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$: commoncoremath.wikispaces.comcommoncoremath.wikispaces.com/file/view/Annotated_DCA... · Web viewInterpret and compute quotients of fractions, and solve word problems involving division$
Common Core Learning StandardsGRADE 6 MathematicsTHE NUMBER SYSTEM

Common Core Learning Standards Concepts Embedded Skills Vocabulary

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

division of fractions

Compute quotients of fractions. Quotient Fraction visual fraction

modelSolve word problems involving the division of fractions.

6.NS.1. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?

Draw a visual fraction model to illustrate the quotient of two fractions.Apply the relationship between multiplication and division to justify your answer.

SAMPLE TASKS

I. Mark has 5 pounds of mozzarella cheese. He is making pizzas and will put 13 pound on each pizza. How many pizzas will Mark be

Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.

able to make?

II. Is 3 x 2 the same as 3 ÷12 ? Justify your answer with words, numbers or pictures.

III. Katrina brought 34 pound of chocolate to a Valentine’s Day party. She gave

18 pound to each of her classmates. If Katrina gave all

of her chocolate away, how many classmates does she have? Show all work.


Compute fluently with multi-digit numbers and find common factors and multiples.

multi-digit division

Fluently divide multi-digit numbers using the standard algorithm.

standard algorithm

dividend divisor remainder quotient6.NS.2.

Fluently divide multi-digit numbers using the standard algorithm.

SAMPLE TASKSI. What is 64,015 divided by 74?

II. There are 1,288 seats in the auditorium. Each of the 23 rows in the auditorium has the same number of seats. How many seats are in each row?

III. Explain how the quotient of 80,000 ÷ 2,000 is different from 80,000 ÷ 200.

IV. Mrs. Smith’s classes brought in their Halloween candy to share. Her first period class containing 25 students brought in 300 pieces of candy total. Her second period class containing 21 students brought in 336 pieces of candy total. Mrs. Smith evenly


distributed the candy among her student within each period. Jose, a student in her first period class, claims that he received more pieces of candy than Elise, a student in her second period class. Is he correct? Justify your answer.


Compute fluently with multi-digit numbers and find common factors and multiples.

operations with decimals

Add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

standard algorithm

decimal place value

6.NS.3. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

SAMPLE TASKSCopyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.

I. What is the sum of 0.8 and 10.339?

II. What is 3.1 subtracted from 4.009?

III. What is the product of 0.05 and 1.2?

IV. What is 3.6 divided by 2.25?

V. Pablo and Jasmine go to the store. They are making Rice Krispy treats for their class. They need 2 boxes of Rice Krispies priced at $2.98 per box, and 3 bags of marshmallows priced at $2.05 per bag. (a) What will be their total cost?(b) If Pablo and Jasmine split the bill evenly, how will Pablo have to pay?

VI. Mark and Jeff are comparing their answers to their homework problem: 0.8 + 10.339. The students’ work is shown below.

Mark Jeff

0.8+10.33910.347

0.800

+10.33911.139

Explain which student’s answer is correct.

VII. Margaret is at the market. She has a $20 bill. She is purchasing items to make a fruit salad. She is buying a bag of apples for $5.65, a watermelon for $6.23, strawberries for $2.45, and two bananas for $0.83 each. As she’s leaving the market, she sees a key lime pie she wants to buy for $5.10. Will she have enough money left to buy the pie?

Common Core Learning Concepts Embedded Skills VocabularyCopyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.

StandardsCompute fluently with multi-digit numbers and find common factors and multiples.

greatest common

factor

Compute the greatest common factor of two whole numbers less than or equal to 100.

greatest common factor

least common multiple

distributive property

compute sum whole

numbers express LCM GCF

Compute the least common multiple of two whole numbers less than or equal to 12.

6.NS.4. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).

Compute the greatest common factor of two whole numbers written as a sum.Apply the distributive property to rewrite the sum with the GCF written outside parentheses and the two whole numbers with no common factor written inside the parentheses.

SAMPLE TASKSI. Find the GCF (greatest common factor) for 18 and 27.II. What is the LCM (least common multiple) of 8 and 6?III. Identify the similarities and differences between factors and multiples. You may use numbers, words or pictures to prove your

answer.IV. There are 16 boys and 12 girls in Mr. Hart’s Science class. He wants to form as many groups as possible for a project. All the

groups must be the same size, with the same number of boys and the same number of girls in each group. What is the greatest number of groups Mr. Hart can make? How many boys and how many girls will be in each group? Explain how you got your answer.

V. An express subway stops at 6th Street every 30 minutes. The local subway stops at the same location every 12 minutes. How often in minutes will both subways stop at 6th Street at the same time?

VI. Identify the greatest common factor of 45 and 54 and use the distributive property to rewrite the sum.



Apply and extend previous understandings of numbers to the system of rational numbers.

positive and negative numbers

Define positive and negative numbers in terms of direction and value.

Positive Negative Opposite Zero Integer Elevation Sea level Credits/debits Deposits/withdrawals Ascend/Descend

Describe real-world situations where positive and negative numbers are used.

6.NS.5. Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

Explain the meaning of 0 with positive and negative integers.

SAMPLE TASKSI. Describe each situation using a positive or negative number.

(a) Seven feet above sea level(b) Withdrawing $40 from a checking account(c) Nick hit eight fewer homeruns than he did last year


(d) The temperature rose 10 degrees Celsius(e) An elevator descended eleven floors(f) A mountain has an elevation of 5,480 feet(g) Depositing $300 into a savings account(h) Draining a battery

II. An eagle is flying fifty-five feet above sea level. A turtle is swimming twelve feet below sea level. Use integers to represent the elevation of the eagle and the turtle. Also, in terms of this situation, describe what an elevation of zero means.

III. Write each integer and its opposite. Then graph them on the number line.

(a) -6(b) 9(c) -7

IV. Does zero have an opposite? Why or why not?



opposite sign numbers

Locate opposite signed numbers on opposite sides of zero on a number line.

opposite sign zero number lineDefine the opposite of the opposite of a number is


the number itself.6.NS.6a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.

Define the opposite of 0 as itself.

SAMPLE TASKSIdentify the opposite of a) -5 b) 8 c) 0

What is the opposite of the opposite of 7?




Graphing ordered pairs

Graph ordered pairs in a coordinate plane. ordered pairs coordinate

plane x-axis y-axis reflection equidistant

Locate positive and negative numbers in a coordinate plane.

6.NS.6b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

Describe that when two ordered pairs only differ by their signs, they are reflections across the x-axis, y-axis, or both axes.Identify the four quadrants on a coordinate plane.

SAMPLE TASKS

I) a) Plot the points:( insert coordinate grid) (2,3) and label A (-2,3) and label B (-2,-3) and label C

(2, -3) and label Db) Name the quadrant that each point is found in.

II) Compare the positive and negative values of the points in each quadrant.III) Reflect a point (3,5) over the y-axis. Name the point and identify the quadrant. IV) Reflect a point (-2, -7) over the x-axis. Name the point and identify the quadrant.V) Reflect a point (4, 1) over both axes. Name the point and identify the quadrant.

Can this be done with only one reflection?




integers and rational

numbers on number lines

and the coordinate

plane

Plot and locate integers and rational numbers on vertical and horizontal number lines.

horizontal number line

vertical number line

integers rational

numbers plot coordinate

plane

Plot and locate integer and rational number pairs on the coordinate plane. 6.NS.6c.

Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

SAMPLE TASKS

Insert vertical and horizontal numbers lines for the following tasks:

I. Graph 12, 12 , 3.5, -6 (horizontal)

II. Graph -8, 16, 434 , -14 (vertical)

III. Display a horizontal integer number line. Have -1 and 0 plotted. Show a point between -1 and 0 and ask name the point.IV. On a coordinate plane plot the following points:

(-5,7) (6, -4) (-3, -2)




numbers on a number line

Compare rational numbers on a number line. Inequality greater than less than equal to

Describe statements of inequality on a number line.

6.NS.7a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.

Plot two numbers on a number line to describe the relationship between them in terms of less than, greater than, or equal to.

SAMPLE TASKSI. Using the following numbers: -10, -5, 4, -7, 17, 0, and 13

(a) Write the following numbers on the number line below.

(b) Using < or >, compare the following:1. 4 ⃝ 132. -10 ⃝ 03. -5 ⃝ -74. 13 ⃝ -10

II . On the number line below, plot -12 and -3.


Then, write an inequality comparing the two numbers. _________________ Explain how you used the number line to write your inequality.



rational numbers in real-world contexts

Write statements of order for rational numbers in real-world contexts.

rational number

temperature debits/credits sea level positive and

negative charge

Explain statements of order for rational numbers in real-world contexts.

6.NS.7b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3° C > –7° C to express the fact that –3° C is warmer than –7° C.

Explain how positive and negative rational numbers are used in real-world contexts.

SAMPLE TASKSI. Buffalo Bills played a football game last Sunday. In the first five plays, Fred Jackson rushed for -4 yards, 5 yards, -12

yards, 25 yards, and a play with no gain, 0 yards. List the yardage in order from least to greatest.

II. Fred, Megan, Jack and Jill all have bank accounts. The table below gives their current balance.Name Fred Megan Jack Jill

Balance $214.34 -$43.50 -$21.18 $75.78(a) Write the account balances in order from greatest to least.


(b) What does a negative account balance represent?(c) Looking at Megan and Jack, who has the greater debt? Explain your reasoning.

III. At a golf tournament, Gretchen scored +6, Mario scored -12, Xavier scored +2, and Bertha scored -5. (a) Using < or >, compare the scores of Bertha and Gretchen. (b) One of the four players won the golf tournament. Who won the tournament? Explain your reasoning.



absolute value of rational

numbers as distances

Define the absolute value of a rational number as a distance from 0 on a number line.

absolute value/distance

magnitude/length positive/negative

quantities

Explain the absolute value of a positive or negative quantity in a real-world situation as magnitude/length.

6.NS.7c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.

SAMPLE TASKS

I. Graph -4 on the number line.

(a) What is the absolute value of -4?(b) Explain how you used the number line to determine your answer to part (a).

II. Negative numbers are less than positive numbers. Does this mean that the absolute value of a negative number must be less Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.

than the absolute value of a positive number? Explain.

III. Explain why -7 and 7 have the same absolute value.

IV. Use absolute value to describe the relationship between a negative credit card balance and the amount owed .



absolute values and ordering

Compare and contrast the absolute value of a rational number to ordering rational numbers.

absolute value distance

Define a number less than a negative number as


rational numbers

having a greater distance from zero.6.NS.7d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.

SAMPLE TASKSI. A football coach recorded the results of his team’s first 4 plays in its last game. The table below shows his data.

Football PlaysPlays Number of Yards

1 102 -43 74 -9

(a) During which play did the team lose the fewest yards? Use what you know about absolute value and the number line below to explain how you determined your answer.

(b) During which play did the team gain the most yards? Use what you know about absolute value and the number line above to explain your answer.

II. Susan received her credit card statement in the mail. It stated that she is greater than $25 in debt. Does Susan owe more than $25 or less than $25? Explain your reasoning.

III. Birch Run Country Club was hosting a golf tournament. Steve scored +4, Julie scored -9, Derek scored +12, and Stacy scored -4. Knowing that “Par” is zero answer the following questions. (a) Which two players’ scores were the same distances from par? Use absolute value to explain your answer. (b) Which player is the furthest from par? Explain using absolute value.(c) Who won the golf tournament?

IV. Last week, Peters bought three stocks, A, B and C, each at the same price. From last week to this week, the change in price of


stock A was -$11, the change in stock B was $19, and there was no change in stock C. (a) Write the stocks in order from least to greatest. (b) Now, write their change in price in order from least to greatest.(c) Using absolute value, explain why the list in (a) differs from the list in (b).



Graphing ordered pairs

distance

Graph points in all four quadrants. ordered pairs coordinate

plane quadrant

Calculate the distance between two points graphed on a coordinate plane (vertical or horizontal lines only).


between two points

6.NS.8. Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

Calculate the distance between two points with the same x-value or the same y-value.

SAMPLE TASKSI. Alice’s garden is in the shape of a rectangle. The vertices are at A(-5, 6), B(3, 6), C(3, -6), and D(-5, -6).

(a) Plot Alice’s garden on the coordinate grid below.

(b) What is the perimeter of her garden? Be sure to show all your work. (Note: 1 unit = 1 yard)

II. Graph and label each


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