This activity prompts students to use number lines to illustrate the decomposition of a fraction into sums of other fractions, reinforcing their work from an earlier lesson. Along the way, students recognize that one way to decompose a fraction greater than 1 is to write it as a sum of a whole number and a fraction less than 1. This insight prepares students to interpret and write mixed numbers in later activities.

• Invite students to share their equations. Record them for all to see.
• Focus the discussion on the second part of the second problem: writing as a sum of a whole number and a fraction. Students likely will write and .
• Explain that can be written as , which we call a mixed number. This number is equivalent to .
• “Why might it be called a mixed number?” (It is a mix of a whole number and a fraction.)

In this activity, students use number lines to represent addition of two fractions and to find the value of the sum. The addends include fractions greater than 1, which can be expressed as a sum of a whole number and a fraction. Students practice constructing a logical argument and critiquing the reasoning of others when they explain which of the strategies they agree with and why (MP3).

• “What do you know about this addition expression?”
• “How can you use the number line to show the value of the addition expression?

• Invite students to share their responses to the first problem.
• “Look at the sums you found. What do you notice about the numbers in each sum? How do they relate to the numbers in the fractions being added?” (The denominator of the sum is 8. The numerator of the sum is the result of adding the numerators of the addends.)
• Select previously identified students to share their responses to the second problem.
• If no students use number lines to make their case, consider sketching or displaying these diagrams.

  

  

  

  Number line. 16 evenly spaced tick marks. First tick mark, 0. Sixth, 1. Eleventh, 2. Arrow, labeled one and two fifths, from first tick mark to eighth tick mark. Arrow, labeled four fifths, from eighth tick mark to point at twelfth tick mark, labeled two and one fifths. 

• Point out that although it is true that , we don’t usually write as the mixed number equivalent to . Because we can make another 1 whole with , or , we instead write .

This optional activity gives students an additional opportunity to practice using number lines to decompose fractions into sums of other fractions and to record the decompositions as equations.

The fractions on the cards (shown here) contain no whole numbers or mixed numbers, but some students may use them to find the second addend (and to avoid counting tick marks on the number line). Some also may choose to label each number line with whole numbers beyond 1 to facilitate their reasoning and equation writing.

• Select previously identified students to share their responses to the first couple of diagrams (or more if time permits). Start with students who used no whole numbers or mixed numbers. Ask them to explain why they chose to write the numbers the way they did.
• Consider discussing the merits and challenges (if any) of expressing the fractions as whole or mixed numbers.