Programa escolar (Estados Unidos)

Matemáticas — 7.º grado

Matemáticas de 7.º grado según los Estándares Estatales Comunes: operaciones con números positivos y negativos, aumento y disminución porcentual, interés simple, ecuaciones de dos pasos, la circunferencia y el área de círculos, prismas y probabilidad. Cada tema incluye hojas de trabajo imprimibles listas para usar.

Crear una hoja de este programa

¿Qué aprende un estudiante en matemáticas de séptimo grado?

En 7.º grado, un estudiante suma, resta, multiplica y divide números positivos y negativos, incluidos los decimales. Los porcentajes describen situaciones reales: recargos, descuentos e interés simple. Un estudiante resuelve ecuaciones de dos pasos, halla la circunferencia y el área de un círculo y el volumen y el área total de prismas, y calcula probabilidades de eventos simples y compuestos.

Estado de la página: exponentes y raíces, notación científica, porcentajes, operaciones con números con signo, ecuaciones y sistemas, funciones lineales, el teorema de Pitágoras, círculos y sólidos, y probabilidad están listos. Las transformaciones y los diagramas de dispersión se añadirán a continuación.

Alcance del currículo

Punto del currículo 7.RP · Contenidos de enseñanza · verificado con el texto oficial

Ratios and Proportional Relationships

CCSS Mathematics (2010), Grade 7, domain Ratios and Proportional Relationships (7.RP)
Punto del currículo 7.RP.A · Contenidos de enseñanza · verificado con el texto oficial

Analizar relaciones proporcionales y usarlas para resolver problemas del mundo real y matemáticos.

nuestra traducción · texto original (EN): Analyze proportional relationships and use them to solve real-world and mathematical problems.

CCSS Mathematics (2010), Grade 7, Ratios and Proportional Relationships, cluster A
Punto del currículo 7.RP.A.1 · Contenidos de enseñanza · verificado con el texto oficial

Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, c ompute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.

CCSS Mathematics (2010), Grade 7, Ratios and Proportional Relationships, standard 7.RP.A.1
Punto del currículo 7.RP.A.2 · Contenidos de enseñanza · verificado con el texto oficial

Reconocer y representar relaciones proporcionales entre cantidades. a. Decidir si dos cantidades están en una relación proporcional, p. ej., comprobando si hay razones equivalentes en una tabla o graficando en un plano de coordenadas y observando si la gráfica es una línea recta que pasa por el origen. b. Identificar la constante de proporcionalidad (tasa unitaria) en tablas, gráficas, ecuaciones, diagramas y descripciones verbales de relaciones proporcionales. c. Representar relaciones proporcionales mediante ecuaciones. Por ejemplo, si el costo total t es proporcional al número n de artículos comprados a un precio constante p, la relación entre el costo total y el número de artículos se puede expresar como t = pn. d. Explicar qué significa un punto (x, y) en la gráfica de una relación proporcional en términos de la situación, con especial atención a los puntos (0, 0) y (1, r) donde r es la tasa unitaria.

nuestra traducción · texto original (EN): Recognize and represent proportional relationships between quantities. a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

CCSS Mathematics (2010), Grade 7, Ratios and Proportional Relationships, standard 7.RP.A.2
Punto del currículo 7.RP.A.3 · Contenidos de enseñanza · verificado con el texto oficial

Usar relaciones proporcionales para resolver problemas de razones y porcentajes de varios pasos. Ejemplos: interés simple, impuestos, recargos y rebajas, propinas y comisiones, tarifas, aumento y disminución porcentual, error porcentual.

nuestra traducción · texto original (EN): Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

CCSS Mathematics (2010), Grade 7, Ratios and Proportional Relationships, standard 7.RP.A.3

Calculating Percent Increase and Decrease (en nuestro curso 7) · Calculating Simple Interest on Savings (en nuestro curso 7) · Finding What Percent One Number Is of Another (en nuestro curso 7) · Finding the Whole Amount from a Percent (en nuestro curso 7)
Punto del currículo 7.NS · Contenidos de enseñanza · verificado con el texto oficial

El sistema de numeración

nuestra traducción · texto original (EN): The Number System

CCSS Mathematics (2010), Grade 7, domain The Number System (7.NS)
Punto del currículo 7.NS.A · Contenidos de enseñanza · verificado con el texto oficial

Aplicar y extender los conocimientos previos sobre operaciones con fracciones para sumar, restar, multiplicar y dividir números racionales.

nuestra traducción · texto original (EN): Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

CCSS Mathematics (2010), Grade 7, The Number System, cluster A
Punto del currículo 7.NS.A.1 · Contenidos de enseñanza · verificado con el texto oficial

Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. d. Apply properties of operations as strategies to add and subtract rational numbers.

CCSS Mathematics (2010), Grade 7, The Number System, standard 7.NS.A.1

Adding and Subtracting Integers (en nuestro curso 7) · Adding and Subtracting Positive and Negative Decimals (en nuestro curso 7)
Punto del currículo 7.NS.A.2 · Contenidos de enseñanza · verificado con el texto oficial

Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world contexts.

CCSS Mathematics (2010), Grade 7, The Number System, standard 7.NS.A.2

Multiplying and Dividing Positive and Negative Numbers (en nuestro curso 7)
Punto del currículo 7.NS.A.3 · Contenidos de enseñanza · verificado con el texto oficial

Resolver problemas del mundo real y matemáticos que involucren las cuatro operaciones con números racionales.

nuestra traducción · texto original (EN): Solve real-world and mathematical problems involving the four operations with rational numbers.

CCSS Mathematics (2010), Grade 7, The Number System, standard 7.NS.A.3

Multiplying and Dividing Positive and Negative Numbers (en nuestro curso 7) · Adding and Subtracting Positive and Negative Decimals (en nuestro curso 7)
Punto del currículo 7.EE · Contenidos de enseñanza · verificado con el texto oficial

Expressions and Equations

CCSS Mathematics (2010), Grade 7, domain Expressions and Equations (7.EE)
Punto del currículo 7.EE.A · Contenidos de enseñanza · verificado con el texto oficial

Usar las propiedades de las operaciones para generar expresiones equivalentes.

nuestra traducción · texto original (EN): Use properties of operations to generate equivalent expressions.

CCSS Mathematics (2010), Grade 7, Expressions and Equations, cluster A
Punto del currículo 7.EE.B · Contenidos de enseñanza · verificado con el texto oficial

Resolver problemas de la vida real y matemáticos utilizando expresiones y ecuaciones numéricas y algebraicas.

nuestra traducción · texto original (EN): Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

CCSS Mathematics (2010), Grade 7, Expressions and Equations, cluster B
Punto del currículo 7.EE.A.1 · Contenidos de enseñanza · verificado con el texto oficial

Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

CCSS Mathematics (2010), Grade 7, Expressions and Equations, standard 7.EE.A.1
Punto del currículo 7.EE.A.2 · Contenidos de enseñanza · verificado con el texto oficial

Comprender que reescribir una expresión de diferentes formas en el contexto de un problema puede aclarar el problema y cómo están relacionadas las cantidades en él. Por ejemplo, a + 0.05a = 1.05a significa que “aumentar en un 5%” es lo mismo que “multiplicar por 1.05”.

nuestra traducción · texto original (EN): Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”

CCSS Mathematics (2010), Grade 7, Expressions and Equations, standard 7.EE.A.2
Punto del currículo 7.EE.B.3 · Contenidos de enseñanza · verificado con el texto oficial

Resolver problemas de varios pasos de la vida real y matemáticos planteados con números racionales positivos y negativos en cualquier forma (números enteros, fracciones y decimales), usando herramientas de manera estratégica. Aplicar las propiedades de las operaciones para calcular con números en cualquier forma; convertir entre formas según sea apropiado; y evaluar la razonabilidad de las respuestas utilizando el cálculo mental y estrategias de estimación. Por ejemplo: Si una mujer que gana $25 por hora recibe un aumento del 10%, ganará 1/10 adicional de su salario por hora, o $2.50, para un nuevo salario de $27.50. Si se quiere colocar un toallero de 9 3/4 pulgadas de largo en el centro de una puerta de 27 1/2 pulgadas de ancho, se necesitará colocar la barra a unas 9 pulgadas de cada borde; esta estimación se puede utilizar como comprobación del cálculo exacto.

nuestra traducción · texto original (EN): Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

CCSS Mathematics (2010), Grade 7, Expressions and Equations, standard 7.EE.B.3
Punto del currículo 7.EE.B.4 · Contenidos de enseñanza · verificado con el texto oficial

Usar variables para representar cantidades en un problema del mundo real o matemático, y construir ecuaciones y desigualdades simples para resolver problemas razonando sobre las cantidades. a. Resolver problemas verbales que conducen a ecuaciones de la forma px + q = r y p(x + q) = r, donde p, q y r son números racionales específicos. Resolver con fluidez ecuaciones de estas formas. Comparar una solución algebraica con una solución aritmética, identificando la secuencia de las operaciones utilizadas en cada enfoque. Por ejemplo, el perímetro de un rectángulo es 54 cm. Su longitud es 6 cm. ¿Cuál es su ancho? b. Resolver problemas verbales que conducen a desigualdades de la forma px + q > r o px + q < r, donde p, q y r son números racionales específicos. Representar gráficamente el conjunto solución de la desigualdad e interpretarlo en el contexto del problema. Por ejemplo: Como vendedor, se le pagan $50 por semana más $3 por venta. Esta semana desea que su pago sea de al menos $100. Escriba una desigualdad para la cantidad de ventas que necesita realizar y describa las soluciones.

nuestra traducción · texto original (EN): Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width? b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.

CCSS Mathematics (2010), Grade 7, Expressions and Equations, standard 7.EE.B.4

Solving Two-Step Equations (en nuestro curso 7)
Punto del currículo 7.G · Contenidos de enseñanza · verificado con el texto oficial

Geometría

nuestra traducción · texto original (EN): Geometry

CCSS Mathematics (2010), Grade 7, domain Geometry (7.G)
Punto del currículo 7.G.A · Contenidos de enseñanza · verificado con el texto oficial

Dibujar, construir y describir figuras geométricas y describir las relaciones entre ellas.

nuestra traducción · texto original (EN): Draw, construct, and describe geometrical figures and describe the relationships between them.

CCSS Mathematics (2010), Grade 7, Geometry, cluster A
Punto del currículo 7.G.B · Contenidos de enseñanza · verificado con el texto oficial

Resolver problemas de la vida real y matemáticos que involucren la medida de ángulos, el área, el área de superficie y el volumen.

nuestra traducción · texto original (EN): Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

CCSS Mathematics (2010), Grade 7, Geometry, cluster B
Punto del currículo 7.G.A.1 · Contenidos de enseñanza · verificado con el texto oficial

Resolver problemas que involucran dibujos a escala de figuras geométricas, incluyendo el cálculo de longitudes y áreas reales a partir de un dibujo a escala y la reproducción de un dibujo a escala a una escala diferente.

nuestra traducción · texto original (EN): Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

CCSS Mathematics (2010), Grade 7, Geometry, standard 7.G.A.1
Punto del currículo 7.G.A.2 · Contenidos de enseñanza · verificado con el texto oficial

Dibujar (a mano alzada, con regla y transportador, y con tecnología) figuras geométricas con condiciones dadas. Centrarse en la construcción de triángulos a partir de tres medidas de ángulos o lados, observando cuándo las condiciones determinan un triángulo único, más de un triángulo o ningún triángulo.

nuestra traducción · texto original (EN): Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

CCSS Mathematics (2010), Grade 7, Geometry, standard 7.G.A.2
Punto del currículo 7.G.A.3 · Contenidos de enseñanza · verificado con el texto oficial

Describir las figuras bidimensionales que resultan de cortar figuras tridimensionales, como en las secciones planas de prismas rectangulares rectos y pirámides rectangulares rectas.

nuestra traducción · texto original (EN): Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

CCSS Mathematics (2010), Grade 7, Geometry, standard 7.G.A.3
Punto del currículo 7.G.B.4 · Contenidos de enseñanza · verificado con el texto oficial

Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

CCSS Mathematics (2010), Grade 7, Geometry, standard 7.G.B.4

Finding the Circumference and Area of Circles (en nuestro curso 7)
Punto del currículo 7.G.B.5 · Contenidos de enseñanza · verificado con el texto oficial

Usar datos sobre ángulos suplementarios, complementarios, opuestos por el vértice y adyacentes en un problema de varios pasos para escribir y resolver ecuaciones simples para un ángulo desconocido en una figura.

nuestra traducción · texto original (EN): Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

CCSS Mathematics (2010), Grade 7, Geometry, standard 7.G.B.5
Punto del currículo 7.G.B.6 · Contenidos de enseñanza · verificado con el texto oficial

Resolver problemas del mundo real y matemáticos que involucren el área, el volumen y el área superficial de objetos bidimensionales y tridimensionales compuestos por triángulos, cuadriláteros, polígonos, cubos y prismas rectos.

nuestra traducción · texto original (EN): Solve real-world and mathematical problems involving area, volume and surface area of two-and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

CCSS Mathematics (2010), Grade 7, Geometry, standard 7.G.B.6

Volume and Surface Area of Triangular Prisms (en nuestro curso 7)
Punto del currículo 7.SP · Contenidos de enseñanza · verificado con el texto oficial

Statistics and Probability

CCSS Mathematics (2010), Grade 7, domain Statistics and Probability (7.SP)
Punto del currículo 7.SP.A · Contenidos de enseñanza · verificado con el texto oficial

Utilizar el muestreo aleatorio para hacer inferencias sobre una población.

nuestra traducción · texto original (EN): Use random sampling to draw inferences about a population.

CCSS Mathematics (2010), Grade 7, Statistics and Probability, cluster A
Punto del currículo 7.SP.B · Contenidos de enseñanza · verificado con el texto oficial

Hacer inferencias comparativas informales sobre dos poblaciones.

nuestra traducción · texto original (EN): Draw informal comparative inferences about two populations.

CCSS Mathematics (2010), Grade 7, Statistics and Probability, cluster B
Punto del currículo 7.SP.C · Contenidos de enseñanza · verificado con el texto oficial

Investigar procesos aleatorios y desarrollar, usar y evaluar modelos de probabilidad.

nuestra traducción · texto original (EN): Investigate chance processes and develop, use, and evaluate probability models.

CCSS Mathematics (2010), Grade 7, Statistics and Probability, cluster C
Punto del currículo 7.SP.A.1 · Contenidos de enseñanza · verificado con el texto oficial

Comprender que las estadísticas se pueden utilizar para obtener información sobre una población al examinar una muestra de la población; las generalizaciones sobre una población a partir de una muestra son válidas solo si la muestra es representativa de esa población. Comprender que el muestreo aleatorio tiende a producir muestras representativas y a respaldar inferencias válidas.

nuestra traducción · texto original (EN): Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

CCSS Mathematics (2010), Grade 7, Statistics and Probability, standard 7.SP.A.1
Punto del currículo 7.SP.A.2 · Contenidos de enseñanza · verificado con el texto oficial

Usar datos de una muestra aleatoria para hacer inferencias sobre una población con una característica de interés desconocida. Generar múltiples muestras (o muestras simuladas) del mismo tamaño para calibrar la variación en las estimaciones o predicciones. Por ejemplo, estimar la longitud media de las palabras en un libro mediante el muestreo aleatorio de palabras del libro; predecir el ganador de una elección escolar basándose en datos de encuestas por muestreo aleatorio. Calibrar qué tan alejada podría estar la estimación o predicción.

nuestra traducción · texto original (EN): Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.

CCSS Mathematics (2010), Grade 7, Statistics and Probability, standard 7.SP.A.2
Punto del currículo 7.SP.B.3 · Contenidos de enseñanza · verificado con el texto oficial

Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.

CCSS Mathematics (2010), Grade 7, Statistics and Probability, standard 7.SP.B.3
Punto del currículo 7.SP.B.4 · Contenidos de enseñanza · verificado con el texto oficial

Usar medidas de centro y medidas de variabilidad para datos numéricos de muestras aleatorias para hacer inferencias comparativas informales sobre dos poblaciones. Por ejemplo, decidir si las palabras en un capítulo de un libro de ciencias de séptimo grado son generalmente más largas que las palabras en un capítulo de un libro de ciencias de cuarto grado.

nuestra traducción · texto original (EN): Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.

CCSS Mathematics (2010), Grade 7, Statistics and Probability, standard 7.SP.B.4
Punto del currículo 7.SP.C.5 · Contenidos de enseñanza · verificado con el texto oficial

Comprender que la probabilidad de un evento aleatorio es un número entre 0 y 1 que expresa la posibilidad de que ocurra el evento. Los números mayores indican una mayor posibilidad. Una probabilidad cercana a 0 indica un evento improbable, una probabilidad alrededor de 1/2 indica un evento que no es ni improbable ni probable, y una probabilidad cercana a 1 indica un evento probable.

nuestra traducción · texto original (EN): Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

CCSS Mathematics (2010), Grade 7, Statistics and Probability, standard 7.SP.C.5

Finding Probabilities of Simple Events (en nuestro curso 7)
Punto del currículo 7.SP.C.6 · Contenidos de enseñanza · verificado con el texto oficial

Aproximar la probabilidad de un suceso aleatorio recopilando datos sobre el proceso aleatorio que lo produce y observando su frecuencia relativa a largo plazo, y predecir la frecuencia relativa aproximada dada la probabilidad. Por ejemplo, al lanzar un dado numérico 600 veces, predecir que saldría un 3 o un 6 aproximadamente 200 veces, pero probablemente no exactamente 200 veces.

nuestra traducción · texto original (EN): Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.

CCSS Mathematics (2010), Grade 7, Statistics and Probability, standard 7.SP.C.6

Predicting How Often an Event Will Happen (en nuestro curso 7)
Punto del currículo 7.SP.C.7 · Contenidos de enseñanza · verificado con el texto oficial

Desarrollar un modelo de probabilidad y usarlo para hallar probabilidades de eventos. Comparar probabilidades de un modelo con frecuencias observadas; si la concordancia no es buena, explicar posibles fuentes de la discrepancia. a. Desarrollar un modelo de probabilidad uniforme asignando igual probabilidad a todos los resultados, y usar el modelo para determinar probabilidades de eventos. Por ejemplo, si se selecciona al azar un estudiante de una clase, hallar la probabilidad de que Jane sea seleccionada y la probabilidad de que una niña sea seleccionada. b. Desarrollar un modelo de probabilidad (que puede no ser uniforme) observando frecuencias en datos generados a partir de un proceso aleatorio. Por ejemplo, hallar la probabilidad aproximada de que una moneda de un centavo que gira caiga con la cara hacia arriba o que un vaso de papel lanzado caiga con el extremo abierto hacia abajo. ¿Parecen ser igualmente probables los resultados para la moneda de un centavo que gira basándose en las frecuencias observadas?

nuestra traducción · texto original (EN): Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected. b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?

CCSS Mathematics (2010), Grade 7, Statistics and Probability, standard 7.SP.C.7

Finding Probabilities of Simple Events (en nuestro curso 7)
Punto del currículo 7.SP.C.8 · Contenidos de enseñanza · verificado con el texto oficial

Hallar probabilidades de eventos compuestos utilizando listas organizadas, tablas, diagramas de árbol y simulación. a. Comprender que, al igual que con los eventos simples, la probabilidad de un evento compuesto es la fracción de resultados en el espacio muestral para los cuales ocurre el evento compuesto. b. Representar espacios muestrales para eventos compuestos utilizando métodos tales como listas organizadas, tablas y diagramas de árbol. Para un evento descrito en lenguaje cotidiano (por ejemplo, “sacar doble seis”), identificar los resultados en el espacio muestral que componen el evento. c. Diseñar y usar una simulación para generar frecuencias para eventos compuestos. Por ejemplo, usar dígitos aleatorios como una herramienta de simulación para aproximar la respuesta a la pregunta: Si el 40% de los donantes tienen sangre tipo A, ¿cuál es la probabilidad de que se necesiten al menos 4 donantes para encontrar uno con sangre tipo A?

nuestra traducción · texto original (EN): Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. b. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event. c. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?

CCSS Mathematics (2010), Grade 7, Statistics and Probability, standard 7.SP.C.8

Finding Probabilities of Two-Stage Events (en nuestro curso 7)

Competencias paso a paso

Finding Probabilities of Simple Events

Students learn to express the likelihood of simple events as numbers from 0 to 1 and calculate probabilities where all outcomes are equally likely.

Punto del currículo 7.SP.C.5 Comprender que la probabilidad de un evento aleatorio es un número entre 0 y 1 que expresa la posibilidad de que ocurra el evento. Los números mayores indican una mayor posibilidad. Una probabilidad cercana a 0 indica un evento improbable, una probabilidad alrededor de 1/2 indica un evento que no es ni improbable ni probable, y una probabilidad cercana a 1 indica un evento probable. (nuestra traducción)
Punto del currículo 7.SP.C.7 Desarrollar un modelo de probabilidad y usarlo para hallar probabilidades de eventos. Comparar probabilidades de un modelo con frecuencias observadas; si la concordancia no es buena, explicar posibles fuentes de la discrepancia. a. Desarrollar un modelo de probabilidad uniforme asignando igual probabilidad a todos los resultados, y usar el modelo para determinar probabilidades de eventos. Por ejemplo, si se selecciona al azar un estudiante de una clase, hallar la probabilidad de que Jane sea seleccionada y la probabilidad de que una niña sea seleccionada. b. Desarrollar un modelo de probabilidad (que puede no ser uniforme) observando frecuencias en datos generados a partir de un proceso aleatorio. Por ejemplo, hallar la probabilidad aproximada de que una moneda de un centavo que gira caiga con la cara hacia arriba o que un vaso de papel lanzado caiga con el extremo abierto hacia abajo. ¿Parecen ser igualmente probables los resultados para la moneda de un centavo que gira basándose en las frecuencias observadas? (nuestra traducción)

Students learn that the probability of a chance event is measured as a number from 0 to 1. A probability near 0 represents an unlikely event, a value around 1/2 indicates an event that is neither unlikely nor likely, and a value near 1 describes a likely event. At this stage, students work with uniform probability models where each individual outcome is equally likely, determining the probability of an event by comparing the count of favorable outcomes to the total number of possible outcomes.

A common mistake occurs when students compare favorable outcomes to unfavorable outcomes instead of the total. For instance, when finding the probability of rolling a specific number, a student might write 1/5 instead of 1/6 because five non-matching faces remain. Others struggle to interpret the scale, confusing a probability of 0 (which means an event will not occur) with low but possible chances.

Practice tasks present simple experiments and ask students to calculate the exact probability of an event or evaluate its likelihood:

Rolling a die: finding the probability of landing on an even number or a number greater than 4.
Drawing a numbered card: calculating the chance of drawing a prime number from a set of cards numbered 1 through 10.
Drawing a ball from a box: determining the likelihood of picking a red ball when the box contains a known mix of colored balls.

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Tipos de ejercicios: Lanzar un dado · Sacar una bola de una caja · Sacar una tarjeta numerada

Calculating Percent Increase and Decrease

Students learn to calculate a new total after increasing or decreasing an initial amount by a given percentage.

Punto del currículo 7.RP.A.3 Usar relaciones proporcionales para resolver problemas de razones y porcentajes de varios pasos. Ejemplos: interés simple, impuestos, recargos y rebajas, propinas y comisiones, tarifas, aumento y disminución porcentual, error porcentual. (nuestra traducción)

Students learn to use proportional relationships to solve multi-step problems where quantities grow or shrink by a percentage. They determine the percentage of an original value and carry out the required operation, adding the change to find an increased total or subtracting it to find a reduced total.

A typical stumbling block occurs when students stop after calculating the percentage itself, confusing the amount of change with the final answer. Another common error is calculating the percentage based on the wrong amount rather than the original starting value.

Practice on this skill centers on two template formats:

Increase by a percent: Students find the percentage of a starting value and add it to find the larger result, such as calculating a price markup.
Decrease by a percent: Students find the percentage of an original value and subtract it to find the lower result, such as applying a discount.

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Tipos de ejercicios: Aumenta en un porcentaje · Disminuye en un porcentaje

Adding and Subtracting Integers

Students learn to add and subtract positive and negative integers using number lines and the rule that subtracting a number is the same as adding its opposite.

Punto del currículo 7.NS.A.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. d. Apply properties of operations as strategies to add and subtract rational numbers.

In 7th grade, students extend arithmetic to include negative whole numbers. They learn to locate values on horizontal and vertical number lines, recognizing that a number and its opposite combine to make 0. They understand addition as moving along the line based on the sign of the number, and they learn to treat subtraction as adding the additive inverse, using the relationship p – q = p + (–q).

A common difficulty occurs when subtracting negative numbers. Students often see a minus sign and intuitively expect the total to decrease, forgetting that taking away a negative value increases the result and moves to the right on a number line. Another frequent misstep is mixing up the signs when adding integers with different signs, such as confusing the rules for addition with the rules for multiplying signs.

Practice tasks on this skill provide direct calculation problems across two formats:

Add integers, where students find sums such as –4 + 9 or –6 + (–3).
Subtract integers, where students evaluate differences such as 5 – (–8) or –7 – 2.

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Tipos de ejercicios: Resta números enteros · Suma números enteros

Adding and Subtracting Positive and Negative Decimals

Students learn to add and subtract positive and negative decimals using number line models and the rule of adding the opposite.

In 7th grade, students extend basic arithmetic to include positive and negative decimals. They learn to represent these operations on horizontal or vertical number lines, interpreting addition as moving along the line and subtraction as finding the distance between two values. A core milestone is understanding additive inverses—knowing that opposite values combine to make zero and that subtracting a number is equivalent to adding its opposite: p – q = p + (–q).

A common mistake occurs when subtracting a negative decimal, such as in the problem 1.4 – (–3.2). Students often see the subtraction symbol and instinctively subtract the smaller absolute value from the larger one, arriving at –1.8. They struggle to recognize that subtracting a negative value reverses direction on the number line, turning the operation into adding a positive amount (1.4 + 3.2 = 4.6).

Worksheets in this area focus on numerical calculations without extra clutter. Practice tasks fall into two clear formats:

Add signed decimals, featuring problems such as –4.5 + 2.1 or –1.8 + (–3.6).
Subtract signed decimals, presenting expressions like 5.3 – 8.7 or –2.4 – (–6.1) that require students to rewrite subtractions as additions of the opposite value.

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Tipos de ejercicios: Resta decimales con signo · Suma decimales con signo

Multiplying and Dividing Positive and Negative Numbers

Students learn to multiply and divide positive and negative numbers by applying the correct sign rules to calculate products and quotients.

Punto del currículo 7.NS.A.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world contexts.
Punto del currículo 7.NS.A.3 Resolver problemas del mundo real y matemáticos que involucren las cuatro operaciones con números racionales. (nuestra traducción)

In 7th grade, students learn to multiply and divide signed numbers, extending their understanding of arithmetic to negative values and fractions. They learn that multiplying or dividing two numbers with the same sign results in a positive value—such as (-1)(-1) = 1—while operating with different signs produces a negative result. In division, students work with non-zero divisors and learn that a negative sign can sit in front of a fraction, in the numerator, or in the denominator without changing the overall value.

A common error occurs when students confuse the sign rules for multiplication and division with the rules for addition. For example, children often overgeneralize the phrase "two negatives make a positive" and incorrectly apply it to addition, or they drop the negative sign entirely when dividing a negative number by a positive one. Another frequent stumbling block is interpreting where the negative belongs in a quotient, leading to confusion when rewriting fractions.

In practice, tasks focus on two templates: Multiply signed numbers and Divide signed numbers. Students solve straightforward calculation problems involving pairs of integers and fractions, such as evaluating products like (-4) × 6 or dividing values where they must correctly place the sign in the final quotient.

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Tipos de ejercicios: Divide números con signo · Multiplica números con signo

Finding the Circumference and Area of Circles

Students learn to use standard formulas to calculate the distance around a circle and the space inside it when given a radius or diameter.

Punto del currículo 7.G.B.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

In 7th grade, students learn to know and use the formulas for the circumference and area of a circle. Working with a circle's dimensions, they identify whether they are given the radius or the diameter, apply the appropriate formula, and calculate the total distance around the boundary or the surface area enclosed within it.

A frequent stumbling block is confusing the radius with the diameter. When given a full diameter, students often plug it directly into the area formula without halving it first. Another common error is mixing up the operations across formulas, such as multiplying the radius by two instead of squaring it when computing area.

Practice tasks are built around two templates: Circumference of a circle and Area of a circle. In these exercises, students look at circle diagrams or word descriptions with specified measurements and calculate the missing circumference or area.

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Tipos de ejercicios: Longitud de la circunferencia · Área del círculo

Finding Probabilities of Two-Stage Events

Students learn to find probabilities of two-step events as fractions by mapping out all possible outcomes using tables and organized lists.

Punto del currículo 7.SP.C.8 Hallar probabilidades de eventos compuestos utilizando listas organizadas, tablas, diagramas de árbol y simulación. a. Comprender que, al igual que con los eventos simples, la probabilidad de un evento compuesto es la fracción de resultados en el espacio muestral para los cuales ocurre el evento compuesto. b. Representar espacios muestrales para eventos compuestos utilizando métodos tales como listas organizadas, tablas y diagramas de árbol. Para un evento descrito en lenguaje cotidiano (por ejemplo, “sacar doble seis”), identificar los resultados en el espacio muestral que componen el evento. c. Diseñar y usar una simulación para generar frecuencias para eventos compuestos. Por ejemplo, usar dígitos aleatorios como una herramienta de simulación para aproximar la respuesta a la pregunta: Si el 40% de los donantes tienen sangre tipo A, ¿cuál es la probabilidad de que se necesiten al menos 4 donantes para encontrar uno con sangre tipo A? (nuestra traducción)

Students learn to find the probability of a compound event by displaying all possible outcomes in an organized list, table, or tree diagram. They understand that the probability is the fraction representing the favorable outcomes divided by the total number of outcomes in the sample space, translating everyday descriptions such as “rolling double sixes” into concrete pairs of results.

A common error occurs when students assume every combined result has an equal chance of happening without listing the full sample space. For example, when adding the values of two rolled dice, a student might assume that a sum of 2 has the same likelihood as a sum of 7. They overlook that different orderings matter—a roll of 3 and 4 is distinct from a roll of 4 and 3—which makes certain totals much more likely than others.

Practice tasks focus on concrete two-step situations:

Sum of two dice: Students use a 6-by-6 table showing all 36 possible pairs of rolls to calculate the probability of getting a specific sum, such as rolling an 8 or rolling doubles.
Two draws with replacement: Students list outcomes for picking an object from a set, returning it, and drawing a second time to determine the fraction of outcomes where a specific combination is picked.

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Tipos de ejercicios: Suma de dos dados · Dos extracciones con devolución

Predicting How Often an Event Will Happen

Students use known probabilities to estimate roughly how many times a specific outcome will occur across repeated trials.

Punto del currículo 7.SP.C.6 Aproximar la probabilidad de un suceso aleatorio recopilando datos sobre el proceso aleatorio que lo produce y observando su frecuencia relativa a largo plazo, y predecir la frecuencia relativa aproximada dada la probabilidad. Por ejemplo, al lanzar un dado numérico 600 veces, predecir que saldría un 3 o un 6 aproximadamente 200 veces, pero probablemente no exactamente 200 veces. (nuestra traducción)

Students learn to use a known probability to estimate the frequency of an event over a large number of trials. For example, when rolling a standard number cube 600 times, a seventh grader can use the one-third probability of rolling a 3 or a 6 to predict that the result will occur roughly 200 times, while understanding that real-world trials will rarely land on exactly 200.

A common stumbling block is treating probability as a rigid guarantee rather than a long-run estimate. Students frequently expect an experiment to produce the exact calculated number—assuming that 600 rolls must yield precisely 200 target outcomes—and struggle to accept the natural variation inherent in chance.

Practice tasks present concrete chance scenarios. In How many times to expect?, students calculate expected counts for events like rolling dice or spinning spinners over hundreds of repetitions. In How many red balls to expect?, they use the ratio of colored items in a container to predict how many times a red ball will be drawn across a given number of attempts.

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Tipos de ejercicios: ¿Cuántas bolas rojas cabe esperar? · ¿Cuántas veces cabe esperar?

Volume and Surface Area of Triangular Prisms

Students learn to calculate the volume and total surface area of right triangular prisms using the dimensions of their faces.

Punto del currículo 7.G.B.6 Resolver problemas del mundo real y matemáticos que involucren el área, el volumen y el área superficial de objetos bidimensionales y tridimensionales compuestos por triángulos, cuadriláteros, polígonos, cubos y prismas rectos. (nuestra traducción)

In this grade, students solve mathematical and real-world problems involving right prisms, focusing specifically on triangular prisms. They determine volume by finding the area of the triangular base and multiplying it by the length of the prism. To find the total surface area, they calculate the area of each individual exterior face—two triangular bases and three rectangular sides—and add them together.

A common mistake occurs when students mix up the height of the triangular base with the overall height of the prism. In surface area calculations, students also frequently assume all three rectangular sides are identical, forgetting that different edge lengths on the triangle produce rectangles with different areas.

Practice tasks present labeled three-dimensional drawings through two main activities: Volume of a triangular prism, where students combine the base area and prism height, and Surface area of a triangular prism, where they account for and sum the areas of all five faces.

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Tipos de ejercicios: Volumen de un prisma triangular · Área total de un prisma triangular

Finding the Whole Amount from a Percent

Students learn to find the original total or whole amount when given a percentage and the partial value it represents.

In 7th grade, students build on proportional relationships to work backward from a known part to find the full 100%. Using ratios, fractions, or decimals, they solve for the missing base quantity when given a percentage and the specific amount that percentage equals.

A frequent mistake occurs when students confuse the part with the whole and multiply instead of divide. Accustomed to calculating a percent of a number, a student faced with a statement like "12 is 25% of what number?" might mistakenly calculate 25% of 12, arriving at 3 instead of recognizing that the starting number must be larger than 12.

Worksheets for this skill feature tasks where students find the whole from a percent. A typical problem presents a real-world scenario or a straightforward equation, such as determining the total price of an item if a $15 discount represents 20% of the original cost.

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Finding What Percent One Number Is of Another

Students learn to determine what percentage one quantity represents of another by setting up a ratio and converting it into a percent.

In 7th grade, students learn to compare two values using proportional relationships to find what percent one number is of another. They write the comparison as a part-to-whole ratio or fraction, divide to find a decimal, and convert that decimal into a percentage out of 100.

A frequent error occurs when students invert the two numbers. When asked what percent 6 is of 24, a student may divide 24 by 6 simply because dividing a larger number by a smaller number feels more familiar. This leads to an answer of 400% instead of the correct 25%.

Worksheets using the What percent is it? template present direct prompts such as "14 is what percent of 70?" or real-world comparison questions. Students calculate the quotient and write the final result using the percent symbol.

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Solving Two-Step Equations

Students learn to solve two-step equations involving positive and negative rational numbers to find the value of an unknown variable.

Punto del currículo 7.EE.B.4 Usar variables para representar cantidades en un problema del mundo real o matemático, y construir ecuaciones y desigualdades simples para resolver problemas razonando sobre las cantidades. a. Resolver problemas verbales que conducen a ecuaciones de la forma px + q = r y p(x + q) = r, donde p, q y r son números racionales específicos. Resolver con fluidez ecuaciones de estas formas. Comparar una solución algebraica con una solución aritmética, identificando la secuencia de las operaciones utilizadas en cada enfoque. Por ejemplo, el perímetro de un rectángulo es 54 cm. Su longitud es 6 cm. ¿Cuál es su ancho? b. Resolver problemas verbales que conducen a desigualdades de la forma px + q > r o px + q < r, donde p, q y r son números racionales específicos. Representar gráficamente el conjunto solución de la desigualdad e interpretarlo en el contexto del problema. Por ejemplo: Como vendedor, se le pagan $50 por semana más $3 por venta. Esta semana desea que su pago sea de al menos $100. Escriba una desigualdad para la cantidad de ventas que necesita realizar y describa las soluciones. (nuestra traducción)

In Grade 7, students learn to fluently solve linear equations in the form ax + b = c. They work with rational numbers—including positive and negative integers, fractions, and decimals—using a sequence of inverse operations to undo addition, subtraction, multiplication, or division and isolate the variable.

A common stumbling block is choosing the wrong order of operations to undo. Students frequently try to divide by the coefficient a before subtracting or adding the constant b, which often leads to calculation mistakes. Errors also frequently arise when working with signs, such as forgetting to apply a negative sign when dividing by a negative coefficient.

Worksheet tasks focus directly on the ax + b = c structure. A typical problem presents an equation such as 3x + 8 = -13 or -5x - 4 = 16, asking students to show each step as they balance the equation and determine the value of x.

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Calculating Simple Interest on Savings

Students learn to use proportional relationships and percentages to calculate the simple interest earned on a savings deposit over time.

In 7th grade, students expand their understanding of ratios and proportional relationships to solve multistep percent problems. When working with simple interest, they calculate the amount of money earned on an initial deposit (the principal) over a period of years by multiplying the principal, the annual interest rate as a decimal or fraction, and the time in years.

A frequent error occurs when converting the percentage rate into decimal form. For example, a student might convert 4% to 0.4 instead of 0.04, which inflates the calculated interest by a factor of ten. Another common slip is answering with just the interest earned when a question asks for the total account balance, forgetting to add the interest back to the original deposit.

Tasks in this topic center on interest on a savings deposit. A typical problem presents a real-world scenario—such as depositing $500 into a savings account that earns 3% annual simple interest—and asks students to determine the interest accumulated after 2 years, or to find the total value of the deposit at the end of that period.

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