2012-13 and 2013-14 Transitional Comprehensive Curriculum
Grade 6
Mathematics
Unit 2: Strengthening Whole Number Division
Time Frame: Approximately two weeks
Unit Description
This unit focuses on reviewing division algorithms for whole numbers that involve various representations of the remainder. The scope of problems extends to dividing a 4-digit by a 2-digit number.
Student Understandings
Students can accurately divide multi-digit whole numbers and can select the appropriate representation of the remainder in real-life situations. Students understand the essential role that estimation plays in solving division problems.
Guiding Questions
1. Can students explain the reasonableness of an answer in a division problem?
2. Can students use different methods to solve division problems?
3. Can students represent remainders in various ways?
4. Can students solve multi-step problems involving multiplication and division of whole numbers?
MATH: Grade 6 Grade-Level Expectations (GLEs) and Common Core State Standards (CCSS)
CCSS for Mathematical ContentCCSS# / CCSS Text
The Number System
6.NS.2 / Fluently divide multi-digit numbers using the standard algorithm.
ELA CCSS
Reading Standards for Literacy in Science and Technical Subjects 6–12
RST.6-8.7 / Integrate quantitative or technical information expressed in words in a text with a version of that information expressed visually (e.g., in a flowchart, diagram, model, graph, or table).
Sample Activities
Activity 1: Understanding Division (CCSS: 6.NS.2)
Materials List: Dividing BLM, pencil
Provide students with several scenarios that require a solution obtained through division. These scenarios should depict situations that help students see division as repeated subtraction and as sharing. In the repeated subtraction or measuring type of division, the product and the number in each group is known. The number of groups must be found. In the sharing type of division, the product and the number of groups is known. The number in each group must be found.
One algorithm for division with a 2-digit divisor is the simple process of repeated subtraction, but starting with multiples of 10.
Example 1: Division as Repeated Subtraction
There are 357 sixth graders registered for Southfork Middle School. The principal wants to know the least number of teachers he has to hire if the classes can be no larger than 27.
Ten classes of students would total 270, but 20 would be 540 students which is too many. Since the number of teachers has to be less that 20, subtracting 270 is a good place to start. .
That leaves 87 students.
Ask students to estimate how many more teachers will be needed. Students may say that 87 is close to 90 and 27 is close to 30 and.
Check the answer, .
That leaves 6 students without a teacher! Discuss with students the best way to deal with a remainder in this situation. This remainder could be written as r6, as 6/27, or as .22… When answering the question, however, the principal will have to hire 14 teachers or 6 students will be left without a teacher.
13 r6
270
87
81
6
Example 2: Division as Sharing
The Saints donated 450 tickets to Magnolia Middle School for next week’s football game. The school has decided to distribute them evenly amongst the sixth grade students at the school. If there are 54 sixth graders, how many tickets will each student receive?
If each student is given 1 ticket, that would be 54 tickets; 2 tickets each, would be 108 tickets; 3 tickets each, would be 162; 4 tickets each, would be 216 tickets; 5 tickets each, would be 270 tickets; 6 tickets each, would be 324; 7 tickets each, would be 378 tickets, 8 tickets each, would be 432 tickets; and 9 tickets each, would be 486 tickets. 486 tickets are more than the Saints donated, so each student would receive 8 tickets and there would be 18 extra tickets. Discuss with students the best way to deal with a remainder in this situation. This remainder could be written as r18, as 18/54, or as .3… In this case, the remainder is ignored. Each student would get 8 tickets. The remaining 18 could be given to the teachers.
432
18
Distribute the Dividing BLM and have them work in groups to solve the problems. These problems will help students better understand the concept of division as well as the process. They will begin to understand that the division algorithm can be a shortcut for using repeated subtraction. Have students write the remainders as fractions. Discuss the answers as a class.
Activity 2: Division Algorithm (CCSS: 6.NS.2, RST.6-8.7)
Materials List: paper, pencil
Students are expected to fluently and accurately divide multi-digit whole number with divisors that are any number of digits. Have students use their understanding of place value to describe what they are doing as they divide. When using the standard algorithm, students’ language should include reference to place value. Have students explain the process in a modified split-page notetaking (view literacy strategy descriptions) format. Model the approach by displaying an example of how to set up the papers for split-page notetaking similar to the example below. Explain the value of taking notes in this format by saying it logically organizes information and ideas; it helps separate big ideas from supporting details; and it allows inductive and deductive prompting for rehearsing and remembering the information.
Tell students to draw a vertical line approximately 2 to 3 inches from the left edge on a sheet of notepaper. They should try to split the page into one third and two thirds. Have students write the algorithm on the left and explain each step on the right side. Continue to periodically model and guide students as they use split-page notetaking to increase their effectiveness with this technique. Assessments should include information that students recorded in their split-page notetaking. In this way, they will see the connection between the taking notes in this format and achievement on quizzes and tests.
There are 200 forty-twos in 9536./ 200 times 42 is 8400.
9536 minus 8400 is 1136.
/ There are 20 forty-twos in 1136.
/ 20 times 42 is 840.
1136 minus 840 is 296.
/ There are 7 forty-twos in 296.
7 times 42 is 294.
/ The remainder is 2. There is not a full forty-two in 2; there is only part of a forty two in 2.
This can also be written as or.
9536 = 227 × 42 + 2
Activity 3: Remainder Game (CCSS: 6.NS.2)
Materials List: Remainder Game BLM, pencil
Provide the students with the opportunity to test various divisors and the impact of those divisors on the quotient and remainder. This activity is played in pairs. Distribute the Remainder Game BLM to the students. The BLM has the numbers 1 to 20 across the top. As they are each used, they will be crossed out and will not be used again.
Begin with the number 100 as the starting number. The first student chooses a divisor from the list and divides the number 100 by that divisor. The remainder is awarded to that player as his/her score. The other student records the division sentence and marks the problem as belonging to the first player. Both players subtract the remainder from the starting number to get the new dividend. The second player now chooses a divisor for the new dividend, divides the problem, and takes the remainder as a score. The first player is charged with recording the division sentence for the second player. Play continues in this fashion with the two players alternating until the starting number is 0, or it is no longer possible for either player to score. The person with the highest score wins. Emphasize to students that points come from remainders, so students should choose divisors carefully to yield the largest remainder possible. If a remainder of 0 is given, the student receives no points, and the opponent repeats the previous starting number but chooses a different divisor.
The Remainder Game1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
Starting #
100 / John
90 / Mary
76 / John
65 / Mary
60 / John
51 / Mary
42 / John
30 / Mary
16 / John
9 / Mary
5 / John
3 / Mary
2 / John
John’s score: 51 Mary’s score: 47
Allow students to play several times until they feel comfortable with the game and have begun developing a strategy to win the game. Ask questions like, “What is the best first move?” “Why?” “What would happen if the starting number were a lower number? A higher number?”
After the students have thoroughly explored this number, give them a new starting number that is larger than 100. In their math learning log (view literacy strategy descriptions), have students explain how the change in the divisor affects the size of the quotient. Give a real-world example to illustrate your answer. Allow students to work in pairs to review their log entries for accuracy. Students should discover that if the dividend stays the same, the larger the divisor, the smaller the quotient.
Activity 4: Dealing with Remainders (CCSS: 6.NS.2)
Materials List: calculators, Remainders BLM, pencil
Allow students to use calculators to solve these division problems where the focus will be on how to handle the remainders. As a class, solve the following problems and discuss how to represent the remainder.
· Dropped remainders: If it takes 8 yards of ribbon to make one bow, how many bows can be made with 22 yards of ribbon? (Division gives 2.75. Since you can’t make 0.75 bow, the correct answer is 2 bows.)
· Remainders written as fractions: 25 mini-pizzas were delivered to school. If there were 20 students in the class, what part would each student receive? ()
· Remainders written as decimals: Mary bought 22 cans of soup. The total cost was $14.96. How much did one can of soup cost? ($0.68)
· Remainders that dictate an increase in the quotient: 150 scouts are going on a camping trip. Each tent will sleep no more than 12 scouts. How many tents will they need? Since you need a whole tent instead of one-half a tent, the answer would be “at least 13 tents.”)
Distribute the Remainders BLM for the students to solve. Have the students work in pairs to solve the problems. Discuss the answers as a class.
Activity 5: Division Problems (CCSS: 6.NS.2)
Materials List: Division BLM, calculator, pencil, paper
Distribute the Division BLM. Have students work the problems by hand. When they are finished, have them use a calculator to check their answers. Check for accuracy.
After discussing the solutions to the problems above, have the students create a text chain (view literacy strategy descriptions). Put students in groups of four. On a sheet of paper, ask the first student to write the opening sentence for the math text chain. For example, a student might begin the chain with this sentence.
Jen had to read a book with 434 pages.
The student then passes the paper to the student sitting to the right, and that student writes the next sentence in the story, which might read:
She read the same number of pages each day for 14 days.
The paper is passed again to the right to the next student who writes the third sentence of the story.
How many pages did she read each day?
The paper is now passed to the fourth student who must solve the problem and write out the answer. The other three group members review the answer for accuracy.
Answer: She read 31 pages each day.
This activity allows students to use their writing, reading, and speaking skills while learning and reviewing the concept of division. When text chains are completed, be sure students are checking them for accuracy and logic. Groups can exchange text chains to further practice reviewing and checking for accuracy.
Activity 6: I Have; Who Has? (CCSS: 6.NS.2)
Materials List: I Have Who Has BLM, card stock, paper, pencil
Cut apart the I Have Who Has BLM cards and distribute them to the students. A few students may have to have more than one card depending on the number of students in the class. One student will start by reading his/her card. For example, the first card might read, “I have 96 divided by 24; who has the quotient?” and then the student with the answer will read his/her card. The process will continue until the last student reads his/ her card which should connect to the first card read.
Sample Assessments
General Assessments
· Have students create a portfolio containing samples of students’ work and game score sheets.
· Have students create their own real-life division problems.
· Make observations to determine student understanding as he/she engages in the various activities.
· Create extensions to an activity by increasing the difficulty or by asking “what if” questions, whenever possible.
· Have students create and demonstrate original math problems by acting them out or using manipulatives to provide solutions on the board or overhead.
· Have students create their own questions.
· Have students show understanding by demonstrating mathematical principles with the use of concrete materials (manipulatives).
· Have students work division problems with 4-digit dividends and 2-digit divisors.
· Have students determine which problems from a given set should be solved using division.