Grade 5
Lesson Title: Multi-Digit Division
Unit 1: Whole Number Computation and Applications Time Frame: 7-12 days
Essential Question: How do you choose different division strategies to divide multi-digit numbers?
Targeted Content Standard(s): / Student Friendly Learning Targets5.NBT.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-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. / I can…
· Divide multi-digit whole numbers.
· Explain how various models can be used to represent and solve problems involving division situations.
Targeted Mathematical Practice(s):
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure.
Look for an express regularity in repeated reasoning
Supporting Content Standard(s): (optional)
5.OA.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
Explanation of Rigor: (Fill in those that are appropriate.)
Conceptual:
Students will model division with multi-digit numbers using manipulatives, area models/arrays, and equations. / Procedural:
Students will use place value strategies and properties of operations to divide multi-digit whole numbers. / Application:
Students will apply their knowledge of the area model of division to solve real-life problems.
Vocabulary:
Divide
Divisor
Dividend
Tens
Hundreds
Thousands / Area model (length, width, area)
Array
Equation
Parentheses
Partial Quotient
Quotient
Evidence of Learning (Assessment):
Pre-Assessment: Division-Subtraction Relationship, Division-Multiplication Relationship
Formative Assessment(s): Dividing with Base 10 Blocks – Observation Checklist
Array/Area Model for Division-Student Practice Sheet
Partial Quotients Division Worksheet
Column division vs. Partial quotients Formative Assessment
Summative Assessment: 5.NBT.6 Base Ten Summative Assessment
Multi-Digit Division Summative Assessment
Self-Assessment: 5.NBT.6 Multi-Digit Division Summative Self-Assessment
Lesson Segments:
1. Assess student understanding of division/subtraction and division/multiplication relationships
2. Manipulative, area and array models for division
3. Solve real-world problems
4. Connecting area models and equations
5. Partial Quotients
6. Column Division
7. Assessment
Procedures:
Segment 1
Approximate Time Frame:
20-30 minutes / Lesson Format:
Whole Group
Small Group
Independent
Modeled
Guided
Collaborative
Assessment / Resources:
Division-Subtraction Relationship
Division-Multiplication Relationship
Focus:
Assess students’ ability to connect area and multiplication. / Modalities Represented:
Concrete/Manipulative
Picture/Graph
Table/Chart
Symbolic
Oral/Written Language
Real-Life Situation
Mathematical Practice Look Fors:
· MP4: Model with mathematics. Students should be making connections between the visual models and equations that they are using to represent multiplication of multi-digit numbers.
· MP7: Look for and make use of structure. Student’s visual models should represent the base 10 structure of numbers. / Differentiation for Remediation:
Have students model multiplication with either base 10 blocks or digi-blocks before making area models.
Differentiation for English Language Learners:
English language learners would benefit from being pre-taught the vocabulary: “area model” and “array” before beginning this segment.
Differentiation for Enrichment:
Have students create real world situation word problems for each of the problems.
Potential Pitfall(s):
Students may have difficulty connecting visual models to justify procedures for multiplying multi-digit numbers. / Independent Practice (Homework):
Procedure:
Have students complete the Division-Subtraction and Division-Multiplication Tasks independently.
These documents are pre-assessments for this lesson. Students have used area models in the previous lesson for multiplication and in Grade 4 for division by 1-digit numbers. Students will build upon this experience to use area models and arrays for division of multi-digit numbers. / Teacher Notes/Reflections:
Segment 2
Approximate Time Frame:
90-100 min. / Lesson Format:
Whole Group
Small Group
Independent
Modeled
Guided
Collaborative
Assessment / Resources:
Base 10 Blocks
Tape to make a Workmat (optional)
Dividing with Base 10 Blocks Worksheet
Dividing with Base 10 Blocks Observation Checklist
Focus:
Students will create manipulative, area and array models for division. / Modalities Represented:
Concrete/Manipulative
Picture/Graph
Table/Chart
Symbolic
Oral/Written Language
Real-Life Situation
Math Practice Look For(s):
MP4: Model with mathematics. Students will use base ten blocks to connect place value concepts to model the process of division.
MP5: Use appropriately tools strategically. Look for students to strategically use the base 10 blocks to represent division. Students should know when to break up the 100 into 10 tens to represent the division situation.
MP7: Look for and make use of structure. Students can see and understand how numbers are put together as parts and wholes. / Differentiation for Remediation: Students may benefit from being reminded how to write numbers in expanded notation to determine where to segment the diagram into sections.
Differentiation for English Language Learners: Permit ELLs to work with partners who speak the same language and discuss their representations in their native language.
Differentiation for Enrichment: Students who are using mental math or multiplication to solve problems should be asked to justify their calculations using visual models. They should also be asked to connect the written equations (using parentheses) to the visual models.
Students can solve problems such as 403 ÷ 31, where it requires regrouping in the dividend. Students have to keep in mind that a rectangular array must be formed with the dividend. Example:
Potential Pitfall(s):
Some students may not choose to use hundreds, but only work with tens and ones.
Students may have difficulty representing the portions of the diagram with equations. They may not know where to divide the diagram into portions.
Students may need to be reminded that the dividend should form a rectangular array. / Independent Practice (Homework):
1) Write an equation for the following base ten blocks:
Answer: 396 ÷ 33 = 21
2) Draw a representation for 231 ÷ 21 =
Procedure:
1. Group students in partners or triads. Give each pair/triad 5 hundreds, 30 tens, 30 ones and a workmat.
2. Have the partners collaboratively build the number 320 out of base 10 blocks.
3. Next, tell them they will need to divide these into equal groups of 20. (If students can do this mentally, have them create the visual model to justify their calculation.)
4. Observe how students are arranging their groups. See if any students arrange their models in an array. Listen to students discussions as they collaborate with their partners to hear their strategies. (Observation Checklist)
5. If any students have arranged their manipulatives into an array, have them share their work. If not, model for students how you can arrange the base 10 blocks into an array using the workmat. (See diagram- The green represents the divisor 20 shown with 2 tens. The blue represents the dividend 320 shown as 2 hundreds and 12 tens. Start with the divisor and then lay the blocks for the dividend to line them up along the divisor blocks. Finally ask what block would be multiplied by 10 to equal 100? What block would be multiplied by 1 to equal 10? Lay the blocks down accordingly.
*Note: If the students have not used base ten blocks to model multiplication, they may need help understanding that the width of the divisor is the width of the dividend and the length of the quotient is the length of the dividend.
*Students need to think about how to arrange the dividend into a rectangular array based upon the number of equal groups that must be created.
6. Once the students model the problem with base ten blocks, they can connect it to an area picture (not drawn to scale). Ask them how this picture connects back to the area model of multiplication.
7. Have students repeat the process with their partners and then ask them to try to solve 440 ÷ 22 using the base ten blocks
*Note: The teacher may initially model how to place the divisor (green) on the workmat first and allow the students to explore how to place the dividend (blue).
on the workmat. Once they have it placed they can determine the quotient (red).
8. Once the students model the problem with base ten blocks, they can connect it to an area picture (not drawn to scale).
9. Again observe and listen using the observation checklist with the groups. Have students share out and then try to solve 132 ÷ 12 with the blocks.
/ Teacher Notes/Reflections:
Teacher Notes/Reflections:
Segment 3
Approximate Time Frame:
45-50 minutes / Lesson Format:
Whole Group
Small Group
Independent
Modeled
Guided
Collaborative
Assessment / Resources:
5.NBT.6 Base Ten Assessment
Focus:
Students will apply their knowledge of base ten blocks and division to solve real world problems. / Modalities Represented:
Concrete/Manipulative
Picture/Graph
Table/Chart
Symbolic
Oral/Written Language
Real-Life Situation
Math Practice Look For(s):
MP1: Make sense of problems and persevere in solving them. Students will be able to plan a solution that makes sense of the problem by connecting their prior knowledge of dividing with base ten blocks.
MP3: Construct viable arguments and critique the reasoning of others. Students will cite evidence and develop a logical argument. / Differentiation for Remediation:
Allow students to use base ten blocks to help solve the problem.
Differentiation for English Language Learners:
Define length, width, and area to assist with the garden problem.
Differentiation for Enrichment:
Have the students create another possible solution using 224 as the dividend. Then either use base ten blocks to model the solution or draw a diagram of the base ten blocks.
Potential Pitfall(s):
In the garden problem, students need to find various unknowns within the dividend and quotient. They need to analyze the problem and determine a strategy to figure out the various problems which includes eliminating unnecessary information. / Independent Practice (Homework):
Procedure:
1. Either assign partners or small groups to work together to solve the assessment.
/ Teacher Notes/Reflections:
Segment 4
Approximate Time Frame:
135-150 minutes / Lesson Format:
Whole Group
Small Group
Independent
Modeled
Guided
Collaborative
Assessment / Resources:
· Array/Area Model for Division-Teacher Notes
· Array/Area Model for Division-Student Practice Sheet
· Division Array/Area Model Formative Assessment
Focus:
Students will connect arrays to area models and use equations to represent the division situations. / Modalities Represented:
Concrete/Manipulative
Picture/Graph
Table/Chart
Symbolic
Oral/Written Language
Real-Life Situation
Math Practice Look For(s):
MP2: Reason abstractly and quantitatively. Students need to reason to appropriate divisors to use when dividing using the array model. Students need to be able to use all of the operations to solve the problems.
MP4: Model with mathematics. Students will use the area model to bridge the understanding of division with base ten blocks to partial quotients.
MP7: Look for and make use of structure. Students can see and understand how the dividend can be broken into smaller parts. / Differentiation for Remediation:
You may want to list the multiples of the divisor from 1-10 instead of just the multiples of 2, 5, and 10 in the help box.
For students that need a visual that is drawn to scale, they may begin with a smaller dividend and draw it to scale on centimeter grid paper.
Differentiation for English Language Learners:
Differentiation for Enrichment:
Encourage the students to create multiple solutions using the area model and then describe which one was the most efficient.
Potential Pitfall(s):
Some people prefer to draw a large rectangle first and divide it in sections as they solve the problem. Some prefer to build the sections, as they solve it. However, the rectangles do not need to be drawn to scale which may lead to some confusion. / Independent Practice (Homework):
A fish tank at the Shedd Aquarium holds 6,358 gallons of water. If every fish needs 13 gallons of water to survive, how many fish can be safely put into the tank?
Use an array model to solve the problem.
Or you can use the Division Array/Area Model Formative Assessment.
Procedure:
1. Review the Array/Area Model for Division-Teacher notes prior to the lesson. It clearly defines how to model the method and offers examples that can be used with the students in the Student Practice Sheet.
2. You may choose to watch the Teacher Tube video listed in the notes by yourself or as a class.
3. The benefits of this division method are that it is a visual representation that connects the multiplication area model and it allows the students to use different factors, which provides differentiation.
4. Prior to solving the problem, it may be beneficial to create a “help box” that offers multiples of the divisor. 2, 5, and 10 were chosen for the “help box” because they are basic factors that can later transfer to mental math.
Work out the following problem on the board.
Problem: 494 ÷ 13
Example 1:
Step 1: Draw a rectangle to represent the area model and record the divisor on the left. Eventually the area of the rectangle will be equal to the dividend.
Step 2: Choose a multiple of 2, 5, or 10 so that 13 x that multiple = a number less than 494. Possible dialogue: “What would be the best number to choose to multiply by 13 to get an answer close to 494? 2, 5, or 10?”