8 Math Curriculum3.docx · Web viewThe size of a TV screen is given as the diagonal length, and you want to find out the maximum diagonal size of a TV that will fit in that space,

Unit Title: Measuring, Representing our measuring, and Reading others’ representations of measuring Grade Level: 8

Timeframe: Marking Period 1Essential Questions

(1) Can we become confident in our knowledge of quantities, units, and values of measuring that are used in relationships?(2) Can we become effective and efficient at representing relationships in all four traditional representations—tables, symbols, graphs, and narratives?(3) Can we become effective and efficient at asking and answering typical mathematical questions? (4) Can we become effective and efficient at talking about relationships given in any representation?(5) Can we become effective and efficient at reading and understanding others’ relationships and questions given in any representation?

New Jersey Student Learning Standards

Standards/Cumulative Progress Indicators (Taught and Assessed):

8.EE.B.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

8.F.B.4. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

8.F.A.2. Compare properties (e.g. rate of change, intercepts, domain and range) of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

8.EE.B.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

8.EE.C.7 Solve linear equations in one variable.

a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, into an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

b. Solve linear equations with rational number coefficients, including equations whose solutions include expanding expressions using the distributive property and collecting like terms.

21st Century Skills Standard and Progress Indicators:

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CRP4. Communicate clearly and effectively and with reason.CRP8. Utilize critical thinking to make sense of problems and persevere in solving them.CRP11. Use technology to enhance productivity. CRP12. Work productively in teams while using cultural global competence.

Instructional Plan ReflectionDiagnostic AssessmentGrade 8 Math – Cumulative Assessment

SLO - SWBAT Student Strategies Formative Assessment Activities and Resources Reflection

CAR © 2009

8.EE.B.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

Do Now – Count around the room. Make combinations. [5 minutes]

Direct Instruction/Modeling

Small Group Instruction

Constructed responses from the Type 2-3 Bank and Carnegie Learning

PARCC Released Items

Performance Tasks

Standards Based Problems Aligned to Touchpoint and End of Unit Assessment

Exit Ticket - Standards Based Problems Aligned to Touchpoint and End of Unit Assessment

8.EE.5 – Touchpoint

Standards Based Constructed Response Examples

The graphs below show the cost of buying pounds of

fruit. One graph shows the cost of buying pounds of

peaches, and the other shows the cost of buying

pounds of plums.

a. Which kind of fruit costs more per pound?

Explain.

b. Bananas cost less per pound than peaches or

plums. Draw a line alongside the other

graphs that might represent the cost of

buying pounds of bananas.

Example 2

The diameter of the Moon is approximately miles. The diameter of the Sun is approximately

miles.

How many times larger is the diameter of the Sun than the diameter of the Moon? Write this value in both scientific notation and in standard form.

Constructed Response ResourcesEdConnect 8.EE.5 - Type 2-3 Bank

Carnegie Learning Course 3

PARCC Released Items#14-15

http://parcc-assessment.org/images/releaseditems/

Grade_08_Math_Item_Set.pdf

Performance Task ResourcesIllustrative Mathematics Performance Tasks 8.EE.5

https://www.illustrativemathematics.org/

content-standards/8/EE/B/5

Direct Instruction ResourcesEngageNY Lessons 8.EE.5

https://www.engageny.org/search-site?search=8.EE.5

Learnzillion Lesson Plans Introduction to Linearity Lessons 1-

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https://www.illustrativemathematics.org/content-standards/8/EE/B/5



http://parcc-assessment.org/images/releaseditems/Grade_08_Math_Item_Set.pdf



Use words, numbers, and/or pictures to show your work.

8.F.B.4. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.






Performance Tasks



8.F.4 – Touchpoint

Standards Based Constructed Response Example

Water Fun Plus charges customers $50 per day to rent a canoe and $25 as a nonrefundable insurance fee.

Part A. Write a function that represents the total cost y of renting a canoe for x days at Water Fun Plus.

Part B. What do the slope and y-intercept in the function for Water Fun Plus represent in terms of the context?

Part C. Make a table of x and y values to show the cost of renting a canoe at Water Fun Plus for the first 5 days. Use this table to determine whether the function is linear. Explain how you know whether or not the function is linear.

Part D. Another company, Outdoor Adventures, charges $70 per day to rent a canoe, but with no insurance fee. Write a function to model the total cost y of renting a canoe for x days at Outdoor Adventures. Explain how this function compares to the function written in part A for Water Fun Plus.


Constructed Response ResourcesEdConnect 8.F.4 - Type 2-3 Bank


PARCC Released Items#22



Performance Task ResourcesIllustrative Mathematics Performance Tasks 8.F.4


content-standards/8/F/B/4

Direct Instruction ResourcesEngageNY Lessons 8.F.4

https://www.engageny.org/search-site?search=8.F.4

Learnzillion Lesson Plans Introduction to Linearity Lessons 2-

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8.F.A.2. Compare properties (e.g. rate of change, intercepts, domain and range) of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.






Performance Tasks




Standards Based Constructed Response ExampleJonah and Noah are saving money to buy a video game system. Each boy saves a different fixed amount of money per week. Jonah made a table and Noah graphed a linear equation through the points (3,55) and (8,90)to show some of the total amounts they each had saved at the end of various weeks.

Part A Which boy saves a greater amount each week, and how much more does he save per week? Show your work. Part B Both boys continue saving money at their respective rates. At the end of 15 weeks, which boy will have saved the greatest amount of money, and


Updated






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how much more will he have saved? Show your work.

One quantity that can be described very usefully by quadratic functions is motion. For example, a table, a graph, or an equation can give the motion of a falling object, the movement of a planet around the Sun, or the swinging of a pendulum. In this problem, you will study the motion of a soccer ball.

Part A. This table shows some points on the path of a soccer ball that was kicked from the ground. The x-values represent the horizontal distance in feet from the starting point, and the y-values give the vertical distance. Plot the points on the graph below and sketch the curve.

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Part B. Based on the graph, at what horizontal distance from the starting point do you think the soccer ball will reach its maximum height? Explain your answer.

Estimate the maximum height from the graph.

Part C. The path of the soccer ball forms a parabola. The equation for the function is

Test the values you found in part B for the maximum point of the graph. See whether those values for the x- and y-coordinates satisfy the equation. If not, adjust your values until they satisfy the equation.

Part D. Test four other points from the table in the equation. Be sure to show your work.

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Part E. Another player kicks the soccer ball. It goes higher than the first ball but not as far. It reaches a maximum height of 36 feet and goes a horizontal distance of 48 feet. At a point when the ball has traveled a horizontal distance of 12 feet from the kicker, its height is 27 feet. When the ball has traveled a horizontal distance of 44 feet from the kicker, its height is 11 feet.

From this description, create a table of values for the

points you know, including the starting point. In addition to the points given in the description, think about other points you can figure out from the information given and the shape of the graph. Explain how you know these points.

Part F. Plot these points and sketch the curve on the same graph in part A.

Part G. Study the pattern in the equation given in part

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C for the path of the first soccer ball,

Which number gives the maximum height of the ball? Which number tells how far the graph is shifted to the right? (Remember

that in the equation the maximum or minimum is on the y-axis.)

Fill in those values to create the equation for the path of the second soccer ball. Then, choose one of the given points and solve to find the value of a. Write the full equation.

Part H. After writing your equation, test three values from your table in part E. Be sure to show your work.

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8.EE.B.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.






Performance Tasks



8.EE.6– Touchpoint

Standards Based Constructed Response ExampleUse the graph to answer the questions below.

Part A. If triangles ABC and EDC are similar and

what is the slope of line l? Show your work.

Part B. What is the equation of line l if it intercepts

the y-axis at Show your work.




Performance Task ResourcesIllustrative Mathematics

Performance Tasks 8.EE.6https://www.illustrativemathematics.org/content-standards/8/EE/B/6



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8.EE.C.7 Solve linear equations in one variable.

a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, into an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

b. Solve linear equations with rational number coefficients, including equations whose solutions include expanding expressions using the distributive property and collecting like






Performance Tasks




Standards Based Constructed Response ExampleKiera is comparing the cost of two swimming pool memberships to determine which she will buy. Pool K charges a $30 membership fee plus $5 per visit. Pool M charges a $10 membership fee plus $10 for every two visits.

Part A. For each pool, write an equation that represents C, the total cost forn visits. Identify the slope and intercept of each equation.

Part B. Use comparative language (greater than, less than, equal to) to describe the relationships between

the slope of the equation representing Pool K and the slope of the equation representing Pool M

the intercept of the equation representing Pool K and the intercept of the equation representing Pool M

Part C. For what value of n, if any, are the costs of both plans equal? Show or explain your work.

Part D. Generalize your response to Part C to apply to any system of equations with slopes and intercepts that have the relationships you found in Part B.


Questions 14-29






Performance Tasks 8.EE.7https://

www.illustrativemathematics.org/content-standards/8/EE/C/7



Learnzillion Lesson Plans Solving Linear Equations

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terms.

Summative Assessment

Grade 8 Math – End of Unit 1 Assessment in EdConnect

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Unit Title: Systems of equations, Expressions and equations with radicals and exponents

Grade Level: 8

Timeframe: Marking Period 2

Essential Questions

(1) Can we become confident in our knowledge of quantities, units, and values of measuring that are used in relationships?(2) Can we become effective and efficient at representing relationships in all four traditional representations—tables, symbols, graphs, and narratives?(3) Can we become effective and efficient at comparing multiple relationships?(3) Can we become effective and efficient at asking and answering typical mathematical questions of comparing relationships? (4) Can we become effective and efficient at talking about relationships given in any representation?(5) Can we become effective and efficient at reading and understanding others’ relationships and questions given in any representation?(6) Can we become effective and efficient at working with very large and very small numbers?



8.EE.C.8 Analyze and solve pairs of simultaneous linear equations.

a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

c. Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

8.F.A.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

8.F.A.3 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

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8.F.B.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

8.EE.A.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3-5 = 3-3 = 1/33 = 1/27.Solve linear equations in one variable.

8.EE. A.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 108 and the population of the world as 7 times 109, and determine that the world population is more than 20 times larger.

8.EE.A.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology



Instructional Plan ReflectionDiagnostic AssessmentGrade 8 Math – Cumulative Assessment

CAR © 2009

Grade 8 Math – End of Unit 1 AssessmentUnit 1 Touchpoints

SLO - SWBAT Student Strategies Formative Assessment Activities and Resources Reflection8.EE.C.8 Analyze and solve

pairs of simultaneous linear equations.a. Understand that

solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

c. Solve real-world and






Performance Tasks




Standards Based Constructed Response Example The Johnson family bought 5 hamburgers and 4 orders of fries for $15.00. The Andrews family bought 3 hamburgers and 2 orders of fries for $8.50. Sales tax was included in the cost of each hamburger and order of fries.

Part AWrite a system of linear equations to represent this scenario. Use h to represent the cost of a hamburger and f to represent the cost of an order of fries.

Part BFind the cost of a hamburger and the cost of an order of fries. Show your work or explain in words how you solved the system of equations.

Part CHow much would it cost to buy 6 hamburgers and 3 orders of fries?

Summer is planning to take aerobics classes at a gym. Non-members pay $8 per class, while members pay $4 per class and a $20 membership fee.

Part AGraph the equations that represent the cost of the two plans.

Part BHow many classes would Summer need to take for the costs of both plans to be the same?


Questions 11-28



Performance Tasks 8.EE.8https://www.illustrativemathematics.org/content-standards/8/EE/C/8



Learnzillion Lesson Plans Systems of Linear Equations

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mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

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8.F.A.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.






Performance Tasks





The following table shows the amount of garbage

that was produced in the US each year between 2002

and 2010 (as reported by the EPA).

(years)

2002

2003

2004

2005

2006

2007

2008

2009

2010

(million tons)

239

242

249

254

251

255

251

244

250

Let's define a function which assigns to an input (a

year between 2002 and 2010) the total amount of

garbage, , produced in that year (in million tons). To

find these values, you can look them up in the table.

a. How much garbage was produced in 2004?

b. In which year did the US produce 251 million

tons of garbage?

c. Does the table describe a linear function?

d. Draw a graph that shows this data.



PARCC Released Items#7-8





content-standards/8/F/A/1



Learnzillion Lesson Plans Functions Lessons 1-3


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8.F.A.3 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2

giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.






Performance Tasks





a. Decide which of the following points are on

the graph of the function :

o

o Find 3 more points on the graph of

the function.

b. Find several points that are on the graph of

the function .

o Plot the points in the coordinate

plane. Is this a linear function?

o Support your conclusion.

c. Graph both functions and list as many

differences between the two functions as

you can.








content-standards/8/F/A/3



Learnzillion Lesson Plans Nonlinear Functions Lessons 1-6

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8.F.B.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.






Performance Tasks



8.F.5– Touchpoint


Antonio and Juan are in a 4-mile bike race. The graph

below shows the distance of each racer (in miles) as a

function of time (in minutes).

a. Who wins the race? How do you know?

b. Imagine you were watching the race and had

to announce it over the radio, write a little

story describing the race.





content-standards/8/F/B/5



Learnzillion Lesson Plans Functions Lessons 4-10


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8.EE.A.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3-5 = 3-3 = 1/33 = 1/27.Solve linear equations in one variable.






Performance Tasks




Standards Based Constructed Response ExamplePart A Melanie stated that the expression

is equivalent to . Explain whether or not Melanie is correct.

Part B What is the value of the expression

Part C Using the equation determine the value of m when n = 10. Explain your answer.




Performance Tasks 8.EE.1https://www.illustrativemathematics.org/content-standards/8/EE/A/1


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https://www.illustrativemathematics.org/content-standards/8/EE/A/1



8.EE. A.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 108 and the population of the world as 7 times 109, and determine that the world population is more than 20 times larger.






Performance Tasks





It is said that the average person blinks about 1000

times an hour. This is an order-of-magnitude

estimate, that is, it is an estimate given as a power of

ten. Consider:

100 blinks per hour, which is about two

blinks per minute.

10,000 blinks per hour, which is about three

blinks per second.

Neither of these are reasonable estimates for the

number of blinks a person makes in an hour. Make

order-of-magnitude estimates for each of the

following:

a. Your age in hours.

b. The number of breaths you take in a year.

c. The number of heart beats in a lifetime.

d. The number of basketballs that would fill

your classroom.



Performance Task Resources Illustrative Mathematics

Performance Tasks 8.EE.3https://

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8.EE.A.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology






Performance Tasks




Standards Based Constructed Response ExampleA study showed that in 2011, 18- to 24-year-olds in

the United States sent or received a total of

text messages per day.

Part A What is the total number of text messages the 18- to 24-year-olds sent or received in 2011? Give your answer in scientific notation. Show or explain

your work.

Part B In 2011, the number of 18- to 24-year-olds who sent or received text messages was

. On average, how many text messages did each of these people send or receive per day? Round your answer to the nearest whole number.

Show or explain your work.






Performance Task Resources Illustrative Mathematics

Performance Tasks 8.EE.4https://www.illustrativemathematics.org/content-standards/8/EE/A/4







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Unit Title: Transformations & the Pythagorean Theorem

Grade Level: 8


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Essential Questions

(1) Can we become confident in our knowledge of quantities, units, and values of measuring that are used in relationships?(2) Can we become effective and efficient at representing relationships in all four traditional representations—tables, symbols, graphs, and narratives?(3) Can we become effective and efficient at asking and answering typical mathematical questions? (4) Can we become effective and efficient at talking about relationships given in any representation?(5) Can we become effective and efficient at reading and understanding others’ relationships and questions given in any representation?



8.F.B.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

8.EE.8 Analyze and solve pairs of simultaneous linear equations.

8.G.1 Verify experimentally the properties of rotations, reflections, and translations

8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

8.G.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

8.G.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

8.G.6 Explain a proof of the Pythagorean Theorem and its converse.

8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

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Instructional Plan ReflectionDiagnostic AssessmentGrade 8 Math – Cumulative AssessmentGrade 8 Math – End of Unit 2 AssessmentUnit 2 Touchpoints

SLO - SWBAT Student Strategies Formative Assessment Activities and Resources Reflection

CAR © 2009

8.G.1 Verify experimentally the properties of rotations, reflections, and translations






Performance Tasks



8.G.1 – Touchpoint


On Your Mark …

This task will explore the use of transformations in interpreting the tracks and paths of several different race cars. Each grid below represents a salt flat with an area of 25 square kilometers that is used by a racing association. Each unit in the grid represents 0.25 kilometer, and the positivey-axis represents movement north.

Part A. The grid below shows the path of a race car that may continue in either direction, represented by line AB. Another race car follows an identical track that is located 1 kilometer north of line AB.

Sketch the figure that represents the new track, CD, on the graph below and label it.

Constructed Response ResourcesEdConnect 8.G.1 - Type 2-3 Bank





Performance Task ResourcesIllustrative Mathematics Performance Tasks 8.G.1


content-standards/8/G/A/1

PMI Math Labshttps://njctl.org/courses/math/8th-

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Direct Instruction ResourcesEngageNY Lessons 8.G.1

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What kind of figure is represented by CD? Explain your answer, including the type of transformation that occurred.

Write an equation to represent CD.

If is a point on line AB, what are the coordinates of the corresponding point on the new figure?

Part B. The track shown by line AB on the grid below goes from northwest to southeast. Another car follows a track that is identical to AB but goes in the direction from northeast to southwest, using the

north-south line at as a line of symmetry.

Sketch the figure that represents the new track, EF, on the graph below and label it.


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What kind of figure is represented by EF? Explain your answer, including the type of transformation that occurred.

Write an equation to represent EF.

If is a point on line AB, what are the coordinates of the corresponding point on the new figure?

Part C. Line segment PQ represents the path of a race car that starts at point P and stops at point Q. When

this path is reflected over the line the path for the race car that starts at point R and stops at point S is obtained.

Sketch the figure that represents the new path,

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RS, on the graph below and label it.

What kind of figure is represented by RS? Explain your answer.

What are the lengths of paths PQ and RS? Show your work.

If is a point on line segment PQ, what are the coordinates of the corresponding point on the new figure?

Part D. Another race car follows a path that is identical to path PQ but is located 2 kilometers west and 0.5 kilometer north.

Sketch the figure that represents the new path, TU, on the graph below and label it.

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What kind of figure is represented by TU? Explain your answer, including the type of transformation that occurred.

What is the length of path TU? Show your work.

If is a point on line segment PQ, what are the coordinates of the corresponding point on the new figure?

Part E. Now consider the path of a race car that starts at point P but goes in a direction that is 90° clockwise from the car that is on path PQ. All other parts of the new path, PZ, are identical to PQ except the direction of travel.

What kind of figure is represented by PZ? Explain your answer, including the type of transformation that occurred.

Use the relationship between the slopes of perpendicular lines to determine the location of

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point Z.

Sketch the figure that represents the new path, PZ, on the graph below and label it.

What is the length of path PZ?

Do you think each point on PZ corresponds to a point on PQ? Explain how you know and give an example to support your answer.

Part F. Use the results from parts A and B and from parts C through E to state a conclusion about the figure that results from translating, reflecting, or rotating a line or line segment. Include an explanation of the type of figure that results from transformation, the length of the figure, and the corresponding points.

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8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.






Performance Tasks





Below is a picture of two rectangles with the same

length and width:

a. Show that the rectangles are congruent by

finding a translation followed by a rotation

which maps one of the rectangles to the

other.

b. Explain why the congruence of the two

rectangles can not be shown by translating

Rectangle 1 to Rectangle 2.

c. Can the congruence of the two rectangles be

shown with a single reflection? Explain.













Learnzillion Lesson Plans Grade 8; Unit 2

https://learnzillion.com/resources/64248-understanding-

congruence-through-transformations


grade-math/2d-geometry/ CAR © 2009



https://learnzillion.com/resources/64248-understanding-congruence-through-transformations













8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.






Performance Tasks





A student reflected across the y-axis to

obtain where X corresponds to X′, Y corresponds to Y , and ′ Zcorresponds to Z .′

Part A What are the coordinates of X′, Y′, and Z′? Explain how you found your answers.

Part B Explain how the reflection across they-axis

affects the side lengths of .

Part C Explain how the reflection across they-axis

affects the angle measures of


Questions 1-6








Learnzillion Lesson Plans Understanding Similarity Lessons 1-


resources/64249-understanding-similarity



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https://learnzillion.com/resources/64249-understanding-similarity










8.G.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.






Performance Tasks





Determine, using rotations, translations, reflections,

and/or dilations, whether the two polygons below

are similar.

The intersection of the dark lines on the coordinate

plane represents the origin (0,0) in the coordinate

plane.












Learnzillion Lesson Plans Understanding Similarity Lessons 9-


resources/64249-understanding-similarity



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8.G.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.






Performance Tasks




Standards Based Constructed Response ExampleMeasures of Angles

In the figure below, lines m and n are parallel and are

intersected by transversals and .

Use this diagram to answer the questions below.

Part A. List all six combinations of angles whose sum

will be equal to 180° formed by lines

Part B. Use one of the combinations of angles you










Learnzillion Lesson Plans Geometric Relationships

https://learnzillion.com/resources/64261-geometric-

relationships



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https://learnzillion.com/resources/64261-geometric-relationships










listed in part A and what you know about angles formed by parallel lines and a transversal to determine the sum of the measures of angles 5, 9, and 11. Explain your work.

Part C. If use your answer in part B to determine the sum of the measures of angles 5 and 11. Explain your work.

Part D. If use what you know about angles formed by parallel lines and a transversal to determine the measure of angle 4. Explain your work.

Part E. Use your answers from the parts above to determine the measures of angles 3 and 5. Explain your work.

Part F. Given that angles 8 and 14 are exterior angles of the triangle containing angles 5, 9, and 11, make statements about the measures of the angles within a triangle and about how the measure of an exterior angle of a triangle relates to the measures of the angles within the triangle.

Part G. Draw line p above line m such that line p is

parallel to line m and intersects lines

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Label the new angle formed by lines p and angle 15

and the new angle formed by lines p and angle 16. Determine the measures of the angles in the new

triangle formed by lines

Part H. How do the measures of each of the angles in this new triangle compare with the measures of angles 5, 9, and 11?

Part I. Cut out the triangles formed by lines

in the diagram. Line up the angles with the same measures in both triangles so that the smaller triangle is on top of the larger triangle. Using the centimeter side of a ruler, measure the length of one of the legs of each triangle that is between angles of the same measure. Then, measure the length of another pair of corresponding legs of the triangles. Determine the ratio of

for both pairs of corresponding legs of the triangles.

Part J. How do the ratios you found in part I compare? What does this tell you about the triangles? Make a statement about corresponding angles and legs within triangles such as this.

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8.G.6 Explain a proof of the Pythagorean Theorem and its converse.






Performance Tasks




Standards Based Constructed Response ExamplePart A

Draw a right triangle. Measure and label the lengths of its legs and hypotenuse.

Part B

Draw three squares:

One square with edges that are the length of one leg of the triangle.

One square with edges that are the length of the other leg of the triangle.

One square with edges that are the length of the hypotenuse.

Part C

Use the areas and dimensions of the figures you drew to explain the relationship between the lengths of the legs and hypotenuse of a right triangle.

Example 2

A Pythagorean triple is a set of three positive whole

numbers which satisfy the equation

A2 + B2 = C2

Many ancient cultures used simple Pythagorean



Performance Task Resources Illustrative Mathematics Performance Tasks 8.G.6


content-standards/8/G/B/6

PMI Math Labshttps://njctl.org/courses/math/8th-grade-math/pythagorean-theorem-

distance-midpoint/



Learnzillion Lesson Plans Pythagorean Theorem

https://learnzillion.com/resources/64252-pythagorean-

theorem

PMI SMART Presentations https://njctl.org/courses/math/8th-grade-math/pythagorean-theorem-

distance-midpoint/

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https://njctl.org/courses/math/8th-grade-math/pythagorean-theorem-distance-midpoint/



https://learnzillion.com/resources/64252-pythagorean-theorem








https://www.illustrativemathematics.org/content-standards/8/G/B/6



triples such as (3,4,5) in order to accurately construct right angles: if a triangle has sides of lengths 3, 4, and 5 units, respectively, then the angle opposite the side of length 5 units is a right angle.

a. State the Pythagorean Theorem and its

converse.

b. Explain why this practice of constructing a

triangle with side-lengths 3, 4, and 5 to

produce a right angle uses the converse of

the Pythagorean Theorem.

e. Explain, in this particular case, why the

converse of the Pythagorean Theorem is

true.

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8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.






Performance Tasks




Standards Based Constructed Response ExamplePythagoras and TVs

Your family is considering buying a certain entertainment center with shelves of various sizes for a TV, books, pictures, knickknacks, and other items.

You are thinking about how you might place different items on the shelves.

The largest opening in the entertainment center is for a TV. It is 46 inches wide and 33 inches high. The size of a TV screen is given as the diagonal length, and you

want to find out the maximum diagonal size of a TV that will fit in that space, leaving at least 3 inches on each side and 3 inches at the top. There will also be a

stand that is about 3 inches high.

Part A. Given the dimensions of the entertainment


Questions 7-13









distance-midpoint/





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center and the restrictions on space, including the space that needs to be left on each side, the top, and for the TV stand, what are the maximum dimensions of a TV that would fit into this entertainment center?

Make sure to include the maximum width, height, and diagonal. Round your answers to the nearest

inch, if necessary.

Part B. The dimensions of the TV screen will be in the ratio 16:9. In other words, for every 16 inches of

width the screen will be 9 inches high. Knowing this, what are the dimensions of the largest TV that will fit into the entertainment center? Make sure to include

the width, height, and diagonal. Show your work using pictures, words, and/or equations.

Part C. You also have a hand-carved spear from South America. It is 34 inches long. One of the

compartments in the entertainment center is 30 inches wide and 15 inches deep. The spear is too long to fit in the compartment straight across. Could it fit diagonally across the bottom of the compartment?

Show your work.

Part D. Would it be possible to fit the spear in the compartment with one end at the upper left front corner and the other end in the lower back right

corner if the shelf is 15 inches high? (See the diagram below.) Explain your answer and show your

theorem


distance-midpoint/

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calculations.

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8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.






Performance Tasks




Standards Based Constructed Response ExampleSarah's town can be mapped on a coordinate grid. On

the map, the store is represented by the point at

Sarah's house is at and her

friend Randi's house is at

Part A

Use the Pythagorean Theroem to calculate the distance Sarah would walk if she traveled in a straight

line from her house to the store.


Questions 5-13






distance-midpoint/





theorem


distance-midpoint/

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Part B

How much farther would she have to walk if she stopped at her friend Randi's house before

continuing on to the store?

Use words, numbers and/or pictures to show your work.

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8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.






Performance Tasks




Standards Based Constructed Response ExampleA cube has a volume of 216 cubic inches.

Part A Write an equation that could be used to determine the edge length, s, of the cube.

Part B What is the edge length of the cube? Be sure to label your answer. Show your work.


Questions 6-8









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Unit Title: Measuring, Representing our measuring, and Reading others’ representations of measuring

Grade Level: 8


Essential Questions

(1) Can we become confident in our knowledge of quantities, units, and values of measuring that are used in measuring populations?(2) Can we become effective and efficient at representing bivariate data as a scatterplot or categorical data as a two way table?(3) Can we become effective and efficient at asking and answering questions about scatter plots? (4) Can we become effective and efficient at talking about relationships given in any representation of a population?(5) Can we become effective and efficient at reading and understanding others’ representations of bivariate data of a population?(6) Can we become effective and efficient at determining whether a real number is rational or irrational? (7) Can we become effective and efficient at using rational approximations of irrational numbers? (8) Can we become effective and efficient at comparing real numbers?(9) Can we become effective and efficient at identifying properties of shapes when presented in scenarios?(10) Can we become effective and efficient calculating values?(11) Can we become effective and efficient answering and answering questions about attributes of shapes?(12) Can we become effective and efficient about talking about attributes of shapes?(13) Can we become effective and efficient about reading about attributes of shapes?



8.SP.A.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

8.SP.A.2 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

8.SP.A.3

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http://www.corestandards.org/Math/Content/8/SP/A/3/



Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

8.SP.A.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?

8.NS.A.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

8.NS.A.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

8.G.C.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.



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http://www.corestandards.org/Math/Content/8/G/C/9/

http://www.corestandards.org/Math/Content/8/NS/A/2/




Instructional Plan ReflectionDiagnostic AssessmentGrade 8 Math – Cumulative AssessmentGrade 8 Math – End of Unit 3 AssessmentUnit 3 Touchpoints

SLO - SWBAT Student Strategies Formative Assessment Activities and Resources Reflection8.SP.A.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.






Performance Tasks



8.SP.1 – Touchpoint

Standards Based Constructed Response Example Jennie surveyed several students in her eighth grade class to determine whether there was a correlation between the number of minutes spent studying for math and the number of minutes spent studying for science the night before a test. The table below shows the number of minutes ten students in Jennie’s class spent studying the night before their last math and science tests.

Constructed Response ResourcesEdConnect 8.SP.1 - Type 2-3 Bank



Performance Tasks 8.SP.1https://

www.illustrativemathematics.org/content-standards/8/SP/A/1


grade-math/data/

Direct Instruction ResourcesEngageNY Lessons 8.SP.1

https://www.engageny.org/search-site?search=8.SP.1


https://learnzillion.com/resources/64256-patterns-of-association-in-bivariate-data


grade-math/data/

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https://njctl.org/courses/math/8th-grade-math/data/









https://www.illustrativemathematics.org/content-standards/8/SP/A/1




Part A. Create a scatter plot to represent the data from Jennie’s class. Make sure to label the axes and include a title for the graph.

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Part B. Is there a correlation between the time students spent studying for their math test and the time students spent studying for their science test? Describe any patterns in the data and identify any outliers.


8.SP.A.2 Know that straight lines are widely used to model

Do Now – Count around the room. Make combinations. [5




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relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

minutes]





Performance Tasks



The scatter plots below display the number of points scored at 10 consecutive games played by team A and team B.









grade-math/data/






grade-math/data/

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Part A. Draw a line of best fit to model the data on each of the scatter plots above.

Part B. Which line of best fit is a better model of the data it represents? Explain your answer.


8.SP.A.3 Use the equation of a linear model to solve problems in the context of bivariate measurement



Standards Based Constructed Response ExampleTo Build a Skyscraper


Questions 1-7


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data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.





Performance Tasks



Construction of a new skyscraper in a large city began in 2006 and was completed in May 2013. The building was designed to have a windowless base that is 200 feet squared and 185 feet high. Above the base, there are the office floors, starting with the 20th floor. At the top of the building, there is a large spire, which was the last part of the skyscraper that was constructed.Part A. The table below shows the progress in the construction of the skyscraper, from the pouring of concrete into the foundation 80 feet below the surface to the placement of the spire. The final height of the building was 1,776 feet.








grade-math/data/






grade-math/data/

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Study the growth of the building from February 2010 until the 100th floor was reached in March 2012. Using the numbers given, estimate the missing heights and fill in the table. Explain your work.

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Part B. In the graph below, each unit on the x-axis represents one quarter of a year, or 3 months, beginning with November 2006. Plot the values from the table on the graph.

Part C. Determine the periods of time when the building had the slowest growth rate and when the building had the most rapid growth rate. Explain what was happening during these time periods, including how this relates to the slope of the graph at that time.

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Part D. Draw a line of best fit on your graph above for both periods of time listed in part C. Then, determine the slopes of these lines, showing all work. What do these slopes represent? Explain how they relate to your answers in part C.

Part E. Write equations for the lines of best fit you drew in part D. Explain the meaning of the y-intercepts of both equations in terms of the context of this situation.

Part F. Explain why you think there were variations in the growth of the skyscraper as it was being built. Why isn't growth always constant? Refer to the equations you wrote in part E and the table in part A.

8.SP.A.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the






Performance Tasks


Standards Based Constructed Response ExampleAt a high school, 200 students were asked whether

they attended the last soccer game. Of the 200 students surveyed, 90 students went to the game, 40 boys went to the game, and 70 girls did not go to the

game.

Part A. Fill in the two-way table below to display the data.


Questions 1-5






grade-math/data/

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two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?



Part B. What percentage of the boys surveyed attended the game? Round your answer to the

nearest percent, if necessary.

Part C. What percentage of the girls surveyed attended the game? Round your answer to the

nearest percent, if necessary.

Part D. Based on the data, do you think there is an association between the boys attending the game

and the girls attending the game? Explain.







grade-math/data/

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8.NS.A.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.






Performance Tasks



8.NS.1 – Touchpoint


Part A Write as a repeating decimal. Show your work.

Part B Explain how you know when the decimal repeats.

Constructed Response ResourcesEdConnect 8.NS.1 - Type 2-3 Bank



Performance Tasks 8.NS.1https://

www.illustrativemathematics.org/content-standards/8/NS/A/1


grade-math/numbers-and-operations-8th-grade/

Direct Instruction ResourcesEngageNY Lessons 8.NS.1

https://www.engageny.org/search-site?search=8.NS.1

Learnzillion Lesson Plans Rational and Irrational Numbershttps://learnzillion.com/

resources/64250-rational-and-irrational-numbers



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https://njctl.org/courses/math/8th-grade-math/numbers-and-operations-8th-grade/



https://learnzillion.com/resources/64250-rational-and-irrational-numbers








https://www.illustrativemathematics.org/content-standards/8/NS/A/1




8.NS.A.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.






Performance Tasks



8.NS.2 – Touchpoint

Standards Based Constructed Response ExampleJuan’s class is going to construct an outdoor garden. The garden will be in the shape of a square. Juan's teacher gave the class three options for the area of

the garden: 30 square feet, 40 square feet, or 50 square feet.

Part A

Without using a calculator, approximate the side length, to the nearest tenth of a foot, for the garden

with an area of 30 square fet. Show your work.

Part B

The other two garden options have approximate side lengths of 6.3 feet and 7.1 feet. Locate and graph the three points on a horizontal number line to show the

approximation of the side length for each option.

Constructed Response ResourcesEdConnect 8.NS.2 - Type 2-3 Bank

Questions 1-9



Performance Tasks 8.NS.2https://

www.illustrativemathematics.org/content-standards/8/NS/A/2



Direct Instruction ResourcesEngageNY Lessons 8.NS.2


Learnzillion Lesson Plans Rational and Irrational Numbershttps://learnzillion.com/

resources/64250-rational-and-irrational-numbers



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8.G.C.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.






Performance Tasks




Standards Based Constructed Response ExampleEbony has an ice-cream cone, shown below, that is 6

inches tall and 2 inches across the diameter.

Part 1. How many cubic inches of ice cream are needed to fill the ice-cream cone? Assume the the ice cream fills the cone, with no gaps, to the very top of the cone, but no ice cream goes above the top of the

cone. Write the answer in terms of

Part 2. If a full scoop of ice cream is in the shape of a sphere, write the formula needed to find the volume

of half a scoop of ice cream. Write the answer in

terms of


Questions 1-6







content-standards/8/G/C/9






https://learnzillion.com/resources/64262-volume-of-cones-

spheres-and-cylinders

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https://learnzillion.com/resources/64262-volume-of-cones-spheres-and-cylinders







https://www.illustrativemathematics.org/content-standards/8/G/C/9






http://www.corestandards.org/Math/Content/8/G/C/9/

Part 3. If Ebony adds half a scoop of ice cream to the top of the already filled cone, how many total cubic inches of ice cream would be used? Assume that the

half scoop of ice cream sits directly on top of the cone and therefore has the same diameter as the

cone. Write the answer in terms of





Grade 8 Math – End of Unit 4 Assessment in EdConnectGrade 8 Math – Cumulative AssessmentDiagnostic Assessment

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Documents

8 Math Curriculum3.docx · Web viewThe size of a TV screen is given as the diagonal length, and you want to find out the maximum diagonal size of a TV that will fit in that space,