARD_RES)

Part 1: Students will use SmartBoard Base Ten Blocks to Divide

1. After reading the story problem, students will use the SmartBoard pens to begin the Four-Step Process.

Main Idea – H. M. $ Henry in savings?

Details / Known – Model drawing will be used to determine the action / operations of the story problem. Record Who, What, and draw a unit bar.

Henry $

Henry saved $3,528 last year working two extra jobs.

Henry $

He spent half of his money on a trip to Disney World and put the other half in savings.

Henry $

How much money did Henry put in savings?

Henry $

2. Ask students:

• What do you see? What do you notice about the model drawing?

• Do we have equal groups in our model drawing? (Yes)

• What words from the story problem told you this? (Half, means to separate or divide into two equal groups.)

• How many equal groups do we have? How do we know this? (Two, Disney World and savings)

• Do we know how much money is in each group? (No)

• What is our question asking? (How much money is in savings?)

• How did we represent this in our model drawing? (With a question mark)

• Which action best represents what is happening in the story? Why? (Share Set Equally, because we are dividing our money into 2 equal groups)

• Which operation would this be? (Division)

3. Write a number sentence to find the amount of money put in savings. Make sure and put labels with each part of the number sentence to represent the story problem. This will help students understand the meanings of the remainders.

3,528 dollars ÷ 2 uses for the money= dollars for each use of the money

4. Have students use the infinity-cloned Base Ten blocks to divide 3,528 into 2 groups.

Note: Regrouping will be necessary.

Students will need to share the thousands. There should be one thousand cube in each group.

Ask students What can we do with this leftover thousand? (The leftover thousand will need to be regrouped into 10 hundreds.)

How many hundreds will we have now? (There will be a total of 15 hundreds, 10 from the thousand and 5 from the original number).

Share the hundreds. There should be 7 hundreds in each group with one left over.

What can we do with this leftover hundred? (The left-over hundred will need to be regrouped into 10 tens.)

How many tens do we have now? (12, 10 from the hundred and 2 from the original number.)

Share the tens. There should be 6 in each group with none remaining.

Share the ones. There should be 4 in each group.

How many did we end up with in each group? (1,764)

5. Continue the Four-Step Process by filling in the How / Justify section.

• Used the base ten blocks to separate 3,528 into 2 equal groups.

• Regrouped when needed.

1. Repeat the same process for dividing 403 into 3 equal groups. Regrouping will be necessary.

Slide 4:

Part II: Students will use base ten block manipulatives to divide while the teacher records the area model

1. After reading the story problem, students will use the SmartBoard pens to begin the Four-Step Process.

Main Idea – H. M. $ one girl earn?

Details / Known – Model drawing will be used to determine the action / operations of the story problem. Record who, what, and draw unit bar.

Girl $

Four best friends worked last Saturday and earned a total of $528.

Girl $

If each girl earned the same amount of money, how much money did one girl earn?

Girl $

2. Ask students:

• What do you see? What do you notice about the model drawing?

• Do we have equal groups in our model drawing? (Yes)

• What words from the story problem told you this? (If each girl earned the same amount of money…)

• How many equal groups do we have? How do we know this? (4, because there were 4 girls)

• Do we know how much money is in each group? (No)

• What is our question asking? (How much money one girl earned?)

• How did we represent this in our model drawing? (With a question mark)

• Which action best represents what is happening in the story? Why? (Share Set Equally, because we are dividing the money between 4 equal groups)

• Which operation would this be? (Division)

3. Write a number sentence to find the amount of money each girl earned. Make sure and put labels with each part of the number sentence to represent the story problem. This will help students understand the meanings of the remainders.

528 total dollars earned ÷ 4 girls = dollars for each girl.

4. Students will use their base ten blocks with a partner to divide 528 into 4 equal groups as the teacher models asking the questions and writing on the SmartBoard with specified colored pens.

Slide 5:

5. This slide will introduce using an area model to find the partial quotients. First, we will set up the area model. Pull the shade down on the right to reveal one question at a time. The SmartBoard eraser can be used to reveal the answers in red.

The labeled area model should look like this.

Slide 6:

6. Explain to students that the area model has been broken into 3 parts; hundreds, tens, and ones. Starting with the hundreds, we will be finding the area of each place value section and will continue to the tens place and ones place.

7. Students (with partners) will build the number 528 with 5 flats, 2 rods and 8 units. Students can draw 4 large sections on their table to represent the 4 girls. The Base Ten blocks will be laid inside each section.

8. As the teacher pulls down the shade, directions are revealed. The students will act out the division by passing out the Base Ten blocks to the rectangles drawn on their tables.

9. Teacher asks: Knowing that we have 4 groups, if we give each group 100, how many will we use?

Students will place one flat in each rectangle.

As the teacher asks the questions he / she is recording here. Be sure and use the blue pen to match the question color. This will show the different place values as we proceed.

Teacher asks: Knowing that we have 4 groups, if we give each group 200, how many will we use?

Students will attempt to place another flat in each rectangle. This will not be possible because only 1 flat remained.

As the teacher asks the questions he / she is recording here. Again, be sure and use the blue pen to match the question color.

Teacher asks: Were we able to give each group 200? Why or Why not? (No, because we only had 5 hundreds and once we gave each group 100 we only had 100 left.)

Teacher draws a line through the 200 (800) to show that this was not possible.

10. Teacher continues with questioning from the SmartBoard.

How many are in each group? Students will respond 100.

Teacher will circle the 100 to indicate that this is the number in each group.

11. Teacher continues with questioning from the SmartBoard.

What would the area be? Why? (400 because 4 groups of 100 is 400 or

4 x 100 = 400)

The teacher will record in two places. First inside the hundreds section of the area model and then below the 528.

12. The teacher will continue with the questioning from the SmartBoard:

How much area is remaining? If the area model represents 528 and we have used 400 to fill in the hundreds section, how much area is left for the rest of the model? How could we find this? (Subtract 400 from 528 to find the difference)

13. Point out that the remaining area, 128, will be left for the tens and ones section of the area model.

14. Ask students: What can we do with the remaining flat? (Break it apart into 10 tens.) Have students trade the flat for 10 tens. How many tens do we have now? Why? (12, because we got 10 tens from the hundred and and we already had 2 tens.)

15. Slide 7 will have all of our previous writing on it in blue. The teacher will now lead the students through the questioning in red to find the area of the tens section.

16. Teacher asks: Knowing that we have 4 groups, if we give each group 10, how many will we use?

Students will place one rod in each rectangle.

As the teacher asks the questions he / she is recording here. Be sure and use the red pen to match the question color.

Teacher asks: Knowing that we have 4 groups, if we give each group 20, how many will we use?

Students will place another ten in each rectangle.

As the teacher asks the questions he / she is recording here. Again, be sure and use the red pen to match the question color.

Teacher asks: Knowing that we have 4 groups, if we give each group 30, how many will we use?

Students will place another 10 in each of the four groups.

As the teacher asks the questions he / she is recording here using the red pen.

Teacher asks: Knowing that we have 4 groups, if we give each group 40, how many will we use?

Students will attempt to place another rod in each rectangle. This will not be possible because all rods have been passed out.

As the teacher asks the questions he / she is recording here with the red pen.

Teacher asks: Were we able to give each group 40? Why or why not? (No, because we didn’t have any more rods to pass out.)

Teacher draws a line through the 40 (160) to show that this was not possible.

17. Teacher continues with questioning from the SmartBoard.

How many are in each group? Students should respond 30.

Teacher will circle the 30 to indicate that this is the number in each group.

What would be area be? Why? 120, because 4 groups of 30 is 120 or

4 x 30 = 120)

The teacher will record in two places. First, inside the tens section of the area model and then below the 128.

18. The teacher continues with questioning from the SmartBoard.

How much area is remaining? If the area model represents 528 and we have used 400 to fill in the hundreds section, and 120 to fill in the tens section, how much is left for the rest of the model? How could we do this? (Subtract 120 from 128 to find the missing part.)

19. Point out that the remaining area, 8, will be left for the ones section of the area model.

20. Slide 8 will have all of our previous writing on it in blue and red. The teacher will now lead the students through the questioning in green to find the area of the ones section.

21. Teacher asks: If we give each group 1, how many will we use?

Students will place one unit in each rectangle and respond 4.

As the teacher asks the questions he/she is recording here. Be sure and use the green pen to match the question color.

Teacher asks: If we give each group 2, how many will we use?

Students will place another unit in each rectangle and respond 8.

As the teacher asks the question he/she records here with the green pen.

Teacher asks: If we give each group 3, how many will we use?

Students will attempt to place another unit in each rectangle. This will not be possible because there are no units remaining.

As the teacher asks the question he/she records here with the green pen.

Teacher asks: Were we able to give each group 3? Why or why not?

(No, because we did not have any units left.)

Teacher draws a line through the 3 (12) to show that his was not possible.

How many are in each group? (2)

Teacher will circle the 2 to indicate that this is the number in each group.

What would the area be? Why? (8, because 4 groups of 2 is 8 or

4 x 2 = 8)

The teacher will record in two places. First, inside the ones section of the area model and then below the 8.

22. The teacher will continue with the questioning from the SmartBoard.

How much area is remaining? We only had 8 left, and now we have used 8 in the ones section. How could we find this? (Subtract 8 from 8 to show that we do not have any left over.)

When we don’t have any of our dividend left over, what is this called? (A remainder of 0)

23. The partial quotients will now be added together to obtain the quotient of 528 and 4.

24. Return to slide 4 to complete the windowpane.

Justify / How: Used partial quotients and the area model to find the quotient of 528 and 4.

Part III: Students will use the Base Ten block manipulatives to divide. Both the teacher and the student will record. Students can use this problem as an example in their IMN.

MATH_4_A_DIVISION WITH PARTIAL QUOTIENTS AND AREA MODELS INTERACTIVE MATH NOTEBOOK_RES

1. After reading the story problem, students will use the SmartBoard pens to begin the Four-Step Process. In order to save room in the IMN, the windowpane will only be shown on the SmartBoard and not in the IMN.

2. Teacher will lead the students through the questioning presented earlier to determine the action of the story.

3. Write a number sentence to find the number of berries Robert gave to each person. Make sure and put labels with each part of the number sentence to represent the story problem. This will help students understand the meanings of the remainders.

437 berries ÷ 3 pails = berries for each (one) pail

4. Students will use their Base Ten blocks with a partner to divide 437 into 3 equal groups as the teacher models asking the questions on the SmartBoard and writing with specified colored pens. Consistency with the questioning is imperative for student understanding. Students will also be recording in their IMN at the same time.

5. Slide 12 will be used to set up the area model. After completing Slide 13, the area model will look like this:

After completing slide 14, the area model will look like this:

After completing slide 15, the area model will look like this:

6. Students will then use slide 17 to add the partial quotients together.

7. What was our remainder in this story? (2) Would it make sense to cut berries up into pieces in order to share them equally? (No) Should the answer to the question change based on our remainder? Why or Why not?

Discuss with students the meaning of the remainder. In this case, the remainder will not have an effect on the answer because if we give one extra berry to two of the people this will mean that one person will have less than the others.

8. Return to the windowpane (slide 11) to complete the Four-Step Process.

How / Justify: Used partial quotients and the area model to find the quotient of 437 and 3.

Part IV: Base Ten blocks will not be used. Both the teacher and students will record the steps on the area model. Students may use regular notebook paper while the teacher will record using the colored pens on the SmartBoard. Students may need to draw the base ten manipulatives in order to complete the area model.

1. After reading the story problem, students will use the SmartBoard pens to begin the Four-Step Process.

2. Teacher will lead the students through the questioning presented earlier to determine the action of the story.

3. Write a number sentence to find the packages of muffins that must be purchased. Make sure and put labels with each part of the number sentence to represent the story problem. This will help students understand the meanings of the remainders.

338 muffins needed ÷ 8 muffins per packages = packages needed

4. The teacher will lead the students through the questioning in order to complete the area model and partial quotients. Which action poster goes along with this story problem? (Take away equal sets). Discuss with students that even though we know the action changed we are still dividing and can continue to use the area model for finding our quotient.

This problem will be somewhat different in that the hundreds section of the area model will be 0. The same questioning and steps will be used regardless.

5. After completing slide 19, the area model should look as follows:

6. The teacher can then use the area model to show how to check with multiplication. If an area model looks like:

This would mean that we have 8 groups of 42 or 8 x 42

8 x 42 + remainder = dividend

7. Teacher asks: What was our remainder in this story? (2 muffins) Should the answer to our question change based on the remainder? Why or Why not? (Yes, because as we can see when we check with multiplication, if we only purchase 42 packages we will only be able to make 336 muffins. An extra package of muffin mix must be purchased in order to make the 2 muffins that are in the remainder. If each package can make 8 muffins and we only need 2 of them, we will have 6 extra muffins available.)

8. Return to slide 18 to complete the Four-Step Process:

How / Justify: Used partial quotients and the area model to find the quotient of 338 and 8

Note: Depending on student needs, other example problems can be done to practice using the area model with partial quotients.

Part V: Student will practice using the area model with partial quotients alone. Students can again draw the base ten manipulatives as needed.

1. After reading the story problem, students will use the SmartBoard pens to begin the Four-Step Process.

2. Teacher will lead the students through the questioning presented earlier to determine the action of the story.

3. Write a number sentence to find the number of bolts in each container. Make sure and put labels with each part of the number sentence to represent the story problem. This will help students understand the meanings of the remainders.

1,084 bolts ÷ 8 equal containers = bolts in each container.

4. The students will attempt the area model with partial quotients on their own. If needed, the teacher will lead the students through the questioning in order to complete the area model and partial quotients. Again, this problem will be somewhat different in that the thousands section of the area model will be 0. The same questioning and steps will be used regardless.

Ask students: How many sections do we need to include in the area model? Why? (4, because we have 4 digits in our dividend.)

5. After completing slide 21, the area model should look as follows:

6. Have students check with multiplication and add the remainder.

7. Teacher asks: How many bolts were in each container? (135) If we have 8 containers with 135 bolts in each, how many bolts is this? (1,082) Should the answer to our question change based on the remainder? Why or Why not? (No, the remainder will not affect our answer because if we give 4 of the containers an additional bolt each container will not be equal.)

8. Return to slide 20 to complete the Four-Step Process:

How / Justify: Used partial quotients and the area model to find the quotient of 1,084 and 8

Note: Depending on student needs, other example problems can be done to practice using the area model with partial quotients.

Part VI: Looking at other Area Models

1. Using slide 22 discuss how area model might look differently depending upon how the dividend was decomposed.

Looking at the first example:

Ask Students:

What do you notice about the area model? (answers may vary)

If we look at the first part of the area model, does 3 x 20 = 60? (Yes)

Does this hold true for the second part of the area model? (Yes,

3 x 20 = 60)

What about the third part? (Yes, 3 x 4 = 12)

What happens when we combine the 3 areas inside the model? (60 + 60 + 12 = 132)

Could this be one possible area model for 132 ÷ 3 ? (Yes)

Continue with this questioning for the other 3 examples:

Which example could not be an area model for 132 ÷3? Why? (The last example because the area does not equal 132.)

Note: Area Models that look like these can also be used as an intervention. Instead of putting the hundreds, tens, and ones in groups, students would use the facts they know with patterns of 10 or 100 to slowly take away from the dividend. For example:

Part VII: Connecting the Algorithm to the Base Ten blocks

Note: The algorithm presented may not resemble the traditional algorithm we as teachers remember and use. Although this algorithm looks different, the meaning is still the same. By using this algorithm, students can make better connections to the Base Ten blocks.

1. Use slide 23 to introduce the algorithm by having students make connections between the area model with partial quotients and the algorithm. Ask students: What do you see? What do you notice?

Points to emphasize:

• The coloring in the algorithm matches the coloring in the area model.

• The subtraction beneath the model matches the algorithm.

• The partial quotients are both recorded on top.

• The partial quotients are both written in expanded form.

2. Slide 24 will show how dividing the Base Ten blocks (done only on the SmartBoard) will tie to the algorithm. Students will share (click and drag) the hundreds equally between the 3 rectangles at the bottom of the screen.

Ask students the follow questions and record as needed:

How many are in each group? (100)

Where would we record this? (At the top of the algorithm)

How is this similar to the area model? ( We also recorded on the top of the area model)

How many did we use? (300, 3 groups of 100)

Where do we record this? (underneath 437)

How is this similar to the area model? (They both are recorded beneath the blocks left over)

How many are remaining? (137) How do we know this? (The Base Ten blocks show that we have 137 remaining.)

How do we show this in our algorithm? (subtract 300 from 437 to get 137)

After completing slide 24, the slide should look similar to this:

3. Slide 25 will show the work previously shown on Slide 24 with the new questioning for sharing the tens equally. In order to share all the tens, the remaining hundred will need to be regrouped to make 10 tens. (Students may use the infinitely cloned rod on the side of the screen to create more tens.) This will give us a total of 13 tens. Each group will get 4 tens (40) with 1 ten (10) remaining. Again, the teacher will lead the students through the questioning on the SmartBoard and record on the algorithm as needed.

After completing slide 25, the slide should look similar to this:

4. Slide 26 will show the work previously shown on Slide 25 with the new questioning for sharing the ones equally. In order to share all the ones, the remaining ten will need to be regrouped to make 10 units. (Students may use the infinitely cloned unit on the side of the screen to create more ones.) This will give us a total of 17 ones. Each group will get 5 ones with 1 one remaining. Again, the teacher will lead the students through the questioning on the SmartBoard and record on the algorithm as needed.

After completing slide 26, the slide should look similar to this:

5. Use slide 28 to provide more practice for students.

Note: Depending on student needs, other example problems can be done to practice using the algorithm.

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

Slide 2:

Slide 2

3,528

3,528

DW

S

3,528

DW

S

?

Slide 3:

Slide 4

G1

G2

G3

G4

528

528

G1

G2

G3

G4

?

Slide 5

4

528

These questions will be answered in upcoming slides.

Slide 6

100 (400)

200 (800)

100 (400)

200 (800)

100 (400)

200 (800

100 (400)

400

400 4 x 100 = 400

- 400

128

10 (40)

20 (80)

10 (40)

30 (120)

20 (80)

10 (40)

40 (160)

30 (120)

20 (80)

10 (40)

40 (160)

30 (120)

20 (80)

10 (40)

40 (160)

30 (120)

20 (80)

10 (40)

120

120

-

120 (4 x 30)

8

1 (4)

2 (8)

1 (4)

3 (12)

2 (8)

1 (4)

3 (12)

2 (8)

1 (4)

R. 0

8

120

400

Slide 10

Teacher will cut apart and give each student one copy of the story problem to put in their IMN. The area model will be drawn by the students as the teacher models.

Slide 11

Slide 18

42

Remainder 2

8

Slide 20

A.

30

30

60

+ 12

132

3 x 10 = 30

3 x 10 = 30

3 x 20 = 60

3 x 4 = 12

B.

C.

75

30

+ 27

132

3 x 25 = 75

3 x 10 = 30

3 x 9 = 27

D.

90

+ 12

102

3 x 30 = 90

3 x 4 = 12

Students are taking away “chunks” at a time based on facts they are familiar with.

132

- 60

72

- 60

12

- 12

0

132

Using slide 27, Students can now add their partial quotients together and check with multiplication. Don’t forget to add the remainder.

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