Unit 1

NUMBER SENSE AND THE NUMBER SYSTEM

Unit Overview

In Part I of this unit, students extend their previous understandings of numbers and the ordering of numbers to the full system of rational numbers, which includes negative rational numbers, and in particular negative integers. They reason about the order and absolute value of rational numbers and about the location of points in all four quadrants of the coordinate plane. Students then apply their understanding of the coordinate plane to graph polygons and find vertical or horizontal distances between points.

In Part 2 of this unit, students build on their understanding of the meaning of fractions, the meanings of multiplication and division, and the relationship between multiplication and division to find quotients of fractions and explain why the procedures for dividing fractions make sense. Students apply their understanding of quotients of fractions to solve real world problems. Students then apply addition, subtraction, multiplication, and division to decimals. Students also write and evaluate numerical expressions involving whole-number exponents.


PART 1:

Understand that the positive and negative representations of a number are opposites in direction and value. Use integers to represent quantities in real-world situations and explain the meaning of zero in each situation.

Extend the understanding of the number line to include all rational numbers and apply this concept to the coordinate plane.

a. Understand the concept of opposite numbers, including zero, and their relative locations on the number line.

b. Understand that the signs of the coordinates in ordered pairs indicate their location on an axis or in a quadrant on the coordinate plane.

c. Recognize when ordered pairs are reflections of each other on the coordinate plane across one axis, both axes, or the origin.

d. Plot rational numbers on number lines and ordered pairs on coordinate planes.

Understand and apply the concepts of comparing, ordering, and finding absolute value to rational numbers.

a. Interpret statements using equal to (=) and not equal to (≠).

b. Interpret statements using less than (<), greater than (>), and equal to (=) as relative locations on the number line.

c. Use concepts of equality and inequality to write and to explain real-world and mathematical situations.

d. Understand that absolute value represents a number’s distance from zero on the number line and use the absolute value of a rational number to represent real-world situations.

e. Recognize the difference between comparing absolute values and ordering rational numbers. For negative rational numbers, understand that as the absolute value increases, the value of the negative number decreases.

Extend knowledge of the coordinate plane to solve real-world and mathematical problems involving rational numbers.

a. Plot points in all four quadrants to represent the problem.

b. Find the distance between two points when ordered pairs have the same x-coordinates or same y-coordinates.


Apply the concepts of polygons and the coordinate plane to real-world and mathematical situations.

a. Given coordinates of the vertices, draw a polygon in the coordinate plane.

b. Find the length of an edge if the vertices have the same x-coordinates or same y-coordinates.


Common Misconceptions

http://mathmistakes.org/

Some students may believe the greater the magnitude of a negative number, the greater the number. To help with this misconception, continue to use the number line. Have the students trace a horizontal number line with a finger starting at a positive number such as 10 and moving left one number at a time. Ask the student each time the finger moves one number left if the number is getting larger or smaller. Continue across 0. By then, a pattern of numbers getting smaller as you move left on the number line should be established.

Some sixth graders do not understand that negative signs change a number to the same distance on the opposite side of zero. Use a tool such as ruler to measure the distance to prove this is true. Some students confuse quadrant labels I through IV going counterclockwise. When introducing the quadrants, have student write the quadrant numbers in the quadrants to help them remember. Some learners may confuse (3,2) and (-3,2), thinking both ordered pairs look the same. Using paper folding or mirrors may help the students understanding the connection between signs on coordinates and their reflections across the axes.

Common misconceptions occur when students are unable to order rational numbers on the number line. Some students incorrectly place -134 between -1 and 0 instead of between -2 and -1. To address this, have students order the opposites. For example, if a student has difficulty placing -134 on the number line, have the student place +134. Discuss with the student how 134 came between 1 and 2. Then use that reasoning to help the student place -134.

Students may have procedural graphing misconceptions and may plot points in spaces rather than intersections. Some sixth graders count intervals on lines rather than x- or y-axes. Provide hands-on experiences for these learners. Have students plot real objects on a coordinate grid while you observe. Then, have them find the distance between the objects and explain how they found it.

Students may not understand that larger negative numbers are smaller in value.

Students may make common errors when plotting points in a coordinate plane (transposing the x- and y-coordinates, interchanging vertical and horizontal lines, struggling visually with the difference between the lines, and confusing the positive and negative parts of the perpendicular number lines when plotting points.)

Students may confuse the absolute value symbol with the number one.

(Common Core Companion: The Standards Decoded, Grades 6 - 8: What They Say, What They Mean, How to Tech Them, Miles and Williams, 2016)

‘I can’ Statements

  • I can describe and give examples of how positive or negative numbers are used to describe quantities having opposite directions or opposite values (R).
  • I can recognize that positive and negative signs represent opposite values and/or directions (K).
  • I can explain that the number zero is the point at which direction or value will change (K).
  • I can use positive and negative numbers along with zero to represent real world situations (S).
  • I can show and explain why every rational number can be represented by a point on a number line (R).
  • I can plot a number and its opposite on a number line and recognize that they are equidistant from zero (K).
  • I can find the opposite of any given number including zero (K).
  • I can use the signs of coordinates to determine the location of an ordered pair in the coordinate plane (K).
  • I can reason about the location of two ordered pairs that have the same values but different signs (R).
  • I can plot a point on a number line or coordinate plane (S).
  • I can read a point from a number line or a coordinate plane (S).
  • I can describe the relative position of two numbers on a number line when given an inequality (S).
  • I can interpret a given inequality in terms of a real world situation (S).
  • I can define absolute value as it applies to a number line (K).
  • I can describe absolute value as the magnitude of the number in a real world situation (K).
  • I can compare between using a signed number and using the absolute value of a signed number when referring to real world situations (R).
  • I can graph points in any quadrant of the coordinate plane to solve real world and mathematical problems (S).
  • I can use absolute values to find the distance between two points with the same x-coordinates or the same y-coordinates (R).

New Academic Vocabulary

  • Integers
  • Quadrant
  • Opposite
  • Zero pair
  • Additive inverse
  • Absolute value
  • Inequality (including ≠)

Resources and Links

Link - Challenging math problems worth solving.

Desmos activity: Integer game

Math Talks - wide variety of prompts

Part 2

Compute and represent quotients of positive fractions using a variety of procedures (e.g., visual models, equations, and real-world situations).

Fluently divide multi-digit whole numbers using a standard algorithmic approach.

Fluently add, subtract, multiply and divide multi-digit decimal numbers using a standard algorithmic approach.

Find common factors and multiples using two whole numbers.

  1. Compute the greatest common factor (GCF) of two numbers.
  2. Compute the least common multiple (LCM) of two numbers.
  3. Express sums of two whole numbers, each less than or equal to 100, using the distributive property to factor out a common factor of the original addends.

Investigate and translate among multiple representations of rational numbers (fractions, decimal numbers, percentages).

Common Misconceptions

http://mathmistakes.org/

Students may incorrectly model division of fractions. Some students may think dividing by 12is the same as dividing in half. Dividing by 12 means to find how many one halves there are in a quantity. Dividing in half means to take quantity and divide it into two equal parts. To address this misconception, ask the students to demonstrate two examples, one that shows dividing by 12 and another that shows dividing in half. For example, 9 divided by 12equals 18. But 9 divided in half equals 412.

For some students, the traditional standard algorithm is difficult simply because of the many steps involved in the procedure. Some sixth graders may focus individually on digits when dividing rather than thinking about the whole number. Other students may ignore the place value and get an incorrect answer. To help students, remind them to describe both the place value as they divide and place value of the digits in the quotients. Ask the students to show the steps of division, one at a time. Provide graph paper to keep the work legible.

Some students may not remember to use the concepts of place value when adding tenths to hundredths. For example, when adding five-tenths to eighty-five hundredths, some students may not realize the answer is one whole and thirty-five hundredths. To help with this misconception, try using decimal blocks or drawing a picture to show how the decimals have been added. Adding zero to 0.5 to write 0.50 before adding 0.85 helps students to put focus on the place values.

Some students may confuse the concepts of factors and multiples. To help with this, use the vocabulary of factors and multiples when working with multiplication and division such as the numbers being multiplied are the factors; the product is the multiple.

Students may incorrectly model division of fractions.

Students may improperly place the decimal in an answer.

Students may interchange the uses for factors and multiples.

When using the distributive property, students may only distribute to the first term.

(Common Core Companion: The Standards Decoded, Grades 6 - 8: What They Say, What They Mean, How to Tech Them, Miles and Williams, 2016)

‘I can’ Statements

  • I can use a visual model to represent the division of a fraction by a fraction (S).
  • I can divide fractions by fractions using an algorithm or mathematical reasoning (S),
  • I can justify the quotient of a division problem by relating it to a multiplication problem (R).
  • I can use mathematical reasoning to justify the standard algorithm for fraction division (R).
  • I can solve real world problems involving the division of fractions and interpret the quotient in the context of the problem (S).
  • I can create story contexts for problems involving the division of a fraction by a fraction (P).
  • I can use the standard algorithm to fluently divide multi-digits numbers (S).
  • I can fluently add and subtract multi-digit decimals using the standard algorithm (S).
  • I can fluently multiply multi-digit decimals using the standard algorithm (S).
  • I can fluently divide multi-digit decimals using the standard algorithm (S).
  • I can find all factors of any given number less than or equal to 100 (S).
  • I can find the greatest common factor of and two numbers, less than or equal to 100 (S).
  • I can create a list of multiples for any number less than or equal to 12 (S).
  • I can use the distributive property to rewrite the sum of two whole numbers when appropriate (e.g., 24+8=8(3+1)(S).
  • I can use the distributive property to rewrite a simple addition problem when the addends have a common factor (S).
  • I can translate among multiple representations of rational numbers:
    • Fraction to decimal.
    • Fraction to percent.
    • Decimal to percent.
    • Decimal to fraction.
    • Percent to fraction.
    • Percent to decimal.

New Academic Vocabulary

  • Reciprocal
  • Inverse
  • Greatest common factor
  • Least common multiple
  • Prime factorization
  • Distributive property
  • Rational number

RESOURCES

Shodor Interactivate - interactive digital tools on a variety of topics

Link - Challenging math problems worth solving.

Desmos activity: Guessing Game

Illuminations activity from NCTM: Product game

Math Talks - wide variety of prompts

DOK Matrix Secondary Maths (with examples)

Learn Desmos - your virtual calculator & graphing utility

HippoCampus - presentations, worked examples & simulations