The Division Process: Division with Remainders

Description:

1. Division Process without Remainders

Introduce the division process through naturally occurring contexts such as a child seeing how many of his friends he can take on a roller coaster ride and a teacher sharing pieces of candy for a class party. Use “tickets” made from construction paper and bags of candy, boxes of cupcakes, boxes of cookies to model “dividing.” (*Remember, the total number of items should be such that they can be divided evenly.)

2. Division with Remainders

Use the same contexts as described above but ensure the total number of items to be “divided” is such that their will be several left over after everyone has received an equal number. *It may be helpful to first replicate the division process without remainders first, and then introduce the division situation with remainders.

Purpose: to assist students to build meaningful connections between what they know about sharing things among their friends to the division process

Learning Objective 2: Divide without remainders using concrete materials within a story problem context - partitive/”sharing” situations.

Materials:

Teacher –

A set of CD’s or CD cases that can be divided evenly among two or three students.
A bag of candy, box of cupcakes, or box of cookies, etc.
A visual display with the word “divide” written on it.
Description:

1.) L ink to students’ prior knowledge of sharing things among their friends.

For Example:

I was at the Music Store yesterday and bought five new CD’s for me and four of my friends. When I bought them, I them gave one CD to each of my friends and to myself. (Demonstrate this process by calling several students up and “acting the story situation out.) Can you think of times when you had a certain number of things and you shared those things with friends so that each friend got the same number of things? (Elicit student examples.)

2.) I dentify the skill students will learn: division.

For Example:

Today, we are going to learn how to “divide.” (Show the visual display with the word “divide” written.) When we divide, we do something very similar to sharing things with our friends. We’re going to first learn how to divide using several different objects like ________ (e.g. candy, cupcakes, cookies). What are we going to learn to do today? (Point to the visual display of “divide” and elicit the response, “divide.”) Yes, we are going to learn how to divide using several different kinds of materials.

3.) P rovide rationale/meaning for learning the division process.

For Example:

Learning how to divide concrete objects will really come in handy. Once, you learn to divide a variety of objects well, you will then be able to share things with your friends, family, and others so that each person gets the same amount and doesn’t feel badly that they received less than somebody else. (Call several students up and simulate this concept by giving them each different amounts of candy, cupcakes, or cookies and ask how each would feel.)

Learning Objective 4: Divide with remainders using concrete materials in measurement/”separating into equal groups” and partitive/”sharing” situations.

Materials:

Teacher –

A bag of candy, box of cupcakes, or box of cookies, etc.
A visual display with the words “division with remainders.”
Description:

1.) L ink to student’s prior knowledge of

For Example:

The past few days, you have been using concrete objects to divide. Let’s do an example together. (Replicate the division process modeled when you divided candy/cupcakes/cookies evenly among the students in the class.) Did everybody get ____ number of candy pieces/cupcakes/cookies? (Elicit the response, “yes.”) Everybody got an equal number of candy pieces/cupcakes/cookies. Do I have any left in my bag/box? (Show students that there are no candy pieces left and elicit the response, “no.”) That’s right there are no candy pieces/cupcakes/cookies left.

2.) I dentify the skill students will learn:

For Example:

Today, you are going to learn about division/”divide” situations where we do have objects left over. We call this division with remainders. (Visually display the words “division with remainders.”) What are we going to learn today? (Point to the display and elicit the response, “division with remainders.”) Yes, we are going to learn division with remainders. What does “division with remainders mean? (Elicit the response, “when we divide and have objects left over.” Cue students as needed.) Yes, division with remainders (Point to the visual display of “division with remainders.”) means situations when we divide and we have objects left over.

3.) P rovide rationale/meaning for learning how to divide in situations where there are remainders.

For Example:

There will be times when you want to share things with your friends, like sharing cookies. (Call up four students to model this situation.) After I have given each friend the same number of cookies (Give each student cookies until you have several left over – 3 or less), I have several cookies left over. There aren't enough cookies left to give everybody one more cookie. (Show students the cookies you have left, count the number of total students there are to make this situation clear) If I gave some of my friends another cookie then others would not get as many. (Demonstrate this by “giving” one more cookie to some of the students.) If I did this, what would those who didn’t get an extra one feel like? (Elicit the response, “they would feel bad/mad.”) Yes. But, if I knew about division with remainders, then I could have told my friends that there will be some cookies left. That way nobody would have felt left out. Also, if I knew I would have several left over before I gave the cookies out, my friends and I could have decided what to do with the left over cookies before I gave them out. That way, everybody will feel good about things.

Purpose: to provide students a clear model of the division process, with and without remainders, using concrete objects.

Learning Objective 1: Divide without remainders using concrete materials.

Materials:

Teacher –

Language cards that have written on them the following phrases: “How many all together?”; “Separate into groups of two.”; “How many groups?”
Concrete materials including discrete counting objects (e.g. beans, counting chips) and containers for grouping where the grouped counting chips are clearly visible to all students (i.e. small paper plates).
A visible platform for demonstrating concrete objects where all students can see the concrete objects and your actions with them.
A. Break down the skill of dividing without remainders using concrete materials.

1.) Identify the “total.”

2.) Identify how many in each group.

3.) Place appropriate number of objects into groups until no objects are left.

4.) Count the number of groups.

Learning Objective 2: Divide without remainders using concrete materials within a story problem context - measurement/”separating into equal groups” situations.

Materials:

Teacher –

Visual display of story problem for division; number phrases representing dividend/total and divisor are color-coded.
Concrete materials that represent objects in story problem (e.g. tickets, candy pieces, string, small paper plates).
Visual displays for the words “dividend,” “divisor,” and “quotient.” ” Color-code the “dend” in “dividend” and the “sor” in “divisor” to match the corresponding number phrases in the story problem.
Markers/chalk for writing.
Description:

A. Break down the skill of division without remainders within a story problem context - measurement/”separating into equal groups” situations.

1.) Introduce story problem.

2.) Read the story problem aloud and then have students read it with you.

3.) Teach how to find the important information in the story problem.

4.) Represent the important information with concrete materials.

5.) Model the division process by “dividing” the objects into equal groups on plates.

6.) Model how to solve the story problem with the concrete objects.

7.) Replicate division process with a variety of concrete objects.

Learning Objective 3: Divide without remainders using concrete materials within a story problem context - partitive/”sharing” situations.

Materials:

Teacher –

Visual display of story problem for division; number phrases representing dividend/total and divisor are color-coded.
Concrete materials that represent objects in story problem (e.g. tickets, candy pieces, string, small paper plates).
Visual displays for the words “dividend,” “divisor,” “quotient.” ” Color-code the “dend” in “dividend” and the “sor” in “divisor” to match the corresponding number phrases in the story problem.
Markers/chalk for writing.
Description:

A. Break down the skill of division without remainders using concrete materials within a story problem context - partitive/”sharing” situations.

1.) Introduce story problem.

2.) Read the story problem aloud and then have students read it with you.

3.) Teach how to find the important information in the story problem.

4.) Represent the important information with concrete materials.

5.) Model the division process by “dividing” the objects evenly among a specified number of groups.

6.) Model how solve the story problem with the concrete objects.

7.) Replicate division process with a variety of concrete objects.

Learning Objective 4: Divide with remainders using concrete materials in measurement/”separating into equal groups” and partitive/”sharing” situations.

Materials:

Teacher –

Visual display of story problem for division; number phrases representing dividend/total and divisor are color-coded.
Concrete materials that represent objects in story problem (e.g. tickets, candy pieces, small paper plates).
Visual displays for the words “dividend,” “divisor,” “quotient,” and “remainder.” Color-code the “dend” in “dividend” and the “sor” in “divisor” to match the corresponding number phrases in the story problem.
Markers/chalk for writing.
Description:

A. Break down the skill of division with remainders using concrete materials in measurement/”separating into equal groups” and partitive/”sharing” situations.

1.) Introduce story problem.

2.) Read the story problem aloud and then have students read it with you.

3.) Teach how to find the important information in the story problem.

4.) Represent the important information with concrete materials.

5.) Model “dividing” the objects in at least one of two ways:

a. dealing objects one-by-one among containers (e.g. small paper plates) until there are fewer objects left over than there are plates – partitive/”sharing”;

b. circling groups of objects based on the “divisor” until there are left over objects that are two few to be grouped – measurement/”separating into equal groups”. Explicitly relate this action to the story context and relevant important information.

6.) Model how to find the solution to the story context with the concrete objects and explicitly naming the “left over” objects as the “remainder.”

7.) Replicate division process with remainders using a variety of discrete counting objects.

*The steps for scaffolding your instruction are the same for each concept you have explicitly modeled and with each Division Situation you teach (Measurement/”Separating into Equal Groups,” & Partitive/”Sharing”). This teaching plan provides you a detailed example of scaffolding instruction for one of the concepts modeled during Explicit Teacher Modeling - Division with Remainders Using Partitive/”Sharing” Situations, emphasizing the essential steps in the problem-solving process. You should scaffold your instruction with each skill/concept you model.

Purpose: to provide students the opportunity to build their initial understanding of the division process, with and without remainders, and to provide you the opportunity to evaluate your students’ level of understanding after your initial modeling of these skills.

Learning Objective 4: Division with Remainders Using Partitive/”Sharing” Situations

Materials:

Teacher –

Visual display of story problem for division; number phrases representing dividend/total and divisor are color-coded.
Concrete counting objects (e.g. unifix cubes, counting chips, small paper plates).
Concrete objects/containers for grouping (e.g. paper plates, note cards, pieces of string)
Visual displays for the words “dividend,” “divisor,” “quotient,” and “remainder.” Color-code the “dend” in “dividend” and the “sor” in “divisor” to match the corresponding number phrases in the story problem.
Markers/chalk for writing.
Students -

Concrete counting objects (e.g. unifix cubes, counting chips, small paper plates).
Concrete objects/containers for grouping (e.g. paper plates, note cards, pieces of string)
Description:

1.) Scaffold Using a High Level of Teacher Direction/Support

a. Choose one or two places in the problem-solving sequence to invite student responses. Have these choices in mind before you begin scaffolding instruction. (Examples of choices are shown in red.)

Introduce the story problem.
“Ok, here we have another story problem.“ I know that I first need to read the story problem. By reading it, I will be able to find the important information that will help me solve the story problem. I know that reading it aloud can help me really hear the words as I read it.

Read the story problem aloud and then have students read it with you.
“I’m going to read the story problem aloud first. Then I’d like you to read it with me the second time. “ (Read the story problem aloud, pointing to the words as you read them.)
Now, I’d like for you to read the story problem with me. (Read the story problem aloud with your students.)

Teach finding the important information in the story problem.
Link process of finding important information of addition, subtraction, and multiplication story problems to division story problems. - “Now, that I’ve read the story problem and you helped me, I know from solving other story problems that involved addition, subtraction, and multiplication, that there is important information in the story problem that will help me find the solution.“
Model finding what you are solving for. – “The first important piece of information that I need to find is what I’m solving for. What do I look for when I want to find what I am solving for? (Elicit the response, “the question/question mark.”) Yes, I look for the question. What do I do when I find the question? (Elicit the response, “underline the question and circle the question mark.) Yes, first I circle the question mark, because it tells me that this statement is a question and then I underline the question. (Circle the question mark and then underline the question.)”
Model finding the number phrases that represent the “dividend/total” and the “divisor.” – “Now, I need to find the number phrases that represent my dividend, or total, and my divisor. (Point to the word cards that represent “dividend” and “divisor.”) I know one strategy that can help me find the number phrases is to read each sentence and ask myself the question, “Is there a number phrase in this sentence?” What question do I ask after I read each sentence? (Elicit the response, “is there a number phrase in this sentence?”) Good. I’ll do that now. (Model this process to find the number phrases that represent the dividend and the divisor. Emphasize the relationship between the number phrase, whether it is the dividend or the divisor and why it is the dividend or the divisor.)”
Model deciding if all the important information is identified. – “When I have finished finding the important information, it is helpful to check back and be sure I have found all of the important information. I have looked for two important things. (Have written on the chalkboard or dry-erase board “question” and “number phrases.”) What two important things do I look for? (Point to the phrases on the board and elicit the response, “question” and “number phrases.”) Good. First, I looked to find what I am solving for. I did this by finding the question. (Point to the circled question mark and the underlined question, then write a check beside the word “question” written on the board.) The second important information I looked for were the number phrases. I read each sentence and asked myself the question, “is there a number phrase in this sentence?” I found two number phrases. (Point to the two number phrases that you circled and then write a check beside “number phrases” written on the board.) Have we found all of the important information? (Point to the checked off phrases and elicit the response, “yes.”)”

Find how many objects are on each plate. - “Ok, in order to solve this, I know I can count the number of objects on each plate. I have ____ counting chips on each plate.” (Count aloud the counting chips on each plate to emphasize that each plate has the same number of objects.)”
Find the remainder. “I have ____ counting chips on each plate, but I have some counting chips left over. Let’s see, I have ___ counting chips left over. Hmm, when I have objects left over, I know I need to make sure there aren’t enough objects to put one more on every plate. Let’s see, I have ___ counting chips left. I have ___ plates. I have fewer counting chips than I do plates. Therefore, I know I don’t have enough counting chips to put one more on every plate. What do I call the left over counting chips? (Point to the word card with “remainder” written, and elicit the response, “the remainder.”) Yes we call the left over objects, the remainder.“
Identify the solution. “I can find the solution to the story problem by counting the number of counting chips on one plate because in the story, ……(Relate the question to the concrete objects.) I have ___ counting chips on each plate. Therefore, each _________ gets ____ ______________. I also have ___ counting chips left over. Therefore, the solution to the story problem is that each ___________ gets ____ ____________ and there are ____ ___________ left over.
b. Maintain a high level of teacher direction/support for another example if students demonstrate mis../../../understanding/non-understanding; move to a medium level of teacher direction/support if students respond appropriately to the selected questions/prompts.

2.) Scaffold Using a Medium Level of Teacher Direction/Support

Choose several more places in the problem-solving sequence to invite student responses. Have these choices in mind before you begin scaffolding instruction.

Read the story problem aloud and then have students read it with you.
“What is the first thing I need to do when I have a story problem? (Elicit the response, “read it.”) Yes. I’ll read the story problem aloud first. Then I’d like you to read it with me the second time. (Read the story problem aloud, pointing to the words as you read them.)”
“Now, I’d like for you to read the story problem with me. (Read the story problem aloud with your students.)”

Teach finding the important information in the story problem.
Link process of finding important information of addition, subtraction, and multiplication story problems to division story problems. - “Now, that I’ve read the story problem and you helped me, I know from solving other story problems that involved addition, subtraction, and multiplication, that there is important information in the story problem that will help me find the solution.“
Model finding what you are solving for. – “The first important piece of information that I need to find is what I’m solving for. What do I look for when I want to find what I am solving for? (Elicit the response, “the question/question mark.”) Yes, I look for the question. What do I do when I find the question? (Elicit the response, “underline the question and circle the question mark.) Yes, first I circle the question mark, because it tells me that this statement is a question and then I underline the question. (Circle the question mark and then underline the question.)”
Model finding the number phrases that represent the “dividend/total” and the “divisor.” – “Now, I need to find the number phrases that represent my dividend, or total, and my divisor. (Point to the word cards that represent “dividend” and “divisor.”) What strategy can I use to find the number phrases? (Elicit the response, “read each sentence and ask, “is there a number phrase in this sentence?”) Good. I’ll do that now. (Model this process to find the number phrases that represent the dividend and the divisor. Emphasize the relationship between the number phrases, whether it is the dividend or the divisor and why it is the dividend or the divisor.)”
Model deciding if all the important information is identified. –“ When I have finished finding the important information, it is helpful to check back and be sure I have found all of the important information. I have looked for two important things. (Have written on the chalkboard or dry-erase board “question” and “number phrases.”) What two important things do I look for? (Point to the phrases on the board and elicit the response, “question” and “number phrases.”) Good. First, I looked to find what I am solving for. How did I find what I was solving for? (Elicit the response, “you looked for the question, circled the question mark and underlined the sentence.”) How did I find the number phrases? (Elicit the response, “you read each sentence and asked, ‘is there a number phrase in this sentence?’”) Yes. How many number phrases did I find? (Elicit the response, “two.”) Great. I found two number phrases. (Point to the two number phrases that you circled and then write a check beside “number phrases” written on the board.) Have we found all of the important information? (Point to the checked off phrases and elicit the response, “yes.”)

Model the division process by “dividing” the objects (separating them into equal groups by dealing objects evenly among small paper plates). Explicitly relate this action to the story context and relevant important information. Re-emphasize the language “dividend” and “divisor.”
“Now that I know all the important information, I can use my concrete materials to ‘act out the story’ and then to solve the problem. “ I’ll do this by reading each sentence and ‘acting’ it out with my concrete objects. (Read each sentence and “act” it out with the concrete objects, thinking aloud what you are doing and why.) (Prompt students to think how to represent the “dividend.”) How do I represent the total? (Elicit the response, “count out that number of counting chips.”) What do we call the total? (Point to the “dividend” word card and elicit the response, “dividend.”) (Represent the divisor with containers such as paper plates. Deal the counting objects one-by-one among the plates. Use visual cueing by pointing to the number phrase that represents the “divisor” as you represent it.)”

Model finding the solution to the story context with the concrete objects.
Review what you are solving for. - “ Now that I’ve ‘acted out’ the story, its time to solve it. Hmm, how do I know what I’m solving for? (Elicit the response, “read the question.”) Yes, the question tells me what I am solving for. What does the question ask? (Point to the question and elicit the appropriate response.)”
Find how many objects are on each plate. - “Ok, in order to solve this, I know I can count the number of objects on each plate. I have ____ counting chips on each plate.” (Count aloud the counting chips on each plate to emphasize that each plate has the same number of objects.)”
Find the remainder. - “I have ____ counting chips on each plate, but I have some counting chips left over. When I have objects left over, what do I need to do? (Elicit the response, “count them to see if there are enough to put one more on every plate.”) Good. This is important to do, because, I want to divide as many counting chips as I can as long as each plate gets the same number. I know I don’t have enough counting chips to put one more on every plate. What do I call the left over counting chips? (Point to the word card with “remainder” written, and elicit the response, “the remainder.”) Yes we call the left over objects, the remainder.“
Identify the solution. - “I can find the solution to the story problem by counting the number of counting chips on one plate because in the story, ……(Relate the question to the concrete objects.) I have ___ counting chips on each plate. Therefore, each _________ gets ____ ______________. I also have ___ counting chips left over. Therefore, the solution to the story problem is that each ___________ gets ____ ____________ and there are ____ ___________ left over.” How many does each __________ get? (Elicit the appropriate response.) How many are left over? (Elicit the appropriate response.) The name we use for the solution to a division problem is “quotient.” (Point to the “quotient” word card.) Therefore, the solution, or quotient (Point to the “quotient” word card.) for this division problem is…..”
b. Maintain a medium level of teacher direction/support for another example if students demonstrate mis../../../understanding/non-understanding; move to a low level of teacher direction/support if students respond appropriately to the selected questions/prompts.

3.) Scaffold Using a Low Level of Teacher Direction/Support

a. When students demonstrate increased competence, do not model the process. Ask students questions and encourage them to provide all responses. Direct students to replicate the process at their desks as you work together.

Introduce the story problem.
“Ok, here we have another story problem.“ What do we need to do? (Elicit the response, “read it.”) Yes, we need to read it. Why will reading the story problem help us? (Elicit the response, “reading it will help us to find the important information that will help solve the story problem.) Great! Should we read it silently or out loud? (Elicit the response, “out loud.”) Yes. Why should we read it out loud? (Elicit the response, ”because it helps us hear the words as we read them. “) Boy, you all are really sharp.”

Read the story problem aloud and then have students read it with you.
“What is the first thing I need to do when I have a story problem? (Elicit the response, “read it.”) Yes. I’ll read the story problem aloud first. Then I’d like you to read it with me the second time. “ (Read the story problem aloud, pointing to the words as you read them.)”
“Now, I’d like for you to read the story problem with me. (Read the story problem aloud with your students.)”

Teach finding the important information in the story problem
“What should we look for in the story problem after we’ve read it? (Elicit the response, “the important information.”) Good. What will “finding the important information” help us to do? (Elicit the response, “find the solution.“) Yes, the important information will help us solve the story problem.”
Model finding what you are solving for. – “What do I look for when I want to find what I am solving for? (Elicit the response, “the question/question mark.”) Yes, I look for the question. What do I do when I find the question? (Elicit the response, “underline the question and circle the question mark.) Yes, first I circle the question mark, because it tells me that this statement is a question and then I underline the question. (Circle the question mark and then underline the question.)”
Model finding the number phrases that represent the “dividend/total” and the “divisor.” – “After we’ve found what we are solving for, what important information do we look for next? (Elicit the response, “the number phrases.”) Yes. What two names do we use for the number phrases in a division story problem? (Elicit the response, “the “dividend” and “divisor.” *Point to the word cards if needed.) What strategy can I use to find the number phrases? (Elicit the response, “read each sentence and ask, “is there a number phrase in this sentence?”) Good. Let’s do that now. (You and the students read through the story problem using this strategy to find the number phrases.)”
Model deciding if all the important information is identified. – “When we’ve finished finding the important information, what do we need to check for? (Elicit the response, “to be sure we’ve found all of the important information.”) That’s correct. What two important things do I look for? (Point to the phrases on the board and elicit the response, “question” and “number phrases.”) Good. First, I looked to find what I am solving for. How did I find what I was solving for? (Elicit the response, “you looked for the question, circled the question mark and underlined the sentence.”) How did I find the number phrases? (Elicit the response, “you read each sentence and asked, ‘is there a number phrase in this sentence?’”) Yes. How many number phrases did I find? (Elicit the response, “two.”) Great. I found two number phrases. (Point to the two number phrases that you circled and then write a check beside “number phrases” written on the board.) Have we found all of the important information? (Point to the checked off phrases and elicit the response, “yes.”)

Model the division process by “dividing” the objects (separating them into equal groups by dealing objects evenly among small paper plates). Explicitly relate this action to the story context and relevant important information. Re-emphasize the language “dividend” and “divisor.”
“Now that I know all the important information, I can use my concrete materials to ‘act out the story’ and then to solve the problem. “How should we ‘act the story out’? (Elicit the response, “by reading each sentence and ‘acting’ it out with our chips.”) Right. (Read each sentence aloud with students and “act” it out with the concrete objects.) (Prompt students to think how to represent the “dividend.”) How do I represent the total? (Elicit the response, “count out that number of counting chips.”) Great! (Ask students to hold up a chip to verify they recognize their counting chips.) What do we call the total? (Elicit the response, “dividend.” *Only point to the word card if additional cueing is needed.) Now, how do I represent the other number phrase? (Elicit the response, “with paper plates.” ) Good. (Ask students to hold up a paper plate to verify they recognize their paper plates.)What do we call the number of paper plates? (Elicit the response, “the divisor. ” *Only point to the word card if additional cueing is needed.”) Great! Now what do we do? (Elicit the response, “deal the chips one-by-one among the plates.”) Yes, we need to deal the counting chips one-by-one on each plate. Let’s all do that now. (Ensure students deal their chips among the plates appropriately.)”
Model finding the solution to the story context with the concrete objects.
Review what you are solving for. - “ Now that I’ve ‘acted out’ the story, its time to solve it. How do I know what I’m solving for? (Elicit the response, “read the question.”) Yes, the question tells me what I am solving for. What does the question ask? (Point to the question and elicit the appropriate response.)”
Find how many objects are on each plate. - “Ok, in order to solve this, what do we need to do? (Elicit the response, “count the number of chips on each plate.”) Yes. Let’s all do that now. (Encourage students to count the number of chips on their plates and ask for responses from several students.)”
Find the remainder. - “We have ____ counting chips on each plate, but we have some counting chips left over. When I have objects left over, what do I need to do? (Elicit the response, “count them to see if there are enough to put one more on every plate.”) Good. Why is this important to do? (Elicit the response, “because we need to know for sure that we’ve divided all the chips we can so each plate has the same number.”) Yes. Are there enough chips to put one more chip on every plate? (Elicit the response, “no.”) How many chips are there? (Elicit the appropriate response.) How many plates are there? (Elicit the appropriate response.) Good. What do I call the left over counting chips? (Point to the word card with “remainder” written, and elicit the response, “the remainder.”) Yes we call the left over objects, the remainder.“
Identify the solution. - “How do we find the solution to the story problem? (Elicit the response, “by counting the number of counting chips on one plate.”) Excellent! Everybody count the number of chips on one plate. How many chips are there? (Elicit the appropriate response. Are their chips left over? (Elicit the response, “yes.”) Right! How many chips are left over? (Elicit the appropriate response.) What is the solution to the story problem? (Elicit the appropriate response.) How many does each __________ get? (Elicit the appropriate response.) How many are left over? (Elicit the appropriate response.) What name do we use for the solution of a division problem? (Elicit the response, “quotient.” *Only point to the word card if additional cueing is needed.”) What is the solution, or quotient for this division story problem? (Elicit the appropriate response, including the number of chips on each plate and the chips left over.)”
b. When you are confident students understand, ask individual students to direct the problem solving process or have the class direct you: Students ask questions and you and the students respond/perform the skill.

*The student practice strategies described below can be used for all skills taught during initial acquisition through Teacher Directed Instruction. A detailed description for providing practice for one of the skills is provided below:

Purpose: to provide students with many opportunities to determine which examples provided reflect the appropriate remainder.

Learning Objective 4: Divide with remainders using concrete materials in measurement/”separating into equal groups” and partitive/”sharing” situations.

Structured Language Experiences

Materials:

Teacher –

Concrete examples of solutions to various division equations with and without remainders.
“Choice” cards (three for each concrete example) that represent possible solutions to each concrete example. One card includes the appropriate solution. Appropriate language is used to represent the solutions (e.g. “four groups of five with two left over” would represent the solution to “22 ¸ 5 = ___.”).
Response sheet with correct solutions for answer key.
Students -

Response sheet numbered according to number of examples provided.
Pencil for writing
*Appropriate accommodations for students with significant writing problems are to have them tape record their responses or have a letter written at the top of each “choice” card. Students can write the letter of the card they choose instead of writing the phrase.

*An appropriate accommodation for students with reading difficulties is to pair them with a classmate who has the ability to read the language “choice” cards.

Description:

Activity:

Students work at a center where there are laid out a variety of concrete examples showing solutions to division equations (For example, the concrete solution to the division equation, “16 ¸ 5 = __” would be three groups of five counting objects and one counting object left over. *It is important to group the counting objects in a distinct fashion so that the “remainder” can be clearly identified.). Above each concrete example are three cards with possible solutions written on them (For example, cards might read, “three groups of five with two left over,” three groups of five with one left over,” “three groups of five with zero left over.”. One card is the correct solution (with remainder). Students select which solution is appropriate and writes it down beside the appropriate number on their response sheet.

“Structured Language Experience” Steps:

1.) Review directions for completing structured language experiences/peer tutoring activity and relevant classroom rules.

2.) Model how to perform the skill(s) within the context of the activity before students begin the activity. Model both how to decide an appropriate choice and model how to write the solution on the response sheet.

3.) Provide time for student questions.

4.) Signal students to begin.

Monitor students as they work. Provide positive reinforcement for both “trying hard,” responding appropriately, and for students using appropriate behavior. Also provide corrective feedback and modeling as needed.

Purpose: to provide students with multiple opportunities to solve division equations involving solutions “with remainders” and to describe the meaning of their concrete representations.

Learning Objective 4: Divide with remainders using concrete materials in measurement/”separating into equal groups” and partitive/”sharing” situations.

Structured Language Experiences/Structured Peer Tutoring

Materials:

Teacher –

Develop learning sheets – Each learning sheet includes division equations involving solutions “with remainders.” With each equation, the following questions structure student responding: How many total chips (or other appropriate concrete object)? How many plates (or other appropriate container)? How many chips on one plate? How many chips are left over?
Appropriate number of counting/discrete concrete objects and containers (e.g. counting chips, unifix cubes and paper plates.)
Master answer key for learning sheet.
Students -

Learning sheets
Appropriate counting/discrete concrete objects and containers
An answer key
Description:

Activity:

Students work in pairs, responding to the learning sheet. The student practice period is separated into two equal time periods. The coach presents the division equation by pointing to it and saying the equation aloud (e.g. points to the equation, 8 ¸ 3 = __, and says, “eight divided by three.”). Then the coach asks each question that follows. The player responds to each question using the appropriate concrete objects and then saying the answer. The coach writes the answer in the appropriate space, checks the answer key and provides appropriate feedback (e.g. positive verbal reinforcement for accurate responses and corrective feedback for inaccurate responses.) For inaccurate responses, the coach provides feedback and the player attempts the question a second time. The first response is crossed out and the second response is recorded. The coach provides appropriate feedback as appropriate. The teacher signals students to “switch roles” at the appropriate time.

“Structured Language Experiences/Structured Peer Tutoring” Steps:

1.) Select pair groups and assign each pair a place to practice (try to match students of varying achievement levels if possible).

2.) Review directions for completing structured language experiences/peer tutoring activity and relevant classroom rules. Practice specific peer tutoring procedures as needed (see step #4).

3.) Model how to perform the skill(s) within the context of the activity before students begin the activity. Model both what the coach does (e.g. reads aloud the division equation) and how the player responds (e.g. using concrete materials).

4.) Divide the practice period into two equal segments of time. One student in each pair will be the player, or “talker/describer” and will solve the division equation using concrete materials and then describe their solution. The other student will be the coach, or “listener/evaluator” and will point to and then say aloud each problem. The coach will then write the response in the appropriate space, check the answer key, and provide feedback regarding the player’s response (e.g. positive verbal reinforcement for accurate responses and corrective feedback for inaccurate responses.) For inaccurate responses, the coach provides feedback and the player attempts the question a second time. The first response is crossed out and the second response is recorded. The “listener/describer” will also tally corrects and incorrects based on the player’s responses.

5.) Provide time for student questions.

Signal students to begin.
Signal students when it is time to switch roles.
Monitor students as they work in pairs. Provide positive reinforcement for both “trying hard,” responding appropriately, and for students using appropriate tutoring behaviors. Also provide corrective feedback and modeling as needed.

Purpose: to provide you with continuous data for evaluating student learning and whether your instruction is effective. It also provides students a visual way to “see” their learning.

Materials:

Teacher –

Appropriate prompts if they will be oral prompts
Appropriate visual cues when prompting orally
Student –

Appropriate response sheet/curriculum slice/probe
Gaph/chart
Description:

Steps for Conducting Continuous Monitoring and Charting of Student Performance:

1.) Choose whether students should be evaluated at the receptive/recognition level or the expressive level.

2.) Choose an appropriate criteria to indicate mastery.

3.) Provide appropriate number of prompts in an appropriate format (receptive/recognition or expressive) so students can respond.

Based on the skill, your students’ learning characteristics, and your preference, the curriculum slice or probe could be written in nature (e.g. a sheet with appropriate prompts; index cards with appropriate prompts), or oral in nature with visual cues (e.g. say aloud the prompt and display it visually with concrete objects, drawings, written language, or numbers and symbols) or a combination of written curriculum slices/probes and oral prompts with visual cues.
4.) Distribute to students the curriculum slice/probe/response sheet/concrete materials.

5.) Give directions.

6.) Conduct evaluation.

7.) Count corrects and incorrects/mistakes (you and/or students can do this depending on the type of curriculum slice/probe used – see step #3).

8.) You and/or students plot their scores on a suitable graph/chart. A goal line that represents the proficiency (for concrete level skills, this should be 100% – 5 out of 5 corrects) should be visible on each students’ graph/chart).

9.) Discuss with children their progress as it relates to the goal line and their previous performance. Prompt them to self-evaluate.

10.) Evaluate whether student(s) is ready to move to the next level of understanding or has mastered the skill at the abstract level using the following guide:

Concrete Level: demonstrates 100% accuracy (given 3 to 5 response tasks) over three consecutive days.

Representational Level: demonstrates 100% accuracy of (given 8-10 response tasks) over two to three consecutive days.

Abstract Level: demonstrates near 100% accuracy (two or fewer incorrects/mistakes) and a rate (# of corrects per minute) that will allow them to be successful when using that skill to solve real-life problems and when using the skill for higher level mathematics that require use of that skill.

11.) Determine whether you need to alter or modify your instruction based on student performance.

Flexible Math Interview

Purpose: to evaluate student conceptual understanding of the division process, with and without remainders.

Materials:

Teacher –

A small notepad to write notes regarding particular student’s understanding as you “interview” them.
Pencil for writing
Students -

Appropriate concrete objects
Description:

As students are working independently or in pairs, ask them to describe their solutions and how they arrived at them. Encourage students to both use concrete materials to do this as well as “talk about” what they are doing with their concrete materials.

Purpose: to provide students with opportunities to maintain their level of mastery of division using concrete materials.

Problem of the Day

Materials:

Teacher –

A written prompt on the chalkboard, dry-erase board, or overhead projector (e.g. a division problem) or a concrete example of some part of the division process (e.g. solution to a division problem that includes a remainder).
Students -

Concrete materials if appropriate
Paper and pencil to record their responses if appropriate
Description:

Teacher presents a “problem of the day” that focuses on a particular skill or conceptual understanding of the division process (e.g. remainders). The problem can be written in nature or be represented with concrete materials. The “problem of the day” is displayed as students enter the room or as the period begins. Students are asked to “solve” the problem and provided necessary directions. After an appropriate amount of time, the teacher and the students “talk through” the problem and its solution. Students can individually describe how they approached the problem. Specific positive verbal reinforcement is provided by the teacher as well as specific feedback regarding misunderstandings students may have. Teacher notes students who seem to be having difficulty for the purpose of reviewing/re-modeling appropriate skills and concepts.