Lesson 7

The purpose of this Notice and Wonder is for students to share what they know about scales and to initiate a discussion about rounding. The weights on the scale total 12.32 ounces, but the scale reads 12.3 ounces. There are different possible explanations for this discrepancy. For example, the scale might be inaccurate. Or the scale might only give readings in tenths of an ounce. In the discussion, students consider the idea that the value shown on the scale is not always exact. It may just show the closest value that it is capable of reading, which is the nearest tenth of an ounce in this case (MP6).

• 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.

• “¿Qué observan acerca de los pesos que están en la balanza y lo que marca la balanza?” // “What do you notice about the weights on the scale and the reading of the scale?” (They aren’t the same. The weights are 12.32 ounces and the scale says 12.3 ounces.)
• “¿Por qué creen que la balanza y los pesos no coinciden?” // “Why do you think the scale and the weights don’t agree?” (The scale could be wrong.)
• “¿Qué dirían si supiéramos que la balanza solo muestra décimas de una onza y no puede mostrar centésimas de una onza?” // “What if the scale only shows tenths of an ounce, and it can’t show hundredths of an ounce?” (The value is still not accurate but it’s the best the scale can do.)
• “En la lección de hoy, vamos a trabajar con balanzas que muestran distintas cantidades de posiciones decimales y vamos a entender cómo influye esto en lo que muestran” // “In today’s lesson we will look at scales that show different numbers of decimals and see how that influences what they show.”

The purpose of this activity is for students to round to the nearest tenth and hundredth. Students have rounded in earlier grades but this is the first time they round to tenths or hundredths. This is a direct extension of rounding to the nearest ten, hundred, thousand, and other whole number values. Locating the numbers on the number line will recall this earlier work.

The launch introduces the context of a doubloon, a major currency in Portugal and Spain in the seventeenth, eighteenth, and nineteenth centuries. Students round the weight of a doubloon to the nearest tenth and hundredth of a gram. In both cases, older doubloons are still heavier after rounding. Rounded to the nearest gram, however, they are the same. This is important from a practical perspective because it is easier to measure a weight to the nearest gram than it is to the nearest tenth of a gram or hundredth of a gram. It is also important because the numbers 6.9 and even 6.87 are not as complex as 6.867, and having fewer digits helps visualize the value more quickly.

• Groups of 2
• Display the image.
• “Este es un doblón. ¿Qué observan? ¿Qué se preguntan?” // “This is a doubloon. What do you notice? What do you wonder?” (I notice 1798. I wonder if that’s the year it was made. I wonder if people actually used these as money. I notice there are words in a foreign language. I wonder what they say. I wonder what language it is.)
• “Los doblones fueron una moneda importante en Portugal y en España durante los siglos diecisiete, dieciocho y diecinueve. Hoy vamos a estudiar los pesos de unos doblones” // “Doubloons were a major currency in Portugal and Spain in the seventeenth, eighteenth, and nineteenth centuries. Today, we are going to study the weights of doubloons.”

• “Tómense un par de minutos para trabajar en el primer problema” // “Take a couple of minutes to work on the first problem.”
• 2 minutes: partner work time
• “¿Qué significa medir un peso a la décima de un gramo más cercana?” // “What does it mean to measure a weight to the nearest tenth of a gram?” (It means you find the closest tenth of a gram to the weight.)
• “¿Cómo decidieron cuál tipo de doblón estaba en la balanza?” // “How did you decide which kind of doubloon was on the scale?” (The older ones are heavier and the tenth they are closest to is 6.9 not 6.8.)
• “El peso del doblón está redondeado a la décima de un gramo más cercana” // “The weight of the doubloon is rounded to the nearest tenth of a gram.”
• “Ahora completen el resto de los problemas” // “Now complete the rest of the problems.”
• 6–8 minutes: partner work time

• Invite students to share their answers for rounding the weights of the doubloons.
• “¿Cómo redondearon el peso del doblón más antiguo a la décima más cercana? ¿Y a la centésima más cercana?” // “How did you round the weight of the older doubloon to the nearest tenth? What about the nearest hundredth?” (It was closer to 6.9 than to 6.8, so I rounded to 6.9 to the nearest tenth. It was closer to 6.87 than to 6.86 so I rounded to 6.87 to the nearest hundredth.)
• “¿Cómo les ayudaron las rectas numéricas a redondear los números?” // “How did the number lines help you round the numbers?” (I could see on the number line which tenth or which hundredth the number was closest to and then knew to round it there.)

The purpose of this activity is for students to examine numbers in different situations and decide if they are exact or approximate. In most cases, there is no definitive answer but it is likely that the numbers are approximate or rounded. Two important reasons for using rounded measurements are

• it is easier to measure a length, for example, to the nearest kilometer or meter than to the nearest centimeter or millimeter
• round numbers can communicate the size of a quantity more clearly than exact numbers

For example, we might say that the classroom is 15 meters long. That number is probably not exact but it gives a good idea of the length. It is possible that 14.63 meters is more accurate but it is also cumbersome. The goal of the activity synthesis is to discuss some of the ways you can tell if a measurement is exact or rounded and what that tells you about the measurement. 

• Groups of 2

• 2 minutes: independent work time
• 6–8 minutes: partner work time
• Monitor for students who
  • recognize that the population of Los Angeles is approximate
  • think that the time is exact
  • think that the time is an estimate

• Invite students to share their reasoning for the population of Los Angeles.
• “¿Por qué el caso de la población de Los Ángeles es diferente al caso de los lápices?” // “Why is the population of Los Angeles different than the pencils?” (There are so many people in Los Angeles that you can’t count them all. The pencils I can count one by one and be sure of my answer.)
• Invite students to share their reasoning for the time of 19.87 seconds.
• “¿Por qué creen que la medida podría ser exacta?” // “Why do you think the measurement could be exact?” (It’s really precise. If it said 20 seconds then that sounds like an estimate but 19.87 seconds looks too precise to be an estimate.)
• “¿Por qué creen que la medida podría ser aproximada?” // “Why do you think the measurement could be approximate?” (It’s like the doubloons. There could be some thousandths of a second and then 19.87 seconds is just an estimate.)