Lesson 8

The purpose of an Estimation Exploration is to practice estimating a reasonable answer based on experience and known information. Students can identify fractions represented by the shaded portions in tape diagrams in which unit or non-unit fractions are marked. To estimate the shaded parts in an unmarked tape, students may rely on the size of benchmark fractions—\(\frac{1}{2}\), \(\frac{1}{3}\), or \(\frac{1}{4}\)—and partition those parts mentally until it approximates the size of the shaded ports. They may also estimate how many copies of the shaded part could fit in the entire diagram.

• Groups of 2
• Display the image.
• “¿Qué estimación sería muy alta?, ¿muy baja?, ¿razonable?” // “What is an estimate that’s too high? Too low? About right?”
• 1 minute: quiet think time

• “Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

Consider asking:

• “¿La estimación de alguien es menor que \(\frac{1}{20}\)? ¿La estimación de alguien es mayor que \(\frac{1}{4}\)?” //  ​“Is anyone’s estimate less than \(\frac{1}{20}\)? Is anyone’s estimate greater than \(\frac{1}{4}\)?”
• “A partir de esta discusión, ¿alguien quiere ajustar su estimación?” // “Based on this discussion does anyone want to revise their estimate?”

In this activity, students examine number lines that have been partitioned into smaller and smaller parts. They see that this strategy can be used to generate many equivalent fractions and to verify if two fractions are equivalent.

Students encounter fractions with 5, 10, 15, and 20 for the denominator. Working with multiples of a number (in this case, 5) allows students to notice structure in how the partitioning of a part on a number line relates to equivalent fractions (MP7). (Students will not be assessed on fractions with denominator 15 or 20.)

If desired and logistically feasible, consider enacting Andre’s reasoning with one or more human number lines.

• Place a strip of masking tape or painter’s tape, at least 25 feet long, on the floor of the classroom or a hallway. 
• Ask a student to stand on each end of the tape. They represent 0 and 1.
• “¿Cómo podemos partir esta recta en quintos?” // “How can we partition this line into fifths?” (Position 4 students on the tape, spaced apart equally between 0 and 1.)
• Ask each student on the line to say the number they represent (0, \(\frac{1}{5}\), \(\frac{2}{5}\), \(\frac{3}{5}\), \(\frac{4}{5}\), 1). Give each student a sign with their fraction. (Consider distinguishing the sign for \(\frac{1}{5}\) with a different color.)

  

  

• Invite 5 students to join the line, each person standing exactly in the middle of two others. 
• “¿Qué fracción representan ahora?” // “What fraction do you represent now?” Ask every student to say the number they represent now (0, \(\frac{1}{10}\), \(\frac{2}{10}\), \(\frac{3}{10}\), . . .  1,). Give the student representing \(\frac{1}{5}\) another sign showing \(\frac{2}{10}\).
• Repeat a couple more times. Each time:
  • Ask 5 additional students to each join a space between two students representing fifths. (The students representing fifths should stay in place, but, to maintain equal intervals, those representing smaller fractional parts may need to shift when others join the line.)
  • Ask students to say aloud the number they represent.
  • Give the student representing \(\frac{1}{5}\) another sign showing a new fraction they represent.
• “¿Pueden explicar por qué el estudiante que representa \(\frac{1}{5}\) también representa \(\frac{2}{10}\), \(\frac{3}{15}\) y \(\frac{4}{20}\)?” //    “Can you explain why the student representing \(\frac{1}{5}\) also ends up representing \(\frac{2}{10}\), \(\frac{3}{15}\), and \(\frac{4}{20}\)?”

• Consider creating a human number line by placing a strip of masking tape or painter’s tape, at least 25 feet long, on the floor of the classroom or a hallway.

• Groups of 2
• Read the opening paragraphs of the activity as a class.
• If possible, consider creating a human number line as outlined in the Activity Narrative.

• “Piensen en el primer problema, en silencio, durante un minuto. Después, discutan con un compañero cómo pensaron. Trabajen juntos en el segundo y en el tercer problema” // “Think quietly for a minute about the first problem. Then, discuss your thinking with a partner and work on the second and third problems together.”
• 1 minute: quiet think time
• 7–8 minutes: partner work time
• Monitor for students who:
  • extend each number line past 1 whole to show \(\frac{6}{5}\), \(\frac{12}{10}\), \(\frac{18}{15}\), and \(\frac{24}{20}\)
  • skip-count by the number in each numerator 6 times—by 2 to get to \(\frac{12}{10}\), by 3 to get to \(\frac{18}{15}\), and by 4 to get to \(\frac{24}{20}\).
  • multiply the numerator (6) and the denominator (5) by 2, 3, and 4 to get \(\frac{12}{10}\), \(\frac{18}{15}\), and \(\frac{24}{20}\).

• Invite a student to share their explanation of Andre’s strategy (first problem). Ask the class if they agree with the explanation or if they would amend it or explain it differently.
• Ask another student to show how the number lines could help us see if two fractions are equivalent (second problem).
• Select previously identified students to share their reasoning for the last problem. Record their reasoning.
• Solicit some initial impressions on how the strategies are alike and different, but save further comparisons for future lessons.

In this activity, students continue to use the idea of partitioning a number line into smaller increments to reason about and generate equivalent fractions. Through repeated reasoning, students begin to see regularity in how the process of decomposing parts on a number line produces the numbers in the equivalent fractions (MP8). The task encourages students to think of the relationship between one denominator and the other in terms of factors or multiples (even if they don’t use those terms which connect to work in a previous unit). 

Partitioning a number line into smaller parts becomes increasingly inconvenient when the denominator gets larger. As students begin to think about the relationship between tenths and hundredths, they see some practical limitations to using a number line to find equivalent fractions and are prompted to generalize the process of partitioning. (Students are not expected to draw a full number line with 100 parts.)

• Groups of 2
• Read the first problem as a class. Ask students to think quietly for a moment about whether what Priya wants to do can be done.
• 30 seconds: quiet think time
• 1 minute: partner discussion

• “Tómense de 7 a 8 minutos en silencio para trabajar en la actividad. Después, discutan sus respuestas con su compañero” // “Take about 7–8 quiet minutes to work on the task. Afterwards, discuss your responses with your partner.”
• 7–8 minutes: independent work time
• 2–3 minutes: partner discussion
• For the first problem, monitor for students who partition the lines by:
  • guessing and checking
  • reasoning multiplicatively (3 times what number gives 9, 10, or 12?) or in terms of multiples (Is 9, 10, or 12 a multiple of 3?)
  • reasoning in terms of division (9 divided by 3 is what number?) or in terms of factors (Is 3 a factor of 9, 10, or 12?)
• For the second and third problem, monitor for students who find equivalent fractions for \(\frac{1}{10}\) by:
  • partitioning the number lines into smaller increments quantifying the new number of parts
  • finding multiples of 1 and 10, and using this strategy to write an equivalent fraction with denominator 100

If students conclude that the answer to the last question is “no, it can’t be done” because they find it impractical to partition each tenth on the number line into so many parts, ask them to visualize the line and think about how they’d partition it. Consider asking: “En la recta, ¿en cuántas partes hay que partir cada sección de un décimo para obtener centésimos? ¿Cómo lo sabes?” // “Into how many parts should each one-tenth section on the line be split to get hundredths? How do you know?”

• Invite previously identified students to share their strategies for answering the first two problems.
• If not done by students in their explanations, consider asking students to revoice their reasoning in terms of factors and multiples.
• See lesson synthesis.