Lesson 11

The purpose of this warm-up is to elicit the idea that multiplication and division can both be used to represent partial quotients. Students compare two representations, both familiar from grade 4, of partial quotients. This will be helpful when students record partial quotients with division expressions later in the lesson.

• Groups of 2
• Display the image.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discutan con su compañero lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• “¿En qué se parecen estas estrategias a la manera en la que encontraron los cocientes en la lección anterior? ¿En qué se diferencian?” // “How are these strategies the same and different as the way you found quotients in the previous lesson?” (They use multiplication and division. They break the problem into smaller parts with numbers that are easier to calculate.)
• “La multiplicación nos puede ayudar a pensar en cocientes parciales” // “Multiplication can help us think about partial quotients.”

The purpose of this activity is to see that some equivalent ways of rewriting a division expression are more helpful than others for finding its value. In particular expressions whose value can be calculated mentally are most helpful. This supports students when they choose their own ways of breaking up the dividend when they use an algorithm that uses partial products in future lessons.

When they identify expressions to find the value of a quotient, students notice that dividends that are readily identifiable as multiples of 14 are most useful for finding the value of the quotient (MP7). 

This activity uses MLR2 Collect and Display. Advances: Conversing, Reading, Writing.

• Create a set of cards from the blackline master for each group of 2.

• Groups of 2
• Display cards for the activity.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?” (They all have a 14 in them. Some of the expressions are repeated. All of them show a number being divided by 14. A lot of the expressions show multiples of 10. A lot of the expressions show multiples of 14. I wonder what we are going to do with these expressions. I wonder why they all divide a number by 14?)
• “Van a escoger expresiones que tengan una suma que sea igual a \(308 \div 14\)” // “You are going to choose expressions that represent a sum that is equal to \(308 \div 14\).”
• “Quédense con los grupos de expresiones que escogieron” // “Keep each set of expressions that you choose.”

• 3–5 minutes: partner work time
• As students play the game, circulate, listen for, and collect the language students use to explain how they know the cards they chose represent a sum that is equal to \(308 \div 14\). Listen for: add, plus, divide, dividend, quotient, equals, “the same as”, and groups of.
• Record students’ words and phrases on a visual display and update it throughout the lesson.
• “Escojan un grupo de expresiones y úsenlo para encontrar el valor de \(308 \div 14\)” // “Choose one set of expressions and use it to find the value of \(308 \div 14\).”
• 3–5 minutes: independent work time
• Monitor for students who use these different combinations to find the value of \(308 \div 14\):
  • \(280 \div 14\), \(28 \div 14\)
  • \(140 \div 14\), \(140 \div 14\), \(28 \div 14\)

If students do not recognize which expressions created by the cards would be helpful to find the value of the quotient \(308 \div 14\), ask, “¿Cuál de estas expresiones tiene un valor que es un número entero? ¿Cómo puedes usar la expresión para encontrar el valor de \(308 \div 14\)?” // “Which of these expressions have a value that is a whole number? How could you use the expression to find the value of \(308 \div 14\)?”

• Refer to the visual display of the words students used during the activity.
• “Estas fueron las palabras que ustedes usaron para explicar cómo supieron que algunas tarjetas representan una suma igual a \(308 \div 14\)” // “These are the words you used to explain how you knew the cards you chose represent a sum that is equal to \(308 \div 14\).”
• “¿Qué otras palabras o frases deberíamos incluir en nuestra presentación?” // “Are there any other words or phrases that are important to include on our display?”
• As students share responses, update the display, by adding (or replacing) language, diagrams, or annotations.
• Remind students to borrow language from the display as needed.
• Display:
  \(308 \div 14\)
• Ask previously selected students to share.
• “¿Por qué escogieron ese grupo de expresiones para encontrar el valor de \(308 \div 14\)?” // “Why did you choose that set of expressions to find the value of \(308 \div 14\)?” (I could find the values of the expressions mentally.)
• Display:
  \(280 \div 14\)
• “¿Por qué nos puede ayudar empezar con la expresión \(280 \div 14\) ?” // “Why is \(280 \div 14\) a helpful expression to start with?” (\(280 \div 14\) is a helpful expression to start with because \(20 \times 14 = 280\) and 280 is close to 308.)
• “¿Cómo sabemos que \(28 \div 14\) es la expresión que corresponde a \(280 \div 14\)?” // “How do we know that \(28 \div 14\) is the expression that matches \(280 \div 14\)?” (\(280 + 28 = 308\))

The purpose of this activity is for students to consider efficient ways to use the partial quotients strategy. The strategy that is most efficient for a student will depend on which products or quotients they can find mentally. Encourage students to try to use larger partial quotients as they work, but to try to find partial quotients which they can calculate mentally.

• Groups of 2
• “En cada problema, escojan con cuál expresión de división van a empezar a encontrar el cociente. Luego, usen la expresión que escogieron para encontrar el valor del cociente” // “For these problems, choose one of the division expressions to use to begin finding the quotient. Then use the expressions you chose to find the value of each quotient."

• 6-8 minutes: independent work time
• 2-3 minutes: partner discussion
• Monitor for students who 
  • use different expressions for \(992\div31\).
  • wrote about changing their mind in the last question.

• Invite selected students to share their reasoning for \(992\div31\) and explain why they started with their expression.
• Invite students to share how they changed their mind about which expressions to use during the activity.