Comparing Fractions with Like and Unlike Denominators: Concrete Level

PHASE 2: Practice Strategies

Receptive/Recognition Level

Purpose: To provide students multiple opportunities to compare a variety of concrete representations of fractions (like or unlike denominators) and choose whether the first concrete representation is greater than, less than, or equal to the second concrete representation. Students also are provided multiple opportunities to describe why one fraction is greater than, less than, or equal to the other fraction.

Learning Objectives 1 - 4: Comparing fractions with like or unlike denominators using concrete materials (Area or Sets Model).

Structured Peer Tutoring/Structured Language Experiences

Materials:

Teacher –

• “Centers” where fractions are represented with the appropriate type of concrete material
• Concrete fraction representations are aligned into pairs to be compared.
• Language cards that “name” each concrete fraction representation are placed underneath each example.
• An “answer key” is prepared for each center that lists the fractional comparisons presented at each center (to be used for checking student response sheets.)

Students -

• A set of “response cards,” one with “greater than,” one with “less than,” and one with “equal to.”
• Response sheets that are numbered according to the number of comparisons provided at the centers. (*response sheets can be coded by center so the teacher can match it with the appropriate answer key.)
• Pencil for writing.

Description:

Activity:

Students work in pairs at centers that have various concrete representations of fractions displayed (e.g. “two-sixths” is represented by two “sixth” circle pieces placed on a circle divided into six equal parts.) Concrete representations are displayed in pairs so that it is clear which ones are to be compared. Fractional names are written underneath each fractional representation (e.g. “two- sixths”). Each student has a set of cards that read “greater than,” “less than,” and “equal to” and a response sheet numbered based on the number of comparisons provided at the centers. Students take turns responding to examples. For each pair of fractions to be compared, one student (“player”) decides whether the first fractional representation is greater than, less than, or equal to the second fractional representation. The student places the appropriate card between the two concrete representations. Then the student describes to their “peer” why they made the comparison they did. The other student (“coach”) checks the “player’s” response and provides appropriate feedback. If a pair has questions, the “coach” raises their hand for help from the teacher. The coach records the player’s response (“greater than,” less than,” or “equal to”) on the player’s response sheet. At the conclusion of the activity, students turn in their response sheets to the teacher. The teacher monitors students as they work in pairs by circulating the room, answering questions, providing specific feedback, providing positive feedback, and modeling as needed.

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. checks player’s response, provides appropriate feedback, writes response on player’s response sheet) and how the player responds (e.g. using cards to identify “greater than,” “less than,” or “equal to;” describe why they chose the response they did.).

4) Direct students to alternate responding to examples and explain roles: the student who is responding is the “player” and the student checking their partner’s responses is the coach. The coach will write the player’s response in the appropriate space on the player’s response sheet 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 player first identifies each fractional part, then uses their cards to identify whether the first fractional part is greater than, less than, or equal to the second fractional part. After this, the player describes to the coach why the first fractional part is greater than, less than, or equal to the second one.

5) Provide time for student questions.

6) Signal students to begin.

7) 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.

8) Review student response sheets and note special difficulties individual students may be having and/or progress they are making.

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Expressive Level

Purpose: to provide students with multiple opportunities for making meaningful connections between what they already know about concrete representations of fractions and comparing them to determine which fraction is greater than, less than, or if two fractions are equivalent.

Learning Objectives 1-4: Comparing fractions with like or unlike denominators using concrete materials (Area or Sets Model).

Planned Discovery Activities

Materials:

Students -

• Planned discovery learning sheets - each sheet has three sections that prompts students to : 1.) represent different fractions that have selected denominators (e.g. “2,” “4’” “6’” & “8”) with concrete materials. At least two different fractions for each chosen denominator should be included (e.g. “1/2” & “2/2”; “2/4” & “3/4;” “3/6 & 5/6;” “5/8” & “3/8”); 2.) represent two fractions and respond whether one is greater than the other or if they are equivalent; 3.) develop “new” comparisons, at least two examples for “greater than,” two examples for “less than,” and two examples for “equal to.”
• Appropriate concrete materials (circle pieces/fraction strips for Area Model; discrete counting objects like unifix cubes or beans for Sets Model).
• “Word strips” that represent each fraction and the phrases “greater than,” “less than,” or “equal to” for student who have writing problems (if appropriate).
• Pencils for writing.

Description:

Activity:

For comparing like denominators:

Students can work in groups, in pairs, or independently. Each student responds to a planned discovery-learning sheet. The sheet prompts them to do three primary things. First, they are prompted to represent different fractions that have selected denominators ( e.g. “2,” “4,” “6,”, and “8”) with concrete materials. At least two different fractions for each chosen denominator should be included. (e.g. “1/2” & “2/2”; “2/4” & “3/4;” “3/6 & 5/6;” “5/8” & “3/8”). As students finish representing their fractions, they raise their hands so the teacher can check their responses (*or another student can check if students are working in pairs or small groups.) After a student’s responses are checked, then they move to the second section of the learning sheet. The second section prompts them to compare those fractions that have like denominators. For each “like denominator,” students compare them and then write whether one fraction is greater than the other or whether the two fractions are equivalent (e.g. “two-halves is greater than one-half; “three fourths is greater than “three-fourths.”) (*For students with writing problems, pre-made word strips can be used that have the appropriate fractions written as well as the phrases “greater than” and “equal to. Another accommodation would be to have students orally compare the fractions by raising their hand and reporting to the teacher or by telling their comparison to a peer.) After students have be “checked off” for each of the comparisons, they respond to the third section of the learning sheet. The third section prompts students to represent and show the comparison of at least two “new” comparisons showing “greater than,” two comparisons showing “less than,” and two comparisons showing “equal to.” Each “new” comparison must incorporate different fractions from those used in the previous section. The same “check off” procedure used for sections one and two can be used to verify student comparisons for section three.

For comparing unlike denominators:

The same process can be used as described for “like denominators” except that the second section of the planned discovery-learning sheet should include comparing unlike denominators. Likewise, the third section prompts students to develop “new” comparisons for unlike denominators.

Planned Discovery Activity Steps:

1) Develop Planned Discovery Activity Learning Sheet as described under Materials.

2) Distribute the Planned Discovery Activity Learning Sheet and provide clear directions for completing the activity, including appropriate behavioral rules.

3) Model how to complete one example for each of the three sections on the Planned Discovery Learning Sheet (and model appropriate behaviors as needed).

4) Provide students with appropriate materials (e.g. appropriate concrete materials).

5) Monitor students as they practice, providing appropriate corrective feedback, prompting student thinking, providing positive reinforcement, and modeling or cueing as needed.

6) At the conclusion of the activity, provide students with solutions to the Planned Discovery Activity Learning Sheet, especially Part 3. Elicit student examples for Part 3, providing students with appropriate feedback and model several examples provided by students. Emphasize why the examples represent “greater than,” “less than,” or “equal to” (e.g. the area represented by the numerator in relationship to the denominator).

7) Review student response sheets and note special difficulties individual students may be having and/or progress they are making.

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