This lesson introduces students to the concept of adding fractions. Students will: 
- write addition equations and inequalities using fractions. 
- develop "operation sense" for adding common fractions. 
- develop estimation skills. 
- solve addition problems with fractions. 

- How are relationships represented mathematically?
- How can mathematics help us communicate more effectively?
- How can mathematics help to quantify, compare, depict, and model numbers?
- What does it mean to analyze and estimate numerical quantities?
- What makes a tool and/or strategy suitable for a certain task?

- Decimal: A number written using base ten place values, includes a decimal point. 
- Decimal Fraction: A numerical fraction with 10 or 100 as its denominator, written to show the fractional place values after a decimal point. 
- Denominator: In a fraction, the number or quantity below the fraction bar. Tells the number of equal parts into which a whole is divided. 
- Fraction: Notation used to represent part of a whole or part of a group by telling the number of equal parts in the whole (denominator), and the number of those parts being described (numerator). 
- Inequality: A mathematical sentence that contains a symbol (<. >, ≤, ≥, or ≠) in which the terms on either side of the symbol are unequal. 
- Mixed Number: The sum of a whole number and a fraction. 
- Numerator: In a fraction, the number or quantity above the fraction bar. Tells the number of parts of a whole being described. 
- Unit Fraction: A fraction with a numerator of 1.

- Fraction Circles, two sets per student (M-4-3-1_Fraction Circle Template)
- Fraction Strips, two sets per student (M-4-3-1_Fraction Strip Template)
- construction paper circles
- scissors
- glue

- Observe during lesson activities and class discussions. Ensure that students do not overlap fraction pieces and that the pieces fit perfectly.

Scaffolding, Active Engagement, Metacognition, Modeling, and Explicit instruction 
W: Students solve a sharing problem by determining the fraction of a cookie each child receives. 
H: Students use fraction circles or strips to practice adding and comparing fractions. 
E: Students learn how to mix fractions to rename them. Students create addition equations and inequalities using fractions. 
R: The class reviews and discusses the correlations discovered while combining fractions. 
E: Next, students use models to subtract fractions. 
T: The lesson can be modified to match student needs. For a simpler challenge, students might use their fraction models to add fractions with the same denominator. Students can increase the difficulty of adding fractions by choosing fractions greater than one. 
O: Students use several fraction models to practice adding and subtracting fractions and demonstrate comprehension of the concepts. 

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

Provide each student with three circles to symbolize the cookies, and enable them to experiment with the circles to replicate the situation. Write the following on the board and ask students to perform the task at their desks. 

Answer Key: 



Allow students to use their circles to demonstrate that \(1 \over 4\) + \(1 \over 4\) = \(1 \over 2\). Have students investigate the question, "Why doesn't \(1 \over 2\) + \(1 \over 4\) = \(2 \over 6\)?" Encourage students to represent their thoughts in many ways, such as: 

relating fractions to \(1 \over 2\) and 1 
using fraction circles or strips 
converting the fractions to a common unit. \(1 \over 4\) + \(1 \over 4\) + \(1 \over 4\) 
relating to other contexts, such as money ($0.50 + $0.25) or time (\(1 \over 2\) hour + \(1 \over 4\) hour). 
Record student responses on the board using a table like the one below. 



Emphasize that \(3 \over 4\) can be expressed by adding \(1 \over 4\) + \(1 \over 4\) + \(1 \over 4\) = \(3 \over 4\) or \(1 \over 2\) + \(1 \over 4\) = \(3 \over 4\) with fraction circles or fraction strips. 

Now that students are familiar with the benchmark fractions of 0, \(1 \over 4\), \(1 \over 2\), and 1, have them determine if the following inequalities are true or false. (You may or may not let students utilize manipulatives.) 

“True or False: \(1 \over 2\) + \(1 \over 4\) ≠ \(2 \over 6\)?” (True) 
“True or False: \(1 \over 4\) + \(1 \over 2\) > 1?” (False) 
“True or False: \(1 \over 2\) + \(1 \over 4\) > \(1 \over 2\)?” (True) 
"We are now going to work in pairs to write number sentences and use fraction circles or strips to model them." Write the four numerals on the board: \(1 \over 2\), \(2 \over 3\), \(3 \over 4\) and 1. Pairs should choose one of these numbers and use fraction models (fraction circles or fraction strips) to illustrate adding two fractions to equal the chosen number. Ask students to list the number of sentences they modeled. Next, ask them to select two alternative fraction models (sections) that add up to the number they chose and write the corresponding addition statement. For example, if students select the number \(2 \over 3\), they may combine \(1 \over 3\) and \(1 \over 3\) to equal \(2 \over 3\), or \(1 \over 3\)+ \(1 \over 3\) = \(2 \over 3\). Another option is to combine \(1 \over 2\) and \(1 \over 6\) to equal \(2 \over 3\), or \(1 \over 2\) + \(1 \over 6\) = \(2 \over 3\). 

When students can easily combine two fraction sections, ask them to combine three, then four. Encourage students to talk about fraction relationships as they investigate the fraction sections. For example, \(1 \over 2\) + \(1 \over 6\) may be written as (\(1 \over 6\) + \(1 \over 6\) + \(1 \over 6\) ) + \(1 \over 6\) = \(2 \over 3\). These should also include combinations of the same fraction sections. Students should investigate equivalent representations for fractions, such as \(1 \over 2\) = \(3 \over 6\). 

Encourage students to repeat the procedure to identify fractional combinations that are greater than one. Use these three mixed numbers: \(1 {1 \over 4} \), \(1 {1 \over 3} \), and \(1 {1 \over 2} \). 

"Now we shall discuss inequalities. Remember that an inequality compares two unequal values. You will perform the same task that you just accomplished, but with an inequality symbol instead of an equal sign." Students should use fraction models to identify three fractions whose sum is less than one (a whole), and then construct the corresponding addition statement for the inequality. For example, \(1 \over 3\) + \(1 \over 4\) + \(1 \over 6\) < 1. After that, ask them to select four fractions with sums greater than 1 and write the resulting inequality statement. 

Bring the class back together and ask students to reflect on their work, concentrating on the qualities of numbers that meet a certain condition. 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 \(1 \over 2\).) 
"If you add a fraction to itself and the sum is greater than \(1 \over 2\), what must be true of the fraction?" (The fraction is greater than \(1 \over 4\).) 
"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 \(1 \over 2\), 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 \(1 \over 2\), what must be true about the fraction?" (The fraction is less than \(1 \over 4\).) 
Optional: Students could collaborate in pairs on the questions to investigate different or higher-level possibilities. Allow students a few minutes to complete their tasks before asking them to share their findings and solutions. Make sure they focus on the concepts rather than the calculation mechanics. This encourages students to consider fractional relationships and estimates. 

Extension: 

Routine: Display one fraction addition problem and instruct students to use their fraction models to express the problem and construct the corresponding equation. 

Small Group: Divide the class into pairs and ask students to write problems like these on an index card: \(2 \over 3\) + \(3 \over 6\) and \(4 \over 6\) + \(1 \over 6\). Students should work together to model each addition sentence using a fraction circle. Repeat the activity with different denominators, like 5 or 8. 

Expansion: For those who have mastered adding fractions, go on to subtracting fractions. Have them begin with a section (fraction circles or fraction strips) representing \(1 \over 2\). Assist them in illustrating the subtraction of \(1 \over 3\) from \(1 \over 2\) by using a \(1 \over 3\) section from their fraction circles or strips and covering as much of the \(1 \over 2\) section as possible. "You used your one-third piece to cover as much of the half-piece as possible. Let's see how much remains to be covered. This is the same as writing \(1 \over 2\) - \(1 \over 3\) = ?. Which of your fractional pieces would cover the remaining section?” (A one-sixth piece.) 

Write on the board: \(1 \over 2\) - \(1 \over 3\) = \(1 \over 6\). Have students complete the following tasks in pairs or small groups. Make sure to include identifying pairs of fractions that differ by more than or less than a certain amount. 
Find two fractions that differ by more than \(1 \over 2\). 
Find two fractions that differ by more than \(1 \over 3\). 
Find two fractions that differ by more than \(3 \over 4\). 
Problems like these can help students build number sense for common fractions by identifying their differences.