Lesson 17

The purpose of an Estimation Exploration is to practice the skill of estimating a reasonable answer based on experience and known information. In this warm-up, students apply what they know about fractions to estimate the length of an insect that is less than 1 inch.

• 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 alguno es menor que _____? ¿La estimación de alguno es mayor que _____?” // “Is anyone’s estimate less than _____? Is anyone’s estimate greater than _____?”
  • “Teniendo en cuenta esta discusión, ¿alguien quiere ajustar su estimación?” // “Based on this discussion, does anyone want to revise their estimate?”

The purpose of this activity is for students to compare two numbers in context, to explain or show their reasoning, and record the results of the comparisons with the symbols >, =, or < (MP2). The numbers may be fractions with the same numerator or the same denominator, or a fraction and a whole number.

Students are likely to generate different comparison statements for the same situation. For example, they may write \(\frac{5}{8} > \frac{3}{8}\) or \(\frac{3}{8}<\frac{5}{8}\) to represent \(\frac{5}{8}\) being the greater fraction. During synthesis, discuss how both statements capture the comparison and are valid.

• Groups of 2
• “Usemos lo que aprendimos acerca de cómo comparar fracciones y cómo reconocer fracciones equivalentes para resolver problemas sobre longitudes” //  “Let’s use what we learned about comparing fractions and recognizing equivalent fractions to solve problems about lengths.”

• “Resuelvan los problemas individualmente. En cada uno, asegúrense de mostrar cómo pensaron y de escribir una afirmación de comparación” // “Work independently to solve the problems. For each one, be sure to show your thinking and to write a comparison statement.”
• 6–8 minutes: independent work time
• “Comparen sus respuestas y su razonamiento con su compañero” // “Share your responses and reasoning with your partner.”
• 2–3 minutes: partner discussion
• Monitor for:
  • different representations or reasoning strategies used for the same problem, such as diagrams, fraction strips, number lines, or explanations in words
  • different statements written for the same problem, such as \(4=\frac{12}{3}\) and \(\frac{12}{3}=4\), or \(\frac{2}{3} > \frac{2}{8}\) and \(\frac{2}{8} < \frac{2}{3}\)

• Select 2–3 students to share their reasoning strategies or representations for at least one of the situations.
• “¿Cómo usaron lo que han aprendido en lecciones anteriores para comparar las fracciones?” // “How did you use what you’ve learned in earlier lessons to compare fractions?” (If the fractions had the same numerator, I thought about the size of the denominators. If they had the same denominator, I compared the numerators.)
• Select students who wrote different but equally valid comparison statements (for instance, \(\frac{2}{8}<\frac{2}{3}\) and \(\frac{2}{3} > \frac{2}{8}\)) to share. 
• Discuss how to read each statement and ask students whether both accurately represent the comparison.
• Emphasize that we can write comparison statements in more than one way, but we need to check that the statements make sense given the numbers we write and the symbols we use. 

The purpose of this activity is for students to generalize what they have learned about comparing fractions to complete comparison statements and to generate new ones, using the symbols <, >, or =. Students first consider all numbers that could make an incomplete comparison statement true. Then, they find a fraction less than, greater than, and equivalent to a given fraction and write statements to record the comparisons. As in the previous activity, students see that there are different ways to record the same comparison of two numbers.

• Groups of 2
• “Ahora que hemos practicado cómo comparar fracciones, inventemos fracciones que sean mayores o menores que una fracción dada o equivalentes a ella” // “Now that we have practiced comparing fractions, let's come up with fractions that are greater than, less than, or equivalent to a given fraction.”

• “Noah estaba trabajando con fracciones cuando se le derramó un jugo. Ahora no logra saber cuáles eran algunos números. Ayúdenlo a averiguar qué estaba escrito antes de que se derramara el jugo” //  “Noah was working with fractions when some juice spilled. Now he can’t tell what some numbers were. Help him figure out what was written before the juice was spilled.”
• 5 minutes: partner work time
• Pause for a discussion and invite students to share the numbers that they think make sense in the first statement (\(\frac{2}{8}< \frac{?}{8}\)).
• Display or write the comparison statements using students’ numbers.
• “¿Todas estas afirmaciones tienen sentido? ¿Cómo lo saben?” // “Do all of these statements make sense? How do you know?”
• “¿Podríamos escribir más afirmaciones?” // “Are there any more statements that we could write?”
• If time permits, repeat with the next two parts.
• “Ahora trabajen individualmente en el último grupo de problemas” // “Now work independently on the last set of problems.”
• 5 minutes: independent work time

If students don’t find more than one number that would make the statements in the first problem true, consider asking:

• “Ya encontraste un número que hace que la afirmación sea verdadera. ¿Cómo encontraste ese número y cómo supiste que hacía que la afirmación fuera verdadera?” // “You found one number that made the statement true. How did you find that number and know that it made the statement true?”
• “¿Cómo puedes usar una estrategia similar para encontrar otro número que haga que la afirmación sea verdadera?” // “How could you use a similar strategy to find another number that would make the statement true?”

• Invite students to share their responses to the last set of problems.
• “¿Cómo encontraron una fracción que fuera menor que la fracción dada (o mayor que la fracción dada o equivalente a ella)?” // “How did you find a fraction that was less than (or greater than or equivalent to) the given fraction?”

The purpose of this activity is for students to use their knowledge of fractions to locate fractions with different denominators on the number line. Students may use a variety of reasoning to locate the fractions, including their knowledge of equivalence, strategies about the same numerator or denominator, or benchmark numbers they are familiar with. The synthesis focuses on the variety of strategies that make sense, and students should be encouraged to use different strategies for different fractions as needed.

Although students have represented fractions on number lines (including those with two different denominators, when reasoning about equivalence), this activity is optional because representing multiple fractions of different denominators on the same number line involves a deeper understanding than required by the standards.

• Groups of 2

• “Individualmente, comiencen a ubicar estas fracciones en la recta numérica” // “Work independently to start placing these fractions on the number line.”
• 3–5 minutes: independent work time
• “Compartan sus estrategias con su compañero y ubiquen juntos las fracciones que les quedan” //  “Share your strategies with your partner and place any fractions you have left together.”
• 5–7 minutes: partner work time
• Monitor for students who:
  • place each fraction separately by partitioning for that single fraction
  • compare the fraction they are placing to others they’ve already placed
  • use equivalent fractions
  • use strategies about same numerators or denominators
  • use benchmarks like whole numbers or halves

• Invite students to share a variety of strategies for placing fractions on the number line.
• Consider asking:
  • “¿Cuáles fracciones fueron más fáciles de ubicar en la recta numérica?” // “Which fractions were easier to place on the number line?”
  • “¿Cuáles fueron más difíciles de ubicar?” // “Which fractions were more difficult?”
  • “¿Alguien usó la misma estrategia, pero la explicaría de otra forma?” // “Did anyone have the same strategy but would explain it differently?”