The purpose of this Warm-up is to remind students of doubling as a strategy for multiplication where a factor in one product is twice a factor in another product. The reasoning that students do here with the factors 2, 4, 8, and 16 will support them as they reason about equivalent fractions and find multiples of numerators and denominators.
Launch
Display the first expression.
“Hagan una señal cuando tengan una respuesta y puedan explicar cómo la obtuvieron” // “Give me a signal when you have an answer and can explain how you got it.”
“¿Cómo les ayudaron las tres primeras expresiones a encontrar el valor de la última expresión?” // “How did the first three expressions help you find the value of the last expression?”
Activity 1
Standards Alignment
Building On
Addressing
4.NF.A.1
Explain why a fraction is equivalent to a fraction by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
The purpose of this activity is to remind students of a key insight from grade 3—that the same point on the number line can be named with fractions that don’t look alike. Students see that those fractions are equivalent, even though their numerators and denominators may be different.
Students have multiple opportunities to look for regularity in repeated reasoning (MP8). For instance, they are likely to notice that:
Fractions that have the same number for the numerator and denominator all represent 1.
In fractions that describe the halfway point between 0 and 1, the numerator is always half the denominator, or the denominator twice the numerator.
In fractions that describe , the denominator is 4 times the numerator.
These observations will help students to identify and generate equivalent fractions later in the unit.
Representation: Internalize Comprehension. Synthesis: Invite students to identify which details were most important to solve the problem. Display the sentence frame, “La próxima vez que los puntos estén en el mismo lugar de varias rectas numéricas, voy a . . .” // “The next time points are in the same place on different number lines, I will . . . .“ Supports accessibility for: Language, Attention
Launch
Groups of 2
Give students access to straightedges. Display the first set of number lines.
“¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?” (I notice each number line has different fractions represented. The first number line has a point that is half-way between 0 and 1 labeled , but if you label all the tick marks you won't have 2 in the denominator for all of them. I wonder what fraction goes on each mark? Can you have a number line with both halves and fourths? How many fourths are at the line?)
1 minute: quiet think time
“Compartan con su pareja lo que observaron y se preguntaron” // “Share what you noticed and wondered with your partner.”
1 minute: partner discussion
Activity Synthesis
Select students to share their responses and reasoning for the first set of questions. Highlight explanations that convey that:
Any fraction with the same number for the numerator and denominator has a value of 1.
Equivalent fractions share the same location or are the same distance from 0 on the number line.
Invite previously selected students to share their responses for the second set of questions.
Consider asking:
“¿Alguien puede expresar el razonamiento de _______ de otra forma?” // “Who can restate _______ 's reasoning in a different way?”
“¿En qué se parecen las maneras de representar los puntos con fracciones? ¿En qué son diferentes?” // “How are these ways for labeling the points the same? How are they different?“
Activity 2
Standards Alignment
Building On
Addressing
4.NF.A.1
Explain why a fraction is equivalent to a fraction by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
In this activity, students reason about whether two fractions are equivalent in the context of distance. To support their reasoning, students use number lines and their understanding of fractions with related denominators (where one number is a multiple of the other). The given number lines each have only one tick mark between 0 and 1, so students need to partition each line strategically to represent two fractions with different denominators on the diagram.
To help students intuit the distance of 1 mile, consider preparing a neighborhood map that shows the school and some points that are a mile away. Display the map during the Launch.
Launch
Groups of 2
“¿Quién ha caminado una milla? ¿Quién ha corrido una milla?” // “Who has walked a mile? Who has run a mile?”
“¿Cuánta distancia es 1 milla? ¿Cómo la describirían?” // “How far is 1 mile? How would you describe it?”
Consider showing a map of the school and some landmarks or points on the map that are 1 mile away.
Activity
6–8 minutes: independent work time
Monitor for the different ways students reason about the equivalence of and . For instance, they may:
Know that 1 fourth is equivalent to 3 twelfths and reason that 3 fourths must be 9 twelfths.
Note that and are both halfway between and 1 on the number line.
Locate and on the same number line (or separate ones) and show that they are in the same location.
2–3 minutes: partner discussion
Han y Kiran planean ir a correr después de la escuela.
Han dice: “Corramos de milla. Es lo mismo que corro hasta mi entrenamiento de fútbol”.
Kiran dice: “Yo solo puedo correr de milla”.
¿Qué distancia deberían correr? Explica tu razonamiento. Usa una o más rectas numéricas para mostrar tu razonamiento.
Mai quiere ir a correr con Han y Kiran. Ella dice: “¿Qué tal si corremos de milla?”.
¿La distancia que propuso Mai es la misma que la que sus amigos querían correr? Explica o muestra tu razonamiento.
Student Response
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Advancing Student Thinking
Activity Synthesis
Invite previously selected students to share their responses and how they knew that is equivalent to but is not.
To facilitate their explanations, ask students to display their work, or display blank number lines for them to annotate.
Consider asking:
“¿Alguien pensó de la misma forma, pero lo explicaría de otra manera?” // “Who reasoned the same way but would explain it differently?”
“¿Alguien lo pensó de otra forma y aun así llegó a la misma conclusión?” // “Who thought about it differently but arrived at the same conclusion?”
Lesson Synthesis
“Hoy representamos fracciones en rectas numéricas y razonamos sobre fracciones equivalentes” // “Today we represented fractions on number lines and reasoned about equivalent fractions.”
Display a labeled diagram of fraction strips and the labeled number lines from the last problem in today’s activities.
“¿Dónde vemos fracciones equivalentes en el diagrama de tiras de fracciones?” // “Where in the diagram of fraction strips do we see equivalent fractions?” (Parts that have the same length are equivalent.)
“¿Dónde vemos fracciones equivalentes en las rectas numéricas?” // “Where on the number lines do we see equivalent fractions?” (Points that are in the same location on the number line, or are the same distance from 0, are equivalent.)
“Supongamos que quieren ayudarle a alguien a entender que es equivalente a . ¿Usarían una recta numérica o una tira de fracciones? ¿Por qué?” // “Suppose you’d like to help someone see that is equivalent to . Would you use a number line or a fraction strip? Why?” (Sample response: Use a number line, because it’s not necessary to show all the tick marks. If using fraction strips, it would mean partitioning each fifth into 10 fiftieths, which would be a lot of work.)
Standards Alignment
Building On
3.OA.B.5
Apply properties of operations as strategies to multiply and divide.Students need not use formal terms for these properties.Examples: If is known, then is also known. (Commutative property of multiplication.) can be found by , then , or by , then . (Associative property of multiplication.) Knowing that and , one can find as . (Distributive property.)
“Trabajen individualmente en la tarea por un momento. Después, discutan su trabajo con su pareja” // “Take a moment to work independently on the task. Then discuss your work with your partner.”
“En los puntos que estén en rectas numéricas diferentes deben escribir fracciones con números diferentes” // “The labels that you write for the points on different number lines should be different.”
7–8 minutes: independent work time
Monitor for students who:
Partition each number line into as many parts as the denominator before naming a fraction for the point on the number line.
Use multiplicative relationships between denominators to name a fraction. (For instance, , so the line showing twelfths has 3 times as many parts as the one showing fourths.)
Estas rectas numéricas tienen fracciones con números diferentes en la marca de más a la derecha.
Explícale a tu compañero por qué en las marcas de más a la derecha se pueden escribir fracciones con números diferentes.
En cada punto, escribe una fracción que lo represente (no escribas ).
Explícale a tu compañero por qué las fracciones que escribiste son equivalentes.
En cada recta numérica, escribe la fracción que el punto representa. Usa una fracción distinta en cada recta numérica. Prepárate para explicar tu razonamiento.
