• Observe during lesson activities and classroom discussion. Make sure that students are not overlapping fraction pieces and that the pieces fit exactly.

Present the following problem: “Sally has three cookies. She wants to share them equally between three friends and herself. How much will each child have?” Ask, “Will the amount be more than one cookie or less than one cookie? How do you know?”

Provide three circles for each student to represent the cookies and allow them to experiment with the circles to simulate the situation. Write the following on the board and have students complete the exercise at their desks.

Answer Key:

Discuss with students that  by allowing them to use their circles as a model. Have students explore the question, “Why doesn’t ?” Have students represent their ideas in different ways, such as:

• relating the fractions to  and 1
• using fraction circles or fraction strips
• changing the fractions to a common unit:
• relating to other contexts, such as money ($0.50 + $0.25) or time ( hour +  hour)

Record student answers on the board using a table like the one that follows.

Emphasize that  could be represented by adding  or  using the fraction circles or fraction strips.

Now that students are familiar with the benchmark fractions of 0, , , and 1, have them identify which of the following inequalities are true or false. (You may or may not allow students to use manipulatives.)

• “True or False:  ?” (True)
• “True or False:  ?” (False)
• “True or False:  ?” (True)

“We are now going to work in pairs to write number sentences and use fraction circles or strips to model them.” Write the following four numbers on the board: , and 1. Have pairs choose one of these numbers and use the fraction models (fraction circles or fraction strips) to represent adding two fractions to equal their chosen number. Ask students to write the number sentences they have modeled. Next, have them find two other fraction models (sections) that add up to the number they chose and write the matching addition sentence. For example, if students chose the number , they might combine  and  to equal , or . Another option would be to combine  and  to equal , or .

When students can easily combine two fraction sections, have them combine three, and then four fraction sections. Encourage students to discuss fraction relationships as they explore the fraction sections. For example,  could be represented as . These should also include combinations of identical fraction sections. Students should explore equivalent representations for the fractions, such as .

Encourage students to repeat the process to find combinations of fractions that are greater than 1. Use the following three mixed numbers: , , and .

“Now we will move on to inequalities. Remember that an inequality compares two unequal values. You will be doing the same task you just completed but using an inequality symbol instead of an equal sign.” Have students use fraction models to find three fractions whose sum is less than 1 (a whole), and then write the matching addition sentence for the inequality. For example, . Afterward, have students find four fractions that have a sum greater than 1 and ask them to write the resulting statement of inequality.

Bring the class back together and have students reflect on their work by focusing on the characteristics of numbers that fit a certain requirement. Allow students to make connections to other contexts such as money (decimals) to help them think about the fractions and estimates. Ask the following questions:

• “If you add a fraction to itself and the sum is greater than 1, what must be true about the fraction?”  (The fraction is greater than .)
• “If you add a fraction to itself and the sum is greater than , what must be true of the fraction?”  (The fraction is greater than .)
• “If two fractions are both greater than 1, what must be true of their sum?”  (The sum must be greater than 2.)
• “If two fractions are both greater than , what must be true of their sum?”  (The sum is greater than 1.)
• “If a fraction is added to itself and the sum is less than , what must be true about the fraction?” (The fraction is less than .)

Optional: Students could work in pairs on the questions, exploring different or higher-level possibilities. Give students a few minutes to work, and then have them share their results and solution methods. Be sure that they focus on the ideas involved and not the computation mechanics. This encourages students to think about fractional relationships and estimation.

Extension:

• Routine: Display one fraction addition problem and have students use their fraction models to represent the problem and write the appropriate equation.
• Small Group: Divide the class into pairs and have students write problems like these on an index card:  and . Have students work together to model each addition sentence using a fraction circle. Repeat the activity using different denominators such as 5 or 8.
• Expansion: For students who have mastered adding fractions, move them into subtracting fractions. Have them start with a section (fraction circles or fraction strips) representing . Guide them in representing the subtraction of  from  by using a  section from their fraction circles or strips to cover as much of the  section as possible. “You used your one-third piece to cover as much of the one-half piece as possible. Let’s find out how much remains to be covered. This is the same as writing . Which of your fractional pieces would cover the remaining section?” (A one-sixth piece.)
  
  Write on the board: . Have students work on the following tasks in pairs or small groups. Be sure to involve finding pairs of fractions that differ by more than or less than a specified amount.

• Find two fractions that differ by more than .
• Find two fractions that differ by more than .
• Find two fractions that differ by more than .

Problems such as these can help students develop number sense about common fractions by determining their differences.