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FRACTIONS FORMULA ADDING SUBTRACTING
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Description: continue to link adding fractions with mixed numbers without drawing to meaningful contexts such as the Pizza Party example described in the Representational/Drawing and Concrete Level Instructional Plans. This is especially important as you teach students to solve story problems without drawing using the FASTDRAW Strategy. Additionally, make explicit links to drawing experiences when teaching the abstract process. Purpose: to assist students to build meaningful connections between what they know about adding fractions with mixed numbers by drawing pictures and adding fractions with mixed numbers without drawing.

Learning Objective 1: Add fractions with mixed numbers using a fraction place value mat and a fraction number line.

Materials:

Teacher

• A platform for writing/drawing so that all students can see (e.g. chalkboard/dry-erase board; chart paper; overhead projector.).
• The written objective: add fractions with mixed numbers without drawing. (*Highlight the words without drawing for cueing purposes.).
• Chalk/markers

Description:

1.) L ink to students prior knowledge of adding fractions with mixed numbers by drawing to adding fractions with mixed numbers without drawing.

For Example:

You now know how to solve story problems and equations that involve addition of fractions with mixed numbers by drawing pictures that represent concrete materials. Lets review how to add fractions with mixed numbers by drawing pictures. (Solve one problem with your students participation.)

2.) I dentify the skill students will learn.

For Example:

Today Im going to show you how to solve problems like this without drawing pictures. (Display the written objective: add fractions with mixed numbers without drawing.) What are we going to learn to do today? (Point to the written objective and elicit the response, add fractions with mixed numbers without drawing.) Thats right. Were going to learn how to add fractions with mixed numbers by drawing. Well learn to add these kinds of problems by using numbers and symbols only. Well begin learning to do this with fractions place value mats and fraction number lines.

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

For Example:

Youve learned how to add fractions with mixed numbers both by using concrete materials and by drawing. These are excellent strategies for solving problems like these. However, it takes quite a bit of time to do this and you may need to be able to do solve these problems more quickly. By the time we finish, youll be very good at solving these kinds of problems in your head using only the numbers and symbols. Purpose: to provide students a clear teacher model of how to add fractions with mixed numbers without using concrete objects and without drawing pictures.

Learning Objective 1: Add fractions with mixed numbers using a fraction place value mat and a fraction number line.

Materials:

Teacher

• Fraction place value mat: Has the same format as a place value mat representing ONES and TENS except the column to the right is labeled Fractions and the column to the left is labeled Wholes.

 Wholes Fractions

Long fraction number lines that represent whole numbers and fractions with like denominators drawn on sentence strips:

Example of fraction number line (fourths and sixths):

*This example shows up to the whole number three. The actual fraction number line should extend to 10 Shorter fraction line strips that depict fractions with like denominators drawn on sentence strips (Each fraction strip should be made in the same proportion as the long fraction number line depicting whole numbers and fractions):

Example of fraction number line strips (fourths and sixths: • A platform for visually displaying the fraction place value mats and fraction number lines so that all students can see.
• Pens/markers for writing.

Description:

A. Break down the skill of adding fractions with mixed numbers using a fraction place value mat and fraction number lines. 1.) Discover the sign/operation.

2.) Read the problem.

3.) Write the wholes and fractions in appropriate columns of place value mat.

4.) Introduce the fraction number line.

5.) Add fractions using fraction number line.

6.) Determine if one or more wholes can be made from adding the fractions by using the fraction number line.

7.) Regroup the whole(s) made to the whole column of the fraction place value mat.

8.) Add the wholes and write the number representing the sum.

9.) Say the sum/total. Learning Objective 2: Solve story problems and equations involving addition of fractions with mixed numbers using the FASTDRAW Strategy without drawing.

Materials:

Teacher

• A visual display of the FASTDRAW and DRAW Strategy (for the A step in DRAW, omit or draw and check.)
F ind what you are solving for.
A sk yourself, What is the important information?
S et up the equation.
T ie down the sign.

D iscover the sign.
R ead the problem.
A nswer the problem.
W rite the answer

• A visual medium for writing (e.g. chalkboard, dry-erase board, chart paper.)
• Markers/pens for writing.
• Prepared story problems and/or equations that represent addition of fractions with mixed numbers. Color code whole numbers and fractions consistent with color-coding used to identify these concepts in story problems/equations used at the concrete level of understanding (e.g. 3 1/3 + 1 2/3 = __.)
• Fraction place value mats and fraction number lines (as needed).

A. Break down the skill of solving story problems and equations involving addition of fractions with mixed numbers using the FASTDRAW Strategy without drawing. 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.
b. Read the problem.
c. Answer without drawing (using fraction place value mats and fraction number lines.)
d. Write the answer.

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

6.) Model how to draw solutions to equations by repeating the steps in#4 and #5 at least two or three more times with different division equations. Purpose: to provide students the opportunity to build their initial understanding of how to add fractions with mixed numbers without concrete materials or drawings, and to provide you the opportunity to evaluate your students level of understanding after you have initially modeled this skill.

Materials:

*Dependent on the skill you are Scaffolding Instruction for (See the materials listed for the specific skill you want to scaffold under Explicit Teacher Modeling)

Description:

*Scaffolding at the abstract level of instruction should occur using the same process as scaffolding instruction at the concrete and representational/drawing levels of instruction. (See the description of Scaffolding Instruction in the Concrete Level Instructional Plan for this math concept.) The steps listed for each skill during Explicit Teacher Modeling should be used as structure for scaffolding your instruction.

A. Scaffold instruction using a high level of teacher direction/support. Dependent on the needs of your students, you may want to continue to associate drawings to the abstract level process during this phase of scaffolding. Move to the next phase of scaffolding only when students demonstrate understanding and ability to respond accurately to your prompts.

B. Scaffold instruction using a medium level of teacher direction/support. If you associated drawings with the abstract process while scaffolding using a high level of teacher direction/support, then do not include drawings during this phase of scaffolding. Move to the next phase of scaffolding only when students demonstrate understanding and ability to respond accurately to your prompts.

C. Scaffold instruction using a low level of teacher direction/support. Students should actually add fractions with mixed numbers as you prompt them during this phase of Scaffolding Instruction. Move students to independent practice of the skill only after they demonstrate the ability to perform the skill with limited prompting from you.  Purpose: to provide students multiple opportunities to practice matching equations to their solution represented on place value mats.

Learning Objective1: Add fractions with mixed numbers using a fraction place value mat and a fraction number line.

Self-Correcting Materials Flash Cards

Materials:

Teacher

• Sets of 5 x 7 note-cards with the following: Front an equation written at the top with two or three examples of place value mats containing the numbers represented in appropriate columns and a solution. A hole is punched next to each choice. Back a star or other cue is drawn next to the hole that represents the correct solution.  *To make the card development process easier, you can develop the equations and choices by writing them on sheets of copy paper to fit the size of the note-cards you are using. For example, the above equation and choices can be made so that if cut out, they can be glued to the front side of a 5 x 7 note-card. Multiple copies of examples and choices can be made to make multiple sets of cards. Code each set of cards by representing each set with a letter and each card with the letter of the set and a number that represents the particular example/problem. Answer keys can be developed based on the letter and number for each card. You can make multiple sets of the same problems or have several sets of note-cards that represent different problems.

• Answer key for each lettered set of cards.

Students -

• One set of note cards.
• Sheet of paper. (Students can number the paper as they respond to the note-cards. They also should put the letter of the set at the top of their response sheet.)
• Pencil

Description:

Activity:

Students work individually with sets of 5 x 7 note cards with appropriate prompts and choices on the front side and the answer provided on the backside. Students read the equation at the top of the card and then examine the three choices underneath the equation. Each choice is a fraction place value mat with the equation represented as well as the solution. Students choose the place value mat that represents the correct solution by placing the point of their pencil in the hole next to their choice. They turn the card over to see if the hole they chose has a star or some other cue indicating the correct solution. Students keep track of their responses by writing their answers on a numbered sheet of notebook paper. Students also write the letter of the set of cards they use found at the upper right hand corner of each card. The teacher monitors students as they practice by providing specific corrective feedback, positive reinforcement, and answering questions as needed. After the activity, students turn their response sheets in and the teacher evaluates them using the answer key for each set of note-cards to check for student understanding.

Self-correcting Materials Steps:

1.) Introduce self-correcting material.

2.) Distribute materials.

3.) Provide directions for self-correcting material, what you will do, what students will do, and reinforce any behavioral expectations for the activity.

4.) Provide time for students to ask questions.

5.) Model responding/performing skill within context of the self-correcting material.

6.) Model how students can keep check their responses.

7.) Have students practice one time so they can apply what you have modeled.       Provide specific feedback/answer any additional questions as needed.

8.) Monitor students as they work.

9.) Provide ample amounts of positive reinforcement as students play.

10.) Provide specific corrective feedback/ re-model skill as needed.

11.) Encourage students to review their individual learning sheets, write the total number of correct responses under the C (Correct) column and do the same for the H (Help) column.

12.) Review individual student performance record sheets. Purpose: to provide students multiple opportunities to practice adding fractions with mixed numbers.

Learning Objective1: Add fractions with mixed numbers using a fraction place value mat and a fraction number line.

Instructional Game Fraction Dice Game

Materials:

Teacher

• Pairs of die: Each pair includes one die that has whole numbers on each side, while the other die has fractions with like denominators on each side. Blank labels can be used to make numbers and fractions. Cut out a piece of label the fits the side of a die and write the appropriate whole number or fraction on it. Glue the numbers and fractions to the sides of the dice. *See Alternative Ideas for Prompts under Description for additional ideas.
• Game board (optional)
• Sets of Chance cards with different consequences written on them (e.g. add 3pts to your score; move ahead 2 spaces; subtract 2 pts from your score, move back 3 spaces.)

Students -

• Two pair of dice; two spinners; two sets of cards depicting whole numbers and fractions with like denominators.
• One game board per small group (optional)
• One set of Chance Cards per small group
• Each student has a copy of a DRAW Strategy Cue Sheet (includes steps of the DRAW Strategy.)
• Each student two sheets of notebook paper with one labeled play and one labeled check.
• Pencil

Description:

Activity:

Students can play in pairs or small groups. Each small group has two pair of dice. One die has whole numbers on each side (e.g. 1, 2, 3, 4, 5, 6) and one die has fractions with like denominators (e.g. , 2/4, , 4/4). Each player roles each pair of dice one at a time. With each role, the player writes the mixed number represented by the pair of dice in the form of an addition equation on their play sheet. The player adds the two mixed numbers using a DRAW Strategy Cue Sheet as needed and writes the answer. The other players in the small group check the players response by solving the equation on their check sheet. For correct responses, each player can either receive a set number of points or move a game piece on a game board. The winner is the student who reaches a designated total of points or who reaches the finish space on the game board. Chance cards can also be included to increase motivation. Students select a chance card after they respond. Each chance card either tells the player to add bonus points to their score/move additional spaces, or it tells them to subtract points from their score/move back a specified number of spaces. The teacher monitors student academic and social behavior as they practice by providing positive reinforcement, providing specific corrective feedback, and answering questions as needed. Students turn in both their play sheets and their check sheets at the end of the activity. The teacher reviews individual sheets to evaluate student understanding.

Alternative Ideas for Prompts:

Two spinners can also be used, where one spinner has whole numbers listed and the other has fractions with common denominators listed (Spinners can be made so they have multiple sets of fractions with like denominators. A circular spinner can be divided into bands of fourths, sixths, and eighths where the outside band has sections with fourths, the middle band has sections with sixths, and the inside band has sections with eighths. Students can be assigned particular fractions with like denominators.) Another alternative is sets of cards that have written whole numbers and fractions. Students pull one card from the whole number pile and one card from the fraction pile for each mixed number.

Instructional Game Steps:

1.) Introduce game.

2.) Distribute materials.

3.) Provide directions for game, what you will do, what students will do, and reinforce any behavioral expectations for the game.

4.) Provide time for students to ask questions.

5.) Model how to respond to the prompts.

6.) Provide time for students to ask questions about how to respond.

7.) Model how students can keep track of their responses, how to check other students responses, and how to give and receive corrective feedback.

8.) Play one practice round so students can apply what you have modeled. Provide specific feedback/answer any additional questions as needed.

9.) Monitor students as they practice by circulating the room, providing ample amounts of positive reinforcement as students play, providing specific corrective feedback/ re-modeling skill as needed.

10.) Play game.

11.) Encourage students to review their individual response sheets, write the total number of correct responses under the C (Correct) column and do the same for the H (Help) column.

12.) Review individual student response sheets to determine level of understanding/proficiency and to determine whether additional modeling from you is needed.  Purpose: to provide you with continuous data for evaluating student learning and for evaluating whether your instruction is effective. It also provides students a way to visualize their learning/progress.

Materials:

Teacher

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

Student

• Appropriate response sheet/curriculum slice/probe
• Graph/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 and point to problems on board as students respond in writing or orally to the same problems written on their individual curriculum slice.).
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 (see step # 10 for abstract level skill proficiency criteria) 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 whether she/he has mastered the skill at the abstract level using the following guide:

Abstract Level: demonstrates near 100% accuracy for adding fractions with mixed numbers (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. *This assessment activity can be used with students who demonstrate difficulty with adding fractions with mixed numbers at the abstract level.
Purpose: to assess where student understanding of adding fractions with mixed numbers is breaking down.
Flexible Math Interview/C-R-A Assessment
Materials:
Teacher
• Appropriate concrete materials for adding fractions with mixed numbers (See Concrete Level Instructional Plan Explicit Teacher Modeling.).
• Appropriate examples for assessment (equations/story problems)
• Paper to record notes.
Description:

Have students solve problems using concrete materials, by drawing, and without concrete materials or drawings. Ask students to explain their answers as they respond. Note where in the problem-solving process students break down; both, at what level they begin having difficulty, and at what point within that level of understanding they demonstrate misunderstanding/non-understanding. Based on where students demonstrate difficulty, provide explicit teacher modeling at that level of understanding and for the particular sub-skill they are having difficulty with. As the student demonstrates understanding, scaffold your instruction until they are ready to practice the skill independently. As students demonstrate mastery of the skill at that level of understanding, then provide explicit teacher modeling at the next level of understanding. Follow this process until students demonstrate mastery at the abstract level.

Key Ideas for Developing Mastery at the Abstract Level
• Students who demonstrate difficulty at the abstract level of understanding may have gaps in their understanding that can be traced back to their representational/drawing level of understanding or even their concrete level of understanding. By providing additional teacher modeling at the level their gap in understanding began and then moving them from a concrete-to-representational-to-abstract level of understanding, you can assist students to become successful at the abstract level of understanding.
• Sometimes students demonstrate difficulty at the abstract level because they did not receive enough practice opportunities at the concrete and representational/drawing levels. The drawing level is a very important step for these students. Some students need continued practice drawing solutions and associating their drawings to the abstract symbols and the mental processes necessary to perform at the abstract level.
• Some students understand the concept, but have difficulty remembering the steps necessary to perform the skill at the abstract level. Providing students with cues they can refer to as they practice at the representational/drawing and abstract levels of instruction is very helpful (e.g. DRAW Strategy). Such cueing provides them the independence to practice. Multiple practice opportunities translate into repetition, and repetition enhances memory. The use of instructional games and self-correcting materials are an excellent way to provide students with multiple opportunities to solve problems.
• Helping your students build their fluency for solving equations can also increase their abstract level problem-solving efficiency. Providing daily one-minute timings and charting student performance is an effective way to do this. It is important to both communicate with students what their learning pictures (charts) mean and to set short-term achievable goals. Seeing what they are striving for and seeing their progress as they move toward a goal is very motivating for children! (See the description of the instructional strategy Continuous Monitoring and Charting Student Performance for more information. This description can be found by clicking on Instructional Strategies on the left panel of the main MathVIDS menu bar.)
• Enhancing the meaningfulness of abstract equations can also aid students who are having difficulty achieving mastery at the abstract level both by providing them a deeper level of conceptual understanding and by enhancing their memory of the problem-solving process. One approach you might try is to reinforce what the numbers and symbols mean using language. By modeling language (and encouraging students to use their own language) that describes what each number and symbol represents, students can gain a deeper level of understanding of the abstract process they are struggling to master.

For Example:
2 1/4 + 3 3/4 = ___
circles combined with circles totals

Provide your students multiple opportunities to use their language as they practice solving equations. As students practice, they have the opportunity to associate meaning to the abstract process. *Maintenance activities at the abstract level of understanding should include concrete and representational/drawing experiences as well as abstract (numbers and symbols only) experiences. By re-visiting previous concrete and representational/drawing experiences, students reinforce the conceptual understanding they acquired during those phases of instruction. Including language experiences during these maintenance activities, where students describe their solutions, also reinforces conceptual understanding students established during their concrete and representational/drawing experiences.
Purpose: to provide students with opportunities to maintain their level of mastery for adding fractions with mixed numbers.
1. Instructional Games & Self-Correcting Materials
Materials:
*Dependent on the Instructional Games or Self-Correcting Materials you implement.
Description:

*Periodically provide students opportunities to practice via self-correcting materials and instructional games. This can be done through centers, in small groups, or as a whole class. Include opportunities to solve problems by using concrete materials and by drawing in addition to abstract level practice opportunities. Even though students master a concept/skill at an abstract level, providing maintenance practice opportunities using concrete materials and by drawing reinforce their conceptual understanding. (See the descriptions for Instructional Games and Self-Correcting Materials for more information of how to implement these student practice strategies. You can access these descriptions through the main MathVIDS navigation bar by clicking "Instructional Strategies" and then accessing the instructional strategies lists.)

2. Problem of the Day
Materials:
Teacher
• A written prompt on the chalkboard, dry-erase board, or overhead projector (e.g. an equation involving addition of fractions with mixed numbers or story problem) or a concrete/drawing example representing a solution to a problem involving addition of fractions with mixed numbers.
Students -
• Paper and pencil to record their responses
Description:
Teacher presents a problem of the day that focuses on a particular skill or conceptual understanding of solving problems involving addition of fractions with mixed numbers. The problem can be written in nature where students solve the problem with concrete materials, by drawing, or at the abstract level only. Students can also be challenged to develop a story problem for an already solved equation. 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 are provided necessary directions. After an appropriate amount of time (e.g. 2-3 minutes), 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. The teacher notes students who seem to be having difficulty for the purpose of reviewing/re-modeling appropriate skills and concepts.
Ideas for Prompts:
1.) Display the concrete or drawing representation of an equation as well as its solution and ask students to represent the equation and the solution using only numbers and symbols.
2.) Display an equation and ask students to represent the equation and the solution with concrete materials or drawings.
3.) Display a concrete, drawing, or abstract representations of an equation and have students develop a story problem for that equation.
4.) Display an equation and solution with concrete materials, by drawing, or with only numbers and symbols with one part of equation missing (e.g. one of the mixed numbers being added) and ask students to determine the missing part.