01-7.NS Teacher's Guide and Lesson - EngageNY

7.NS

EduTron Corporation Draft for NYSED NTI Use Only

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RRAATTIIOONNAALL NNUUMMBBEERRSS OONN TTHHEE NNUUMMBBEERR LLIINNEE Understand p - q = p + (-q) for rational numbers

DRAFT 2012.11.29

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Teacher’s Guide: Common Core Mathematics 7th

Grade Subtracting Signed Numbers of 15

© 2012 EduTron Corp. All Rights Reserved. (781)729-8696 [email protected]

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I. Overview 3

II. Advanced Content Knowledge for Teachers 5

1. Can the difference of two numbers be negative? 5

2. Can the distance between two points on the number line be negative? 6

3. What’s subtle about negative mixed numbers? 8

4. How are additive inverses connected to each other? 9

5. Can the following be true? Why or why not? |a|=-a, where a is a number. 9

6. Reasons behind “Direction of Movement” on the number line when adding 10

7. Reasons behind “Direction of Movement” on the number line when subtracting 11

8. Some mathematicians say, “p – q = p + (-q) IS the definition of subtraction.” What do you think?

11

III. Lesson 8

A. Assumptions about what students know and are able to do coming into this lesson

8

B. Objectives 9

C. Anticipated Student Preconceptions/ Misconceptions 9

D. Assessments 10

E. Instructional Tools 10

F. Lesson Sequence and Description 11

G. Closure 14

H. Teacher Reflection 14

IV. Worksheets {To be developed} 15

A. Class Practice 15

B. Homework 15

V. Worksheets with Answers {To be developed} 15

A. Class Practice 15

B. Homework 15




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II.. OOvveerrvviieeww Essential Questions to be addressed in the lesson

How is subtracting signed numbers related to adding signed numbers?

Standards to be addressed in this lesson

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.

According to the Common Core Standards Map, in 7.NS.1, a, b, c, and d are strongly related; this progression should be reflected in the teaching sequence (See highlight in the Common Core Standards Map, Figure 1). Figure 1.




This standard is closely connected to the following standards:

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. 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.

a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

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.

c. [Current Lesson] d. Apply properties of operations as strategies to add and subtract rational numbers. Solve

real-world and mathematical problems

Standards of Mathematical Practice Although all eight Standards of Mathematical Practice should be instilled in students in these topics, four of them were chosen to be highlighted. They are denoted with these symbols: MPX.

MP1: Make sense of problems and persevere in solving them. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for and make use of structure.

The three components of rigor in the Common Core Standards (Computation Fluency, Conceptual Understanding and Problem Solving) will be denoted by Fluency Concept Application . Additional materials should be used to make the Rigor and Mathematical Practice Standards come alive. (For example, see a separate document Challenging Problems and Tasks.) Here is an overview of how the MP’s will play out in this lesson. MP1. Make sense of problems and persevere in solving them. Using their past experience of adding signed numbers, the students will make connections to the subtraction of signed numbers. With different problems, they will eventually realize that subtracting a negative number is the same as adding a positive number, which manifests itself




on the number line as moving to the right. They will also realize that subtracting a positive number is the same as adding a negative number, which means moving to the left on the number line. With continued practice and perseverance through solving the problems, they will be able to explain the connection between adding and subtracting a signed number and applying these skills to solve real-life problems. Concept Application

MP5. Use appropriate tools strategically. Students will use provided number lines to solidify conceptual understanding and to find answers to the problems. They will then use calculators to confirm that their answers are correct. The number lines will assist the students to make the necessary connection between addition and subtraction of signed numbers. As the students become proficient in the skill,

Fluency they will no longer need the number line because they will understand that subtracting a signed number is the same as adding its opposite. MP6. Attend to precision. As the students become proficient in subtracting signed numbers, they will be able to communicate the concept clearly using both the number line and algorithm rules. They will be able to calculate accurately and efficiently, and express numerical answers with a degree of precision appropriate for the problem context. MP7. Look for and make use of structure. Using the examples given in the lesson description, the students will notice that 5 – 9 has the same direction on the number line and solution as 5 + (-9). They will also notice the structure of how the other examples from class description are related to the pre-assessment question. The students should also notice that the first number in the numerical expression of subtraction (minuend) does not change when the statement is re-written as an addition problem. Using this structure, the students should also be able to re-write addition problems as subtraction problems where necessary.

II. Advanced Content Knowledge 7th



IIII.. AAddvvaanncceedd CCoonntteenntt KKnnoowwlleeddggee 1. Can the difference of two numbers be negative? Concept

In common language the difference of two quantities is always positive (or zero); it is not true in mathematics. In mathematics, the difference of two numbers, a and b, means a – b, in that order. The difference of two numbers, b and a, is b – a. For example, the difference of 2 and 7 is 2 – 7 = -5.

2. Can the distance between two points on the number line be negative? Concept

Students need to understand that the distance between two numbers a and b is ba . The

distance involves absolute values and it cannot be negative.

3. What’s subtle about negative mixed numbers?

In the notation of mixed numbers, e.g., 24

3, an addition sign “+” is implied, but not written

out explicitly, between the whole number part (2) and the fraction part (4

3). For a positive

mixed number, both the whole number and the fraction part are positive. This means that

24

3 = 2 +

4

3. For a negative mixed number, both the whole number part and the fraction

part are negative because the negative sign distributes into both parts. -24

3 = - (2 +

4

3) = -2

+ -4

3. Another example of written convention will be described here: For 2 3 , a

multiplication sign “x” is implied, but not written out explicitly, between the rational part (2)

and the irrational part ( 3 ).

4. How are additive inverses connected to each other?

Additive inverses or opposite numbers have the same distance from zero. One number is positive and the other is negative; the sum of them is 0. They have the same absolute

value, e.g., 3

22

3

22 because the two numbers have the same distance from zero on the

number line.

II. Advanced Content Knowledge 7th



5. Can the following be true? Why or why not? aa , where a is a number. Concept

This equation is true if, and only if, a is less than or equal to 0.

6. Reasons behind “Direction of Movement” on the number line when adding

a. Adding a positive number means moving to the right on the number line because the value of the sum resulting from the addition is increased.

b. Adding a negative number means moving to the left on the number line because the value of the sum resulting from the addition is decreased.

7. Reasons behind “Direction of Movement” on the number line when subtracting

Subtracting a number is equivalent to adding its additive inverse. This means that p – q = p + (-q), where p and q are signed numbers. This means that: a. Subtracting a positive number means moving to the left on the number line since this is

the inverse operation of adding a positive number. (See last question 6.a.) The value of the difference resulting from the subtraction is decreased.

b. Subtracting a negative number means moving to the right on the number line (since this is inverse operation of 6.b. adding a negative number) and the value of the difference resulting from the subtraction is increased—just like adding a positive number.

8. Some mathematicians say, “p – q = p + (-q) IS the definition of subtraction.” What do you

think?

Yes. You can say that.

III. Lesson Teacher’s Guide 7th



IIIIII.. LLEESSSSOONN Note to teachers: Although this plan is for one lesson, students will need more than one lesson on subtracting signed numbers before they are considered proficient in the concept. The follow-up lesson should have similar problems to those provided in this lesson so that the students have an opportunity to practice to attain fluency and better understanding.

A. Assumptions about what students know and are able to do coming into this

lesson

a. Students can add and subtract positive numbers that are whole numbers, fractions, or decimals with positive sums and difference. Examples:

1. 302 + 197

2. 302 – 197

3. 9

82

4

14

4. 9

82

4

14

5. 397.05 + 23.88

6. 397.05 – 23.88

b. Students know what additive inverses or opposite numbers are. The students should

also understand that the sum of additive inverses is zero. Examples: i. The additive inverse of -4 is 4. That means -4 + 4 = 0.

ii. The additive inverse of 4

3 is -

4

3. That means 0

4

3

4

3 .

iii. The additive inverse of 24

3

is -2

4

3 = - (2 +

4

3) = -2 + -

4

3. That means

04

32

4

32

.

c. Students know what inverse operations are. They should know that addition and

subtraction are inverse operations and so are multiplication and division.

d. Students know what absolute values are. They should know that absolute value is the distance from zero to the number and that it is always non-negative. Examples:

i. The absolute value of six, denoted as 6 = 6, because it is six units away from

zero.

ii. 5.23 = 23.5




B. Objectives By the end of the lesson, the students will know that: Concept Subtracting a number is the same as adding its additive inverse (or its opposite). By the end of the lesson, the students will be able to: Fluency Re-write a subtraction problem as an addition problem and then solve it.

C. Anticipated Student Preconceptions/ Misconceptions

1. Students will frequently forget the direction to move when adding on a number line. Remedy: It is advisable to start with smaller numbers that they are familiar with before giving problems with larger numbers or with fractions, or decimals.

2. When interpreting a negative mixed number, the students frequently assume that the

whole number part is negative and the fraction part is positive instead of considering the whole mixed number as negative, both the whole number and the fraction part (See Advanced Content Knowledge 3).

Remedy: Just as students are taught that 23 means 20 + 3, and that 24

3 means 2 +

4

3,

teachers should explicitly explain what -24

3 means. They should lead the students to

understand that it means (-2 + -4

3) and not (-2 +

4

3).

3. Students often make the mistake of assuming that signed numbers mean only integers. Remedy: They should be exposed to exercises that include signed fractions and decimals to curb this mistake.

4. When dealing with addition and subtraction rules, students often make the mistake of changing the sign of the first number instead of leaving it as it is and then changing the subtraction sign and changing the second number to its additive inverse. Remedy: Students should spend more time working on addition and subtraction using the number line so that they may have a strong foundation and understanding of the reason that subtraction changes to addition and the second number is changed to its additive inverse.




D. Assessments

Pre- assessment/ Formative Post – assessment/ Formative

Discussion questions: Fluency Concept MP3 1. What are opposite numbers and their

relationships? 2. What is the absolute value of a number? 3. What are inverse operations? 4. Using a number line solve the following

problems: a. 5 + 9 b. -5 + 9 c. -5 + (-9) d. 5 + (- 9)

Think-pair-share Fluency Concept MP3 Write the equivalent addition problem and then evaluate:

a. -10 – 7 b. -10 – (-7) c. 119 – (-136)

d. -5

7

12 - (-6

7

15)

e. 23.45 – 36.33 f. Then inter-mix these with

addition problems.

E. Instructional tools A number line that is either projected onto a screen or drawn on the board, a worksheet with several numbered number lines, and a calculator. The students will only use the calculators to confirm their answers after they have used the number lines.




F. Lesson Sequence and Description

The following lesson sequence provides a baseline for building foundation skills and concepts. Additional materials should be used to make the practice standards come alive. (See a separate document Challenging Problems and Tasks.) The class discussion starts with the pre-assessment so that misconceptions may be cleared. The lesson proceeds with the student solving four subtraction problems using number lines. The problems should have the same numbers as the pre-assessment problems, such as 5 and 9. Other non-number line solution approaches, if mathematically correct, should be accepted. The connection among different interpretations/approaches should be explicitly established.

a. 5 – 9

b. -5 – 9

c. 5 – (-9)

d. -5 – (-9)

Instruction leads the students to establish (or discover) the relationship between the pre-assessment and the subtraction questions here (involving both positive and negative subtrahends). [Teacher’s note: In 5 – 3, 5 is the minuend and 3 is the subtrahend]. Repetition is essential in acquiring fluency here. Use additional application problems to connect the topic to the real world. (See examples in “Application questions”.) The conclusion is that subtracting a number is equivalent to adding its additive inverse. Extension questions for class work or homework: Fluency Concept

1. Without using a number line, re-write the problem as an addition problem and then

solve it: a. 10 – 13 b. 10 – (-3)

c. – 10 – 13 = d. – 10 – (-3)




e.

53

4 21

5 = f.

53

4 (2

1

5)

g. – 11.2 – 0.39 h. – 11.2 – (-0.39) i. –285 – (-124) – (-321) j. –379 – 128 – 251

Application questions: Concept Application

1. The temperature in Albany, NY, on January 7th at 11 a.m. was 13 o F. By 9 p.m., the

temperature had dropped 21 o . What was the temperature at 9 p.m.?

2. The highest point in the world, Mt. Everest, is about 8,850 meters above sea level. The lowest point in the world, Dead Sea Shore, is about 418 meters below sea level. What is the difference of heights of the two points?

3. At noon, the temperature of a certain place is registered as 35 o C. At 1 a.m. the next

day, the temperature was -5 o C. a. What was the difference in temperature?

b. It was found that the temperature after 1 a.m. further decreased by 0.5 o C every hour. What was the temperature at 4 a.m.?

4. The table below shows a company’s profits and losses for the first six months of a

certain year.

January $5000 profit April $1000 profit

February $8000 loss May ?

March ? June $3000 loss

a. Re-write this table using positive numbers for profits and negative numbers for

losses. b. What is the total profit/loss for the first two months of the year? c. There is a net loss of $5000 in the first quarter (3 months) of the year. What is the

profit/loss in March? d. Using the answer in part (c), what is the profit/loss in May if the company “broke

even” (net loss = net profit = $0) in the first 6 months?

5. The difference of two numbers is -10.

a. Can both numbers be positive? If not, explain. If so, what do you notice about the numbers?




b. Can one of the numbers be positive and the other be negative? If not, explain. If so, what do you notice about the numbers?

c. Can both numbers be negative? If not, explain. If so, what do you notice about the numbers?

6. The sum of two numbers is -10. a. Can both numbers be positive? If not, explain. If so, what do you notice about the

numbers? b. Can one of the numbers be positive and the other be negative? If not, explain. If so,

what do you notice about the numbers? c. Can both numbers be negative? If not, explain. If so, what do you notice about the

numbers?

d. Can the following be true? Why or why not? |a| = -a, where a is a number.

7. Are the following correct? Why or why not? p and q are signed numbers. a. p – q = p + (-q) b. p + q = p – (-q) c. p – q = p – (-q) d. p + q = p + (-q)

8. A Beryllium atom is made up of a nucleus at the center and 4 electrons on the outside. Note: Each electron has 1 negative electric charge; each proton has 1 positive electric charge; neutrons are neutral and do not have any electric charge. (a) What is the nucleus made up of? (b) What is the net electric charge of the nucleus itself? (c) What is the net electric charge of the whole atom (nucleus + electrons)? (d) The outer electrons can be knocked off so that the “atom” becomes an “ion” with

some net electric charge. This process is called ionization. If we take away the 2 outer electrons from the Beryllium atom, it becomes a Beryllium ion. What will this ion’s electric charge be? [Teacher’s note: One of the ways is to start from neutral atom in part c and take away 2 * (-1). That is, 0 – (-2) = 2.]




G. Closure

Closure to the lesson using the post-assessment activity (See table in lesson description): Use the think-pair-share activity so that the students will demonstrate their level of understanding of, and fluency concerning, the concept. They will discuss with the peer partner and share their answer to the whole class. They will be required to justify why their answer is correct by demonstrating it on a number line.

H. Teacher Reflection

Were the objectives realized?

Did the students adequately demonstrate understanding of, and fluency concerning,

subtraction on a number line?

Did the students adequately demonstrate understanding of, and fluency concerning, re-

writing subtraction problems as addition problems?

How would I improve instruction to achieve greater success?

IV. Worksheets Teacher’s Guide 7th



IV. Worksheets A. Class Practice

01-7.NS Teacher's Guide and Lesson - EngageNY

Documents

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