Grade 4 - richland.k12.la.us
Grade 4
Mathematics
Unit 3: Performing Multi-Digit Whole Number Subtraction and Division
Time Frame: Approximately four weeks
Unit Description
This unit develops a more complete understanding of the operations of subtraction and division. Students learn the subtraction algorithm for 4-digit minus 4-digit numbers and solve division problems with 4-digit dividends and 1-digit divisors, with and without remainders. This unit develops students’ use of division as sharing, repeated subtraction, and division as related to whole numbers and fractions. These skills build on the understanding of place value, the distributive property, arrays, as well as number facts and the concepts of multiplication and division.
Student Understandings
Students will use division as repeated subtraction, as the sharing principle, as work tied to equal groups and as area to develop a basis for mastery of the basic facts for division through 144 ÷ 12. They will successfully complete subtraction and division activities with and without manipulatives. Students will understand the concept of subtraction and its algorithm. Students will develop an understanding of the concept of division and will be able to divide 4-digit numbers by 1-digit numbers, with and without remainders, using a variety of methods.
Guiding Questions
1. Can students model and represent subtraction with objects and verbal situations?
2. Can students model and represent division with objects and verbal situations?
3. Can students show mastery of the basic division facts through 144 ( 12?
4. Can students solve simple whole number sentences having whole number solutions related to the division facts?
5. Can students model and solve division of 4-digit numbers by a 1-digit number with and without a remainder?
.
Unit 3 Grade-Level Expectations (GLEs) and Common Core State Standards (CCSS)
|Grade-Level Expectations |
|GLE # |GLE Text and Benchmarks |
|Number and Number Relations |
|10. |Solve multiplication and division number sentences including interpreting remainders (N-4-E) (A-3-E) |
|Algebra |
|15. |Write number sentences or formulas containing a variable to represent real-life problems (A-1-E) |
|CCSS for Mathematical Content |
|CCSS # |CCSS Text |
|Operations and Algebraic Thinking |
|4.OA.2 |Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and |
| |equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative |
| |comparison from additive comparison |
|4.OA.3 |Solve multistep word problems posed with whole numbers and having whole-number answers using the four |
| |operations, including problems in which remainders must be interpreted. Represent these problems using |
| |equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental|
| |computation and estimation strategies including rounding. |
|4.OA.4 |Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of |
| |each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given |
| |one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite. |
|Number and Operations in Base Ten |
|4.NBT.2 |Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare |
| |two multi-digit numbers based on meanings of the digits in each place, using >. =, and < symbols to record |
| |the results of the comparisons. |
|4.NBT.4 |Fluently add and subtract multi-digit whole numbers using the standard algorithm. |
|4.NBT.6 |Find whole number quotients and remainders with up to four-digit dividends and one-digit divisors, using |
| |strategies based on place value, the properties of operations, and/or the relationship between |
| |multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, |
| |and/or area models. |
|ELA CCSS |
|CCSS # |CCSS Text |
|Writing Standards |
|W.4.2d |Write informative/explanatory texts to examine a topic and convey ideas and information clearly. Use precise|
| |language and domain-specific vocabulary to inform about or explain the topic. |
|Speaking and Listening Standards |
|SL.4.1 |Engage effectively in a range of collaborative discussions (one-on-one, in groups, and teacher-led) with |
| |diverse partners on grade 5 topics and texts, building on others’ ideas and expressing their own clearly. |
| |Follow agreed-upon rules for discussions and carry out assigned roles. |
| |Pose and respond to specific questions by making comments that contribute to the discussion and elaborate on|
| |the remarks of others. |
Sample Activities
2013 - 2014
Activity 1: Types of Subtraction Problems (CCSS: 4.OA.3)
Materials List: Types of Subtraction Problems BLM, paper, pencils
Using the Types of Subtraction Problems BLM, students will complete a modified word grid (view literacy strategy descriptions) for a variety of subtraction problems. Discuss the differences among the problems. Ask students what the questions are asking and what the different methods are for answering the questions. Have students write number sentences containing a variable to solve the word problem. Work together with students to complete the word grid so that students will better understand subtraction problems. Once the word grid is formed, give students other problems. Have students draw the problems, solve them, and mark on their word grid which type of problem it was. In this way, the completed grid can serve as a review tool for students in preparation for quizzes and other assignments.
2013 - 2014
Activity 2: The Subtraction Algorithm (CCSS: 4.NBT.4, SL.4.1b, SL.4.1.c)
Materials List: base-ten blocks, pencil, paper
Students will use their knowledge of place value to complete the standard algorithm for subtraction. Provide the students with base-ten blocks. Begin with 2-digit minus 2-digit problems such as 34 – 17 = n. Have students write the problem using the variable, n, to stand for the unknown difference. Explain to students that a variable is used to stand for an unknown number. Have students use three tens and four ones to make 34. Model solving the problem having students use their blocks to follow along. Subtract 17 from 34 by renaming one ten as 10 ones. Show that 10 ones + 4 ones = 14 ones. Subtract 7 ones from 14 ones. Use paper and pencil to show how to record this in the standard algorithm.
Have students work in pairs to solve 42 – 26 = n using base ten blocks and recording their work on paper. Have each pair orally explain to another pair how they got their answer.
Make sure to include modeling the subtraction of 3-digit numbers and then 4-digit numbers. Give the problem 2,456 – 827 = n. Model for the students how to solve this problem using base-ten blocks and how to explain the process using split-page notetaking (view literacy strategy descriptions).
Example:
| |I have the problem 2,456 – 827 = n |
|2,456 | |
|- 827 | |
| 4 16 |There are not enough ones to take 7 ones from 6 ones, so I have to rename one ten as 10 ones. I am making|
|2, 4 5 6 |a mark through the 5 tens and rewriting it as a 4 because there are 4 tens now instead of 5 tens. I am |
|- 8 2 7 |making a mark through the 6 in the ones place because there are now 16 ones since I added 10 ones to the |
| |6 ones that were already there. |
| 4 16 |16 ones minus 7 ones is 9 ones. I am writing a 9 in the ones place of the answer. |
|2, 4 5 6 | |
|- 8 2 7 | |
|9 | |
| 4 16 |4 tens minus 2 tens is 2 tens. I am writing a 2 in the tens place of the answer. |
|2, 4 5 6 | |
|- 8 2 7 | |
|2 9 | |
|1 14 4 16 |There are not enough hundreds to take 8 hundreds from 4 hundreds so I have to rename one thousand as 10 |
|2, 4 5 6 |hundreds. I am making a mark through 2 thousands and rewriting it as 1 thousand because I used 1 thousand|
|- 8 2 7 |to make 10 hundreds. I am making a mark through 4 hundreds and rewriting it as 14 hundreds because I |
|2 9 |added 10 hundreds from the 1 thousand to the 4 hundreds that were already there. |
|1 14 4 16 |14 hundreds minus 8 hundreds is 6 hundreds. I am writing a 6 in the hundreds place of the answer. |
|2, 4 5 6 | |
|- 8 2 7 | |
|6 2 9 | |
|1 14 4 16 |I have 1 thousand left. Since I did not have to subtract any thousands, I am writing a 1 in the thousands|
|2, 4 5 6 |place of the answer. |
|- 8 2 7 | |
|1 6 2 9 | |
|2,456 – 827 = 1, 629 |2,456 – 827 = 1,629. n = 1,629. |
Give the students additional subtraction problems to solve using base-ten blocks and split-page notetaking. Have students reference their notes to study independently or with a partner for other class activities, homework, or quizzes.
Activity 3: Understanding Division (CCSS: 4.NBT.6)
Materials List: Understanding Division BLM, paper, pencil
Have students maintain a modified vocabulary self-awareness chart (view literacy strategy descriptions) for division. Provide this list of words to the students and have them complete a self-assessment of their knowledge of the words using a chart like the one below. Do not give students definitions/descriptions or examples at this stage. Ask students to rate their understanding of each word with either a “+” (understands well), a “√” (limited understanding) or a “–“ (don’t know). Over the course of this unit, students should be told to return to the chart and add new information to it. The goal is to replace all the check marks and minus signs with a plus sign. Since students continually revisit their vocabulary charts to revise their entries, they have multiple opportunities to practice and extend their understanding of key division terms.
|Word |+ |√ |– |Example |Definition/Description |
|division | | | | | |
|divisor | | | | | |
|quotient | | | | | |
|dividend | | | | | |
|division notations | | | | | |
|division as sharing | | | | | |
|division as whole | | | | | |
|numbers | | | | | |
|division as | | | | | |
|fractions | | | | | |
|division as repeated| | | | | |
|subtraction | | | | | |
Activity 4: Dividing with Tiles (CCSS: 4.OA.4)
Materials List: counters or tiles, Dividing with Tiles BLM, learning log
Students, working in pairs, will use the Dividing with Tiles BLM to begin exploring number patterns of division. Students will sort tiles into groups of twos and record their answers on the table according to the directions on the Dividing with Tiles BLM. Students should begin to recognize a pattern of numbers divisible by two. (Numbers that end in 2, 4, 6, 8, or, 0 are divisible by 2.) Continue using numbers that are exactly divisible by 5 (numbers that are multiples of five). Have the students sort the tiles into groups of fives. Students should be able to identify a pattern for numbers that are exactly divisible by 5. (Numbers that end in 5 or 0 are divisible by 5) Students progress to sorting tiles into groups of tens, then threes. Students should be able to recognize and explain patterns that emerge for numbers that are divisible by 2, 3, 5, and 10. Students should record the patterns that they have discovered in their math learning logs (view literacy strategy descriptions). This information will serve as a reference for future division problems and serve as a key for unlocking many simple division problems. After the students have recorded their observations in their math logs, discuss the divisibility rules they have discovered. Check that all number patterns for divisibility have been found.
(A number is divisible by another number if there is no remainder when dividing.)
• A number is divisible by 2 if the number is an even number. (Even numbers end with 2, 4, 6, 8, or 0.)
• A number is divisible by 5 if the number ends in 5 or 0.
• A number is divisible by 10 if the number ends in 0.
• A number is divisible by 3 if sum of the digits of the number are divisible by 3.
Repeat this activity on another day and include numbers that are evenly divisible by a given number, as well as numbers that will have remainders. Conduct a class discussion on how you are able to predict whether a division problem will have a remainder. If they cannot figure this out on their own, have them refer to their math logs and the divisibility rules.
Activity 5: Division Dollars: Division as Sharing (CCSS: 4.NBT.6)
Materials List: paper money, paper, pencil, learning log
Tell students they will begin keeping a record in their math learning log (view literacy strategy descriptions) of different ways division can be used. This will create a quick reference that they will be able to use to help clarify problems. It will also serve as a concrete way to see division as more than an algorithm. Have them label this section, Division as Sharing. Give each group of two students a zip lock bag with paper money. Give them this problem to complete. You and a friend have been washing cars. Together you have made $34. How can you share the $34 between the two of you? You only have $10 bills and $1 bills.
Solution: They can share one $10 bill with each of the two people and trade the remaining one $10 bill in for ten $1 bills. These ten $1 bills, plus the original four $1 bills, are shared, giving seven $1 bills to each person; hence, 34 ( 2 = 17.
If the original problem had involved a remainder, the remainder would have represented dollars left over. Additional problems can be made that involve paper money for $100, $10, and $1 bills. After students can easily perform these fair shares, they are ready to move to the algorithm, using the dollar bill place value understandings to develop the standard algorithm.
Activity 6: Division as Repeated Subtraction (CCSS: 4.NBT.6)
Materials List: paper, pencil, newspapers or magazines, calculators
Present a division problem as a repeated subtraction situation such as “Sally has 56 stickers and would like to give her friends 4 stickers each. How many friends will get 4 stickers?” This can be solved by repeatedly subtracting 4 from 56 until the remainder is either less than 4 or zero. Have students use the calculator to perform the repeated subtraction. Have them record each time they subtract by using tally marks. In this case, 4 can be subtracted from 56 fourteen times. So, Sally can give 14 of her friends 4 stickers each.
Have students work in groups of four and use pictures from the newspaper or magazines to create a text chain (view literacy strategy descriptions) that lends itself to repeated subtraction.
For example: One group may cut out a bag of candy. Have the students in that group work to write a story about the bag of candy. Have each student contribute one sentence.
The first student writes--It was Christmas and the class was going to have a party.
The second writes--John brought a bag of candy that had 96 pieces in it to share with his class.
The third writes--His teacher said he could give four pieces to each student.
The fourth writes--There were 23 students in the class. Was there enough candy for everyone?
Have students solve their problem. Have them record their text chain in their math learning log (view literacy strategy descriptions) and label this section Division as Repeated Subtraction.
Activity 7: Writing Division Comparison Problems (GLE: 15; CCSS: 4.OA.2)
Materials List: paper, pencils
Give the students the following problem:
A video game costs $48. That is 3 times more than a board game. How much does a board game cost?
Review with students what multiplication and division comparison is. Have students identify which quantity is being divided (the $48 video game) and which number tells how many times it is divided (3 times). Have students write a number sentence to match the word problem (48 ( 3 = n).
Give the students the following information: The quantity being divided is 36 and the number of times it is divided is 4. Have students work in pairs to write a word problem that matches the criteria. Have the students share their word problems with the class. Listen to see if the students matched the numbers to the criteria correctly.
Give the students other criteria to write their own word problems. Have the pairs give their problems to other groups. Have the groups identify which quantity is being divided and which number identifies how many times. Have the students write a number sentence to match the word problem.
Activity 8: Division Letters (GLE: 10; CCSS: 4.NBT.6, W.4.2d)
Materials List: base-ten blocks, paper, pencils
Give the students the following problem: A 4th grade teacher bought 4 new pencil boxes. She has 260 pencils. She wants to put the pencils in the boxes so that each box has the same number of pencils. How many pencils will there be in each box? Teach students four ways to solve this division problem: using base-ten blocks, place value, multiplication, and arrays.
For example:
Using base-ten blocks: Build 260 with base-ten blocks and distribute them into 4 equal groups. Rename the 2 hundreds as 20 tens so that 6 tens can be given to each group. Rename the remaining 2 tens as 20 ones and give 5 ones to each group. Each group gets 6 tens and 5 ones or 65.
Using place value: 260 ( 4 = (200 ( 4) + (60 ( 4) = 50 + 15 or 65.
Using multiplication: 4 ( 50 = 200, 4 ( 10 = 40, 4 ( 5 = 20; 50 + 10 + 5 = 65; so 260 ( 4 = 65.
Using a rectangular array or area model: Make a rectangle and write 4 at the top. Students need to think of the rectangle as representing a total of 260.
4 4 4 4
Example:
50 200 50 200 50 200
260
10 40 10 40
5 20
.
1. Think aloud: 4 times what number is close to 260? 4 ( 50 is 200 so I will record 50 as a factor and partition the rectangle into 2 rectangles. I will label the area aligned to the factor of 50 as 200. I’ve only used 200 out of the 260 so I still have 60 left.
2. Think aloud: 4 times what number is close to 60? 4 ( 10 is 40, so I will record 10 as a factor and partition the rectangle again. I will label the area aligned to the factor of 10 with 40. I’ve only used 240 of the 260, so I have 20 left.
3. Think aloud: 4 times what number is close to 20? 4 ( 5 is 20, so I will record 5 as a factor for the last rectangle. I will label the area aligned to the factor of 5 with 20.
4. Express your calculation in 2 ways:
a) Expressing the division using subtraction
260
- 200 (4 ( 50)
60
- 40 (4 ( 10)
20
- 20 (4 ( 5) 260 ( 4 = 50 + 10 + 5 = 65
0
b) Expressing the division using addition
260 ( 4 = (200 ( 4) + (40 ( 4) + (20 ( 4) = 65
Once students are familiar with each method, divide them into four groups. Assign each group a method of solving division problems. Each group will participate in RAFT writing (view literacy strategy descriptions). For this assignment, the RAFT will be:
R – The dividing method they were assigned
A – One of the other three dividing methods
F – Letter
T – Teaching the other method how to solve a division problem based on their own methods. Discuss with them how the two methods are similar and different.
Have students work in the group to write their RAFT explaining to them another division method of how to solve a division problem using their method and how the two methods are similar and different. When they finish, have students share their RAFTs with a group or the whole class. Have students listen for accuracy and logic. Listen to the students’ thoughts on the RAFT’s being read to assess the students’ understandings of the division method.
Activity 9: A Remainder of One (GLEs: 10, 15; CCSS: 4.NBT.6)
Materials List: A Remainder of One, manipulatives (black beans or red beans make great “ants”), dry erase marker, whiteboard
Supply 25 manipulatives for each student and allow students to recreate the corresponding groups of ants from the book, A Remainder of One, by Elinor Penczes, as it is read. In this book, the main character, Joe, has to continue rearranging the number of lines of his squadron so that there are no bugs left out (the remainder). As Joe arranges the lines, have students write the number sentence for the situation and predict what will happen. Have them give an estimated answer for the solution before using the manipulatives to prove the results.
As an extension of this book or if the book is unavailable, the students can play the game “Mingle.” To play “Mingle,” call a given number of students to the front of the room. Have them walk around until a number is called. When the number is called, they must stop walking and quickly put themselves in groups of that designated number. Have a student write the corresponding division sentence on the board. Continue the game with the same group of children but call different numbers to form other groups. Example: Thirteen children are mingling and the teacher calls 3. The students stop and make groups of 3. After all groups are formed, 1 is left over (remainder). Have another student write the corresponding division sentence 13 ÷ 3 = 4 R 1 on the board.
2013-14
Activity 10: Meaningful Remainders (CCSS: 4.OA.3)
Materials List: Divide and Ride, paper, pencil
Read, discuss, and have students act out Divide and Ride by Stuart Murphy to illustrate division with remainders in a real-life situation. In Divide and Ride, children go to an amusement park as one large group. Once in the park, they have to divide into different sized groups in order to fit onto the different rides. Have students explore how many students are in each group as well as how many students are left over. Provide students with several division word problems that include remainders. Remainders in real-life problems can be handled in different ways.
Some remainders must be included: A class trip is planned where transportation must be arranged by using parent volunteers and their vehicles. If only 4 students can ride in a car and there are 30 students in the class, how many cars will they need? Discuss this situation with the students and help them see that 7 cars will not be enough and so the remainder of 2 students requires that 8 cars be used.
Other remainders do not need to be included: Mrs. Honeysuckle has 86 pieces of candy. She wants to give the candy to 10 students. If everyone must get the same amount of candy, how much candy does each student get? Discuss this situation with the students and help them see that each student gets 8 pieces of candy. The remainder of 6 candies is ignored.
Provide students with additional problems to practice with including multi-digit division.
Sample Assessments
General Assessments:
• Maintain portfolios containing samples of student work that illustrates the various approaches to solving division problems.
• Record anecdotal notes on students as they complete tasks.
• Give prompts such as the ones that follow, and have students record their thoughts in their personal math journals.
o Your class is planning a party. You want each person to have at least three
cookies. There are 48 cookies in a bag. There are 24 children in your class.
Will one bag be enough? Why or why not?
o Explain why 12 R2 is not reasonable for 242 ÷ 2.
o Your class is making gingerbread boys to sell. How many gingerbread boys
will you be able to make with 112 raisins for eyes?
Activity-Specific Assessments
• Activity 1: Give the students a variety of subtraction problems. Have them solve the problem, illustrate what the question is asking them to do, and explain which type of subtraction problem it represents.
• Activities 5, 6: Give the students real-life division problems. Have the students work the division problems in more than one way. (Repeated subtraction, sharing, drawing pictures, using multiplication to work backwards, etc.)
• Activities 9, 10: Create a thematic bulletin board display that illustrates real-life division with remainder scenarios. Use problems from the unit to serve as examples. Have students write ways to solve problems when there is a remainder.
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