The purpose of this Warm-up is to elicit strategies and understandings students have for adding, subtracting, and multiplying fractions and mixed numbers. The series of equations prompt students to use properties of operations (associative and commutative properties in particular) in their reasoning, which will be helpful when students solve geometric problems involving fractional lengths (MP7).
Launch
Display one equation.
“Hagan una señal cuando sepan si la ecuación es verdadera o no, y puedan explicar cómo lo saben” // “Give me a signal when you know whether the equation is true and can explain how you know.”
1 minute: quiet think time
Activity
Share and record answers and strategy.
Repeat with each statement.
Decide si la afirmación es verdadera o falsa. Prepárate para explicar tu razonamiento.
Student Response
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Advancing Student Thinking
Activity Synthesis
“¿Qué estrategias les parecieron útiles para sumar o restar estos números con fracciones?” // “What strategies did you find useful for adding or subtracting these numbers with fractions?” (Sample response:
Add whole numbers separately than fractions.
Notice that is 10 and use that fact to add or subtract fractions.
Combine fractions that add up to 1 (such as and ).
In the second equation, add up the fractions and subtract the sum from 10, instead of subtracting each fraction individually.)
Consider asking:
“¿Alguien puede expresar el razonamiento de _____ de otra forma?” // “Who can restate _____’s reasoning in a different way?”
“¿Alguien usó la misma estrategia, pero la explicaría de otra forma?” // “Did anyone have the same strategy but would explain it differently?”
“¿Alguien pensó en la expresión de otra forma?” // “Did anyone approach the expression in a different way?”
“¿Alguien quiere agregar algo a la estrategia de _____?” // “Does anyone want to add on to _____’s strategy?”
Activity 1
Standards Alignment
Building On
Addressing
4.G.A.3
Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.
Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
Previously, students reasoned about the perimeter of two-dimensional figures based on given side lengths and known attributes of the figures, including symmetry. In this activity, students find unknown side lengths given the perimeter, some side lengths, and information about the symmetry of the figures. Students have opportunities to practice adding, subtracting, and multiplying numbers with fractions, as not all of the given measurements are whole numbers.
MLR8 Discussion Supports. Synthesis: Create a visual display of the figures. As students share their strategies, annotate the display to illustrate connections. For example, next to each figure, write expressions and draw the lines of symmetry. Advances: Speaking, Representing
Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select 2 of the 4 figures to work with. Offer feedback that emphasizes effort and time on task, and invite them to try another figure if time allows. Supports accessibility for: Organization, Attention, Social Emotional Functioning
Launch
Groups of 2
Give a ruler or a straightedge to each student.
Provide access to patty paper.
Activity
5 minutes: independent work time
2–3 minutes: partner discussion
Monitor for students who:
Can clearly articulate how lines of symmetry help them determine unknown side lengths.
Can explain how they know that all four sides of Q are equal.
Write expressions to show their reasoning.
Estas son 4 figuras y lo que sabemos sobre ellas.
Cada figura tiene un perímetro de 64 pulgadas.
P, R y S tienen, cada una, 1 línea de simetría.
Q tiene 4 líneas de simetría.
Dibuja las líneas de simetría de cada figura.
En cada figura, encuentra la longitud del lado marcado con “?”. Explica o muestra cómo razonaste.
Activity Synthesis
Select students to share their responses and reasoning.
“¿Cómo les ayudan las líneas de simetría de P, R y S a encontrar las longitudes de lado desconocidas?” // “How do the lines of symmetry in P, R, and S help you find the unknown side lengths?” (The lines of symmetry tell us the lengths of unlabeled sides that mirror labeled sides, making it possible to find the length of the side with a question mark.)
“¿Y las líneas de simetría de Q?” // “What about the lines of symmetry in Q?” (The vertical line of symmetry tells us the left and right sides have the same length and the horizontal one tells us the top and bottom sides are of equal length. So, all sides have the same length.)
Activity 2
Standards Alignment
Building On
Addressing
4.G.A.3
Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.
Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
The purpose of this activity is for students to practice completing a geometric drawing given half of the drawing and a line of symmetry, and reason about the perimeter of a line-symmetric figure. While a precise drawing is not an expectation here, if no students considered using tools and techniques—such as using patty paper or by folding—to complete the drawing precisely, consider asking how it could be done (MP5).
Monitor for and select students with the following approaches for determining whether Lin has enough tape to share in the Activity Synthesis:
Add 5 segments on one side of the line of symmetry and double it: .
Multiply each segment by 2 and find the sum of those products: .
List the length of each segment in an addition expression or vertical list and use the associative property to find sums that are easier (for example, add 19 and 6 to get 25, then multiply 25 and by 4: ).
The approaches will later be displayed side by side to help students connect ways of reasoning about side lengths and symmetry to different expressions for the perimeter of the design. As students practice adding and multiplying fractional side lengths, they will use properties of operations in their reasoning. Students will also have opportunities to look for and make use of structure (MP7) to expedite their calculation. Aim to elicit both key mathematical ideas and a variety of student voices, especially students who haven't shared recently.
Launch
Groups of 2
Give a ruler or a straightedge to each student.
Provide access to patty paper.
Display the image for all to see.
“¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
1 minute: quiet think time
1 minute: discuss observations and questions.
Activity
3–5 minutes: independent work time
2 minutes: partner discussion
As you monitor for the approaches listed in the Activity Narrative, consider asking:
“¿Qué observaron acerca de esta figura?” // “What did you notice about this figure?”
“¿Cómo usaron lo que observaron como ayuda para encontrar el perímetro?” // “How did you use what you noticed to help you find the perimeter?”
Lin tiene 145 pulgadas de cinta decorativa para adornar el contorno de un diseño.
Esta es la mitad del diseño. La línea punteada es la línea de simetría del diseño completo.
Dibuja el diseño de Lin completo.
¿Lin tiene suficiente cinta para todo el contorno? Explica o muestra cómo razonaste.
Si te queda tiempo: Lin tiene una hoja de papel decorativo que puede recortar para cubrir el interior del diseño. El papel es un rectángulo de 30 pulgadas por 18 pulgadas. Los ángulos del diseño son ángulos rectos. ¿Lin tiene suficiente papel para cubrir el interior del diseño? Explica o muestra cómo razonaste.
Student Response
Activity Synthesis
Invite previously selected students to display their equations for the perimeter side by side without sharing their thinking.
“Tómense un momento para examinar estas formas de determinar si Lin tiene suficiente cinta” // “Take a minute to look at each of these ways of determining whether Lin has enough tape.”
Connect students’ approaches by asking:
“¿En qué se parecen las ecuaciones y en qué son diferentes?” // “How are the different equations the same and different?” (They show the same side lengths. They show doubling side lengths. The order and how the numbers are grouped is different. The numbers added or multiplied are different.)
Connect students’ approaches to the learning goal by asking:
“¿Por qué tener lados de la misma longitud nos ayuda a encontrar el perímetro?” // “How does having sides of the same length help us find the perimeter?” (You can look for ways to use multiplication instead of having to add a bunch of different lengths.)
Lesson Synthesis
“Hoy usamos las características de las figuras para razonar sobre sus longitudes de lado y su perímetro” // “Today we used attributes of figures to reason about their side lengths and perimeter.“
Display:
“Estas son dos figuras. Supongamos que sabemos que el perímetro de cada figura es 48 unidades. La figura A tiene una línea de simetría y la figura B no tiene líneas de simetría” // “Here are two shapes. Suppose we know the perimeter of each shape is 48 units. Shape A has a line of symmetry and B has none.”
“¿Cómo nos puede ayudar la línea de simetría de A a encontrar las longitudes de lado desconocidas?” // “How can knowing the line of symmetry in A help us find the unknown side lengths?” (The line of symmetry tells us that the longer unlabeled side is 14 and the two shorter sides are equal. We can subtract from 48 and divide the result by 2 to get the shorter sides.)
“La figura B no tiene líneas de simetría. ¿Podemos descubrir las longitudes desconocidas?” // “Shape B has no line of symmetry. Can we figure out the unknown lengths?” (No. There isn’t enough information. We'd need to know if some of the sides are the same length.)
If students argue that they can tell that one of the other sides must also be 15 units long, ask: “Sin medir, ¿qué necesitan saber para estar seguros de que uno de los lados que están marcados con un signo ‘?’ también mide 15 unidades?” // “Without measuring, what would you need to know to be sure one of the labeled sides is also 15 units long?”
“Supongamos que sabemos que B es un paralelogramo. ¿Nos ayudaría esto a encontrar las longitudes desconocidas? ¿Por qué sí o por qué no?” // “Suppose we know that B is a parallelogram. Would that help us find those lengths? Why or why not?” (Yes. Opposite sides of a parallelogram have the same length, so we know the unlabeled sides are 15 and 9.)
Student Section Summary
Usamos características, como longitudes de lado, ángulos, líneas de simetría y lados paralelos, para resolver problemas sobre el perímetro de figuras.
Aprendimos que si una figura tiene ciertas características, podemos usarlas para encontrar su perímetro, incluso cuando no sabemos cuáles son las longitudes de todos sus lados. También aprendimos que si conocemos el perímetro de una figura y tenemos suficiente información sobre sus características, podemos encontrar las longitudes de sus lados.
Por ejemplo, estas son 2 figuras:
Sabemos que el perímetro de cada figura es 48 unidades.
Si sabemos que la línea punteada que atraviesa la figura A es una línea de simetría, podemos encontrar las 3 longitudes de lado desconocidas.
Debido a la simetría, el lado opuesto al que mide 14 unidades también mide 14 unidades.
, entonces la suma de las longitudes de los otros 2 lados es 12 unidades.
, entonces cada uno de los otros 2 lados mide 6 unidades.
Si sabemos que las longitudes de los lados opuestos de la figura B son iguales, podemos encontrar las 3 longitudes de lado desconocidas.
El lado opuesto al que mide 15 unidades también mide 15 unidades.
, entonces la suma de las longitudes de los otros 2 lados es 18 unidades.
, entonces cada uno de los otros 2 lados mide 9 unidades.
Standards Alignment
Building On
Addressing
4.NF.B.3.c
Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
If students find the unknown sides lengths for Figures P and Q, but say they need more information to find the unknown length for Figures R and S, consider asking:
“¿Qué información tienes que te puede ayudar a encontrar la longitud desconocida? ¿Qué información te falta?” // “What information do you have to help you find the unknown length? What information do you still need?”
“¿Cómo puedes usar la línea de simetría para marcar más lados con su longitud?” // “How can you use the line of symmetry to label more of the sides?”
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Advancing Student Thinking
If students begin to add each side length after sketching Lin’s design and lose track of some side lengths, consider asking:
“¿Qué observaste acerca de las longitudes de lado que dibujaste y las longitudes de lado que ya estaban dibujadas y marcadas?” // “What did you notice about the side lengths you drew and the side lengths that were already drawn and labeled?”
“¿Cómo podrías usar eso para encontrar el perímetro?” // “How could you use that to find the perimeter?”
