Lesson 11

The purpose of this Number Talk is for students to demonstrate strategies and understandings they have for adding and subtracting a fraction and a whole number. For these problems, students do not need to focus on a common denominator as the numbers either have the same denominator or one of the numbers in the sum is a whole number. Their strategies for thinking about the sums and differences will be helpful throughout the lesson as they calculate more complex differences involving mixed numbers. 

• Display one problem.
• “Hagan una señal cuando tengan una respuesta y puedan explicar cómo la obtuvieron” // “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

• Record answers and strategy.
• Keep problems and work displayed.
• Repeat with each problem.

• “¿Cómo encontraron el valor de \(1\frac{5}{8} + \frac{6}{8}\)?” // “How did you find the value of \(1\frac{5}{8} + \frac{6}{8}\)?” (I made a fraction from the mixed number and then added the numerators. I added on to get 2 and then added the rest of the eighths.)

The purpose of this activity is for students to subtract a fraction or mixed number from a mixed number. There are multiple strategies available and the differences are selected in order to highlight these strategies:

• subtracting the whole number and fraction parts of the numbers separately 
• rewriting the mixed number to facilitate subtraction
• adding on to find the difference

In each case, students will need to choose a common denominator in their calculation. Students first identify expressions that are equivalent to the mixed number that appears in all of the differences they calculate. These expressions are deliberately chosen to support the listed techniques to find the value of the subtraction expressions. The goal of the activity synthesis is to compare and connect several different strategies and consider the benefits and challenges of each strategy.

When students adapt their subtraction strategy to the numbers, they look for and make use of structure (MP7).

This activity uses MLR7 Compare and Connect. Advances: Conversing.

• Groups of 2

• 10 minutes: independent or group work 
• monitor for students who:
  • add on to find the value of \(3\frac{5}{8}-1\frac{15}{16}\)
  • use an equivalent expression that has a fraction greater than 1, such as \(2\frac{13}{8}\), to find the value of \(3\frac{5}{8}-1\frac{12}{16}\)

• “Creen una presentación visual que muestre cómo pensaron al encontrar el valor de la expresión \(3\frac{5}{8}-1\frac{15}{16}\). Incluyan detalles, como notas, diagramas o dibujos, para ayudar a los demás a entender sus ideas” // “Create a visual display that shows your thinking about \(3\frac{5}{8}-1\frac{15}{16}\). You may want to include details such as notes, diagrams or drawings to help others understand your thinking.”
• 2 minutes: independent or group work
• 5 minutes: gallery walk 

• “¿En qué se parecen y en qué son diferentes estas estrategias?” // “What is the same and what is different between the strategies?” (Some people added on or used a number line. Some people changed the mixed number to a whole number plus a fraction greater than one. Some people changed the mixed number to a fraction. Some people got different, equivalent answers.)
• Display: \(3\frac{5}{8}=2\frac{13}{8}\)
• “¿Cómo sabemos que esta ecuación es verdadera?” // “How do we know this is true?” (There are 8 eighths in 1 so I can break up the 3 as 2 and 1 and put the 1 with the \(\frac{5}{8}\) to make \(\frac{13}{8}\).)
• Invite previously selected students to share how they found the value of \(3\frac{5}{8} - 1\frac{12}{16}\).
• “¿Cómo les ayudó reescribir \(3\frac{5}{8}\) como \(2\frac{13}{8}\) en este cálculo?” // “How was rewriting \(3\frac{5}{8}\) as \(2\frac{13}{8}\) helpful in this calculation?” (I could take 1 from 2 and take \(\frac{12}{16}\) from \(\frac{13}{8}\) after rewriting it as \(\frac{6}{8}\).)
• Invite previously selected students to share how they found the value of \(3\frac{5}{8}-1\frac{15}{16}\).
• “¿Por qué decidieron usar esa estrategia?” // “Why did you decide to use that strategy?” (I noticed that \(\frac{15}{16}\) was really close to 1 so it was easy for me to count up.) 
• “¿Cómo les ayudó reescribir \(3\frac{5}{8}\) como \(3\frac{10}{16}\) en este cálculo?” // “How was rewriting \(3\frac{5}{8}\) as \(3\frac{10}{16}\) helpful in this calculation?” (I needed to add a sixteenth to get to 2 and it’s easy to combine sixteenths and sixteenths.)
• “En la siguiente actividad, vamos a encontrar los valores de más diferencias de números mixtos y fracciones” // “We are going to find the values of more differences of mixed numbers and fractions in the next activity.”
• Keep displays available for students to refer to during activity 2.

The purpose of this activity is for students to find the value of differences of mixed numbers. The numbers are chosen to encourage a variety of strategies that were highlighted in the previous activity. Students should be encouraged to find the differences in a way that makes sense to them. This may mean choosing a different strategy depending on the problem but it could also mean writing each difference as a difference of fractions and then finding a common denominator. 

• Groups of 2

• 5 minutes: independent think time
• 5 minutes: small-group work time
• Monitor for students who:
  • add on to find the value of \(9\frac{1}{8}-8\frac{8}{9}\)
  • rewrite \(\frac{10}{4}\) as 2\(\frac{1}{2}\) to find the value of 3\(\frac{1}{2}-\frac{10}{4}\)
  • use an equivalent expression with a fraction greater than one, such as \(3\frac{24}{15}-1\frac{10}{15}\), to find the value of 4\(\frac{3}{5}-1\)\(\frac{2}{3}\)

If students do not have a strategy to find the value of 4\(\frac{3}{5}-1\)\(\frac{2}{3}\), write some expressions that are equivalent to \(4\frac{3}{5}\), such as \(4\frac{9}{15}\) and \(3\frac{24}{15}\) and ask, “¿Cómo sabes que estas expresiones son equivalentes?” // “How do you know these expressions are equivalent?”

• Ask previously selected students to share in the given order.
• “¿Por qué decidieron sumarle algo a \(8\frac{8}{9}\)?” // “Why did you decide to add on to \(8\frac{8}{9}\)?” (\(\frac{8}{9}\) is really close to 1.)
• “¿También pueden restar en 2 pasos para encontrar el valor de \(9\frac{1}{8} - 8\frac{8}{9}\)?” // “Can you also subtract in 2 steps to find the value of \(9\frac{1}{8} - 8\frac{8}{9}\)?” (Yes, I can subtract \(\frac{1}{8}\) from \(9\frac{1}{8}\) to get 9 and then take away \(\frac{1}{9}\) more to get \(8\frac{8}{9}\). That’s the same idea as adding on.)
• “¿Por qué decidieron usar \(2\frac{1}{2}\) en vez de \(\frac{10}{4}\)?” // “Why did you decide to use \(2\frac{1}{2}\) instead of \(\frac{10}{4}\)?” (It is easy to subtract \(3\frac{1}{2}-2\frac{1}{2}\).)