# 7.1.2A Applying & Making Sense of Rational Numbers

7
Subject:
Math
Strand:
Number & Operation
Standard 7.1.2

Calculate with positive and negative rational numbers, and rational numbers with whole number exponents, to solve real-world and mathematical problems.

Benchmark: 7.1.2.1 Arithmetic Procedures

Add, subtract, multiply and divide positive and negative rational numbers that are integers, fractions and terminating decimals; use efficient and generalizable procedures, including standard algorithms; raise positive rational numbers to whole-number exponents.

For example: $3^{4}\times \left ( \frac{1}{2} \right )^{2}=\frac{81}{4}$.

Benchmark: 7.1.2.2 Explain Arithmetic Procedures

Use real-world contexts and the inverse relationship between addition and subtraction to explain why the procedures of arithmetic with negative rational numbers make sense.

For example: Multiplying a distance by -1 can be thought of as representing that same distance in the opposite direction. Multiplying by -1 a second time reverses directions again, giving the distance in the original direction.

Benchmark: 7.1.2.3 Calculators & Rational Numbers

Understand that calculators and other computing technologies often truncate or round numbers.

For example: A decimal that repeats or terminates after a large number of digits is truncated or rounded.

Benchmark: 7.1.2.6 Absolute Value

Demonstrate an understanding of the relationship between the absolute value of a rational number and distance on a number line. Use the symbol for absolute value.

For example: |-3| represents the distance from -3 to 0 on a number line or 3 units; the distance between 3 and $\frac{9}{2}$ on the number line is |3-$\frac{9}{2}$| or $\frac{3}{2}$.

## Overview

Big Ideas and Essential Understandings
##### Standard 7.1.2 Essential Understandings

In this standard, students will develop a unified understanding of number, recognizing fractions, decimals (that have a finite or a repeating decimal representation), and percents as different representations of rational numbers. Students extend addition, subtraction, multiplication and division to all rational numbers, maintaining the properties of operations and the relationships between addition and subtraction, and multiplication and division. By applying these properties, and by viewing negative numbers in terms of everyday contexts (e.g., amounts owed or temperatures below zero), students explain and interpret the rules for adding, subtracting, multiplying and dividing with negative numbers.

The focus of instruction at the 7th grade level is on being able to comfortably translate between decimal and fractional forms of a number for both positive and negative values. Students should be able to compare numbers and manipulate the values to derive other forms of the numbers to make comparing less inhibiting and more accessible. Students will also use ratios and proportional reasoning to solve problems in various contexts. Students will be able to use information given to help find missing values. Their knowledge of equivalent fractions and scaling will enable them to use ratios and solve proportions.

##### All Standard Benchmarks

7.1.2.1
Add, subtract, multiply and divide positive and negative rational numbers that are integers, fractions and terminating decimals; use efficient and generalizable procedures, including standard algorithms; raise positive rational numbers to whole-number exponents.

7.1.2.2
Use real-world contexts and the inverse relationship between addition and subtraction to explain why the procedures of arithmetic with negative rational numbers make sense.
7.1.2.3
Understand that calculators and other computing technologies often truncate or round numbers.
7.1.2.4
Solve problems in various contexts involving calculations with positive and negative rational numbers and positive integer exponents, including computing simple and compound interest.
7.1.2.5
Use proportional reasoning to solve problems involving ratios in various contexts.
7.1.2.6
Demonstrate an understanding of the relationship between the absolute value of a rational number and distance on a number line. Use the symbol for absolute value.

Benchmark Cluster
##### 7.1.2 Group A - Applying and Making Sense of Rational Numbers

7.1.2.1
Add, subtract, multiply and divide positive and negative rational numbers that are integers, fractions and terminating decimals; use efficient and generalizable procedures, including standard algorithms; raise positive rational numbers to whole-number exponents.
7.1.2.2
Use real-world contexts and the inverse relationship between addition and subtraction to explain why the procedures of arithmetic with negative rational numbers make sense. For example, multiplying a distance by -1 can be thought of as representing that same distance in the opposite direction. Multiplying by -1 a second time reverses directions again, giving the distance in the original direction.
7.1.2.3
Understand that calculators and other computing technologies often truncate or round numbers.
7.1.2.6
Demonstrate an understanding of the relationship between the absolute value of a rational number and distance on a number line. Use the symbol for absolute value.
For example, |-3| represents the distance from -3 to 0 on a number line or 3 units; the distance between 3 and $\frac{9}{2}$ on the number line is |3 - $\frac{9}{2}$| or $\frac{3}{2}$.

##### What students should know and be able to do [at a mastery level] related to these benchmarks:
• Perform basic operations (add, subtract, multiply and divide) with all positive and negative rational numbers, including 2 or 3 (up to 5) operations in one problem;
• Use a number line that extends to the right and left of 0;
• Understand the concept of absolute value as the distance from 0 on a number line;
• Order negative and positive rational numbers;
• Use positive whole number exponents;
• Demonstrate awareness, knowledge and use of the absolute value symbol;
• Understand the concept of addition and subtraction with negative rational numbers;
• Understand how calculators operate when numbers need to be either rounded or truncated.
##### Work from previous grades that supports this new learning includes:
• Know basic facts (multiplication/division) through 12's;
• Change from fractional form to decimal form;
• Know how to calculate with positive integer exponents;
• Locate positive numbers on a number line;
• Plot points in quadrant 1 on the coordinate grid;
• Add, subtract, multiply and divide with positive numbers;
• Compare numbers using greater-than and less-than symbols;
• Convert among decimals, fractions, mixed numbers and percents;
• Demonstrate appropriate calculator usage;
• Demonstrate understanding and use of inverse operations;
• Model addition and subtraction of positive numbers in a variety of ways;
• Use estimation strategies with fractions and decimals;
• Use fractions, decimals and mixed numbers;
• Calculate percent of a number and what percent one number is of another;
• Solve real-world and mathematical problems involving arithmetic with fractions, decimals and mixed numbers.
Correlations
##### NCTM Standards

Understand numbers, ways of representing numbers, relationships among numbers, and number systems:

• Work flexibly with fractions, decimals, and percents to solve problems;
• Understand and use ratios and proportions to represent quantitative relationships.

Understand meanings of operations and how they relate to one another:

• Understand the meaning and effects of arithmetic operations with fractions decimals, and integers;
• Understand and use the inverse relationships of addition and subtraction, multiplication and division, and squaring and finding square roots to simplify computations and solve problems.

Compute fluently and make reasonable estimates:

• Select appropriate methods and tools for computing for computing with fractions and decimals from among mental computation, estimation, calculators or computers, and paper and pencil, depending on the situations and apply the selected methods;
• Develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results.
##### Common Core State Standards (CCSS)

7.NS (The Number System ) Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

• 7.NS. 1. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram;
• 7.NS.1.b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts;
• 7.NS.1.c. Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts;
• 7.NS.1.d. Apply properties of operations as strategies to add and subtract rational numbers;
• 7.NS.2. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers;
• 7.NS.2a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts;
• 7.NS.2.c. Apply properties of operations as strategies to multiply and divide rational numbers;
• 7.NS.3. Solve real-world and mathematical problems involving the four operations with rational numbers.

6.NS (The Number System) Apply and extend previous understandings of numbers to the system of rational numbers.

• 6.NS.7c. Understand ordering and absolute value of rational numbers. 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.

6.EE (Expressions and Equations) Apply and extend previous understandings of arithmetic to algebraic expressions.

• 6.EE.1. Write and evaluate numerical expressions involving whole-number exponents.

7.EE (Expressions and Equations) Use properties of operations to generate equivalent expressions.

• 7.EE.3. Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations as strategies to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.

7.RP (Ratios and Proportional Relationships) Analyze proportional relationships and use them to solve real-world and mathematical problems.

• 7.RP.3. Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

## Misconceptions

Student Misconceptions
##### Student Misconceptions and Common Errors
• Students may think the absolute value symbols mean "opposite of."
• Student may think that bars make things positive. For example, |-4 - 5| becomes |-4 + 5|.
• When computing with absolute value bars, students want to change subtraction to addition and add the opposite WITHIN the absolute value bars. For example: |-4| - |-4|. Students want to change it to |-4| + (+4).
• Students may think that adding two negatives equals a positive (confusion of procedures with multiplication).
• Students have a hard time "seeing" negative numbers unless real world applications are used, such as temperatures, money, elevation, etc.
• Students may not understand that in the equation -22= 4, the exponent is only squaring what is immediately in front of it, which is the 2, NOT the -2. By writing 0 - 22, students often see that that IS 0 - 4, which is -4, not +4.
• Students may think that a negative times a negative will yield a negative answer.
• Students mistakenly think that -2 - 2 = 0 because when they add opposites they get 0.
• When working with a real world problem and students know that subtraction could be done to find the answer, they sometimes don't make the connection that the answer could also be found by a number problem that is adding a negative value.
• When given an absolute value problem, students sometimes forget that there are sometimes two possible values for answers. For example, in the equation |x| = 5 what could x be? Students may say x = 5 but forget that the answer could also be x = - 5.
• Students frequently aren't familiar with the word 'truncate.'
• Students often think that when they subtract opposites the answer will also be 0.

## Vignette

##### Vignette.1

This vignette explores positive and negative rates and the use of a number line.

Teacher: Today we are going to explore both positive and negative rates. Imagine a rate that is positive when a person moves from left to right in front of the number line and negative when that person moves right to left.

Allow time for student comments to indicate that they understand the idea.

Teacher: Next, imagine continuing the rate for one minute, two minutes, and so forth.

Do some examples at the chalkboard of rates, such as:

+10 yards/minute × 1 minute
+10 yards/minute × 2 minutes
+10 yards/minute × 3 minutes

Repeat the same examples for:

-10 yards per minute × 1 minute
-10 yards per minute × 2 minutes
-10 yards per minute × 3 minutes

Left-to-right movement in front of the number line is the model for forward rates. Right-to-left movement denotes negative rates.

Initially, the following situations are demonstrated:

positive time x positive rate = positive distance
positive time x negative rate = negative distance

Present the other two situations as opposites of the former two:

negative time x negative rate = positive distance
negative time x positive rate = negative distance

Student: How could you have a negative time? Time moves only forward.

Clearly, the students may be discomfited by the use of a model that accounted for only half the multiplication products. Allow students the chance to discuss this problem in small groups of two or three. Try to prompt questions such as the following:

Student: Maybe it's like a film that is rewinding.

Next, select two student volunteers. One student may operate the video camera while another student is walking approximately fifteen yards forward carrying a green "I am walking forward" sign. After that, the student will walk backward carrying a pink "backward" sign.

This lesson plan is adapted from the article:
Cooke, M.B.  (November 1993). A Videotaping Project to Explore the Multiplication of Integers. Arithmetic Teacher, 41(3), 170-171.

##### Vignette.2

In this vignette, students focus on the definition of absolute value.

Teacher: The concept of absolute value has many uses, but you probably won't see anything interesting for a few more classes yet. There is a technical definition for absolute value, but you could easily never need it. For now, you should view the absolute value of a number as its distance from zero. Let's look at the number line:

Teacher: The absolute value of x, denoted "| x |" (and which is read as "the absolute value of x"), is the distance of x from zero. This is why absolute value is never negative; absolute value only asks "how far?" not "in which direction?" This means not only that

| 3 | = 3, because 3 is three units to the right of zero, but also that | -3 | = 3, because -3 is three units to the left of zero.

Teacher: But here's a warning: The absolute-value notation is bars, not parentheses or brackets. Be sure to use the proper notation; the other notations do not mean the same thing. For example, the absolute value bars do NOT work in the same way as do parentheses. Whereas -(-3) = +3, this is NOT how it works for absolute value. Try this example:    Simplify -| -3 |.

Teacher: Given -| -3 |, let's first handle the absolute value part, taking the positive and converting the absolute value bars to parentheses:

-| -3 | = -(+3)

Teacher: Now let's take the negative through the parentheses:

-| -3 | = -(3) = -3

Teacher: As this illustrates, if you take the negative of an absolute value, you will get a negative number for your answer.

http://www.purplemath.com/modules/absolute.htm

## Resources

Instructional Notes
##### Teacher Notes
• Students should be encouraged to write down every step in solving absolute value problems so they can see what they are doing. If they are taught to drop the signs outside of the absolute value bars straight down, and calculate within the absolute value bars first, they may be less likely to make errors. Students know that subtracting a number is the same as adding its opposite, so students need to be taught to NOT do this step first if absolute value bars are in the problem.
• In the equation -22= -4, the exponent is only squaring what is immediately in front of it, which is the 2, NOT the -2. By writing 0 - 22, students often see that that IS 0 - 4, which is -4, not +4.
• Here is a way of thinking that could possibly help students remember the patterns for multiplying and dividing integers:
##### Rules for multiplying and dividing integers

It is good (+) to love (+), and it is bad (-) to hate (-), so:

If you love to love, that is good. (positive x positive = positive)
If you love to hate, that is bad. (positive x negative = negative)
If you hate to love, that is bad. (negative x positive = negative)
If you hate to hate, that is good. (negative x negative = positive)

• Hot air balloon model: Students have a hard time understanding adding and subtracting with positive and negative numbers. Many can memorize the algorithm for them, but not the "why." One way to help with this is by giving each student a copy of a number line and a hot air balloon illustration. The students tear out the hot air balloon off the number line paper. The hot air will go up - numbers increasing - with puffs of air (adding positives), or by taking away sandbags (subtracting a negative). The hot air balloon will go down - numbers decreasing - by adding a sandbag (adding a negative), or by taking away a puff of air (subtracting a positive). After using this illustration, many students catch on to the rules; the illustration may stick with them longer than the algorithms. Remind them to pull them out when they are working on any problems involving addition and subtraction.

• The idea that subtracting a negative number gives the same result as adding the opposite of the negative number (adding a positive) is difficult for many students. This must be developed over time and with the use of models, manipulatives and observations.
• An example of a virtual manipulative is Color Chips - a good interactive tool to use to demonstrate subtraction of integers. This website uses plus-minus chips to demonstrate adding positive and negative values.
• Multiplying two negatives and getting a positive number is difficult for students; it goes against their usual model of repeated adding.
• Given the problem |x| = 5, what could the value of x be? Have students make the connection to a number line and the distance away from zero. This question is asking what value(s) are 5 units away from 0. Make sure they make the connection that there are TWO answers to this problem and that they can recognize this problem on a number line.
• Students need to realize that |3 - $\frac{9}{2}$| represents the distance between 3 and 9/2 on a number line. Help students realize this distance could also be represented as |$\frac{9}{2}$ - 3|. Most students will understand the distance between two positive numbers. Also examine examples that include finding the distance between a positive and a negative number using absolute value. For example, have students use absolute value to represent the distance between 3 and -2. So distance d = |3 - (-2)| = |3 + 2| = 5. To make sure students understand the concept, give them an absolute value problem and have them translate the problem into one about the distance  between two numbers.
• Students need to understand that calculators only have so much space on the display screen in order to display the number. One way to have this discussion is to ask what decimal is equal to $\frac{2}{3}$. Most students will recognize the decimal equivalent is $0.\overline{6}$ and understand that the six is repeating. Then have the students divide 2 by 3 on their calculators. Ask the students to explain why the answer may be 0.666666667 and then brainstorm other rational numbers for which the calculator would behave in this way. Use proper vocabulary in explaining the concept (repeating, truncated and rounding).
Instructional Resources
• Multiplying Integers Using Videotape
In this lesson, students experience beginning-algebra concepts through discussion, exploration and videotaping. The concept of multiplication of integers is presented in a format which encourages understanding, not simply rote memorization of facts.
This lesson plan is adapted from the article:
Cooke, M.B.  (November 1993). A Videotaping Project to Explore the Multiplication of Integers. Arithmetic Teacher, 41(3), 170-171.
• Dynamic Paper
The Dynamic Paper tool allows you to create such things as a pentagonal pyramid that's six inches tall, a number line that goes from 18 to 32 by 5's, and a set of pattern blocks where all shapes have one-inch sides. The tool allows for the placement of images which are then exported as a PDF activity sheet or as a JPEG image for use in other applications or on the web. The site allows users to customize to meet the needs of their lesson and their learners.
• Using an Elevator to Evaluate Signed Number Expressions
Using the idea of an elevator moving up and down, students will be given a small set of rules to add and subtract signed integers. The goal of this method is that the number of rules that students have to memorize is minimal.
• Absolute value sample problems and hints
• National Lab of Virtual Manipulatives: Color Chips - Subtraction
Use color chips to illustrate subtraction of integers.
• National Lab of Virtual Manipulatives: Number Line Bounce
Number line addition and subtraction game.
• Related lessons
• Card "War" with multiplication
Students are with a partner and have a deck (or stack) of cards. All the cards are dealt out, and the students keep them in a pile facedown. Both students flip their cards over at the same time. Then they multiply the numbers on the cards, with red suits counted as negatives and the black suits counted as positives. The first person to say the product correctly gets the cards. Play continues until all the cards are used, or until the teacher tells them to stop.
• Provide two colored counters to model addition and subtraction. Let yellow be positive and red be negative. Pull out any zero pairs, which are the combination of one yellow and one red, since a positive (yellow) and a negative (red) equal zero. The remaining counters add up to the answer.

Example: 3 + -5
Y Y Y (represents positive 3)
R R R R R (represents negative 5)
There are three Y/R pairs which are equivalent to zero. Two reds are left;
Have students relate the actions to the algorithm.
Example: 3 - ( -4)
There are no negatives (R) to subtract so zero pairs (Y/R) must be added.
Y Y Y
YR YR YR YR
When the four negatives are removed, seven positives (Y) remain.
3 - (-4) is the same as 3 + 4 = 7
Source: Bounds, Ph.D., H.M. et al. (2007). 2007Mississippi Mathematics Framework Revised Strategies. Jackson, MS: Mississippi Department of Education. (p. 34).

•  Circle 0
The goal of this computer-based puzzle is to put three numbers inside of each circle so that they add up to 0. The puzzle is solved by dragging the black numbers to the blank spaces; blue numbers cannot be moved. When the three numbers inside any circle add up to 0, the circle changes color.
New Vocabulary
• absolute value: the absolute value of a number is its distance from 0 on a number line. It can be thought of as the value of a number when its sign is ignored.
Example: The numbers -3 and 3 both have an absolute value of 3.
• exponent: the number of times a number or expression (called base) is used as a factor of repeated multiplication. Also called the power.

Professional Learning Communities
##### Reflection - Critical Questions regarding the teaching and learning of these benchmarks
• How should the instruction be scaffolded for students?
• Are the assigned tasks connected to underlying concepts or focus on memorization?
• How can it be demonstrated that students have reached this learning goal?
• How was the lesson differentiated?
• Do students understand the difference between "opposite of" a number and "absolute value of" a number?
• Are students able to model their mathematical thinking using chips or a number line?
• What can students conclude about an answer they derive at on their calculators? (rounded/terminated/repeating)
• Are students aware of why two numbers can have the same absolute value?
• What computational errors are students still making?
• What strategies are students using to solve computational problems?
References

## Assessment

##### 1.

DOK: Level 1
Minnesota Grade 7 Mathematics MCA-III Item Sampler Item, 2011, Benchmark 7.1.2.1

##### 2.

DOK: Level 1
Minnesota Grade 7 Mathematics Modified MCA-III Item Sampler Item, 2011, Benchmark 7.1.2.1

##### 3.

DOK: Level 3
Minnesota Grade 7 Mathematics MCA-III Item Sampler Item, 2011, Benchmark 7.1.2.2

##### 4.

DOK: Level 2
Minnesota Grade 7 Mathematics MCA-III Item Sampler Item, 2011, Benchmark 7.1.2.6

##### 5.

DOK: Level 2
Massachusetts Comprehensive Assessment System Release of Spring 2010 Test Items

##### 6.

DOK: Level 2
Massachusetts Comprehensive Assessment System Release of Spring 2010 Test Items

7.

DOK: Level 2
Massachusetts Comprehensive Assessment System Release of Spring 2010 Test Items

## Differentiation

Struggling Learners
• Provide number lines and red/black chips so the students can see what they are doing when working with positive and negative numbers.
• Absolute value bars are often mistaken for meaning the "opposite" of the value within the bars or another form of brackets. Remind students that the vertical bars mean the distance from 0 on a number line.
• Fraction division using models
In this lesson, students represent the division of fractions using manipulatives, such as freezer pops, candy bars, and models, such as drawings squares. Students develop algorithms from these examples and solve problems using fractions.
This virtual manipulative uses plus-minus chips to demonstrate adding positive and negative values.
• Absolute value video
This video segment covers absolute value and the real world use of absolute value.
English Language Learners
• Absolute value bars are often mistaken for meaning the "opposite" of the value within the bars or another form of brackets. Remind students that the vertical bars mean the distance from 0 on a number line.
• Review the use of the term inverse operations, meaning "operations that undo each other." Addition and subtraction are inverse operations. For example, start with 7. Subtract 4. Then add 4.You are back to the original number 7. Thus, 7 - 4 + 4 = 7. Multiplication and division are inverse operations. For example, start with 12. Multiply by 2.Then divide by 2.You are back at the original number 12. Thus, (12 * 2) / 2 = 12.
Extending the Learning
• Include problems that have multiple steps with absolute value.
• Students could examine the effect of negative exponents.