The Division Process: Division with Remainders: Representational Level

More Teaching Plans on this topic: Concrete, Abstract

 Introduction Phase 1 Initial Acquisition of Skill Phase 2 Practice Strategies Phase 3 Evaluation Phase 4 Maintenance Videos

PHASE 1: Initial Acquisition of Skill

Teach Skill with Authentic Context

Description:  Drawing solutions to division problems should continue to be taught within the framework of story situations that resonate with students given their age and interests.

Build Meaningful Student Connections

Purpose: to assist students to build meaningful connections between what they know about solving division problems (with and without remainders) using concrete objects and drawing pictures to solve division problems (with and without remainders.)

Learning Objectives 2: Drawing solutions to division equations using the DRAW Strategy.

Materials:

Teacher -

• Appropriate counting objects and containers.
• Visual display of an appropriate division problem.
• A visual display that identifies the learning objective.

Description:

1) L ink to students’ prior knowledge of solving division problems with concrete materials.

For Example:

The last few days, you have learned how to use concrete materials such as _______________ to solve division problems. You’ve used ____________ to represent the total, or dividend, in division problems (Hold up the corresponding concrete materials), and you have used _____________ to group, or “divide” the total number of objects by the divisor (Hold up the corresponding containers used). These concrete materials have been very helpful for solving division problems. Let’s solve another division problem together using these concrete materials. (Solve a division problem with your students using previously used concrete materials, highlighting the dividend, divisor, quotient, and remainder (if appropriate).

2) I dentify the skill students will learn: Drawing pictures to solve division problems.

For Example:

Today we are going to learn how to draw pictures to solve division problems instead of using these concrete materials. I will teach you how to draw simple pictures that represent the concrete objects you have been using the past few days. The pictures we will draw to solve division equations will be very similar to those you learned to draw for multiplication problems.

3) P rovide rationale/meaning for drawing pictures to solve division problems.

For Example:

Drawing pictures to solve problems is a lot like using our concrete materials. When we use concrete materials like __________________ (counting objects) and _________________ (containers), we can see what it is we are solving. Moving the concrete objects around, like when we group counting chips onto plates, also helps us problem solve because it helps us make the numbers and symbols of a division problem “come alive.” Drawing pictures helps in the same way. We can see the pictures we draw and we also can group pictures much like we did with our concrete materials. Drawing pictures is also faster than using concrete objects so it will help you become even more talented at solving division problems.

Provide Explicit Teacher Modeling

Purpose: to provide students a clear model of how to draw solutions to division problems with and without remainders.

Learning Objective 1: FASTDRAW Strategy (to solve division story problems by drawing pictures).

Materials:

Teacher -

• A visual display of the “FASTDRAW” Strategy (Color-code the “FAST” and the “DRAW” in “FASTDRAW.”) *The FASTDRAW strategy comes from Mercer & Mercer (1998)
• A format for visually displaying division story problems (e.g. chalk-board, dry-erase board, chart & chart paper).
• Story problems that both depict division situations and contain color-code phrases that represent the dividend and the divisor.
• A format to visually display division equations and drawings.
• Appropriate writing utensil (e.g. chalk, markers).
• Cue cards/visual displays for the language “dividend,” “divisor,” ‘quotient,” & “remainder.” (Color-code the “dend” in “dividend” and the “sor” in “divisor” to correspond with the color of the corresponding number phrases.)

Description:

A. Break down the skill of teaching the FASTDRAW Strategy.

1) Introduce students to the concept of a Learning Strategy.

2) Introduce students to the “FASTDRAW” Strategy.

3) Describe the purpose of the “FASTDRAW” Strategy.

4) Teach the purpose for “FAST” and the steps “FAST”.

a. Find what you are solving for.

b. Ask yourself, what is the important information (circle it).

c. Set up the equation.

d. Tie down the sign.

5) Teach the purpose of “DRAW” and the steps “DRAW”.

a. Determine the sign.

c. Answer, or draw and check.

Learning Objective 2: Draw solutions to division story problems using the “FASTDRAW Strategy.”

Description:

A. Break down the skill of drawing solutions to division story problems using the FASTDRAW Strategy.

1) Introduce story problem.

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

3) Teach finding the important information in the story problem and setting up an equation using the steps “FAST” from the “FASTDRAW” Strategy.

a. Find what you are solving for.

b. Ask yourself, what is the important information (circle it).

c. Set up the equation.

d. Tie down the sign.

4) Teach drawing solutions using the steps “DRAW” from the “FASTDRAW” strategy.

a. Determine the sign.

c. Answer, or draw and check.

5) Model how to solve the story problem by relating the “answer” to the division equation back to the story problem context.

6.) Model how to draw solutions to division equations by repeating the steps in#4 and #5 at least two or three more times with different division equations.

Scaffold Instruction

*This teaching plan provides a description of how to scaffold instruction for using “DRAW” to solve division equations without and with remainders. The same basic process can be used for scaffolding instruction for other skills/concepts explicitly modeled during Explicit Teacher Modeling. First, break the skill/concept into learnable parts (e.g. use those “parts” taught during Explicit Teacher Modeling) and then fade your direction in three phases:

1) High level of teacher support; 2.) Medium level of teacher support; 3.) Low level of teacher support. Scaffolding Instruction should occur for each skill/concept taught during Explicit Teacher Modeling before providing student practice.

Purpose: to provide students the opportunity to build their understanding of how to draw solutions to division story problems and equations, 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 2: Drawing solutions to division story problems using the FASTDRAW Strategy – Solving division equations using DRAW.

Materials:

Teacher -

• Appropriate division equations represented visually on chalkboard, dry-erase board, chart/chart paper, overhead projector.
• Chalk, markers for writing and drawing

Students -

• Paper with appropriate division equations to practice drawing solutions to during 3rd phase of Scaffolding Instruction – Low Level of Scaffolding Instruction.
• Pencils for drawing and writing answers.

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. *Color-code the dividend and the divisor in each equation during this phase of instructional scaffolding.

Review steps of “DRAW”

"I have an equation here. Hmm, I know there is a strategy that can help me draw pictures to solve this equation. What is the name of the strategy? Oh, yes, it is called ‘DRAW’. (Point to ‘DRAW’ in the FASTDRAW Strategy or display ‘DRAW’ and its steps separately.) What is the name of the strategy that can help me draw pictures to solve this equation? (Elicit the response, “DRAW.”) I know each letter in ‘DRAW’ stands for a step in solving an equation. The ‘D’ stands for ‘Discover the sign.’ (Point to the appropriate phrase as you say it.) What does the ‘D’ stand for? (Elicit the response, “Discover the sign.”) Yes, ‘D’ stands for ‘Discover the sign.’ (*Repeat this process for each step of ‘DRAW’.)"

Model the “D” step, “Discover the sign.”

"Now that I know I can use the ‘DRAW’ Strategy to help me solve this equation, I can begin by completing the first step, ‘D’. What does ‘D’ stand for? (Point to the ‘D’ step and elicit the response, “Discover the sign.”) Yes, I first need to discover the sign. I do this by finding the symbol that tells me what math operation to use. (Point to the division sign.) Hmm, this sign has a line in the middle and a dot on the top and a dot on the bottom. It also looks like a sideways face. I see a long nose with two eyes. (Point to the relevant features of the division sign.) What can I do to help me remember what math operation I need to use? (Elicit the response, “circle it.”) Yes, I can circle it to help me remember what math operation to use. (Circle the division sign.)"

Model the “R” step, “Read the problem.”

“I’ve discovered the sign, and know I need to divide. The next step is ‘R.’ What does ‘R’ stand for? (Point to the ‘R’ step and elicit the response, “Read the problem.”) Yes, I need to read the problem. “ When I read a division problem, I know I need to read the dividend or total first. Usually, the dividend will be the number that has the higher value because it represents the total. In this equation, the dividend must be __ because it has the higher value. (Point to the dividend and say it aloud.) I know that I have to divide because the sign I discovered is a division sign. (Point to the division sign.) Last, I find the divisor. This number must be my divisor because it is the remaining number. It also represents a lower value. The divisor usually is a number that has a lower value than the dividend. Now that I know all the parts of this equation, I’ll read it. (Read the equation aloud.) Now, read the problem with me. (Encourage students to read the problem aloud with you as you point to each part of the equation.)”

Model the “A” step, “Answer, or draw and check.”

Draw pictures to represent the dividend. - "Now that I know what the problem is and what math operation I need to use, I need to complete the “A” step. What is the “A” step? (Elicit the response, “Answer, or draw and check.”) Yes, I need to answer the equation. I know I can draw pictures to solve a division equation. Hmm, I remember using concrete materials to do this. When I did this, I first represented the dividend by counting out that number of objects. I can do the same thing by drawing pictures instead of using concrete objects. What kind of pictures can I draw? (Elicit the response, “tallies or dots.”) Yes, I can represent the dividend by drawing tallies or dots. I’m going to draw tallies. My dividend is __ so I need to draw __ tallies. (Draw the appropriate number of tallies.)"

Draw circles around pictures to represent dividing them into equal groups based on the divisor.-"Now I have to divide or separate these tallies into groups. I know the divisor tells me how many tallies belong in each group. (Point to the divisor and say aloud how many tallies should be in each group.) I can put the tallies into groups by drawing circles around them. How many tallies do are in each group? (Elicit the appropriate response.) Yes, I need to circle __ tallies at a time. I’ll do that now. (Circle the tallies until you have tallies “left over.”) I have __ tallies left over. That is not enough to put in a group so I know this is my remainder."

Model checking your drawings.-“Now that I have finished drawing pictures, I need to check them and be sure I drew them correctly. I do this by counting my tallies to be sure they total the dividend. (Count aloud the tallies and compare the total to the dividend.) I know I have the correct number of tallies, so now I need to check to see if I have my dividend drawn correctly. The dividend is represented by the tallies in each circle or group. I can check this by counting the number of tallies in each group and be sure they equal the divisor. (Count aloud the tallies in each group and compare the total in each group to the divisor.) Last, I check my remainder by being sure there are fewer tallies left over than there are in each group. (Count aloud the remaining tallies and compare the total to the divisor/the number of tallies in each group.)”

Model how to find the answer to the equation.-“Now, to answer the equation, I count the number of groups. (Count aloud the groups.) How many groups do I have? (Elicit the appropriate response.) Yes. And how many tallies do I have left over? (Elicit the appropriate response.)”

Model the ‘W’ step, “Write the answer.”

“ I have found my answer by drawing pictures and I have checked my drawings to be sure they are accurate. Now I can finish solving the problem by completing the ‘W’ step. What is the ‘W’ step? (Elicit the response, “Write the answer.”) Great, after I have found my answer by drawing, I write the answer. I know the answer to a division equation should be written here. (Point to the appropriate space.) What is my answer? (Elicit the appropriate response.) Yes, my answer is _____. I know this because I have ___ groups (Point to each circled group of tallies and count them aloud.) and I have ___ tallies left over (Point to the remaining tallies and count them aloud.) I’ll write the answer here. (Point to the appropriate space and write the answer.) “

b. Maintain a high level of teacher direction/support for another example if students demonstrate misunderstanding/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

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

Review steps of “DRAW”

"I have another equation. What is the name of the strategy that can help me draw pictures to solve this equation? (Elicit the response, “DRAW.”) I know each letter in ‘DRAW’ stands for a step in solving an equation. The ‘D’ stands for ‘Discover the sign.’ (Point to the appropriate phrase as you say it.) What does the ‘D’ stand for? (Elicit the response, “Discover the sign.”) Yes, ‘D’ stands for ‘Discover the sign.’ (*Repeat this process for each step of ‘DRAW’.)"

Model the “D” step, “Discover the sign.”

“Now that I know I can use the ‘DRAW’ Strategy to help me solve this equation, I can begin by completing the first step, ‘D’. What does ‘D’ stand for? (Point to the ‘D’ step and elicit the response, “Discover the sign.”) Yes, I first need to discover the sign. How do I do this? (Elicit the response, “by finding the symbol that tells what math operation to use.”) Good. (Point to the division sign.) What is the sign? (Elicit the response, “division.”) Good. How do you know it is a division sign? (Elicit the response, it has a line in the middle and a dot on the top and a dot on the bottom/it also looks like a sideways face. I see a long nose with two eyes.)” (Point to the relevant features of the division sign.) Excellent thinking! What can I do to help me remember what math operation I need to use? (Elicit the response, “circle it.”) Yes, I can circle it to help me remember what math operation to use. (Circle the division sign.)"

Model the “R” step, “Read the problem.”

“I’ve discovered the sign, and know I need to divide. The next step is ‘R.’ What does ‘R’ stand for? (Point to the ‘R’ step and elicit the response, “Read the problem.”) Yes, I need to read the problem. “ When I read a division problem, I know I need to read the dividend or total first. Usually, the dividend will be the number that has the higher value because it represents the total. What is the dividend in this problem? (Elicit the appropriate response.) Yes. (Point to the dividend and say it aloud.) I know that I have to divide because the sign I discovered is a division sign. (Point to the division sign.) Last, I find the divisor. This number must be my divisor because it is the remaining number. It also represents a lower value. The divisor usually is a number that has a lower value than the dividend. Now that I know all the parts of this equation, I’ll read it. (Read the equation aloud.) Now, read the problem with me. (Encourage students to read the problem aloud with you as you point to each part of the equation.)”

Model the “A” step, “Answer, or draw and check.”

Draw pictures to represent the dividend. - “Now that I know what the problem is and what math operation I need to use, I need to complete the “A” step. What is the “A” step? (Elicit the response, “Answer, or draw and check.”) Yes, I need to answer the equation. I know I can draw pictures to solve a division equation. What kind of pictures can I draw? (Elicit the response, “tallies or dots.”) Yes, I can represent the dividend by drawing tallies or dots. I’m going to draw tallies. My dividend is __ so I need to draw __ tallies. (Draw the appropriate number of tallies.)“

Draw circles around pictures to represent dividing them into equal groups based on the divisor. – “Now I have to divide or separate these tallies into groups. What number tells me how many tallies belong in each group. (Elicit the response, “the divisor.”) Good. (Point to the divisor and say aloud how many tallies should be in each group.) I can put the tallies into groups by drawing circles around them. How many tallies are in each group? (Elicit the appropriate response.) Yes, I need to circle __ tallies at a time. I’ll do that now. (Circle the tallies until you have tallies “left over.”) I have __ tallies left over. Can I circle these tallies? (Elicit the response, “no.”) Why? (Elicit the response, “because there are not enough/they are fewer than the divisor.”) That’s right, there are fewer tallies left than are represented by the divisor.”

Model checking your drawings. – “Now that I have finished drawing pictures, I need to check them and be sure I drew them correctly. How do I check the dividend? (Elicit the response, “count the total number of tallies to be sure they total the dividend.”) Yes. (Count aloud the tallies and compare the total to the dividend.) I know I have the correct number of tallies, so now I need to check to see if I have my dividend drawn correctly. The dividend is represented by the tallies in each circle or group. I can check this by counting the number of tallies in each group and be sure they equal the divisor. (Count aloud the tallies in each group and compare the total in each group to the divisor.) Last, I check my remainder by being sure there are fewer tallies left over than there are in each group. (Count aloud the remaining tallies and compare the total to the divisor/the number of tallies in each group.)”

Model how to find the answer to the equation. – “Now, to answer the equation, I count the number of groups. (Count aloud the groups.) How many groups do I have? (Elicit the appropriate response.) Yes. And how many tallies do I have left over? (Elicit the appropriate response.)” What do we call the left over tallies? (Elicit the response, “remainder.”) Yes, they represent the remainder.”

Model the ‘W’ step, “Write the answer.”

“ I have found my answer by drawing pictures and I have checked my drawings to be sure they are accurate. Now I can finish solving the problem by completing the ‘W’ step. What is the ‘W’ step? (Elicit the response, “Write the answer.”) Great, after I have found my answer by drawing, I write the answer. Where to I write the answer? (Elicit the appropriate response.) Great thinking! I know the answer to a division equation should be written here. (Point to the appropriate space.) What is my answer? (Elicit the appropriate response.) Yes, my answer is _____. I know this because I have ___ groups (Point to each circled group of tallies and count them aloud.) and I have ___ tallies left over (Point to the remaining tallies and count them aloud.) I’ll write the answer here. (Point to the appropriate space and write the answer.) “

b. Maintain a medium level of teacher direction/support for another example if students demonstrate misunderstanding/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.

Review steps of “DRAW”

“ I have another equation. What is the name of the strategy that can help me draw pictures to solve this equation? (Elicit the response, “DRAW.”) What does each letter in ‘DRAW’ stand? (Elicit the response, “the steps for solving an equation.”) Yes. What does the ‘D’ stand for? (Elicit the response, “Discover the sign.”) Yes, ‘D’ stands for ‘Discover the sign.’ (*Repeat this process for each step of ‘DRAW’.)"

Model the “D” step, “Discover the sign.”

“Now that I know I can use the ‘DRAW’ Strategy to help me solve this equation, I can begin by completing the first step, ‘D’. What does ‘D’ stand for? (Point to the ‘D’ step and elicit the response, “Discover the sign.”) Yes, I first need to discover the sign. How do I do this? (Elicit the response, “ by finding the symbol that tells what math operation to use.”) Good. (Point to the division sign.) What is the sign? (Elicit the response, “division.”) Good. How do you know it is a division sign? (Elicit the response, it has a line in the middle and a dot on the top and a dot on the bottom/it also looks like a sideways face. I see a long nose with two eyes.” (Point to the relevant features of the division sign.) Excellent thinking! What can I do to help me remember what math operation I need to use? (Elicit the response, “circle it.”) Yes, I can circle it to help me remember what math operation to use. (Circle the division sign.)"

Model the “R” step, “Read the problem.”

Model the “A” step, “Answer, or draw and check.”

Draw pictures to represent the dividend. - “Now that we know what the problem is and what math operation I need to use. What is the next step? (Elicit the response, “the “A” step.) What is the “A” step? (Elicit the response, “Answer, or draw and check.”) Yes, I need to answer the equation. I know I can draw pictures to solve a division equation. What kind of pictures can I draw? (Elicit the response, “tallies or dots.”) What do we draw first? (Elicit the response, “___ tallies.”) Why do we draw ___ tallies? (Elicit the response, “because that is the dividend/total.”) Yes, we represent the dividend first by drawing tallies or dots. I’m going to draw ___ tallies and you draw them on your paper (Draw the appropriate number of tallies.)"

Draw circles around pictures to represent dividing them into equal groups based on the divisor. – “How do we divide or separate these tallies into groups? (Elicit the response, “circle them.”) What number tells me how many tallies belong in each group. (Elicit the response, “the divisor.”) Good. (Point to the divisor and say aloud how many tallies should be in each group.) We can put the tallies into groups by drawing circles around them. How many tallies do are in each group? (Elicit the appropriate response.) Yes, we need to circle __ tallies at a time. Let’s do that now. (I’ll circle tallies here and you do the same on your paper.) Do we have any tallies left over? (Elicit the appropriate response.) Yes, we have __ tallies left over. Can we circle these tallies? (Elicit the response, “no.”) Why? (Elicit the response, “because there are not enough/they are fewer than the divisor.”) That’s right, there are fewer tallies left than are represented by the divisor. What do we call the left over tallies? (Elicit the response, “remainder.”) Yes, they represent the remainder.”

Model checking your drawings. – “Now that I have finished drawing pictures, what do I need to do? (Elicit the response, “check the drawings.”) Yes, we need to check them and be sure we drew them correctly. How do we check the dividend? (Elicit the response, “count the total number of tallies to be sure they total the dividend.”) Yes. Let’s count them. You count yours and I’ll count mine. How many tallies should you have? (Elicit the appropriate response.) We know we have the correct number of tallies. What do we check next? (Elicit the response, “to see if we have the dividend drawn correctly.”) Yes. How do we know if the dividend is drawn correctly? (Elicit the response, “make sure the number of tallies we circled is the same as the divisor.”) Excellent thinking! I can check this by counting the number of tallies in each group and be sure they equal the divisor. What is the divisor? (Elicit the appropriate response.) You count yours and I’ll count mine. What do we check last? (Elicit the response, “the remainder.”) Yes, we have to check the remainder. How do we do this? (Elicit the response, “count them to see that they are less than the divisor/less than the number of tallies in each group.”) Right. Count your remaining tallies and I’ll count mine. Are there enough to make another group of __? (Elicit the response, “no.”)”

Model how to find the answer to the equation. – “Now, to answer the equation, I count the number of groups. (Count aloud the groups.) How many groups do I have? (Elicit the appropriate response.) Yes. And how many tallies do I have left over? (Elicit the appropriate response.)”

Model the ‘W’ step, “Write the answer.”

“ I have found my answer by drawing pictures and I have checked my drawings to be sure they are accurate. What do I do next? (Elicit the response, “the ‘W’ step.”) Yes. What is the ‘W’ step? (Elicit the response, “Write the answer.”) Great, after I have found my answer by drawing, I write the answer. Where to I write the answer? (Elicit the appropriate response.) Great thinking! I know the answer to a division equation should be written here. (Point to the appropriate space.) What is my answer? (Elicit the appropriate response.) Yes, my answer is _____. How do you know this? (Elicit the response, “ because there are ___ groups and there are ___ tallies left over.”) Great. I’ll write the answer and you write yours on your paper.“

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.

Videos

Learning Objective 1:
view  Clip 1
FASTDRAW Strategy (to solve division story problems by drawing pictures).

Learning Objective 2: view  Clip 1, Clip 2
Draw solutions to division story problems using the “FASTDRAW Strategy.”