Lesson 23

The purpose of this True or False is to elicit strategies and understandings students have for finding differences between two numbers. These understandings help students build fluency in addition and subtraction, while preparing them to think about distances between two points.

Students may use estimation or place value understanding to solve the problems (MP7).

• 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 saber si cada ecuación es verdadera sin hacer ningún cálculo?” // “How can we tell if each equation is true without calculating?”
• “Podríamos usar el algoritmo estándar para encontrar todas las diferencias y luego decidir si la ecuación es verdadera o falsa. ¿Sería una buena idea? ¿Por qué sí o por qué no?” // “We could use the standard algorithm to find each difference and then decide if the equation is true or false. Would that be a good idea? Why or why not?”

This activity prompts students to interpret and represent situations about distances and use multiple operations to solve problems. In problem 3, a piece of information is withheld. Students will need to make sense of what’s missing and find out that information before the question could be answered. Throughout the activity, students reason abstractly and quantitatively as they interpret the diagram and use the information to solve problems (MP2).

• Groups of 2
• Give students access to grid paper.
• Read opening paragraphs as a class. Display the diagram.
• “¿Qué preguntas matemáticas pueden hacer sobre esta situación?” // “What mathematical questions can you ask about this situation?”
• 1–2 minutes: quiet think time
• 1–2 minutes: partners compare questions
• Share and record responses.

• 6–8 minutes: independent work time
• 3–4 minutes: partner discussion
• When requested, tell students that 1 mile is equal to 5,280 feet.
• Monitor for:
  • the different equations students write for the first question (see examples in Student Responses)
  • the assumptions students make when answering the last question (as noted in the narrative)
  • the different ways of reasoning (as noted in the narrative)

• Consider collecting and displaying the different expressions students wrote for the first question and discussing what they tell us about the way the writer reasoned about the problems. For example:
  • \(2 \times (157 + 134 + 162 + 275)\) suggests that the writer found the distance one way then doubled it.
  • \((2 \times 157) + (2 \times 134) + (2 \times 162) + (2 \times 275)\) suggests that the distances between classes were doubled first before being added.
• “¿Cómo averiguaron si la prima de Mai recorría 2 millas cada semana? ¿Qué hicieron primero?, ¿qué hicieron después?” // “How did you find out if Mai’s cousin traveled 2 miles each week? What did you do first? What did you do next?”
• Select students to share their responses and reasoning.
• Highlight how the strategies are alike and different.

This activity gives students another opportunity to use multiple operations to model the quantities in a situation and to solve problems involving large numbers. Students interpret the quantities in context, reason about them abstractly as they perform computations, and then return to the context to interpret the results. As they do so, students are reasoning quantitatively and abstractly (MP2).

Students may choose to answer the first problem by dividing a five-digit number by a one-digit divisor. Though finding a quotient of a five-digit dividend is not an expectation, this particular number ends in a 0. Students can use the division strategies they’ve learned so far and what they know about the structure of numbers in base ten to find the quotient (MP7).

• Groups of 2
• Give students access to grid paper.
• “A veces, las personas compiten para ver quién puede dar la mayor cantidad de pasos en un día, en una semana o en un mes” // “Sometimes people compete to see who can walk the most steps in a day, week, or month.”
• “Estas competencias se llaman retos” // “These competitions are called challenges.”
• “Esta actividad se trata de un reto físico sobre la cantidad de pasos” // “This activity involves a fitness challenge involving steps.”

• 6–8 minutes: independent work time
• Monitor for the different ways students reason about each question.

• Invite students to share their responses and reasoning.
• Focus the discussion on how students found the number of steps Han took each day and how they found the last number in the table.
• “¿Cómo saben si su respuesta a la primera pregunta es correcta?” // “How would you know if your answer to the first question is correct?” (Multiply 4,650 by 7 to see if it equals 32,550.)
• “¿Cómo saben si su respuesta a la pregunta 3 es razonable?” // “How would you know if your answer to problem 3 is reasonable?” (Add it to the other three numbers and see if the sum is 120,000.)