Lesson 4

The purpose of this True or False is for students to demonstrate the strategies and understandings they have for dividing by powers of 10. In this lesson they will convert from a smaller metric unit to a larger unit which means dividing by an appropriate power of 10. The problems here are selected so that students can begin to see how the value of the digits in a number change when that number is divided by 100 or 1,000.

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

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

• “¿Cómo encontraron el valor de \(1,\!328 \div 1,\!000\)?” // “How did you find the value of \(1,\!328 \div 1,\!000\)?” (I thought of it as 1,328 thousandths and then wrote that as a decimal.)

The purpose of this activity is for students to convert from a smaller metric length unit to a larger metric length unit. In this activity, students divide a three- or four-digit number by 100. The measurements in centimeters are whole numbers, but the converted measurements in meters are decimals. To find the value of the quotients, students may reason about the meaning of each place value and how that changes when the value is divided by 100. They may also reason that each place value has \(\frac{1}{10}\) the value of the place to its left so shifting two places to the right gives a value of \(\frac{1}{100}\) the value for each place (MP2, MP7).

• Groups of 2
• Display the table shown in the student workbook.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?” (Who are those people? What is a javelin? What is a shot put? What is the long jump? Some of the numbers are close. Why are the long jump numbers so much smaller?)
• Record responses for all to see.
• “Esta tabla muestra los resultados de unas pruebas de atletismo” // “This table shows the results from track and field events.”

• 2 minutes: quiet think time
• 10 minutes: partner work time
• Monitor for students who observe that when they convert the whole number distances in centimeters to meters they get decimals to the hundredth.

If students do not have a strategy, ask:

• “¿Cuántos centímetros hay en un metro? ¿Cuántos metros es 200 centímetros?” // “How many centimeters are in a meter? How many meters is 200 centimeters?”
• “¿Qué expresión puedes escribir para representar esta relación?” // “What expression can you write to represent this relationship?”

• Display:
  \(727 \div 100 = 7.27\)
  \(4,\!566 \div 100 = 45.66\)
  \(1,\!580 \div 100 = 15.80\)
• “¿Qué patrones observan?” // “What patterns do you notice?” (The measurements in centimeters are whole numbers but the measurements in meters are in hundredths. All the digits in the numbers are the same but they shifted 2 places to the right. So the 7 hundreds in centimeters is 7 ones in meters.)
• “¿Por qué podemos encontrar cuántos metros hay en los centímetros dados si dividimos entre 100?” // “Why does dividing by 100 represent how many meters are in the given centimeters?” (I know there are 100 centimeters in each meter so each centimeter is \(\frac{1}{100}\) of a meter.)
• “¿Cómo encontraron el resultado de 727 dividido entre 100?” // “How did you find the result of 727 divided by 100?” (I went place by place. 7 hundreds divided by 100 is 7 ones. 2 tens divided by 100 is 2 tenths. 7 ones divided by 100 is 7 hundredths.)
• Invite students to share which unit of measurement they found most informative for these distances and why.

This activity continues the work of the previous activity as students convert from a smaller metric length unit to a larger one. Students are given several different conversions and multiple numbers for each set of conversions. They observe that dividing by 100 shifts each digit two places to the right while dividing by 1,000 shifts each digit three places to the right when they convert different units. This allows students to solidify their understanding that dividing by powers of 10 shifts each digit one place to the right for every power of 10 in the divisor (MP8). 

This activity uses MLR1 Stronger and Clearer Each Time. Advances: Reading, Writing.

• Groups of 2
• “Trabajen con su compañero en los tres primeros problemas” // “Work with your partner on the first three problems.”

• 5–7 minutes: partner work time
• “Ahora van a trabajar individualmente para mostrar o explicar si piensan que la estrategia de Tyler siempre va a funcionar” // “Now, you are going to work independently to show or explain whether you think Tyler’s strategy will always work.”
• 3–5 minutes: independent work time

• “Compartan con su compañero su respuesta a si la estrategia de Tyler siempre va a funcionar o no. Por turnos, uno habla y el otro escucha. Si es su turno de hablar, compartan sus ideas y lo que han escrito hasta ese momento. Si es su turno de escuchar, hagan preguntas y comentarios que ayuden a su compañero a mejorar su trabajo” // “Share with your partner your response as to whether or not Tyler’s strategy will always work. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
• 3–5 minutes: structured partner discussion
• Repeat with 1–2 different partners.
• If needed, display question starters and prompts for feedback.
  • “¿Puedes dar un ejemplo que ayude a mostrar . . . ?” // “Can you give an example to help show . . . ?”
  • “¿Puedes agregar un diagrama, tabla o representación para mostrar . . . ?” // “Can you add a diagram, table, or representation to show . . . ?”
• “Ajusten su borrador inicial basándose en los comentarios que les hicieron sus compañeros” // “Revise your initial draft based on the feedback you got from your partners.”
• 2–3 minutes: independent work time