Grade 4 - Richland Parish School Board
Grade 4
Mathematics
Unit 8: Building Mathematical Fluency
Time Frame: Approximately three weeks
Unit Description
This unit provides an opportunity for a more in-depth analysis of addition, subtraction, multiplication and division by building on the understanding of the algorithms learned in previous units as they analyze the mathematical basis for the algorithms. Comparisons are made between standard algorithms and alternative methods and those methods are used to solve multi-step word problems.
Student Understandings
Students will use place value and their previous knowledge of addition, subtraction, multiplication, and division to explore the standard algorithms of each operation and compare them to alternative methods. Students will use their knowledge of place value and the four operations to critique computational mistakes. Students will also solve multi-step word problems involving all four operations with additional attention paid to remainders.
Guiding Questions
1. Can students model and represent addition, subtraction, multiplication, and division with objects and verbal situations?
2. Can students fluently add and subtract multi-digit whole numbers using the standard algorithm?
3. Can students solve for an unknown quantity in addition, subtraction, multiplication, and division number sentences?
4. Can students model multiplication of 2-digit by 2-digit numbers?
5. Can students use the multiplication algorithm to solve problems involving up to 4-digit by 1-digit numbers, and 2-digit by 2-digit numbers?
6. Can students use alternative methods to solve division problems?
7. Can students interpret remainders?
Unit 8 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) |
|19. |Solve one-step equations with whole number solutions (A-2-E) (N-4-E) |
|CCSS for Mathematical Content |
|CCSS # |CCSS Text |
|Operations and Algebraic Thinking |
|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. |
|Number and Operations in Base Ten |
|4.NBT.4 |Fluently add and subtract multi-digit whole numbers using the standard algorithm. |
|4.NBT.5 |Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit |
| |numbers, using strategies based on place value and the properties of operations. Illustrate and explain the |
| |calculation by using equations, rectangular arrays, and/or area models. |
|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. |
| |Review the key ideas expressed and draw conclusions in light of information and knowledge gained from the |
| |discussions. |
Sample Activities
2013 – 2014
Activity 1: One Answer - Multiple Methods (CCSS: 4.NBT.4, SL.4.1b, SL.4.1c, SL.4.1d)
Materials List: One Answer- Multiple Methods BLM, pencil, paper, learning logs
Give each student the One Answer – Multiple Methods BLM. Have students look at the problem 574 + 869 = n. Working in groups, have students solve the problem using method #1 on the BLM. Have students discuss how they got 1,300 in step 1, 130 in step 2, and 13 in step 3. Have students describe their thoughts in their learning logs (view literacy strategy descriptions). Have students solve the problem using method #2. Have students discuss how they got 13 in step 1, 130 in step 2, and 1,300 in step 3. Have them describe their thoughts in their learning logs. Ask students to solve the problem using method #3. Have students discuss why there is a 1 above the tens place in step 1 and why there is a 1 above the hundreds place in step 2. Have students describe their thoughts in their learning logs.
Have students participate in a fishbowl discussion (view literacy strategy descriptions). In a fishbowl discussion, students can improve their learning and comprehension of mathematical material by participating in a dialog about the class topics. Select a group of four students to discuss how method #1 and method #2 are similar and different. Encourage the students to use mathematics vocabulary and place value to discuss the topic. While the group is discussing, have the rest of the class listen but not contribute to the conversations of the students in the “fishbowl.” When the students finish discussing, have the onlookers discuss their reactions to the conversations they observed. Allow them to make additional comments about the comparisons that were left out from the original conversation.
Split the class into groups of four. Have them discuss how method #2 and method #3 are similar and different. Engage students in a classroom discussion after the groups finish.
Have the students move to new groups to discuss how method #1 and method #3 are similar and different. Conclude with a class discussion after the groups finish.
2013 – 2014
Activity 2: What If Addition and Subtraction (CCSS: 4.NBT.4, W.4.2d)
| |
Materials List: What If Addition and Subtraction BLM, paper, pencils
Review with students how to solve addition and subtraction problems. Give students 3-digit plus 3-digit problems such as 456 + 372 = n. Have students solve the problem and use place value concepts to explain to their partners how to solve it (Reference Unit 2, Activity 2 as an example). Engage students in SPAWN writing (view literacy strategy descriptions). SPAWN writing provides students with meaningful opportunities to write about mathematics content. For this SPAWN, students will be writing based on the what if? prompt. In this what if? prompt, prompt students write about how a sum or difference changes if the value of one of the digits changes. For example, the original problem, 456 + 372 = n, n = 828. Prompt the students to write what would happen if the 6 in 456 became a 9. Have students write their responses from a place value perspective noting the changes to the ones, tens, and hundreds places of the sum. Have them consider questions such as these: Does the ones place change? Do I need to compose ten ones into one ten? Does the tens place change? And so on. Have students read aloud their writing. Discuss the similarities and differences of students’ responses. Continue to give students what if? prompts for different addition problems.
Review with students how to use place value to solve subtraction problems. Give students 3-digit minus 3-digit subtraction problems such as 833 – 295 = n. Have students solve the problem and use place value concepts to explain to their partners how to solve the problem (Reference Unit 3, Activity 2 as an example). Give students what if? scenarios prompting them to write about how the difference changes as a result in a change in value of one of the digits. Provide students with additional subtraction problems and what if? prompts. Have students read aloud their writing. Discuss the similarities and differences of students’ responses.
Have students practice fluently adding and subtracting multi-digits using the standard algorithm. Provide students with the What If Addition and Subtraction BLM. Have students solve the problem and then change one of the digits in each number as they did in the What if prompts. Have students solve the new problem and explain how the change impacted their answer.
2013 – 2014
Activity 3: Finding the Mistake (CCSS: 4.NBT.4, W.4.2.d)
Materials List: Finding the Mistakes BLM, paper, pencil
Give students the problem 618 + 943 = n and have them solve it (n = 1,561). Tell students that someone thought that n = 1,551. Have students determine where the mistake was made and explain the mistake using mathematical vocabulary and place value concepts. Provide students with the Finding the Mistakes BLM. After students have completed the BLM, lead a discussion about the mistakes.
Have students participate in RAFT writing (view literacy strategy descriptions). For this assignment, the RAFT will be:
R – Role of Writer – the teacher checking student’s work
A – Audience – the student whose work is being checked
F – Form – letter
T – Topic – the mistake that the student made solving the problem
Have students work with partners to write their RAFT. Have the students choose at least two addition problems and two subtraction problems. When they finish, have students share their RAFTs with another group or the whole class. Have students listen for accuracy and logic. Listen to students’ RAFTs to assess the students’ understandings of addition, subtraction, and place value.
2013 - 2014
Activity 4: Adding and Subtracting using the Standard Algorithm (CCSS: 4.NBT.4)
Materials List: Monitoring Fluency BLM, stopwatch, paper, pencils
Have students work in pairs to practice adding and subtracting using the standard algorithm. Provide each pair with the Monitoring Fluency BLM. Have students monitor the time it takes them to accurately complete the addition and subtraction problems using the standard algorithm. Have one student work a problem. Have the partner record the time it takes the partner to finish the problem. Have students switch roles, complete the same problem and record the time it took to complete the problem. If the two answers differ, have both students rework the problem to determine the error. Have students continue in this format to work all of the problems. Monitor the students for accuracy and efficiency. Have students repeat this activity over the course of the unit to monitor the growth of their fluency. Use problems similar to those on the BLM. Chart students’ accuracy and efficiency over the course of the unit. As students become more fluent, extend the number of digits in each problem to 1,000,000.
2013 - 2014
Activity 5: Addition and Subtraction Multi-step Word Problems (CCSS: 4.OA.3, 4.NBT.4)
Materials List: Addition and Subtraction Multi-step Word Problems BLM, paper, pencil
Review with students the types of addition and subtraction problems (reference Unit 2, Activity 15 and Unit 3, Activity 1 for examples). Provide students with the Addition and Subtraction Multi-step Word Problems BLM. Have students read the first problem and make a diagram of what the numbers in the problem represent. The diagram will help students organize their thinking for the multiple steps. Have the students write the number sentences using variables to represent the unknown quantity as they add or subtract from their diagrams. For example:
28 children were playing on the playground in the morning. In the afternoon, 15 more children came. Later in the day, 9 children left. How many children are at the park at the end of the day?
28 children at the playground
15 children came
28 children + 15 children = t children
28 + 15 = t, so t = 43
9 children left
43 children – 9 children = c children
43 – 9 = c; so c = 34
There were 34 children at the end of the day.
After the students have found their answers, have them using rounding to estimate the reasonableness of their answers (30 + 20 = 50; 50 – 10 = 40; 34 is a reasonable answer).
Tell students that drawing a picture is a great way to help them understand how to solve the problem. Give students the following word problems.
• Problem 1: Jefferson Elementary has 479 students. Madison Elementary School has 347 students. Franklin Middle School has 648 more students than both schools combined. How many students does Franklin Middle School have? (479 + 347 = 826. 826 + 648 = 1,474 students)
• Problem 2: The Annual Spring Fair is Friday, Saturday, and Sunday. Theo’s hot dog stand sold 4,390 hot dogs on Friday and 1,476 fewer hot dogs on Saturday than Friday. If they sold 9,503 hot dogs over the three-day weekend, how many hot dogs did they sell on Sunday? (Possible solution: 4,390 – 1,476 = 2,914 hot dogs on Saturday; 9,503 – 4,390 – 2,914 = 2,199 hot dogs on Sunday)
Have students draw pictures to solve the problems and have a few students explain what they did to solve the problem.
Have students work the rest of the problems on the BLM. Lead a discussion of how they solved the problems.
Activity 6: Multiplication Fact Fun (GLEs: 10, 15, 19)
Materials List: decks of playing cards, paper, pencils
Have students work in pairs to use playing cards to practice their multiplication facts. Remove the jacks, queens, and kings from each deck. Have students shuffle the deck and split it between the two students with their pile facing down. At the same time, have each player turn over the top card in his/her deck. Have Player 1 say the number sentence that the factors represent, give an example of a real-life situation in which those numbers would be multiplied, and find the product. For example, if the cards turned over are a 3 and an 8, Player 1 might respond by saying 3 times 8 equals n, 3 bags of candy with 8 pieces in each bag, 3 times 8 equals 24. Player 2 must check Player 1. If Player 1 is correct, he/she gets the 2 cards. If the answer is incorrect, Player 2 must correct the errors and then he/she gets the cards. Have players rotate answering and checking. The object of the game is to gather the most cards.
Activity 7: Shortcut Multiplication Method for Multi-Digit Multiplication (GLE: 10; CCSS: 4.NBT.5)
Materials List: base-ten blocks, paper, pencils
Give students the problem 43 × 56 = n. Review with students how multiplication can be modeled using an array (Unit 2, Activity 9). Have students use base-ten blocks to set up the arrays and have them draw their model in their learning logs (view literacy strategy descriptions).
Teacher Note: If you do not have enough thousands blocks for the entire class, you can substitute strawberry crates. Explain to the students that while the strawberry crates represent thousands cubes, they are not the exact same size as a base-ten thousands block. Another method would be to use pictures of a thousands block. Their models should look like this:
50 6
40 40 × 50 40 × 6
3 3 × 50 3 × 6
2000 + 240 + 150 + 18 = 2408
Explain to students that the shortcut multiplication method is similar, but it only has two numerical rows. For example:
43
50 43 × 50
6 43 × 6
2150 + 258 = 2408
Students are using their knowledge of the distributive property when visualizing the problem this way.
43 × 56 = 43(50 + 6) = 43 × 50 + 43 × 6.
Have students use their knowledge of 2-digit by 1-digit multiplication to multiply 6 × 43.
43
× 56
Ask students which numbers are multiplied first (6 × 3). Ask a student to explain what needs to be done with the 18 ones. (18 ones is the same as 1 ten and 8 ones. The 8 ones remain in the ones column and the 1 ten is placed above the tens column.)
1
43
× 56
8
Ask students which numbers are multiplied next (40 × 6). Ask students what should happen with the 1 ten that was placed above the tens column (Add the 1 ten to the product (240) to get 250.) Explain to students that since this is the shortcut method, the 250 goes on the same line as the 8. Since 250 has a 0 in the ones place, the 8 remains in the ones place from the first step. Tell students to use what they had at the bottom, but to put it into the paragraph. They could also leave it at the bottom.
1
43
× 56
258
Ask students what numbers need to be multiplied now (50 × 3). Ask students why multiply by 50. (Since the 5 in 56 is in the tens place, they are multiplying 50 × 3 and not 5 × 3.) Have students multiply 50 × 3 (150). 150 is the same as 1 hundred + 5 tens + 0 ones. Explain that the 0 ones and 5 tens are written in the second row and the 1 hundreds is written above the 4 to be added in the next step when 50 is multiplying 40.
1
1
43
× 56
258
50
Ask students what is left to multiply (40 × 50). Ask students what should happen with the 1 hundred that was placed above the tens column. (The product (2,000) adds 1 hundreds from the previous step to become 2,100. The 2 is written in the thousands column; the 1 is written in the hundreds column, and since there are zeros in the tens and ones column of 2,100, the 5 tens and 0 ones remains the same.)
1
1
43
× 56
258
2,150
Ask students what is the last thing needed to be done to find the product of 43 × 56. (Add the section (or partial) products together: 258 and 2,150.)
1 1 1
1 1 1 1 1
43 43 43 43 43
× 56 × 56 × 56 × 56 × 56
8 258 258 258 258
50 2,150 + 2,150
2,408
Provide students with multiple opportunities to find the unknown product in a number sentence by practicing the shortcut multiplication method using 2-digit by 2-digit factors. Giving the students the problem in the form of a number sentence helps them develop algebraic thinking.
Activity 8: Comparing Expanded and Shortcut Methods (GLE: 10; CCSS: 4.NBT.5)
Materials List: Comparing Expanded and Shortcut Multiplication Methods BLM, base-ten blocks
Give each student the Comparing Expanded and Shortcut Multiplication BLM. Have students look at the problem 39 × 82 = n. Have students use base-ten blocks to solve the problem using method #1 shown on the BLM. Have students discuss how they got 18 in step 1, 60 in step 2, and 720 in step 3, and so on. Have students describe their thoughts in their learning logs (view literacy strategy descriptions). Ask students to use base-ten blocks to solve the problem using method #2. Have students discuss why there is a 1 above the tens place in step 1 and what the 7 stands for above the tens place in step 3. Tell students to write their conclusions in their learning logs.
Have the students use the graphic organizer (view literacy strategy descriptions) on the Comparing Expanded and Shortcut Multiplication Methods BLM to compare and contrast expanded and shortcut multiplication methods. Graphic organizers are visual displays that students can use to organize information so that it is understandable to them. In a Venn diagram, students place information that is unique to each topic in the outside section and the information that is shared between the two topics in the inside, intersecting section. For example, students might write that the two methods are similar because they both start with the ones digits. Students may say that they are different in that the expanded method writes the value of each step below the problem, whereas the value is written both above and below in the shortcut method.
Have students work in pairs to label one section as expanded method and the other section as shortcut method. Have the students use their conclusions from their learning logs to determine how the two methods are similar and how they are different. Have students present their graphic organizers to the class. Ask students to use it as a study aid as they continue developing their understanding of 2-digit by 2-digit multiplication.
Have students continue to find the unknown product in a number sentence by using the expanded and shortcut methods. Give students problems in which they are finding the unknown product when there are 2-digit by 2-digit factors, 3-digit by 1-digit factors, and 4-digit by 1-digit factors.
Activity 9: Rectangle Section Division (GLE: 10; CCSS: 4.NBT.6)
Materials List: base-ten blocks, paper, pencils
One way that students learn division is through a process known as rectangle section division in which students use a rectangle to find the unknown length of the rectangle when given the area and one length. Give students the problem 460 ( 5 = n. Have students draw a rectangle with the area inside the rectangle and the divisor as the vertical length.
5 460
Ask students, “5 times what number of tens gives an answer closest to 460 without going over?” (5 × 9 tens = 45 tens or 5 × 90 = 450. 5 × 10 tens = 50 tens or 5 × 100 = 500. 90 is the answer closest to 460 without going over.) Write 90 at the top of the rectangular section. Multiply 5 × 90 and write the product, 450, under the dividend, 460, inside the rectangle.
90
5 460
450
Have students subtract 460 – 450 and write the difference, 10, at the bottom of the rectangle section.
90
5 460
- 450
10
Draw a second rectangle to the right of the first rectangle. Write 10 in the second rectangle. Ask students, “5 times what number of ones gives an answer closest to 10 without going over?” (5 × 2 = 10) Write 2 at the top of the rectangular section. Multiply 5 by 2 to get 10. Write this 10 below the 10 in the second rectangular section.
90 2
5 460 10
- 450 10
10
Have students subtract 10 – 10 and write the difference, 0, at the bottom of the right rectangle section. For larger dividends, the students would continue putting the difference in another section to the right until the difference is less than the divisor.
90 2
5 460 10
- 450 - 10
10 0
Have students add the quotients above the rectangle section. (90 + 2 = 92)
90 + 2 = 92
5 460 10
- 450 - 10
10 0
460 ( 5 = 92.
Provide students with additional division problems in which they have to find the unknown quotient of a number sentence. Have them practice finding the quotient using the rectangle section method with 3-digit and 4-digit dividends. For example, give them the problem 4365 ( 7.
600 20 3 R4
4365 165 25
7 - 4200 - 140 - 21
165 25 4
4365 ( 7 = 600 + 20 + 3R4 = 623R4
Activity 10: Patterns in Remainders (GLE: 10; CCSS: 4.NBT.6)
Materials List: Patterns in Remainders BLM
Provide students with the Patterns in Remainders BLM. Have students independently work to solve the nine division problems. Pair students and have them discuss the pattern that they saw. (As the dividend increased by 1, the remainder increased by 1 until the divisor divided evenly again.) Have students relate the division problems to addition and multiplication problems. For example, 41 ( 8 = 5 r1 becomes 5 × 8 = 40. 40 + 1 = 41. Have students relate all the division problems to multiplication and addition number sentences. Ask students for dividing by 8, what were the remainders? (1, 2, 3, 4, 5, 6 and 7) Ask why they think there was not a remainder greater than 7. (If the remainder was 8 or greater, a larger number should have been used in the quotient.) Give them examples of different division problems to determine the remainder and see how large the remainder can be. Ask students to predict the possible reminders for other divisors. Ask why 30 ( 6 did not have a remainder. Have students conclude that the remainder is always smaller than the divisor.
2013 - 2014
Activity 11: Division Word Problems with Remainders (CCSS: 4.OA.3)
Materials List: Division Word Problems with Remainders BLM, pencil, paper
Provide each student with the Division Word Problems with Remainders BLM. Have students work in pairs to discuss the problems. The students should realize that each question has a numerical solution of 49 ( 6 = 8 r1. Assign each pair of students a question on which to focus. Have students discuss what each remainder means in each question. Have students report to the class the meaning of the remainders in each question (Question 1: the remainder is ignored; Question 2: the remainder causes the answer to be rounded up; Question 3: the remainder is the only number that matters.) Students should realize that the result they get from solving the algorithm does not always answer what the question is asking. Provide students with additional division word problems with remainders to discuss and solve.
Activity 12: Practicing Multiplication and Division Algorithms (GLE: 10, 19; CCSS: 4.NBT.5, 4.NBT.6)
Materials List: Practicing Multiplication and Division Algorithm BLM, decks of playing cards, calculators, pencils
Have students work in groups of four to practice the multiplication and division algorithm for 4-digit by 1-digit multiplication, 2-digit by 2-digit multiplication, and 4-digit by 1-digit division. Remove the jack, queen, and king from a deck of playing cards. Provide each pair of students with the playing cards and a calculator to check their answers. Have the students shuffle the playing cards. To practice multiplication of a 4-digit number by a 1-digit number, have one student turn over the top 5 cards. Have each student try to make the largest product possible with the 5 cards arranged as a 4-digit number times a 1-digit number. Display page 1 of the BLM to show them the format to use. Have each student write the number sentence for the multiplication problem and have them solve the problem. Have students compare their products to see who created the number sentence that produced the largest product. Ask students to use the calculator to check their work. Have students repeat the process with 2-digit by 2-digit multiplication and then division with 4-digit dividends by 1-digit divisors.
Sample Assessments
General Assessments
• Maintain portfolios containing student work.
• Record anecdotal notes on students as they complete tasks.
• Give prompts such as the one that follows, for students to record their thoughts in their personal math learning logs.
o Ask students to demonstrate comprehension of addition, subtraction, multiplication, and division concepts in real-world problems.
o Ask students to explain how to interpret the remainder in division problems.
• Give prompts such as the ones that follow, and have students record their thoughts in their personal math journals.
o Explain how you can use rectangle sections to solve division problems
o Write a story problem that shows when it would be easier to multiply than to add.
Activity-Specific Assessments
• Activity 3: Provide students with several addition and subtraction problems that contain errors. Have students solve the problems correctly and explain the errors made using place value concepts.
• Activity 5: Provide students with addition and subtraction multi-step word problems. Have students solve the problems using the diagrams or pictures representing the numbers in the word problem and the situation. Have students write number sentences as they draw the pictures.
• Activity 7: Give each student a 2-digit by 2-digit multiplication problem. Have them solve it using shortcut multiplication. Have students explain, using sentences and place value concepts, how each step of shortcut multiplication works.
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