• 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
  

• 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.

• A platform for visually displaying the fraction place value mats and fraction number lines so that all students can see.
• Pens/markers for writing.

• 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. 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.

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.

• 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.

• Answer key for each lettered set of cards.

• 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

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.

• 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.)

• 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

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.

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

• Appropriate response sheet/curriculum slice/probe
• Graph/chart

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.

• 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.

• 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.

• 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.

• Paper and pencil to record their responses