Grade 6 Pre-AP - Math 2016-17 Unit 1 - cleburne.k12.tx.us · (6.2D) Which list shows the ... apply...

Grade 6 Pre-AP - Math 2016-17 Unit 1

Grade 6 Pre-AP Math- Unit 1 Updated August 8, 2016 of 19

Title Suggested Time Frame Numbers and Operations *CISD Safety Net Standards: 6.3DE, 6.11A

1st Six Weeks 29 days

Big Ideas/Enduring Understandings Guiding Questions Students will understand that: • Rational numbers include whole numbers, integers, fractions, and

decimals. • The coordinate plane consists of four quadrants using ordered pairs of

rational numbers. • Mathematics can be used in everyday experiences, investigations and

activities in and outside of school. • Mathematics can be communicated through informal and mathematical

language, representations and models.

• How are rational numbers used in real world situations? • What does it mean to multiply and divide fractions and decimals? • What happens when you add, subtract, multiply, and divide integers?

Vertical Alignment Expectations

*TEKS one level below* *TEKS one level above* TEA Vertical Alignment 5th Grade – Algebra I

Sample Assessment Question 6.3D 2016 STAAR #12 2016 STAAR #30

http://projectsharetexas.org/resource/vertical-alignment-charts-revised-mathematics-teks



6.3E 2016 STAAR #2 2016 STAAR #38 (6.3E) A grocery store sells steak for $6.10 per pound. What would be the cost of 2 3/5 lb of steak? A. $14.03 B. 8.70 C. $15.86 D. $12.06 (6.2D) Which list shows the numbers below in order from least to greatest? -6 ½ , 7.25 , 45/7 , 6 ¼ , -5.58 , 84/9 A. -6 ½ , -5.58 , 6 ¼ , 46/7 , 7.25 , 84/9 B. -5.58 , 6 ¼ , 46/7 , -6 ½ , -5.58 , 84/9 C. – 6 ½ , 6 ¼ , 5.58 , 46/7 , 7.25 , 84/9 D. -5.58 , 6 ½ , 6 ¼ , 46/7 , 7.25 , 84/9



6.11A 2016 STARR #5 2016 STAAR #41

The resources included here provide teaching examples and/or meaningful learning experiences to address the District Curriculum. In order to address the TEKS to the proper depth and complexity, teachers are encouraged to use resources to the degree that they are congruent with the TEKS and research-based best practices. Teaching using only the suggested resources does not guarantee student mastery of all standards. Teachers must use professional judgment to select among these and/or other resources to teach the district curriculum. Some resources are protected by copyright. A username and password is required to view the copyrighted material. Some district specificity and examples are a product of the Austin Area Math Supervisors TEKS Clarifying documents as found on Region XI Mathematics Livebinder.



Ongoing TEKS

(A) apply mathematics to problems arising in everyday life, society, and the workplace; • Focus is on application

(B) use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;

(C) select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;

• Students should assess which tool to apply rather than trying only one or all

(D) communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate;

(E) create and use representations to organize, record, and communicate mathematical ideas;

• Students should evaluate the effectiveness of representations to ensure they are communicating mathematical ideas clearly

• Students are expected to use appropriate mathematical vocabulary and phrasing when communicating ideas

(F) analyze mathematical relationships to connect and communicate mathematical ideas; and • Students are expected to form conjectures based on patterns or sets of examples and non-examples

(G) display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication

• Precise mathematical language is expected.

MAT.6.01 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:



Knowledge and Skills with

Student Expectations

District Specificity/ Examples

Vocabulary

Suggested Resources Resources listed and categorized to indicate suggested uses. Any

additional resources must be aligned with the TEKS.

6.02A The student is expected to classify whole numbers, integers, and rational numbers using a visual representation such as a Venn diagram to describe relationships between sets of numbers.

Supporting Standard

6.02B The student is expected to identify a number, its opposite, and its absolute value. Supporting Standard

6.02A 6.02B

In 6th grade, students extend the number line to represent all rational numbers and recognize that number lines may be either horizontal or vertical (i.e. thermometer) which facilitates the movement from number lines to coordinate grids. Students recognize that a number and its opposite are equidistance from zero (reflections about the zero). The opposite sign (–) shifts the number to the opposite side of 0. For example, – 4 could be read as “the opposite of 4” which would be negative 4. In the example, – (–6.4) would be read as “the opposite of the opposite of 6.4” which would be 6.4. On a number line model, the number is represented by an arrow drawn from zero to the location of the number on the number line; the absolute value is the length of this arrow.

• integers • negative • nonnegative • nonpositive • positive • rational numbers • sets of numbers • Venn diagram • whole numbers • absolute value • opposite numbers • decimal • equal (=) • fraction • greater than (>) • greatest to least • least to greatest • less than (<) • number line • percentages • denominator • dividend • division • divisor • fraction notation • numerator • quotient • symbols for division

𝑎𝑎𝑏𝑏 b≠0, a÷b,

Textbook Resources: McGraw Hill Chapter 1 Web Resources: Region XI: Livebinder NCTM: Illuminations

http://www.livebinders.com/play/play?id=930542

http://illuminations.nctm.org/



Students recognize the distance from zero as the absolute value or magnitude of a rational number. Students need multiple experiences to understand the relationships between numbers, absolute value, and statements about order. Students understand absolute value as the distance from zero and recognize the symbols | | as representing absolute value. Example 1: Which numbers have an absolute value of 7

Solution: 7 and –7 since both numbers have a distance of 7 units from 0 on the number line.

Example 2: a. Use an integer to represent 25 feet below sea level b. Use an integer to represent 25 feet above sea level. c. What would 0 (zero) represent in the scenario above?

Example 3:

What is the | –3 |

Solution: 3

In real-world contexts, the absolute value can be used to describe size or magnitude. For example, for an ocean depth of 900 feet, write | –900| = 900 to describe the distance below sea level.

When working with positive numbers, the absolute value (distance from zero) of the number and the value of the number is the same; therefore,

•



6.02C The student is expected to locate, compare, and order integers and rational numbers using a number line. Supporting Standard

ordering is not problematic. However, negative numbers have a distinction that students need to understand. As the negative number increases (moves to the left on a number line), the value of the number decreases. For example, –24 is less than –14 because –24 is located to the left of –14 on the number line. However, absolute value is the distance from zero. In terms of absolute value (or distance) the absolute value of –24 is greater than the absolute value of –14. For negative numbers, as the absolute value increases, the value of the negative number decreases.

6.02C

In working with number line models, students internalize the order of the numbers; larger numbers on the right (horizontal) of the number line and smaller numbers to the left (horizontal) of the number line. They use the order to correctly locate integers and other rational numbers on the number line. By placing two numbers on the same number line, they are able to write inequalities and make statements about the relationships between two numbers.

Students will use common models to represent and compare integers including number line models, temperature models and the profit- loss model.

The number line can also be viewed as a thermometer where each point of on the number line is a specific temperature. In the profit-loss model, a positive number corresponds to profit and the negative number corresponds to a loss.

Each of these models is useful for examining values but can also be used in performing operations with integers.

In working with number line models, students internalize the order of numbers; larger numbers on the right (horizontal) or top (vertical) of the number line and smaller numbers to the left (horizontal) or bottom (vertical) of the number line. They use the order to correctly locate



integers and other rational numbers on the number line. By placing two numbers on the same number line, they are able to write inequalities and make statements about the relationships between two numbers.



6.02D The student is expected to order a set of rational numbers arising from mathematical and real-world contexts. Readiness Standard

6.02D

Students write statements using < or > to compare rational number in context. However, explanations should reference the context rather than “less than” or “greater than”.

Example 1: The balance in Sue’s checkbook was –$12.55. The balance in John’s checkbook was



–$10.45. Write an inequality to show the relationship between these amounts. Who owes more?

Solution: –12.55 < –10.45, Sue owes more than John. The interpretation could also be “John owes less than Sue”.

Example 2: One of the thermometers shows -3°C and the other shows -7°C. Which thermometer shows which temperature? Which is the colder temperature? How much colder? Write an inequality to show the relationship between the temperatures and explain how the model shows this relationship.

Solution:

The thermometer on the left is -7; right is -3 The left thermometer is colder by 4 degrees Either -7 < -3 or -3 > -7

Example 3:

A meteorologist recorded temperatures in four cities around the world. List these cities in order from coldest temperature to warmest temperature:

Albany: 5°

Anchorage: -6°

Buffalo: -7°



6.02E The student is expected to extend representations for division to include fraction notation such as a/b represents the same as a ÷ b where b ≠ 0. Supporting Standard

Junea:-9°

Reno:12°

Solution: Junea, Buffalo,Anchorage, Albany, Reno

6.02E

Students should connect fractions with division, understanding that 5 ÷ 3

= . Students should explain this by working with their understanding of

division as equal sharing.

Students should also create story contexts to represent problems involving division of whole numbers.

Example 1: If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? This can be solved in two ways. First, they might partition each pound among the 9 people, so that each person

gets 50 x = pounds.



Second, they might use the equation 9 x 5= 45 to see that each person can be given 5 pounds, with 5 pounds remaining. Partitioning the remainder

gives 5 pounds for each person.

Students need ample experiences to explore the concept that a fraction is a way to represent the division of two quantities. Students are expected to demonstrate their understanding using concrete materials, drawing models, and explaining their thinking when working with fractions in multiple contexts.

They read as “three fifths” and after many experiences with sharing

problems, learn that can also be interpreted as “3 divided by 5.”

Example 2: Ten team members are sharing 3 boxes of cookies. How much of a box will each student get? When working this problem, a student should recognize that the 3 boxes are being divided into 10 groups, so s/he is seeing the solution to the following equation, 10 x n = 3 (10 groups of some amount is 3 boxes) which can also be written as n = 3 ÷ 10. Using models or diagram, they divide each box into 10 groups, resulting in each team member getting 3/10 of a box. Two after school clubs are having pizza parties. For the Math Club, the teacher will order 3 pizzas for every 5 students. For the student council, the teacher will order 5 pizzas for every 8 students. Since you are in both groups, you need to decide which party to attend. How much pizza would you get at each party? If you want to have the most pizza, which party should you attend? The six fifth grade classrooms have a total of 27 boxes of pencils. How many

boxes will each classroom receive? Students may recognize this as a whole



number division problem but should also express this equal sharing problem

as . They explain that each classroom gets boxes of pencils and

can further determine that each classroom get boxes of

pencils.

6.03A The student is expected to recognize that dividing by a rational number and multiplying by its reciprocal result in equivalent values. Supporting Standard

6.03B The student is expected to determine, with and without computation, whether a

6.03A

This concept must be introduced and explored through models and visuals.

(example: dividing by 2 and multiplying by , dividing by 4 and multiplying by

, dividing by 3 and multiplying by .)

6.03B

This standard asks students to examine how numbers change when we multiply by fractions. Students should have ample opportunities to examine

• denominator • division • equivalent • expressions • multiplication • numerator • product • quotient • rational number • reciprocal • decreased • factor • improper fraction • increased

McGraw Hill Chapter 2 Chapter 3 Web Resources: Region XI: Livebinder NCTM: Illuminations





quantity is increased or decreased when multiplied by a fraction, including values greater than or less than one. Supporting Standard

6.03C The student is expected to represent integer operations with concrete models and connect the actions with

both cases in the standard: a) when multiplying by a fraction greater than 1, the number increases and b) when multiplying by a fraction less the one, the number decreases.

This concept must be introduced and explored through models and

visuals. (example: dividing by 2 and multiplying by , dividing by 4 and

multiplying by , dividing by 3 and multiplying by .)

Example 1: Mrs. Bennett is planting two flower beds. The first flower bed is 5 meters long

and meters wide. The second flower bed is 5 meters long and meters wide.

How do the areas of these two flower beds compare? Is the value of the area larger or smaller than 5 square meters? Draw pictures to prove your answer. 6.03C & D Students add, subtract, multiply and divide integers. Visual representations such as color counters and number lines are necessary for students to gain understanding of operations with integers. When using color counters, student should be able to represent “zero pair”

• mixed number • quantity • unit fraction • addends • addition • difference • dividend • divisor • subtraction • sum • positive rational

numbers



the models to standardized algorithms. Supporting Standard *CISD Safety Net* 6.03D The student is expected to add, subtract, multiply, and divide integers fluently. Readiness Standard

using color counters. The students will need to be able to match an integer equation to the model as well as match the model the integer equation. Example 1: Use a number line to add -5 + 7. Solution 1: Number Line Students find -5 on the number line and move 7 in a positive direction (to the right). The stopping point of 2 is the sum of this expression. In the example, 7 – 5, the difference is the distance between 7 and 5, or 2, in the direction of 5 to 7 (positive). Therefore the answer would be 2. Solution 2: Color Counters

Students should use color counters and combine the zero pairs. Once the zero pairs are combined, notice that there are 2 positive counters remaining. The answer is 2. The equation is -5 + 7 = 2. Example 2: Use a number line to subtract: -6 – (-4) Solution: This problem is asking for the distance between -6 and -4. The



distance between -6 and -4 is 2 and the direction from -4 to -6 is left or negative. The answer would be -2. Note that this answer is the same as adding the opposite of -4: -6 + 4 = -2 Example 3: Use a number line to illustrate: p – q ie. 7 – 4 p+(-q) ie. 7+(–4) Is this equation true p – q = p + (-q)? Students explore the above relationship when p is negative and q is positive and when both p and q are negative. Is this relationship always true? Example 4: Morgan has $4 and she needs to pay a friend $3. How much will Morgan have after paying her friend? Solution: 4 + (-3) = 1 or (-3) + 4 = 1

Using a number line, students should begin at 0 and follow the arrows to get the solution. Students understand that multiplication and division of integers is an extension of multiplication and division of whole numbers. Students recognize that when division of rational numbers is represented with a fraction bar, each number can have a negative sign.

Example 5: Which of the following expressions is equivalent to -4 ÷ 5? Explain your reasoning. a. 4 ÷ -5 b. -16 ÷ 20 c. -4 ÷ -5 Solution:



A and B are correct. When dividing a positive and a negative, the answer will be negative.

Example 6: Examine the family of equations in the table below. What patterns are evident? Create a model and context for each of the products. Write and model the family of equations related to 3 x 4 Using long division from elementary school, students understand the difference between terminating and repeating decimals.



*CISD Safety Net* 6.03E The student is expected to multiply and divide positive rational numbers fluently. Readiness Standard

Example 7: Equation Number Line Model Context 2•3=6 Selling two packages of apples at $3.00 per pack 2 • -3 = -6 Spending 3 dollars each on 2 packages of apples -2 • 3 = -6 Owing 2 dollars to each of your three friends -2 • -3 = 6 Forgiving 3 debts of $2.00 each Using long division, express the following fractions as decimals. Which of the following fractions will result in terminating decimals; which will result in repeating decimals? Identify which fractions will terminate. 6.03E Provide ample opportunities in context to determine a need to multiply or divide to help student’s computation/procedural fluency. Estimation is important to determine reasonableness of solutions. Problems in context will foster understanding of when to multiply or divide. Example 1: How many 1 ½ foot boards can be cut from a board that is 7 ½ feet long? What is the total length of 7 boards that are 1 ½ feet long? If the bill at a restaurant is $9.50, how much will each person pay if 5 people share the bill?

*CISD Safety Net* 6.11 The student is expected to graph points in all four quadrants using

6.11A

The x-axis and y-axis are extending to include negatives, so students begin to work with the Cartesian Coordinate system. Students worked with Quadrant I

• coordinate plane • graph • ordered pair • origin

McGraw Hill Chapter 1



ordered pairs of rational numbers. Readiness Standard

in elementary school. As the x-axis and y-axis are extending to include negatives, students begin to with the Cartesian Coordinate system. Students recognize the point where the x-axis and y-axis intersect as the origin. Students identify the four quadrants and are able to identify the quadrant for an ordered pair based on the signs of the coordinates. For example, students recognize that in Quadrant II, the signs of all ordered pairs would be (–, +).

Students understand the relationship between two ordered pairs differing only by signs as reflections across one or both axes. For example, in the ordered pairs (-2, 4) and (-2, -4), the y-coordinates differ only by signs, which represents a reflection across the x-axis. A change is the x-coordinates from (-2, 4) to (2, 4), represents a reflection across the y-axis. When the signs of both coordinates change, [(2, -4) changes to (-2, 4)], the ordered pair has been reflected across both axes. Students understand the relationship between two ordered pairs differing only by signs as reflections across one or both axes. For example, in the ordered pairs (-2, 4) and (-2, -4), the y-coordinates differ only by signs, which represents a reflection across the x-axis. A change is the x-coordinates from (-2, 4) to (2, 4), represents a reflection across the y-axis. When the signs of both coordinates change, [(2, -4) changes to (-2, 4)], the ordered pair has been reflected across both axes. Students are able to plot all rational numbers on a number line (either vertical or horizontal) or identify the values of given points on a number line.

• point • quadrant • rational numbers • x-axis (horizontal

axis) • x-intercept • y-axis (vertical axis) • y-intercept

Web Resources: Region XI: Livebinder NCTM: Illuminations



Grade 6 Pre-AP - Math 2016-17 Unit 1 - cleburne.k12.tx.us · (6.2D) Which list shows the ... apply...

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