Lesson 9

The purpose of this True or False is for students to demonstrate strategies they have for using equivalent fractions to add and subtract fractions with different denominators. These mental calculations prepare students for working with more complex common denominators during this lesson. 

• Display one statement.
• “Hagan una señal cuando sepan si la afirmación es verdadera o no, y puedan explicar cómo lo saben” // “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

• Share and record answers and strategy.
• Repeat with each statement.

• “¿Cómo podemos encontrar el valor correcto de \(\frac{3}{4}-\frac{1}{2}\)?” // “How can we find the correct value of \(\frac{3}{4}-\frac{1}{2}\)?” (\(\frac{1}{2}=\frac{2}{4}\) so \(\frac{3}{4}-\frac{2}{4}=\frac{1}{4}\).)

The purpose of this activity is for students to identify equivalent sums of fractions and use them to find the value of sums of fractions with different denominators. In a previous course, students learned to use factors and multiples to generate and identify equivalent fractions. They recall that technique here and then use those equivalent fractions to find sums. This helps reinforce the idea that it is helpful, when adding two fractions, if the fractions have the same denominator while also recalling how to find an equivalent fraction with a different denominator.   
When students identify that equivalent fractions with the same denominator help to find the value of a sum they notice and take advantage of the meaning and structure of fractions (MP7).

• Groups of 2

• 5–8 minutes: independent work time
• 1–2 minutes: partner discussion
• Monitor for students who:
  • use multiplication to explain why the expressions are equivalent. For example, multiply  \(\frac{2 \times 4}{3 \times 4}\) to show why \(\frac{2}{3}=\frac{8}{12}\)
  • use division to explain why the expressions are equivalent. For example, divide \(\frac{10 \div 2}{12 \div 2}\) to show why \(\frac{10}{12}=\frac{5}{6}\)

If a student needs help getting started, suggest they draw 2 different number lines to represent \(\frac{2}{3}\). Then, for each number line, ask, “¿Cómo puedes modificar el diagrama para mostrar \(\frac{8}{12}\)?, ¿y \(\frac{4}{6}\)?” // “How can you adapt the diagram to show \(\frac{8}{12}\)? \(\frac{4}{6}\)?”

• Invite previously selected students to share how they know the expressions \(\frac{8}{12} + \frac{10}{12}\) and \(\frac{4}{6} + \frac{5}{6}\) are equivalent to \(\frac{2}{3} + \frac{10}{12}\).
• “¿Cómo saben que \(\frac{8}{12} + \frac{10}{12} = \frac{2}{3} + \frac{10}{12}\)?” // “How do you know that \(\frac{8}{12} + \frac{10}{12} = \frac{2}{3} + \frac{10}{12}\)?” (I can divide each \(\frac{1}{3}\) into 4 equal parts. Those parts are \(\frac{1}{12}\)s and there are 8 of them.)
• “¿Por qué la expresión \(\frac{8}{12} + \frac{10}{12}\) nos ayuda a encontrar la suma?” // “Why is the expression \(\frac{8}{12} + \frac{10}{12}\) helpful for finding the sum?” (It’s all twelfths. I have 8 of them and 10 more so that's \(\frac{18}{12}\).)
• “¿Cuál expresión escogieron para encontrar la suma?” // “Which expression did you choose to find the sum?” (Sample response: I used \(\frac{4}{6} + \frac{5}{6}\) because the numbers were smaller.)

This activity builds on the previous activity where students saw how equivalent expressions can be a valuable tool to add or subtract fractions. The purpose of this activity is for students to generate an equivalent expression in order to find the value of a difference of fractions. Monitor for students who:

• find equivalent fractions with smaller numerators and denominators than the given fractions
• find equivalent fractions with larger numerators and denominators than the given fractions

• Groups of 2

• 5 minutes: independent work time
• 5 minutes: partner discussion

If students try to use \(1\frac{2}{6}-\frac{3}{6}\) to find the value of \(\frac{16}{12}-\frac{3}{6}\) and do not get the correct value, ask, “¿Cómo puedes usar una recta numérica para representar la expresión \(1\frac{2}{6}-\frac{3}{6}\)?” // “How can you use a number line to represent the expression \(1\frac{2}{6}-\frac{3}{6}\)?”

• Invite previously selected students to share how they found the value of \(\frac{16}{12} - \frac{3}{6}\).
• “¿En qué se parecen las estrategias para encontrar el valor de la expresión?” // “How are the strategies for finding the value of the expression the same?” (They both change one of the fractions to an equivalent fraction so the fractions have the same denominator.)
• “¿En qué son diferentes las estrategias para encontrar el valor de la expresión?” // “How are the strategies for finding the value of the expression different?” (To make the denominator bigger I multiply by a whole number. To make the denominator smaller I divide by a whole number.)
• “¿Por qué es importante que las fracciones tengan el mismo denominador?” // “Why is it important to have the same denominator?” (Then I can add or subtract the number of parts because they are the same size.)

The purpose of this activity is for students to solve a problem that involves finding the difference of fractions. Students may use addition or subtraction to solve the problem. Either way they will need to find a common denominator for the fractions. One of the numbers is a mixed number so students may:

• convert the mixed number to a fraction
• find the difference in steps, adding on or subtracting

When students recognize mathematical features of objects in the real world, they model with mathematics (MP4).

• Groups of 2

• 5 minutes: independent work time
• 1–2 minutes: partner discussion

• Continue to lesson synthesis.