8th Grade Math

Learning Opportunities to Refresh, Review, and Reinforce

1

Concept/Skill: Cubes and Square Roots

Activity: Cube & Square Root Expressions

8th Grade Math

8th Grade Math

Supplies needed: Pencil, if you have it your math journal page with the blue foldable, a basic calculator.

Concept/Skill:

8th Grade Math

2 Squares and Cube Roots

Activity: Exponent Puzzle

Supplies needed: pencil, scissors, glue or tape, sheet of scratch paper, blue foldable from your math journal or a basic calculator

Concept/Skill: Perimeter, Area, and Similar Shapes

8th Grade Math

3 Activity: Polly Gone

Problem of the Month

Polly Gone

Level A Polly works in a zoo and needs to build pens where animals can live and be safe. The walls of the pens are made out of cubes that are connected together. Polly has 40 cubes and wants to make the largest pen possible, so the animals can move around freely but not get loose. Build the largest area using all 40 cubes. Your walls must: • Be fully enclosed, with no doors or windows so Polly’s animals can’t get out. • Have a height of one cube. • Be joined cube face to cube face. Help Polly by making pens of several shapes and determine which pen provides the largest area for the animals. You might want to build the pen on the grid paper first, so that it will be easier to determine the area. Use the grid paper to show the shape of the pen. Explain to Polly why you believe your pen is the largest one that can be made.

8th Grade Math

Level B The large triangle is made up of five shapes and is drawn on graph paper. Name each of the five shapes and determine the area of each.

Rearrange the shapes to find all possible parallelograms of any size using any number of them. Draw a picture of each parallelogram that you have found, and then determine its area. How did you find all of them? How do you know you found them all?

8th Grade Math

Level C In November 1958, the magazine, Scientific American, showed this diagram on its cover.

Each of the interior rectangles is a square. If square D is 81 square units and square C is 64 square units, what is the area of the other seven squares? What is the area of the entire figure? What is the perimeter of the entire figure? Explain your solutions.

8th Grade Math

Level D A new arena is going to be constructed at a local university. A study is being done to find the best performance or playing area design. Since the arena will be used for many different sports, as well as shows and concerts, the designers want a seating arrangement that allows spectators to be as close as possible to the action. They also want to seat as many front row spectators as possible around the performance area. They have also decided that the boundary of the performance area needs to have straight sides, no curves, due to the building materials they are using. The goal is to have front row seats not more than 20 meters from the center of the performance or playing area. They want to hire you as a consultant to investigate this matter and explain to them which design would best suit their needs. They need to see several examples of possible performance area designs that will fit their constraints. The final recommendation must explain the advantages of the design in terms of the size of the playing area and the number of people they can seat in the front row.

8th Grade Math

Level E Catherine said to Rebecca, “I need to draw an octagon and I want it to be accurate.” Rebecca replied, “I have an easy way to draw an octagon. Start with a large square. Find the midpoint of each side. Now draw a line segment from each midpoint to the two opposite vertices. In the center of the drawing, an octagon will be formed.” “That’s a great method, Rebecca, but I want to make my octagon a certain size. How big do I need to make the original square in terms of area to get an octagon of a certain area?” Catherine asked. Please help Catherine and Rebecca determine these relationships. Fully explain your reasoning. “After you have drawn your octagon, you will see that it comes out as a beautiful regular octagon,” Rebecca exclaimed. “Well, it may be beautiful, but I don’t think it is regular,” challenged Catherine. Who is right? Determine your answer using mathematics.

Problema del mes que Polly Gone

Los problemas del mes (POM) se utilizan de varias maneras para promover la solución de problemas y fomentar la primera practica estándar de matemáticas común que cumple con los Estándares Estatales Centrales: "Darle sentido a los problemas y perseverar en resolverlos". Los POM pueden ser utilizado por un maestro para promover la solución de problemas y para abordar las necesidades diferentes de sus alumnos. Un departamento o nivel de grado puede involucrar a sus estudiantes en un POM para mostrar la solución de problemas como un aspecto clave de las matemáticas. Los POM también pueden usarse en toda la escuela para promover un tema de solución de problemas en una escuela. El objetivo es que todos los estudiantes tengan la experiencia de atacar y resolver problemas no rutinarios y desarrollar sus habilidades de razonamiento matemático. Aunque el objetivo es obtener y justificar soluciones a los problemas, el proceso de aprender a resolver problemas es aún más importante.

El problema del mes está estructurado para proporcionar tareas razonables a todos los estudiantes en una escuela. El POM está diseñado de tal forma para que todos los estudiantes puedan participar, luchar y perseverar de una manera productiva. La versión

8th Grade Math

primaria está diseñada para que todos los estudiantes puedan ingresar a ella y especialmente como un desafío clave para los grados de Kindergarten y Primero. El Nivel A será un desafío para la mayoría de los alumnos de Segundo y Tercer grado. El nivel B puede ser el límite en donde los estudiantes de Cuarto y Quinto grado tienen éxito y comprensión. El nivel C puede compartirse a los estudiantes de Sexto y Séptimo grado.

El Nivel D puede desafiar a la mayoría de los estudiantes de Octavo y Noveno grado, y el Nivel E debería ser un desafío para la mayoría de los estudiantes de secundaria. Estas expectativas de nivel de grado son solo estimaciones y no deben usarse como una expectativa mínima absoluta o como una limitación máxima para los estudiantes. La solución de problemas es una habilidad aprendida, y los estudiantes pueden necesitar mucha experiencia para desarrollar sus habilidades de razonamiento, enfoque, estrategias y tener perseverancia para lograr éxito. El problema del mes se basa en niveles secuenciales de comprensión. Todos los estudiantes deben experimentar el Nivel A y luego avanzar a través de las tareas para profundizar lo más que puedan en el problema. Habrá aquellos estudiantes que no tendrán acceso ni siquiera al Nivel A. Los educadores deben sentirse libres de modificar la tarea para permitir el acceso en algún nivel.

Visión general En el problema del mes Polly Gone, los estudiantes usan polígonos para resolver problemas relacionados con el área. Los temas matemáticos que se encuentran en el POM son los atributos de medición lineal, medición cuadrada, geometría bidimensional, perímetro, área y justificación geométrica. El problema les pide a los estudiantes que exploren los polígonos y la relación de sus áreas en diversas situaciones problemáticas. En el primer nivel del POM, a los estudiantes se les presentan 40 cubos y se les pide que hagan todas las regiones rectangulares posibles usando los cubos como borde. Luego se les pide a los estudiantes que determinen el área de las regiones interiores e identifiquen el rectángulo con el área más grande para hacer un corral para animales en un zoológico. En el Nivel B, se les presenta a los estudiantes una forma triangular en papel cuadriculado compuesta por cinco polígonos más pequeños. Se les pide a los estudiantes que nombren y determinen el área de cada polígono. También se les pide reorganizar las formas para construir tantos paralelogramos diferentes como sea posible. En el Nivel C, los estudiantes reciben un rectángulo que se subdivide en nueve cuadrados más pequeños. Los estudiantes reciben el área de dos de los cuadrados y se les pide que determinen el área de los siete cuadrados restantes. En el Nivel D, los estudiantes exploran conceptos para maximizar el área dado un perímetro fijo. Los estudiantes determinaran qué polígono producirá el área más grande y mantendrán una distancia constante desde el perímetro para diseñar la superficie de juego en una arena deportiva. En el Nivel E, se les pide a los estudiantes que construyan una figura geométrica a partir de un cuadrado. Se produce un octágono dibujando segmentos de línea desde cada vértice hasta sus puntos medios opuestos. Se les pide a los estudiantes que determinen el área del octágono en relación con el área del cuadrado. Se les pide a los estudiantes que justifiquen sus soluciones.

8th Grade Math

8th Grade Math

Problema del mes

Polly Gone

Nivel A Polly trabaja en un zoológico y necesita construir corrales donde los animales puedan vivir y estar seguros. Las paredes de los corrales están hechas de cubos que están conectados entre sí. Polly tiene 40 cubos y quiere hacer el corral más grande posible, para que los animales puedan moverse libremente pero sin escaparse. Construye el área más grande usando los 40 cubos. Tus paredes deben: • Estar completamente cerradas, sin puertas ni ventanas para que los animales de Polly

no puedan salirse. • Tener una altura de un cubo

• Unir cara a cara en cada cubo. Ayuda aPolly haciendo corrales de varias formas y determina qué corral proporciona el área más grande para los animales. Es posible que primero desees construir el corral usando un bolígrafo y un papel cuadriculado, para que te sea más fácil determinar el área. Usa el papel cuadriculado para mostrar la forma del corral. Explica porque Polly cree que su corral es la más grande que puede construirse.

Nivel B

8th Grade Math

El triángulo grande se compone de cinco formas y se dibuja en papel cuadriculado. Nombra cada una de las cinco formas y determina el área de cada una.

Reorganiza las formas para encontrar todos los paralelogramas posibles de cualquier tamaño usando cualquier número de ellos.

Dibuja la imagen de cada paralelogramo que hayas encontrado y luego determina su área. ¿Cómo los encontraste? ¿Cómo sabes que pudiste encontrar todos?

8th Grade Math

Nivel C

En Noviembre de 1958, la revista Scientific American mostró este diagrama en su portada.

Cada uno de los rectángulos interiores es un cuadrado. Si el cuadrado D es de 81 unidades cuadradas y el cuadrado C es de 64 unidades cuadradas, ¿cuál es el área de los otros siete cuadrados? ¿Cuál es el área de toda la figura? ¿Cuál es el perímetro de toda la figura? Explica tus soluciones.

8th Grade Math

Nivel D

Se construirá una estadio nuevo en una universidad local. Se está realizando un estudio para encontrar el mejor rendimiento o diseño del área de juego. Dado que el estadio se utilizará para muchos deportes diferentes, así como también para espectáculos y conciertos, los diseñadores desean una disposición de asientos que permita a los espectadores estar lo más cerca posible de la acción. También quieren acomodar tantos espectadores en primera fila como sea posible alrededor del área de actuación. También han decidido que el límite del área de actuación debe tener lados rectos, sin curvas, debido a los materiales de construcción que están utilizando. El objetivo es tener asientos en la primera fila a no mas de 20 metros del centro del área de actuación o área de juego.

Quieren contratar a un consultor para investigar este asunto y explicarles qué diseño se adaptará mejor a sus necesidades. Necesitan ver varios ejemplos de posibles diseños de las diferentes áreas que se ajusten a sus necesidades. La recomendación final debe explicar las ventajas del diseño en términos de tamaño del área de juego y el número de personas que

pueden sentarse en la primera fila.

8th Grade Math

Nivel E

Catherine le dijo a Rebecca: "Necesito dibujar un octágono y quiero que sea preciso". Rebecca respondió: "Tengo una manera fácil de dibujar un octágono. Comienza con un cuadrado grande. Encuentra el punto medio de cada lado. Ahora dibuja un segmento de línea desde cada punto medio hasta los dos vértices opuestos. En el centro del dibujo, se formará un octágono ". "Ese es un gran método, Rebecca, pero quiero que mi octágono tenga un tamaño determinado. ¿Qué tan grande necesito hacer el cuadrado original en términos de área para obtener un octágono de una área determinada? Catherine preguntó. Ayuda a Catherine y Rebecca en como puden determinar estas relaciones. Explica completamente tu razonamiento. "Después de haber dibujado tu octágono, verá como sale un hermoso octágono regular", exclamó Rebecca.

"Bueno, puede ser hermoso, pero no creo que sea regular", desafió Catherine.

¿Quién tiene la razón? Determina tu respuesta usando las matemáticas.

Supplies needed: scissors, graph paper (optional), pencil, basic calculator

8th Grade Math

4

Concept/Skill: Linear Pattern Work

Activity: How Much Does a 100 x 100 In-N-Out Burger Cost

Linear Pattern work:

How Much Does A 100×100 In-N-Out Cheeseburger Cost?

20 X 20

The Situation

In-N-Out ordinarily sells hamburgers, cheeseburgers, and Double-

Doubles (two beef patties and two slices of cheese). While they don’t

advertise it, they have a secret menu which includes a burger where

you can order as many extra beef patties and cheese slices as you

like. The prices and nutrition information are not listed though. The

most common orders are 3×3’s (read as “three by three”) and 4 by 4’s

(read as “four by four”) that contain three and four layers of beef and

cheese, respectively. However some people have ordered 20×20’s

(pictured above) and even a 100×100 (pictured below)!

8th Grade Math

8th Grade Math

The Challenge(s)

How much money does a 3 x 3 cost?

How many calories is a 3 x 3?

How much money does a 20 x 20 cost?

How many calories is a 20 x 20?

How much money does a 100 x 100 cost?

How much money does an N x N cost?

How many calories is an N x N?

Create a table, graph, and an equation. Question(s) To Think About These questions may be useful in helping your down the problem solving path:

How would you describe what ingredients are in a 3 x 3? Did you realize it is a cheese burger with all the cheese burger toppings plus 2 additional layers of beef patties and cheese?

How can we figure out how much an additional beef pay and cheese slice cost?

What are the differences between a cheeseburger and a Double-Double?

Where do you see the cost of the extra layer mentioned in the words in the table, graph, and symbols?

Where do you see the cost of the cheeseburger in the words in the table, graph, and symbols?

How would you answer change if you had started with a Double-Double and added 98 layers versus a Cheeseburger and added 99 layers?

8th Grade Math

A picture of a cheeseburger (to establish that it has one cheese slice and one burger

patty):

A picture of a Double-Double (to establish that it has two cheese slices and two

burger patties):

8th Grade Math

What is the problem you are trying to solve?

What do you already know about

the problem?

What do you need to know to

solve the problem?

Your conclusion and work to back it up. (Table, graph, equation and

detailed explanation.

Supplies needed: graph paper, scratch paper, and a basic calculator

8th Grade Math

5

Concept/Skill: Solving Systems of Equations

Activity:

SOLVING SYSTEMS OF EQUATIONS: SUBSTITUTION

MILD EXAMPLE

Example a) 𝑦 = 3𝑥 + 6

b) 𝑦 = 5𝑥 − 2

● Step 1: Set the equations equal ● Step 2: Solve the equation.

○ Move x’s to one side by using inverse operations

○ Move numbers to other side by using inverse operations

● Step 3: Substitute x value into equations to find y value

● Step 4: Solve for y ● Step 5: write the solution as an ordered

pair (x,y)

● Step 1: 3𝑥 + 6 = 5𝑥 − 2 ● Step 2: 3𝑥 + 6 = 5𝑥 − 2

−3𝑥 − 3𝑥

6 = 2𝑥 − 2

+2 + 2

8 = 2𝑥

4 = 𝑥

● Step 3&4: 𝑦 = 3𝑥 + 6 𝑦 = 3(4) + 6

𝑦 = 12 + 6

𝑦 = 18

● Step 5: (4,18)

MEDIUM EXAMPLE

8th Grade Math

8th Grade Math

SPICY EXAMPLE

8th Grade Math

MILD PRACTICE

8th Grade Math

MEDIUM/SPICY PRACTICE

8th Grade Math

8th Grade Math

8th Grade Math

NOTES ON ELIMINATION

Solving Systems with

ELIMINATION

(Addition and Subtraction)

1. Look for a variable in both equations with opposite coefficients (same number—one positive, one negative).

Example: 3x and -3x

2. Add the equations. The variable with opposite coefficients will cancel out.

● If one of the variables has the same coefficient (same number, same sign), you will need to multiply one of the equations by -1. This will change the sign of every term in that equation.

● Now, add the equations.

3. Solve for the variable that DID NOT cancel out.

4. Substitute your answer back into the original equations. If you get the same answer for the second variable, you have solved the system correctly! If not, go back and check your work for a mistake.

5. Write the solution as a coordinate pair.

( , )

X Y

Solving Systems with ELIMINATION

(Multiplication)

1. If you cannot eliminate (cancel out) a variable by adding or subtracting, you need to use multiplication.

2. THE GOAL: Multiply one or both equations by a number to get one of the variables (x or y) to have opposite coefficients (same number—one positive, one negative).

Example: 5y and -5y

3. After using the Distributive Property to multiply through the parentheses, rewrite the equations.

4. Box the variables with opposite coefficients and cancel them out.

5. Add the equations to solve for the variable that DID NOT cancel out.

6. Substitute your answer back into the original equations. If you get the same answer for the second variable, you have solved the system correctly! If not, go back and check your work for a mistake.

7. Write the solution as a coordinate pair.

( , )

X Y

8th Grade Math

EXAMPLES:

Addition Subtract (Add opposite)

Multiplication (1) Multiplication (2)

-----------------------------

We can add to cancel out y’s (same number different signs)

−9𝑥 = 9

𝑥 = −1

-----------------------------

Then use substitution to get other variable

−5𝑥 − 4𝑦 = −11

−5(−1) − 4𝑦 = −11

5 − 4𝑦 = −11

−5 − 5

−4𝑦 = −16

𝑦 = 4

-----------------------------

Write coordinates

(x,y)

(-1, 4)

-----------------------------

We can subtract to cancel out y’s (same number, same signs)

7𝑥 − 9𝑦 = 5

−1(−4𝑥 − 9𝑦 = −17)

----------------------------

7𝑥 − 9𝑦 = 5

4𝑥 + 9𝑦 = 17

-----------------------------

11𝑥 = 22

𝑥 = 2

-----------------------------

Then use substitution to get other variable

7𝑥 − 9𝑦 = 5

7(2) − 9𝑦 = 5

14 − 9𝑦 = 5

−14 − 14

−9𝑦 = −9

𝑦 = 1

-----------------------------

-----------------------------

We need to multiply to be able to cancel out y’s

8𝑥 + 14𝑦 = 4

2(−6𝑥 − 7𝑦 = −10)

-----------------------------

8𝑥 + 14𝑦 = 4

−12𝑥 − 14𝑦 = −20

-----------------------------

−4𝑥 = −16

𝑥 = 4

-----------------------------

Then use substitution to get other variable

8𝑥 + 14𝑦 = 4

8(4) + 14𝑦 = 4

32 + 14𝑦 = 4

−32 − 32

14𝑦 = −28

𝑦 = −2

-----------------------------

-----------------------------

We need to multiply both equations to cancel out x’s

3(−5𝑥 + 5𝑦 = −25)

5(3𝑥 + 2𝑦 = 10)

-----------------------------

−15𝑥 + 15𝑦 = −75

15𝑥 + 10𝑦 = 50

-----------------------------

25𝑦 = −25

𝑦 = −1

-----------------------------

Then use substitution to get other variable

−15𝑥 + 15𝑦 = −75

−15𝑥 + 15(−1)= −75

−15𝑥 − 15 = −75

+15 + 15

−15𝑥 = −60

𝑥 = 4

-----------------------------

8th Grade Math

Write coordinates

(x,y)

(2, 1)

Write coordinates

(x,y)

(4, -2)

Write coordinates

(x,y)

(4, -1)

8th Grade Math

8th Grade Math

MEDIUM/SPICY PRACTICE

KEY:

8th Grade Math

Supplies needed: pencil, scratch paper, colored pencil or crayons, basic calculator, math journal if you have it at home.

6

Concept/Skill: Pythagorean Theorem

Activity: Bird and Dog Race

Illustrative Mathematics

1

8.G Bird and Dog Race Alignments to Content Standards: 8.G.B Doug is a dog, and his friend Bert is a bird. They live in Salt Lake City, where the streets are 1/16 miles apart and arranged in a square grid. They are both standing at 6th and L. Doug can run at an average speed of 30 mi/hr through the streets of Salt Lake, and Bert can fly at an average speed of 20 mi/hr. They are about to race to 10th and E.

8th Grade Math

a. Who do you predict will win, and why?

b. Draw the likely paths that Doug and Bert will travel. c. What will you need to compare, in order to determine the winner? d. Devise a plan to calculate these, without measuring anything. e. Who will win the race?

8th Grade Math

8.G Carrera de pájaros y perros

Doug es un perro y su amigo Bert es un pájaro. Ambos viven en Salt Lake City, donde las calles están a 1/16 millas de distancia entre ellas y arregladas en hileras decuadros como una cuadricula. Ambos están ubicados en la esquina de 6ta y L. Doug puede correr a una velocidad promedio de 30 millas/hora por las calles de Salt Lake, y Bert puede volar a una velocidad promedio de 20 millas/hora. Ambos competiran para llegar a la esquina de 10th y E.

a) ¿Quién crees que ganará y por qué? b) Dibuja los posibles caminos que recorrerán Doug y Bert. c) ¿Qué necesitarás comparar para determinar a el ganador? d) Diseña un plan para hacer tus calculos, sin medir nada, por ejemplo: ¿Quién ganará la carrera?

Supplies needed: something to write with

Concept/Skill: Pythagorean Theorem

Activity: Pythagorean Theorem Mazes

8th Grade Math

7

Pythagorean Theorem

Watch this cool video to see the idea

applied in real life

The Pythagorean Theorem - Science of NFL Football https://www.youtube.com/watch?v=Grzy

-ZAotB0

Use this review information to refresh

yourself on how to solve for the missing

sides of a missing right triangle and then

challenge yourself with the mazes on the

following sheets. Begin at start and solve

to find the correct answer – follow the

path!

8th Grade Math

8th Grade Math

8th Grade Math

Supplies needed: Pencil, if you want to watch the video you will need access to the internet but you don’t need it to complete the mazes

8

Concept/Skill: Systems of Equations, Linear Equations, Transformations, Solving Equations

Activity: Khan Academy Instructions 8th grade math and Boost

To create your account, go to the Khan Academy homepage. If you click Parents, start here, you will be guided through the process of adding children. No matter what you select, your account will still allow you the option to learn, coach, or parent.

If you’re looking to parent or coach a student, once you've created your account, you can access Khan Academy's coaching tools by clicking your name at the top right of your screen and then selecting either Your students or Your children.

8th Grade Math

Once you've created an account for yourself, you can create accounts for your students or children. If your child is under 13, their account will have special privacy considerations.

For instructions, choose the appropriate guide below.

If you're a parent and you already have a Khan Academy account, you can visit this page and click the Add your child button. (If you don't have an account for yourself yet, go to this link to create one.)

Once you click Add your child, you will be asked to enter your child's birthday:

Your child's birthday is needed to determine permissions. Accounts for students younger than 13 are managed by a parent to protect the child's privacy. You can find more information about accounts for students younger than 13 here.

If Your Child is Younger Than Age 13

You will get a form to fill in for your child (note that the account is a restricted child account):

8th Grade Math

Fill in the form and click Next to create the account.

If Your Child is Age 13 or Older

You will be asked if your child has an email address. If your child has an email address you may either send your child an invitation to join Khan Academy, or you could create the account for your child:

8th Grade Math

If your child does not have an email address (or you choose to create the account yourself) you will get a form similar to the form if your child were younger than 13:

8th Grade Math

Fill in the form and click Next.

Multiple Children

If you have more children you can continue creating accounts now by clicking Add another child.

In addition, you can click Add your child from your parent homepage at any time.

After Account Creation

After you have created your child account(s) you can return to your homepage and log out and your child can log into his/her account.

Note: Always remember to log out of your account when you are done using it and teach your child to do the same! We frequently get reports of progress being made on the wrong account because parents or children forgot to log out.

After you have your accounts set up you can then begin working in your courses. Please follow the list below depending on your math class. We will be assigning additional topics in a few weeks. We are encouraging students that took Boost to continue to review Boost material. Students should aim to work on math for 40 minutes per day. The goal of our learning opportunities will be to review, reinforce and refresh!

8th Grade Math Boost

1. From the courses tab choose 8th grade math

1. From the courses tab choose Algebra 1

2. Under the courses summary choose systems of equations.

2. Under the course summary choose systems of equations.

3. Begin working through Systems of Equations subtopics.

3. Begin working through Systems of Equations subtopics.

4. Once completed, you can then proceed to Geometry Transformations.

4. Once completed, you can then proceed to Inequalities (systems & Graphs)

5. Proceed to Solving Equatioins with One Unknown

5. Students may then proceed to Quadratics (Multiplying and Factoring)

8th Grade Math

6. Once completed, students can move to Linear Equations and Functions

6. Proceed to Quadratic Functions and Equations

7. Last Unit: Geometry 7. Last Unit: Exponents and Radicals

Supplies needed: computer with internet access