Lesson 16

The purpose of this True or False is for students to demonstrate strategies they have to estimate the size of a product. Students can find the value of \(\frac{4}{5} \times 100\) and thereby solve all of the problems but the exact value is not needed to make the comparisons. Throughout the next several lessons, students will investigate different ways to compare a product like this to one of the factors (100 in this case).

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

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

• Display: \(\frac{4}{5} \times 100 < 100\)
• “¿Cómo sabemos que esto es verdadero sin encontrar el valor de \(\frac{4}{5}\times100\)?” // “How do we know this is true, without finding the value of \(\frac{4}{5}\times100\)?” (Since \(\frac{4}{5}\) is \(\frac{1}{5}\) less than a whole, \(\frac{4}{5} \times 100\) is less than \(1 \times 100\). I know that \(\frac{4}{5} \times 100\) is 80 and that's less than 100.)

The purpose of this activity is for students to compare the size of different products where one factor stays the same, allowing students to focus on the size of the varying factor. Students should be encouraged to use whatever strategies and representations make sense to them. Monitor for students who

• use number lines
• use multiplication to compute the total distances
• reason about the relationship between the fraction of the trail and the total distance without performing any computations

This activity uses MLR2 Collect and Display. Advances: Reading, Speaking.

• Groups of 2

• 3–5 minutes: independent work time
• 3–5 minutes: partner discussion

MLR2 Collect and Display

• Circulate, listen for and collect the language students use to explain their reasoning for each of the problems.
• When students describe the order of the distances that Elena, Noah, and Kiran ran, listen and look for:
  • “_____ del sendero es más largo que _____ del sendero” // “_____ of the trail is longer than _____ of the trail.”
  • “_____ del sendero es más corto que _____ del sendero” // “_____ of the trail is shorter than _____ of the trail.”
  • gestures or diagrams that represent comparisons of the fractions
• When students describe the numbers that could go in the blanks, listen for:
  • “El número tiene que estar entre _____ y _____” // “The number has to be between _____ and _____.”
  • “El número tiene que ser más grande que ______ porque . . .” // “The number has to be larger than ______ because . . . .”
  • “El número tiene que ser más pequeño que ______ porque . . .” // “The number has to be smaller than _____ because . . . .”
  • gestures or diagrams that represent comparisons of the fractions
• Record students’ words and phrases on a visual display and update it throughout the lesson.

If students do not start to solve the problems during the independent work time, draw a diagram to represent the trail and ask students to label the distance that each student ran.

• “¿Qué otras palabras o frases importantes deberíamos incluir en nuestra presentación?” // “Are there any other words or phrases that are important to include on our display?”
• As students share responses, update the display by adding (or replacing) language, diagrams, or annotations.
• Remind students to borrow language from the display as needed.
• Display: \(1\frac{1}{4} \times 5\) miles
• “¿Quién corrió la distancia que está representada por esta expresión?” // “Whose distance does this expression represent?” (Kiran)
• “¿Qué expresión de multiplicación representa el número de millas que corrió Noah?” // “What multiplication expression represents the number of miles Noah ran?” (\(\frac{1}{2}\times 5\))
• “¿Cómo decidieron de qué forma ordenar las distancias que corrieron los estudiantes?” // “How did you decide how to order the lengths that each student ran?” (The length of the trail is always 5 so we can just compare the fraction factors.)
• Display the problem about Tyler.
• “¿Qué números tienen sentido? ¿Por qué?” // “What numbers make sense? Why?” (I can use any fraction that is bigger than \(\frac{1}{2}\) but less than \(\frac{3}{4}\). It has to be bigger than \(\frac{1}{2}\) so that Tyler runs farther than Noah. It has to be less than \(\frac{3}{4}\) so that Elena runs farther than Tyler.)

The purpose of this activity is for students to compare a fractional amount of a whole number with that same whole number. Students may calculate, draw a diagram, or reason about the size of the factor. When students choose their own numerator or denominator to make equations and inequalities true, monitor for students who:

• experiment with different numerators and multiply the fraction by the whole number to see if it makes the statement true
• choose a numerator of 1 and use their understanding of unit fractions as part of a whole
• explain why more than one answer makes sense

• Groups of 2

• 6–8 minutes: independent work time
• “Comparen sus soluciones con las de su compañero. Si es necesario, ajusten sus ideas” // “Check in with your partner and compare solutions. Revise your thinking, if necessary.”
• 3–5 minutes: partner work time

If students multiply to determine whether an expression is greater or less than a given number, draw a number line diagram with the whole number labeled and ask them to explain the approximate location of the expression.

• Display the inequality: \(\frac{\boxed{\phantom{{5^3}^3}}}{9} \times 50 < 50\)
• Invite students to share their responses.
• “¿Cómo saben que \(\frac{1}{50} \times 50 < 50\)?” // “How do you know \(\frac{1}{50} \times 50 < 50\)?” (Because \(\frac{1}{50} \times 50 = 1\).)
• “¿Cómo saben que \(\frac{10}{50} \times 50 < 50\)?” // “How do you know \(\frac{10}{50} \times 50 < 50\)?” (Because it is 10 of 50 parts, there are 40 other parts.)
• Display inequality: \(\frac{9}{\boxed{\phantom{5^{3^{3^3}}}}} \times 50 < 50\)
• Invite students to share their responses.
• “¿Cómo saben que \(\frac{9}{90} \times 50 < 50\)?” // “How do you know \(\frac{9}{90} \times 50 < 50\)?” (Because \(\frac{9}{90}\) is less than 1. \(\frac{9}{90} \times 50\) is equal to 5.)
• Display the equation: \(\frac{\boxed{\phantom{{5^3}^3}}}{9} \times 50 = 50\)
• “¿Cómo encontraron el número que hace que esta ecuación sea verdadera?” // “How did you find the number that makes this equation true?” (It has to be equal to 1 and \(\frac{9}{9}=1\).)