Adding and Subtracting Fractions with Mixed Numbers: Concrete Level

More Teaching Plans on this topic: Representational, Abstract

PHASE 1: Initial Acquisition of Skill

Teach Skill with Authentic Context

Description: Determining how much pizza is left over after a class party serves as a context for introducing and providing explicit teacher modeling for addition of fractions with mixed numbers. “Pizzas” made from tag-board are used to re-create the story context that provides the problem solving situation.

Build Meaningful Connections

Purpose: to assist students to build meaningful connections between what they know about adding common fractions with concrete materials to adding fractions with mixed numbers.

Learning Objective 1: Combine two sets of concrete materials that represent fractions with mixed numbers.

Materials:

Teacher –

• Concrete materials that represent fractions (e.g. circle pieces, fraction strips, cuisenaire rods.)
• Visual platform for showing concrete materials (e.g. floor where children can circle around; place magnetic strips on back of concrete materials and demonstrate them on the chalkboard or dry-erase board.)
• Language card that reads: “mixed numbers.” (*by including simple drawings of concrete materials that represent several examples of mixed numbers students with reading difficulties can be provided a meaningful cue.)
• Written display of learning objective: “Add fractions with mixed numbers using concrete materials.” (*Highlight “mixed numbers” in the written objective.)

Description:

1) L ink to students’ prior knowledge of concrete representations of fractions.

For Example:

Let’s review some things about fractions using concrete materials. (Hold up a circle piece.) What is this? (Elicit the response, “a circle.”) Yes, it’s a circle. Does this circle represent a whole or a fractional part of a whole? (Elicit the response, “a whole.”) Yes, this circle represents a whole. (Hold up a “1/2” piece and place it on top of the circle.) What does this piece represent? (Elicit the response, “one-half.”) Yes, this piece represents “one-half.” It takes up one-half of the space of the whole circle. (*Continue this process for several more fraction pieces – e.g. “1/4,” and “1/8.”)

You already know how to add fractions like “one-fourth” plus “two-fourths.” Let’s review this using circle pieces. (Place down a “one-fourth” piece and then place down two “one-fourth” pieces.) I have “one-fourth” (Point to the “one-fourth” piece) and I want to add or combine it with “two-fourths” (Point to the two one-fourth pieces.) To combine them, I simply put them together to make as much of a circle as I can. (Place the “one-fourth” piece adjacent to the pieces representing “two-fourths” so that they represent three-fourths of a circle. How many fourths do we have altogether? (Elicit the response, “three-fourths.”) Yes, “one-fourth” plus “two-fourths” equals “three-fourths.” I know that my fraction pieces equal three fourths because they make up three fourths of a whole circle. (Place the three “fourth” pieces on top of a whole circle to show/review this relationship.)

2) I dentify the skill students will learn: Adding fractions with mixed numbers using concrete materials.

For Example:

We know how to add two fractions that represent parts of a whole, like “one-fourth plus two-fourths.” Today we are going to learn how to add or combine concrete materials that represent both wholes and fractional parts of wholes. Let me show you what I mean. (Show two groups of concrete materials where each group has whole pieces and a fractional part – e.g. two circles and a one-fourth piece.) When we have groups like these that have both wholes and fractional parts, we call them “mixed numbers” (Point to a written display that reads, “mixed numbers.”) When we add concrete materials that represent mixed numbers, we add fractions with mixed numbers. (Point to a written display of the learning objective, “add fractions with mixed numbers using concrete materials.”) What are we going to learn to do today? (Point to the learning objective and elicit the response, “addition of fractions with mixed numbers.”). That’s right, we are going to learn how to add fractions with mixed numbers.

3) P rovide rationale/meaning for adding and subtracting fractions with mixed numbers.

For Example:

Being able to add fractions helps us in a lot of situations. You already know situations where adding fractions can come in handy, like when you and some friends buy….. Being able to add fractions with mixed numbers is also a helpful thing because sometimes you have situations where you have both wholes and fractional parts you need to add. For example, say we have a pizza party in class as a celebration for our hard work learning about adding fractions with mixed numbers. After the party, we will probably have pizza left over. We may have left over both individual slices of pizza as well as several whole pizzas left over. Being able to add them to see how much pizza we have total would allow us to determine whether there was enough pizza left over to give to another class

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Provide Explicit Teacher Modeling

Purpose: to provide students a clear teacher model of how to add fractions with mixed numbers using concrete materials.

Learning Objective 1: Combine two sets of concrete materials (circle pieces) representing mixed numbers with like denominators.

Materials:

Teacher –

• Concrete materials (area model) to represent fractions (e.g. circle pieces).
• A platform for displaying concrete materials so all students can see (e.g. floor; chalkboard/dry-erase board with concrete materials that have adhesive magnetic strips attached to their backs).
• Prepare “sets” of concrete objects that represent fractions with mixed numbers that will be combined.

Description:

A. Break down the skill of combining two sets of concrete materials representing mixed numbers.

1) Identify the fractional parts.

2) Combine the “wholes.”

3) Combine the fractional parts into “wholes.” - View Video

4) Add the “wholes.”

5) Add the fractional parts that remains.

6) Say what the total/sum means.

Learning Objective 2: Solve story problems involving addition of mixed numbers.

Materials:

Teacher –

• Concrete materials that represent fractional parts (i.e. circles and circle pieces).
• A platform for demonstrating the use of concrete materials that is clearly visible to all students.
• Prepared story problems/contexts representing addition of fractions with mixed numbers.

Description:

A. Break down the skill of solving story problems involving addition of mixed numbers into teachable/learnable steps.

1) Read the story problem/context.

2) Identify what is to be solved for. - View Video

3) Identify the important information.

4) Represent the mixed numbers with concrete objects.

5) Combine the “wholes.”

6) Combine the fractional parts into “wholes.” - View Video

7) Add the “wholes.”

8) Add the fractional part remaining.

9) Answer the story problem.

Learning Objective 3: Solve equations involving addition of fractions with mixed numbers using concrete objects.

Description:

A. Break down the skill of solving equations involving addition of fractions with mixed numbers using concrete materials into teachable/learnable parts.

1) Discover the sign/operation.

2) Read the equation and identify the wholes and fractional parts.

3) Represent the wholes and fractional parts for each mixed number with concrete materials

4) Combine the whole pieces.

5) Combine the fractional pieces to make as many wholes as possible.

6) Combine “new wholes” made with original whole pieces.

7) Combine the remaining fractional pieces.

8) Say what the total/sum means and write the solution.

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Scaffold Instruction

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 you have initially modeled these skills.

Learning Objective 1: Combine two sets of concrete materials that represent fractions with mixed numbers.

Materials:

Teacher –

• Concrete materials (area model) to represent fractions (e.g. circle pieces).
• A platform for displaying concrete materials so all students can see (e.g. floor; chalkboard/dry-erase board with concrete materials that have adhesive magnetic strips attached to their backs).
• Prepare “sets” of concrete objects that represent fractions with mixed numbers that will be combined.

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.)

Model how to identify the fractional parts.

“I have two groups of circle pieces here. The first thing I need to do is identify the fraction pieces in each group. Let’s see, in the first group I have two whole circles and two circle pieces. What do the whole circles represent? (Elicit the response, “two wholes.”) Yes, the two circles represent two “wholes.” How can I find out what the circle pieces represent? Oh, I know how to do this. I remember that when I learned about fractions, I found I could determine what part of a whole circle a circle piece was by placing it on top of a whole circle piece. I’ll take my two circle pieces and do that now. (Model placing the two one-fourth pieces on top of the whole circle piece one at a time.)

What do the two fraction pieces represent together? (Elicit the response, “one-half.”) Yes, two of my pieces equals half of my whole circle. If I had two more pieces like these pieces then I could cover the whole circle. If four equal parts equal a whole, what do we call each part? (Elicit the response, “one-fourth.”) That’s right. That means my two circle pieces in this group are one-fourth pieces. (*Follow this same process for the second group of circles and circle pieces. Elicit student responses at the same points as for the first set of concrete objects.)”

Model how to combine the “wholes.”

“Now that I have identified my circle pieces, I need to combine them since we are adding these circle pieces. An easy way to do this is to put similar pieces together. Well, I have three whole circles and five one-fourth pieces. Which pieces should I combine first, the whole pieces or the pieces that represent fractional parts. (Elicit the response, “the whole pieces.”) Good thinking, I can combine my whole circle pieces first. I’ll do that now. (Move the three circle pieces in one group.) Count them aloud with me. (Point to each circle as you count them aloud with your students.) Ok, I have three whole circle pieces. How many whole circle pieces do I have? (Elicit the response, “three.”) That’s correct, I have three whole circle pieces.”

Model how to combine the fractional parts into “wholes.”

“Now, I need to combine my circle pieces that represent ‘one-fourth’. I can do that by placing them one by one, making as many whole circles as I can. I’ll do this now. (Place the one-fourth pieces one-by-one in appropriate positions to make a whole circle. You should have one ‘one-fourth’ piece remaining.)”

Model how to add the “wholes.”

“Now that I’ve combined the whole circle pieces together and the one-fourth pieces together, I need to check to see if I created any more whole circles with my one-fourth pieces. Did I make another whole circle with four of my one-fourth pieces? (Point to the four one-fourth pieces that make the whole circle counting them aloud, and elicit the response, “yes.”) Yes, I did make another ‘whole’ with my one-fourth pieces. (Move your finger around the four pieces to show they make a whole circle.) I can combine this “additional” whole circle with the three original whole circles by moving it over here. Now I have four whole circles altogether. How many whole circles are there now? (Elicit the response, “four.”) That’s right, I have four whole circles now. Where did my fourth whole circle come from? (Elicit the response, “from the four one-fourth pieces you put together.”) Excellent thinking! I made this fourth circle (Point to the circle.) by combining four of the one-fourth pieces I started with.”

Model how to add the fractional part that remains.

“Ok. We have four whole circles. How many one-fourth pieces do we have remaining? (Point to the remaining one-fourth piece and elicit the response, “one.”) That’s right, we have one ‘one-fourth’ piece remaining.”

Model saying what the total/sum means.

“When we have finished combining our circle pieces, then we can say what they represent. I have four whole circles and one one-fourth piece. (Point to the circles and count them aloud and then point to the one-fourth piece and count it aloud.) This represents the total after I combined my original circle pieces. What is the total after I combined the circle pieces? (Elicit the response, “four whole circles and one one-fourth piece.) Correct. I have a sum of four whole circles and one one-fourth piece.”

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 continue to scaffold your instruction. (Examples of choices are shown in red.)

Model how to identify the fractional parts.

“I have two groups of circle pieces here. What do I need to do first if I am going to add or combine the groups? (Elicit the response, “identify the fraction pieces in each group.”) Excellent thinking! Let’s see, in the first group I have two whole circles and two circle pieces. What do the whole circles represent? (Elicit the response, “two wholes.”) Yes, the two circles represent two “wholes.” How can I find out what the circle pieces represent? (Elicit the response, “place it on top of a whole circle piece.” I’ll take my two circle pieces and do that now. (Model placing the two one-fourth pieces on top of the whole circle piece on at a time.)

What do the two fraction pieces represent together? (Elicit the response, “one-half.”) Yes, two of my pieces equal one-half of my whole circle. If I had two more pieces like these pieces then I could cover the whole circle. If four equal parts equal a whole, what do we call each part? (Elicit the response, “one-fourth.”) That’s right. That means my two circle pieces in this group are one-fourth pieces. (*Follow this same process for the second group of circles and circle pieces. Elicit student responses at the same points as for the first set of concrete objects.)”

Model how to combine the “wholes.”

“Now that I have identified my circle pieces, what do I do? (Elicit the response, “combine them.”) Yes, I need to combine them since we are adding these circle pieces. Which pieces should I combine first, the whole pieces or the pieces that represent fractional parts. (Elicit the response, “the whole pieces.”) Good thinking. I can combine my whole circle pieces first. I’ll do that now. (Move the three circle pieces in one group.) Count them aloud with me. (Point to each circle as you count them aloud with your students.) How many whole circle pieces do I have? (Elicit the response, “three.”) That’s correct, I have three whole circle pieces.”

Model how to combine the fractional parts into “wholes.”

“What do I add or combine now? (Elicit the response, “the one-fourth pieces.”) Great. How do I place the pieces when I combine them? (Elicit the response, place them one by one, making as many whole circles as you can.) That’s right. I’ll do this now. (Place the one-fourth pieces one-by-one in appropriate positions to make a whole circle. You should have one ‘one-fourth’ piece remaining.)”

Model how to add the “wholes.”

“Now that I’ve combined the whole circle pieces together and the one-fourth pieces together, what do I need to check for? (Elicit the response, “to see if you made any more whole circles with your one-fourth pieces.”) Excellent! Did I make another whole circle with four of my one-fourth pieces? (Point to the four one-fourth pieces that make the whole circle counting them aloud, and elicit the response, “yes.”) Yes, I did make another ‘whole’ with my one-fourth pieces. (Move your finger around the four pieces to show they make a whole circle.) How can I combine my newly made ‘whole’ with the other whole circles? (Elicit the response, move it with the other circles.”) Good. (Move the ‘new’ circle.) How many whole circles are there now? (Elicit the response, “four.”) That’s right, I have four whole circles now. Where did my fourth whole circle come from? (Elicit the response, “from the four one-fourth pieces you put together.”) Excellent thinking! I made this fourth circle (Point to the circle.) by combining four of the one-fourth pieces I started with.”

Model how to add the fractional part that remains.

“How many one-fourth pieces do we have remaining? (Point to the remaining one-fourth piece and elicit the response, “one.”) That’s right, we have one ‘one-fourth’ piece remaining.”

Model saying what the total/sum means.

“When we’ve finished adding/combining our circle pieces, what do we do? (Elicit the response, “say what they represent.”) Yes. What is our total? (Elicit the response, “four whole circles and one one-fourth piece.”) Correct. I have a sum of four whole circles and one one-fourth piece. Another way to say that is to say ‘four and one-fourth’ (Point to the four circles and then the one-fourth piece as you say this.) What is another way to say four whole circles and one ‘one-fourth’ piece? (Elicit the response, “four and one-fourth.”) Yes, we can say four and one-fourth.”

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.

Model how to identify the fractional parts.

“What do you need to do first (Elicit the response, “identify the fraction pieces in each group.”) Excellent thinking! What’s in the first group? (Elicit the response, “two whole circles and two circle pieces.”) Good. Hold up the two circles. (Check to see all students hold up the appropriate pieces.) Great. Hold up the circle pieces. (Check to see all students hold up the appropriate pieces.) What do the whole circles represent? (Elicit the response, “two wholes.”) Yes, the two circles represent two “wholes.” Show me how you can I find out what the circle pieces represent.”

What do the two fraction pieces represent together? (Elicit the response, “one-half.”) Yes, two of my pieces equal half of my whole circle. If four equal parts equal a whole, what do we call each part? (Elicit the response, “one-fourth.”) That’s right. (*Follow this same process for the second group of circles and circle pieces. Elicit student responses at the same points as for the first set of concrete objects.)”

 

Model how to combine the “wholes.”

“Now that you have identified your circle pieces, what do you do? (Elicit the response, “combine them.”) Yes. Which pieces should wecombine first, the whole pieces or the pieces that represent fractional parts. (Elicit the response, “the whole pieces.”) Good thinking. Everybody do that now. (Check to see all students move the circles into one group. How many whole circle pieces do we have? (Elicit the response, “three.”) Good, you have three whole circle pieces.”

Model how to combine the fractional parts into “wholes.”

“What do you add or combine now? (Elicit the response, “the one-fourth pieces.”) Great. How do you place the pieces when you combine them? (Elicit the response, place them one by one, making as many whole circles as you can.) That’s right. Do this now. (Check to see all students respond appropriately.)”

Model how to add the “wholes.”

“Now that you’ve combined the whole circle pieces together and the one-fourth pieces together, what do you need to check for? (Elicit the response, “to see if we made any more whole circles with our one-fourth pieces.”) Excellent! Did you make any more whole circles? (Elicit the response, “yes.”) How many wholes did you make? (Elicit the response, “one.”) Good. How do you combine your newly made ‘whole’ with the other whole circles? (Elicit the response, move it with the other circles.”) Good. Everybody do that now. (Check to see all students respond appropriately.) How many whole circles are there now? (Elicit the response, “four.”) Fantastic!"

Model how to add the fractional part that remains.

“How many one-fourth pieces do we have remaining? (Elicit the response, “one.”) That’s right, you have one ‘one-fourth’ piece remaining. Everybody hold up the remaining one-fourth piece. (Check to see all students respond appropriately.)”

Model saying what the total/sum means.

What is your total? (Elicit the response, “four whole circles and one one-fourth piece.”) Correct. What is another way to say four whole circles and one ‘one-fourth’ piece? (Elicit the response, “four and one-fourth.”) Yes, we can say four and one-fourth.

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.

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Videos

Learning Objective 1: view  Clip 1
Combine two sets of concrete materials (circle pieces) representing mixed numbers with like denominators.

Learning Objective 2: view  Clip 1, Clip 2
Solve story problems involving addition of mixed numbers.

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