Linden Community Schools

 

 

STRAND VI: Probability and Discrete Mathematics

 

Content Standard 1:  Probability- Students develop an understanding of the notion of certainty and of probability as a measure of the degree of likelihood that can be assigned to a given event based on the knowledge available, and make critical judgements about claims that are made in probabilistic situations.

                                                                                   

Grade Level:  6th Grade

 

Benchmark

Performance Description

Recommended Activities

Recommended Assessments

Benchmark 1
Describe events as likely or unlikely and give qualitative and quantitative descriptions of the degree of likelihood.

D.PR.06.01 Express probabilities as fractions, decimals or percentages between 0 and 1; know that 0 probability means an event will not occur and that probability 1 means an event will occur.

D.PR.06.02 Compute probabilities of events from simple experiments with equally likely

outcomes, e.g., tossing dice, flipping coins, spinning spinners, by listing all possibilities and finding the fraction that meets given conditions.

 

 

Collect examples of chance and uncertainty from everyday situations and discuss the likelihood of various events (e.g., the sun will rise tomorrow, the 1st snow in Michigan will fall in October).

 

Students will write their own examples of chance.

Benchmark 2
Describe probability as a measure of certainty ranging from 0 to 1 and conduct activities that allow them to express probabilities of simple events in mathematical terms.

D.PR.06.01 Express probabilities as fractions, decimals or percentages between 0 and 1; know that 0 probability means an event will not occur and that probability 1 means an event will occur.

D.PR.06.02 Compute probabilities of events from simple experiments with equally likely outcomes, e.g., tossing dice, flipping coins, spinning spinners, by listing all possibilities and finding the fraction that meets given conditions.

 

Conduct experiments with simple devices and use the observed results to describe the likelihood of various outcomes (e.g., you will toss a coin twice and get two heads, you will roll a 6 on a single die compared to a total of 6 on two dice).

 

Students should engage in activities in which they describe the likelihood an event in number terms (fractions). Independent and dependent events will be used. Knowing the number of possible outcomes such as 36 for rolling 2 number cubes can help to quantify degree of likelihood. Decimals, fractions, and percentages may be used to indicate probability. They should compare events as equally likely or one more likely than the other. Students may be given a probability and asked to match an event such as rolling an even number on a number cube to the probability.

Benchmark 3
Conduct experiments and give examples to illustrate the difference between dependent and independent events.

D.PR.06.02 Compute probabilities of events from simple experiments with equally likely outcomes, e.g., tossing dice, flipping coins, spinning spinners, by listing all possibilities and finding the fraction that meets given conditions.

 

 

See benchmark 2 above

See benchmark 2 above

Benchmark 5
Conduct probability experiments and simulations to model and solve problems.

D.PR.06.02 Compute probabilities of events from simple experiments with equally likely outcomes, e.g., tossing dice, flipping coins, spinning spinners, by listing all possibilities and finding the fraction that meets given conditions.

 

Conduct experiments with simple devices and use the observed results to describe the likelihood of various outcomes (e.g., you will toss a coin twice and get two heads, you will roll a 6 on a single die compared to a total of 6 on two dice).

 

Students should engage in activities in which they describe the likelihood an event in number terms (fractions). Independent and dependent events will be used. Knowing the number of possible outcomes such as 36 for rolling 2 number cubes can help to quantify degree of likelihood. Decimals, fractions, and percentages may be used to indicate probability. They should compare events as equally likely or one more likely than the other. Students may be given a probability and asked to match an event such as rolling an even number on a number cube to the probability.

Linden Community Schools

 

 

STRAND V: Numerical and Algebraic Operations and Analytical Thinking

 

Content Standard 1:  Operations and their Properties- Students understand and use various types of operations (e.g., addition, subtraction, multiplication, division) to solve problems.         

                                                           

Grade Level:  6th Grade

 

Benchmark

Performance Description

Recommended Activities

Recommended Assessments

Benchmark 2
Compute with integers, rational numbers and simple algebraic expressions using mental computation, estimation, calculators and paper-and-pencil; explain what they are doing and how they know which operations to perform in a given situation.

N.FL.06.02 Given an applied situation involving dividing fractions, write a mathematical

statement to represent the situation.

N.FL.06.04 Multiply and divide any two fractions, including mixed numbers, fluently.

N.FL.06.09 Add, subtract, multiply, and divide integers between -10 and 10; use number line

and strip models for addition and subtraction.

N.FL.06.10 Add, subtract, multiply and divide positive rational numbers fluently.

N.FL.06.12 Calculate part of a number given the percentage and the number.

N.FL.06.14 For applied situations, estimate the answers to calculations involving operations

with rational numbers.

 

 

4/5 of a cubic foot of copper weighs 440 pounds.  What is the weight of 1 cubic foot of copper?

 

Add, subtract, multiply or divide any combination of positive and negative integers.

 

Add, subtract, multiply or divide any combination of positive whole numbers.

 

Given the regular price of an object, find a sale price. (Find the sale price of a $30 shirt that is discounted at 20%.)

 

Develop strategies for estimating computations (e.g., recognize that 2.47 x 3.93 is approximately 10 by rounding the factors to 2.5 x 4).

 

Apply mental estimation to consumer situations (e.g., reason that the bill for 12 assorted candy bars each less than 50 cents must be less than $6).

 

 

A picture is shown of a house and the proposed fencing of its backyard. Privacy fencing costs $49 for a six-foot section and costs $27 for 3 one-foot sections. Write number sentences to model the amount of fencing needed to enclose the backyard. Calculate the cost and explain your reasoning.

 

Benchmark 4
Efficiently and accurately apply operations with integers, rational numbers and simple algebraic expressions in solving problems.

N.FL.06.13 Solve word problems involving percentages in such contexts as sales taxes and

tips, and involving positive rational numbers.

N.FL.06.15 Solve applied problems that use the four operations with appropriate

decimal numbers.

 

 

Use mathematical operations and their properties to efficiently obtain the result of a specific problem (e.g., 5 • 3 • 4 • 6 = ? , 23 + 18 + 17 + 12 = ?).

Examine the results of several computations and determine and explain which answers are correct and which are not (e.g., 3 + 6 ÷ 2 = 4.5 is incorrect whereas 6 ÷ 2 + 3 = 1.2 is correct).

From a list of problems, choose a method for computing them and explain you choice. (mental computation, paper and pencil, or calculator).

Write real-world situations that would best be solved using mental computation; using paper and pencil; using a calculator.

Read and discuss problem situations and explain what mathematical operations will be needed and why.

Predict the approximate result of a computation before actually carrying it out.

Examine the results of several computations and, without calculating, determine which answers are reasonable and which are not.

 

A young mathematician named Johann Carl Friedrich Gauss in 1784 when asked by his teacher to add all of the numbers from 1 to 100 was able to do so faster than anyone in his class. What method could he have used? Hint: ( 1 + 2 + 3 + ••• + 98 + 99 + 100) = (1 + 100 + 2 + 99 + ••• + 50 + 51).

The temperature has been decreasing at a constant rate of 3 degrees per hour and is expected to continue this rate for another 5 hours. It is now 7 degrees.

When will the temperature be below 0 degrees? (A little over two hours from now.)

When was the temperature 16 degrees? (3 hours ago)

Five hours from now, what will the temperature be? (-8 degrees)

A 22 ounce value drink costs $1.59. A 15 ounce regular drink costs $.79. What is the cost per ounce for both drinks? Is the value drink a better bargain? Why? (22 oz costs $0.07 per oz, 15 oz costs $0.05 per oz. The smaller drink is a better value because it costs less per ounce.)

 

 

Content Standard 2:  Algebraic and Analytic Thinking- Students analyze problems to determine an appropriate process for solution, and use algebraic notation to model or represent problems.                             

Grade Level:  6th Grade

 

Benchmark

Performance Description

Recommended Activities

Recommended Assessments

Benchmark 1
Read and write algebraic expressions; develop original examples expressed verbally and algebraically; simplify expressions and translate between verbal and algebraic expressions; and solve linear equations and inequalities.

N.MR.06.01 Understand division of fractions as the inverse of multiplication, e.g.,

if  4/5 ÷2/3 =  , then  2/3 x     = 4/5, so

     = 4/5 • 3/2 = 12/10

N.FL.06.02 Given an applied situation involving dividing fractions, write a mathematical statement to represent the situation.

N.MR.06.03 Solve for the unknown in equations such as:  ¼ ÷    = 1, ¾ ÷    = ¼  and  ½ = 1x

A.FO.06.03 Use letters, with units, to represent quantities in a variety of contexts,

e.g., y lbs., k minutes, x cookies.

A.FO.06.04 Distinguish between an algebraic expression and an equation.

A.FO.06.05 Use standard conventions for writing algebraic expressions, e.g., 2x + 1 means “two times x, plus 1” and 2(x + 1) means “two times the quantity (x + 1).”

A.FO.06.06 Represent information given in words using algebraic expressions and equations.

A.FO.06.07 Simplify expressions of the first degree by combining like terms, and evaluate using specific values.

A.RP.06.10 Represent simple relationships between quantities, using verbal descriptions, formulas or equations, tables, and graphs, e.g., perimeter-side relationship for a square, distance-time graphs, and conversions such as feet to inches.

A.FO.06.11 Relate simple linear equations with integer coefficients to particular contexts, and solve, e.g., 3x = 8 or x + 5 = 10.

 

 

Express the total number of candies in a bag of Skittles with 12 reds, 8 purples, and 4 yellows as an algebraic expression.

Create a real life-situation that could be represented by 15x + 10.
Possible answer: Horseback riding costs $15.00 per person plus $10.00 for the group guide.

Use a balance to illustrate algebraic equalities and inequalities (e.g., 3 + 4 = 7, 3 + 3 > 5).

Write a situation modeled by an algebraic expression; e.g., 3 x + y + 6z could model total points for x field goals, y extra points and z touchdowns in a football game.

A parent has budgeted $60 for a movie party to celebrate a child’s birthday. Four adults will accompany the children in the theater. How many children can be invited in addition to the birthday child?

 

Given an algebraic expression, write a situation that it models.

Given a real-world mathematical situation, write it as an expression, equation or inequality and solve, if possible.

Solve the inequality 3x - 4 > 22 by adding 4 to each side, then dividing each side by three. You get x > 6. You may wish to have students show answer on a number line

Example
The school is taking a trip to a nearby ski resort which normally charges $22.00 for a student lift ticket and $28.00 for an adult lift ticket. However, the resort gives a 10% discount to all people in school groups. Choose the correct expression that represents the total lift ticket cost for x adults and y students.

A. 0.10(28x + 22y)
B. 10(28x + 22y)
C. 0.90(28x + 22y)
D. 90 (28x + 22y)

Answer C

 

Benchmark 3
Solve linear equalities and inequalities using algebraic and geometric methods, and use the context of the problem to interpret and explain their solutions.

A.FO.06.12 Understand that adding or subtracting the same number to both sides of an equation creates a new equation that has the same solution.

A.FO.06.14 Solve equations of the form ax + b = c, e.g., 3x + 8 = 15 by hand for positive integer coefficients less than 20, using calculators otherwise, and interpret the results.

 

 

Students should engage in activities such as these.

 

Use number lines to model solutions to linear equations and inequalities in one variable.
(For example, the solution to 2x + 3 > 7 and the graph of its solution set is shown.)

Use geometric representations, including graphs in the Cartesian plane, to model solutions of linear equations in two variables.

Given the equation y = x - 1, find the value of x when y = 4. Solve 4 = x - 1.
(Find the x location where the graph of y = x - 1 has risen to a height of 4.)

y = x -1

Use graphing calculators to find solutions to linear equations.

 

Write both the perimeter and the area of the largest rectangle in this figure.

Area = (x + 4) • 5 or 5x + 20
Perimeter = 2 • (x + 4) + 2 • (5) or 2x + 18

Have students model the following expression geometrically: 3x + 8.

Have students experiment with different linear (x to the fist power only) equations using the graphing calculator. For ease of exploring, ask them to use integers from -10 to 10 and to use addition, not subtraction (y = 2x + 3, y = 4x + -2, y = -7x + 1, y = -3x + 5, etc.). Ask students to identify and explain which part of the equation determines how steep the line is.

Ask students to identify and explain which part of the equation determines where the line crosses the y-axis. Ask them what causes a line to slant. Answer: The coefficient of x indicates how steep a line is (the slope). If that number is negative, it will slant down; if positive, slant up. The closer to 0, the more horizontal the line will be (y = -2x + 1 will slant down and be less steep than y = -6x + 8).

Benchmark 4
Analyze problems modeled by linear functions, determine strategies for solving the problems and evaluate the adequacy of the solutions in the context of the problems.

A.PA.06.09 Graph and write equations for linear functions of the form y = mx, and solve related problems, e.g., given n chairs, the “leg function” is f(n) = 4n; if you have 5 chairs, how many legs?; if you have 12 legs, how many chairs?

When given a graph, table, or equation of a linear function, students identify the x-value when the y-value is given (e.g., from a temperature conversion chart from Celsius to Fahrenheit degrees, find the temperature after the effect of the wind chill factor is applied).

Discuss whether a solution is acceptable in the context of a problem. (For example, is 11.69 an appropriate solution to ; (1) when computing the average cost of 39 items ordered from a catalog, (2) when computing the number of 39-passenger buses needed to transport 456 students, (3) when determining the number of boxes that can be filled when 456 cookies are packaged in boxes of 39?

Discuss how individual students understand and represent selected problems and compare such multiple representations.

Devise and explain strategies for solving problems

 

In Japan, Sumo wrestlers are weighed in kilograms. There are approximately 2.2 pounds in 1 kilogram. Which would be the best equation to find the weight in pounds, p, of a sumo wrestler whose weight is given in k kilograms?

A. k • p = 2.2
B. p ÷ k = 2.2
C. p = k • 2.2
D. p ÷ 2.2 = k

Answer C

[B, C, and D are correct relationships, but when you are given the amount of kilograms and want to find the number of pounds, C will give you the answer directly without any additional changing of the equation.]

The number of times a cricket chirps depends on the temperature. It is such a dependable relationship, that there is even a formula: n = 4t - 160, where n is the number of times the cricket chirps per minute and t is the temperature in degrees Fahrenheit.

In the context of the problem explain what it means that when t = 50 that n = 40. [NOTE: At 50° a cricket chirps 40 times in one minute.]

When t is 40, n is 0. What does this mean? [NOTE: A cricket does not chirp at all when the temperature is 40°.]

 

Benchmark 5
Explore problems that reflect the contemporary uses of mathematics in significant contexts and use the power of technology and algebraic and analytic reasoning to experience the ways mathematics is used in society.

A.PA.06.01 Solve applied problems involving rates including speed, e.g., if a car is going 50 mph, how far will it go in 3 ½  hours?

 

Solve problems using rates including mph, ft/min, beats/sec., etc.

Marty works in a canning factory. He must order the paper for the labels to be placed around the cans. The height of each can is 16 cm and the diameter is 7.5 cm. Marty can order rolls of paper in 16 cm widths. What is the length of the roll of paper required to label 500 cans? Show your work and explain.
Answer: The width of the roll of paper is the height of the can. The circumference of the can top or bottom is the length needed for each can. Circumference = 3.14 x 7.15 = 23.55. For 500 cans, 23.55 • 500 = 11,775 cm long for the roll of paper. The label would probably include an overlap for each can for gluing purposes, perhaps to 1 cm. Also the 3.14 approximation for p is an underestimate of the actual value of . An overestimate of 3.15 or 3.142 for would be more appropriate in this case.

 

 

Linden Community Schools

 

STRAND III: Data Analysis and Statistics

 

Content Standard 1: Collection, Organization and Presentation of Data- Students collect and explore data, organize data into a useful form, and develop skill in representing and reading data displayed in different formats.

                       

Grade Level:  6th Grade

 

Benchmark

Performance Description

Recommended Activities

Recommended Assessments

Benchmark 3
Present data using a variety of appropriate representations and explain why one representation is preferred over another or how a particular representation may bias the presentation.

 A.RP.06.08 Understand that relationships between quantities can be suggested by graphs and tables.

A.RP.06.10 Represent simple relationships between quantities, using verbal descriptions, formulas or equations, tables, and graphs, e.g., perimeter-side relationship for a square, distance-time graphs, and conversions such as feet to inches.

 

Conduct a survey to determine if students in your school are interested in buying bagels and juice on Fridays.

Record your data.

From the survey develop an appropriate table, chart, or graph to display results.

Represent your data from the bagel and juice survey in your choice of 2 graphic forms (e.g., pie, boxplots, stem and leaf, histogram) and explain why.

Decide an appropriate way to present data in various situations and explain choices (For example, showing why a line graph is preferred to a bar graph or a pie chart in a given setting.).

Generate examples of situations in which one would choose a particular form of presentation, such as examples of data sets that favor presentation via box plots.

Discuss the ways in which data are presented in newspaper and magazine articles, and identifying questions that can be answered from the data.

Use a spreadsheet to generate several different presentations from the same data set.

 

Construct a graph that will compare the length of the main spans of the 5 longest bridges in the chart.


Justify your choice of graph.

 

 

 

Linden Community Schools

 

 

STRAND II:  Geometry and Measurement

 

Content Standard 1:  Shape and Shape Relationships- Students develop spatial sense, use shape as an analytic and descriptive tool, identify characteristics and define shapes, identify properties and describe relationships among shapes.

Grade Level:  6th Grade

 

Benchmark

Performance Description

Recommended Activities

Recommended Assessments

Benchmark 4
Construct familiar shapes using coordinates, appropriate tools (including technology), sketching and drawing two- and three-dimensional shapes.

 M.PS.06.02 Draw patterns (of faces) for a cube and rectangular prism that, when cut, will cover the solid exactly (nets).

G.SR.06.05 Use paper folding to perform basic geometric constructions of perpendicular lines, midpoints of line segments and angle bisectors, and justify informally.

 

Create 3-dimensional geometrically shaped buildings that have faces that are octagonal, rectangular and triangular using toothpicks and gumdrops.

Construct 3-dimensional shapes using paper-folding patterns. These may be decorated and used as picture holders or ornaments.

 

Students should engage in activities in which they sketch or plot familiar 2- and 3-dimensional shapes from descriptions or coordinates. Sketching does not involve construction with a protractor or compass. Students may be asked to make accurate shapes from given angle and side length measures with a protractor and ruler.

 

 

Benchmark 5
Combine, dissect and transform shapes.

G.TR.06.03 Understand the basic rigid motions in the plane (reflections, rotations, translations), relate these to congruence, and apply them to solve problems.

G.TR.06.04 Understand and use simple compositions of basic rigid transformations, e.g., a translation followed by a reflection.

 

Combine all the tangram pieces to form a rectangle, a parallelogram, and a triangle.

 

Determine which of 5 shapes supplied by the instructor tessellates the plane. Create an original shape that tessellates the plane.

Create a quilt design and indicate the portion that is repeated.

 

Benchmark 6
Generalize about the common properties of similar, congruent, parallel and perpendicular shapes and verify their generalizations informally.

G.GS.06.01 Understand and apply basic properties of lines, angles, and triangles, including:

v triangle inequality

v relationships of vertical angles,

  complementary angles, supplementary

  angles

v congruence of corresponding and

    alternate interior angles when parallel

lines are cut by a transversal, and that

such congruencies imply parallel lines

v locate interior and exterior angles of

any triangle and use the property that a exterior angle of a triangle is equal to

  the sum of the remote (opposite)

interior angles know that the sum of

the exterior angles of a convex polygon is 360

G.GS.06.02 Understand that for polygons, congruence means corresponding sides and angles have equal measures.

 

Draw, trace, fold and cut a variety of shapes and use them to develop concepts of congruence and similarity.

Cut the tangram diagram into its 7 pieces.

    • How do the size and shape of the 2 small triangles compare to the size and shape of the 3 larger triangles?
    • Are any pairs of triangles similar?
    • Are any pairs of triangles congruent?
    • Give an argument to prove your answer.

 

Identify corresponding sides and angles given similar figures. [Be sure some have been reflected or transformed in some way.]

 

 

Content Standard 2: Position- Students identify locations of objects, identify location relative to other objects, and describe the effects of transformations (e.g., sliding, flipping, turning, enlarging, reducing) on an object.             

                                                                       

Grade Level:  6th Grade

 

Benchmark

Performance Description

Recommended Activities

Recommended Assessments

Benchmark 2
Locate and describe objects in terms of their orientation and relative position, including coincident, collinear, parallel, perpendicular; differentiate between fixed (e.g., N-S-E-W) and relative (e.g., right-left) orientations; recognize and describe Examples of bilateral and rotational symmetry.

 G.SR.06.05 Use paper folding to perform basic geometric constructions of perpendicular lines, midpoints of line segments and angle bisectors, and justify informally.

Use physical objects, graphs or sketches to represent positions such as “parallel to”, “at the midpoint”, or “equidistant from”.

 

Create a figure with rotational symmetry (e.g., 30°, 60°, 90°, 120°, 150° or 180°).

 

Benchmark 3
Describe translations, reflections, rotations and dilations using the language of transformations, and employ transformations to verify congruence of figures.

G.TR.06.03 Understand the basic rigid motions in the plane (reflections, rotations, translations), relate these to congruence, and apply them to solve problems.

G.TR.06.04 Understand and use simple compositions of basic rigid transformations, e.g., a translation followed by a reflection.

 

Transform a triangle with given coordinates by translating, reflecting/rotating.

 

Draw with paper and pencil and with computer drawing tools, the resulting image when given a description of the transformation of a shape or object (e.g., translate 10 centimeters at an angle of 30° from the horizontal, reflect over the y-axis, rotate 60° about the vertex).

 

Benchmark 4
Locate the position of points or objects described by two or more conditions; locate all the points (locus) that satisfy a given condition.

 A.RP.06.02 Plot ordered pairs of integers and use ordered pairs of integers to identify points in all four quadrants of the coordinate plane

 

Play the game Battleship by Milton Bradley.

 

Create a city map with the following buildings at the specified locations. Each unit represents one block. Position and label the following points.

    • A school in the center of the grid.
    • A post office 4 blocks south and 3 blocks west of the school.
    • A library 2 blocks north and 4 blocks east of the school.
    • A convenience store 5 blocks walking distance from the library and 3 blocks from the school.

 

Benchmark 5
Use concepts of position, direction and orientation to describe the physical world and to solve problems.

G.TR.06.03 Understand the basic rigid motions in the plane (reflections, rotations, translations), relate these to congruence, and apply them to solve problems.

Lead another student to a desired location with written or oral directions.

 

Students should engage in activities in which they use position and direction by (e.g. left and right, compass directions, latitude and longitude, placement on a coordinate graph). The problem may ask to move an object according to specific directions or criteria, and ask for the new position. The problem could also ask students to describe the position of a certain object.

 

 

Content Standard 3:  Measurement- Students compare attributes of two objects, or of one object with a standard (unit) and anlyze situations to determine what measurement(s) should be made and to what level of precision

Grade Level:  6th

Benchmark

Performance Description

Recommended Activities

Recommended Assessments

Benchmark 5
Use proportional reasoning and indirect measurements to draw inferences.

M.UN.06.01 Convert between basic units of measurement within a single measurement system, e.g., square inches to square feet.

M.TE.06.03 Compute the volume and surface area of cubes and rectangular prisms given the lengths of their sides, using formulas.

Give students notes and conversion rates.

 

Use centimeter cubes to calculate volume and surface area.

Given formulas, students will calculate volume and surface area of solids.

 

Linden Community Schools

 

 

STRAND IV: Number Sense and Numeration

 

Content Standard 1: Concepts and Properties of Numbers- Students experience counting and measuring activities to develop intuitive sense about numbers, develop understanding about properties of numbers, understand the need for and existence of different sets of numbers, and investigate properties of special numbers.    

Grade Level:  6th Grade

 

Benchmark

Performance Description

Recommended Activities

Recommended Assessments

Benchmark 1
Develop an understanding of integers and rational numbers and represent rational numbers in both fraction and decimal form.

N.ME.06.06 Represent rational numbers as fractions or terminating decimals when possible, and translate between these representations.

N.ME.06.07 Understand that a fraction or a negative fraction is a quotient of two integers,e.g., - 8/3 is -8 divided by 3.

N.ME.06.17 Locate negative rational numbers (including integers) on the number line; know that numbers and their negatives add to 0, and are on opposite sides and at equal distance from 0 on a number line.

N.ME.06.18 Understand that rational numbers are quotients of integers (non-zero

denominators), e.g., a rational number is either a fraction or a negative fraction.

 

 

Students should engage in activities such as these.

 

Use physical models of quantities (include whole, fractional, decimal, and mixed numbers both positive and negative): number lines, circles, grids, and area models (not limited to familiar polygons). For instance, ask students to model of a trapezoid using pattern blocks.

 

Develop a knowledge of integers using models (e.g., the rising and descending of hot air balloons, elevator movements, above and below sea level, receiving checks and paying bills, temperature readings).

 

Use physical models of fractional quantities to represent rational quantities such as fractions close to zero, , or 1.

 

Express numbers using a variety of equivalent representations.

 

After counting votes, students reported that 130% of the 6th graders favored extending the lunch hour 15 additional minutes. Explain why the vote was questioned.

Sam’s shoe store offers shoes for  off the regular price. Foot World has shoes for the same original price, but they are on sale for 30% off. Which is the better deal? Why?

Karina has  cups of flour. This is enough for of a recipe of brownies. How much flour would she need for the whole recipe? Explain.

 

Benchmark 2
Extend their understanding of numeration systems to include decimal numeration, scientific numeration and non-decimal numeration systems.

N.ME.06.16 Understand and use integer exponents, excluding powers of negative numbers; express numbers in scientific notation.

 

 

Investigate real-life examples of very large and very small numbers (For instance, calculator displays of 1 e-12, computer terms, space exploration, parts per million in environmental quality issues, advanced technology, the national debt). As a result of investigating these numbers, the importance and practicality of writing these numbers in scientific notation should be noted.

Explore various numeration systems (e.g., Roman, Egyptian, Mayan, Greek) as a basis for understanding the power and convenience of a place-value system. Identify the similarities and difference of these systems. How do they differ from our place-value system?