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### Overview

This hands-on, minds-on activity helps students use what they already know about customary measurement (CCRS 2010 #18 [5.MD.1]: Convert among different-sized standard measurement units within a given measurement system.) to help them add and subtract fractions with like and unlike denominators.

## UP:MA19.5.9

### Vocabulary

• Fraction
• Benchmark fraction
• Denominator
• Fraction model
• Estimate
• Reasonableness
• Equation
• Unlike denominator
• Unlike units

### Knowledge

Students know:
• The meaning and magnitude of fractions expressed in units of halves, fourths, eighths, thirds, sixths, twelfths, fifths, tenths, and hundredths.
• Strategies to find sums of two or more fractions with like denominators.
• Strategies to find the difference of two fractions with like denominators.
• How to decompose a fraction greater than 1 and express as a mixed number.
Example: 7/3 = 3/3 + 3/3 + 1/3 = 2 1/3.

### Skills

Students are able to:
• Solve real-word problems involving addition and subtraction of fractions with unlike denominators.
• Represent problems using fraction models or equations.
• Assess reasonableness of answers using estimation and benchmark fractions.

### Understanding

Students understand that:
• solving word problems involving addition and subtraction of fractions with unlike units
• Require strategies to find equivalent fractions in a common unit, and the sum or difference will be expressed in the common unit.
• Can be assessed for reasonableness of answers using estimation strategies.

## UP:MA19.5.10

### Vocabulary

• Fraction
• Denominator
• Numerator
• Visual Model
• Sum
• Difference
• Equivalence
• Unlike denominators
• Unlike units

### Knowledge

Students know:
• Strategies to determine if two given fractions are equivalent.
• How to use a visual model to illustrate fraction equivalency.
• Contextual situations for addition and subtraction.

### Skills

Students are able to:
• Use fraction equivalence to add and subtract fractions and mixed numbers with unlike denominators.

### Understanding

Students understand that:
Addition and subtraction of fractions and mixed numbers with unlike units,
• Require strategies to find equivalent fractions in a common unit, and the sum or difference will be expressed in the common unit.
• Can be assessed for reasonableness of answers using estimation strategies.

### Primary Learning Objectives

Students will represent fractions and add fractions with like and unlike denominators using pieces of Fraction Friend and their own body.

Mathematical Practice Standards:

1. Make sense of problems and preserver in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4.  Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

### Procedures/Activities

ENGAGE

1.) Students will add today’s lesson to their Table of Contents and head their new entry page.   The students will write down three simple problems on their new entry page:  (1) 1/16 + 1/4 = ____, (2) 1/8 + 1/4= ___, and (3) 1/8 + 1/16 = ___.  If I have 1/16 cup of sugar and add another ¼ cup, how much sugar will I have? (Ask similar questions for problems 2 and 3.)

2.) Review Gallon Man with students. Integrate the following learning styles: (1)  Verbal/Linguistic (aural) what we hear, (2)  Visual /Spatial (handout, slides) what we see, (3)  Bodily/Kinesthetic (produce, reproduce) what we do, (4)  Logical/Mathematical what we think, (5)  Rhythm/Music (mnemonics, repetition, beat, meter), (6)  Naturalistic (observe, experience), (7)  Existential (feel, know intuitively), (8)  Intrapersonal (internal, self-study/reflection), and (9) Interpersonal (external, group-study/interaction)

Ask the students to stand and point to each part of their gallon: (1) tummy--one gallon, (2) vest- two half-gallons, (3) upper arms and  thighs--four quarts, (4)  use fingers and point to two bones in the lower arms and lower legs--eight pints, and (5) eight fingers and eight toes (tuck in thumbs and forget big toes)--16 pints.  Then, ask them questions, which require them to think about, touch, wiggle, say the name for each relevant part(s) of their bodies. These activities are a good review for all nine classes of learners.

3.) Tell the students that, by the end of the lesson, they will be able to mentally add the fractions they wrote in their journals. Tell them that they will always be able to do this for the fractions: fourths, eighths, and sixteenths, by referring to what they already know about Gallon Man.

EXPLORE

4.) Ask the students to be seated. Tell them that today we are going to build Fraction Friend.  Tell them that Gallon Man is going to be transformed into another helper: Fraction Friend.

5.) Call the students attention to the pieces of colored paper on their desks. Tell them they will be using this paper to build Fraction Friend, exactly as they have constructed Gallon Man.

Ask the students to touch their gallon, half-gallons, quarts, pints, and cups. Then, ask what would that fraction be for each part? This requires the students to make connections: transfer knowledge about physical things to mathematical ideas. It may take some time and a few extra iterations to bring the whole class to understanding.

As the students identify each colored sheet/body part/liquid metric/fraction, have the students tear their paper into the appropriate number of sections, corresponding to the fractional part.

The responses you are looking for are as follows:

Purple- 1 gallon – 1 whole

Red- ½ gallon- ½ + ½ = 1 whole

Orange- 4 quarts- ¼ + ¼ + ¼ + ¼ = 1 whole

Blue- 8 pints- 1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 = 1 whole

Yellow- 16 cups- 1/16 + 1/16 + 1/16 + 1/16 +1/16 + 1/16 + 1/16+ 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16= 1 whole

This part will be a little time consuming. The teacher can be setting up the ELMO for students to show their work and interact with the class members during this time.

6.) Adding Fractions – Write on the board fraction problems for the students to solve. Ask the students to explain what they are doing as they are changing the fractions into parts that they can work with easily.

Demonstrate by using Fraction Friend parts to solve  2/8 + 1/4 =

(¼ is equal to 2/8, so 2/8 + 2/8 is 4/8.  Some will know that they can simplify this into 1/2.)

Formative Assessment:  Watch students to make sure they are not using parts of the original problem to solve the equation. Ask them to explain how they are solving their problems. Are the students able to make equivalent fractions?  Are the students able to add the fractions?

EXPLAIN

7. Ask the students to look back at the problems they wrote down at the beginning of the lesson. Tell them that they will be working with a partner to solve them, using both of their Fraction Friend parts.

8.  Have students work through the problems they wrote in their journals. The teacher should walk around to each table and make sure that the students understand why and can tell others how they are able to use the smaller fractions to make up the larger fraction.

EXTEND

9. Use equity sticks to call on students to share what they have learned by asking them to solve more Fraction Friend problems, using the ELMO to display their thinking.

10.  In the following days, the students will work on solving problems with different unlike denominators not found in Fraction Friend.  Please visit the following websites:

http://www.aaamath.com/fra66kx2.htm

www.mathplayground.com/howto_fractions_diffden.htm

ENGAGE

1.) Students will add today’s lesson to their Table of Contents and head their new entry page.   The students will write down three simple problems on their new entry page:  (1) 1/16 + 1/4 = ____, (2) 1/8 + 1/4= ___, and (3) 1/8 + 1/16 = ___.  If I have 1/16 cup of sugar and add another ¼ cup, how much sugar will I have? (Ask similar questions for problems 2 and 3.)

2.) Review Gallon Man with students. Integrate the following learning styles: (1)  Verbal/Linguistic (aural) what we hear, (2)  Visual /Spatial (handout, slides) what we see, (3)  Bodily/Kinesthetic (produce, reproduce) what we do, (4)  Logical/Mathematical what we think, (5)  Rhythm/Music (mnemonics, repetition, beat, meter), (6)  Naturalistic (observe, experience), (7)  Existential (feel, know intuitively), (8)  Intrapersonal (internal, self-study/reflection), and (9) Interpersonal (external, group-study/interaction)

Ask the students to stand and point to each part of their gallon: (1) tummy--one gallon, (2) vest- two half-gallons, (3) upper arms and  thighs--four quarts, (4)  use fingers and point to two bones in the lower arms and lower legs--eight pints, and (5) eight fingers and eight toes (tuck in thumbs and forget big toes)--16 pints.  Then, ask them questions, which require them to think about, touch, wiggle, say the name for each relevant part(s) of their bodies. These activities are a good review for all nine classes of learners.

3.) Tell the students that, by the end of the lesson, they will be able to mentally add the fractions they wrote in their journals. Tell them that they will always be able to do this for the fractions: fourths, eighths, and sixteenths, by referring to what they already know about Gallon Man.

EXPLORE

4.) Ask the students to be seated. Tell them that today we are going to build Fraction Friend.  Tell them that Gallon Man is going to be transformed into another helper: Fraction Friend.

5.) Call the students attention to the pieces of colored paper on their desks. Tell them they will be using this paper to build Fraction Friend, exactly as they have constructed Gallon Man.

Ask the students to touch their gallon, half-gallons, quarts, pints, and cups. Then, ask what would that fraction be for each part? This requires the students to make connections: transfer knowledge about physical things to mathematical ideas. It may take some time and a few extra iterations to bring the whole class to understanding.

As the students identify each colored sheet/body part/liquid metric/fraction, have the students tear their paper into the appropriate number of sections, corresponding to the fractional part.

The responses you are looking for are as follows:

Purple- 1 gallon – 1 whole

Red- ½ gallon- ½ + ½ = 1 whole

Orange- 4 quarts- ¼ + ¼ + ¼ + ¼ = 1 whole

Blue- 8 pints- 1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 = 1 whole

Yellow- 16 cups- 1/16 + 1/16 + 1/16 + 1/16 +1/16 + 1/16 + 1/16+ 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16= 1 whole

This part will be a little time consuming. The teacher can be setting up the ELMO for students to show their work and interact with the class members during this time.

6.) Adding Fractions – Write on the board fraction problems for the students to solve. Ask the students to explain what they are doing as they are changing the fractions into parts that they can work with easily.

Demonstrate by using Fraction Friend parts to solve  2/8 + 1/4 =

(¼ is equal to 2/8, so 2/8 + 2/8 is 4/8.  Some will know that they can simplify this into 1/2.)

Formative Assessment:  Watch students to make sure they are not using parts of the original problem to solve the equation. Ask them to explain how they are solving their problems. Are the students able to make equivalent fractions?  Are the students able to add the fractions?

EXPLAIN

7. Ask the students to look back at the problems they wrote down at the beginning of the lesson. Tell them that they will be working with a partner to solve them, using both of their Fraction Friend parts.

8.  Have students work through the problems they wrote in their journals. The teacher should walk around to each table and make sure that the students understand why and can tell others how they are able to use the smaller fractions to make up the larger fraction.

EXTEND

9. Use equity sticks to call on students to share what they have learned by asking them to solve more Fraction Friend problems, using the ELMO to display their thinking.

10.  In the following days, the students will work on solving problems with different unlike denominators not found in Fraction Friend.  Please visit the following websites:

http://www.aaamath.com/fra66kx2.htm

www.mathplayground.com/howto_fractions_diffden.htm

### Assessment Strategies

EVALUATE

Formative Assessments: This is for the introductory lesson given in detail above.

The continuous assessments mentioned in the proposal above provide the teacher a constant flow of individual student learning.  These combine to help optimally structure the fluid classroom environment.

An “Exit Pass” protocol is used in the effort to keep the students focused over-night and help the teacher prepare for the next class period. Using the ELMO, show the students four problems on different colored sticky notes:  (1) red  ¼ + 5/16 =, ( 2) blue  ¾ + 1/8 =, (3) yellow 3/8 + ½ =, and (4) green 4/16 + 2/8 =. Ask the students to copy these four problems and write their initials on the back of each sticky note.  Then have them solve each problem and post their notes as they leave the room. The students will continue their discussion about the Fraction Friend lesson as they compare their answers and talk about how much fun it is and how easy it is. The teacher will use this student specific information to guide the next day’s instruction.

Summative Assessment: A traditional pencil and paper test.  Note:  This test should be administered only after the students have had ample time to explore adding fractions with unlike denominators. http://math.about.com/od/fractionsrounding1/ss/FractionTest.htm.

### Acceleration

Students can act as a peer tutor, and/or design other whole objects that contain natural divisions to solve fraction problems. The students can also make connections to proportions and ratios.

### Intervention

Instruction and support for each child will be determined through formative assessment.

Some examples of remediation:

Students may require more practice adding fractions with like denominators.  Premade fraction parts should be made available to those students.

Students may work with a peer tutor or teacher.

### Total Duration

31 to 60 Minutes

### Background/Preparation

Students need to understand customary capacity. Students should be able to convert gallons to quarts, pints, and cups.  Group students heterogeneously in groups of 2 to 4 so that they may work with a partner and discuss their findings as a group.

http://www.educationworld.com/a_lesson/dailylp/dailylp/gallon_man.shtml

### Materials and Resources

Gallon Man (commercial or student made), construction paper (five colors for each child: purple, red, orange, blue, and yellow), and magic markers, students’ journals and pencils.

### Technology Resources Needed

Computer with Internet access and digital document camera (ELMO)