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Computing and Problem Solving with Negative Numbers

Unit Plan

Computing and Problem Solving with Negative Numbers

Objectives

Students will learn how to compare and order negative numbers (integer and rational). They will:

  • understand the relationship between an integer and its additive inverse.
  • develop models and computational strategies for computing with negative numbers and solve problems in contexts that involve negative numbers.
  • extend understanding of operations and their properties for all rational numbers, including negative integers.
  • explain why the rules for adding, subtracting, multiplying, and dividing with negative numbers make sense.

Essential Questions

  • How is mathematics used to quantify, compare, represent, and model numbers?
  • How can mathematics support effective communication?
  • How are relationships represented mathematically?
  • How can expressions, equations and inequalities be used to quantify, solve, model, and/or analyze mathematical situations?
  • What makes a tool and/or strategy appropriate for a given task?

Related Unit and Lesson Plans

Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

Formative Assessment

  • View

    Multiple-Choice Items:

     

    1. To describe the prevailing wind direction in one location, a meteorologist assigned positive values (+) for east and negative values (−) for west.

    The table shows the speed and direction of the wind over a one-week period.

    Prevailing Wind Direction for One Week

    Day

    Direction

    Average Speed (mph)

    Monday

    east

    18

    Tuesday

    east

    11

    Wednesday

    west

    15

    Thursday

    east

    6

    Friday

    west

    8

    Saturday

    west

    22

    Sunday

    east

    9

     

    For this week, what was the net wind speed?

    1. 99
    2. 44
    3. −1
    4. −45

     

    1. You are going on a hiking trip. The overnight temperature may be as low as −10°C. You have sleeping bags with temperature ratings of 0°C, −5°C, 10°C, and −15°C. Which bag should you take?
    1. the 0°C-rated bag
    2. the −5°C-rated bag
    3. the 10°C-rated bag
    4. the −15°C-rated bag

     

     

    1. Two kilometers above ground, the temperature is 0°C. For each kilometer of height above ground, the temperature drops 7 degrees. What is the temperature 13 kilometers above ground?
      1. 0°C
      2. −77°C
      3. 91°C
      4. −91°C

     

    1. Scott thought he had $27 in his checking account, and he wrote a check for $12 to buy a new CD. On the way home, Scott remembered that he had withdrawn $20 the day before. What is the minimum amount Scott must quickly deposit in his account to make sure there is enough money for the check he wrote?
    1. $3
    2. $5
    3. $12
    4. $19

     

    1. An aircraft flying 1000 feet above ground began climbing at a rate of 300 feet per minute for 5 minutes. Then the aircraft descended at 100 feet per minute for
      18 minutes. How many feet above ground was the aircraft flying after climbing and descending?
    1. 300 feet
    2. 700 feet
    3. 1000 feet
    4. 2300 feet

     

    1. The temperature on Mars ranges between −94°F at night and 72°F during the day. What is the temperature range in a single day?
      1. 18°F
      2. 22°F
      3. 72°F
      4. 166°F

     

     

    1. Simplify   
    1. −2
    2. 2
    3. −20
    4. 20

     

    1. Simplify    (−18)(−2)
    1. −36
    2. −9
    3. 9
    4. 36

     

    1. Simplify    (14)(−3) + (4)(−12)
    1. −90
    2. −6
    3. 3
    4. −50

     

    Multiple Choice Answer Key:

    1. C

    2. D

    3. B

    4. B

    5. B

    6. D

    7. C

    8. D

    9. A

     

     

     

     

    Short-Answer Items:

    10. Simplify the expression  . Show each step in the process. Explain each step.

     

     

    11. Write a short paragraph explaining what kinds of real-life situations and applications negative numbers can represent. Back up your explanation with a specific example showing how negative numbers affect the real-life situation.

     

     

    12. Write a story problem that would represent the following calculation:
    7 + (−10) + 4 + 20 + (−25). Show the solution to your story problem.

     

     

     

    Short-Answer Key and Scoring Rubrics:

    10. Simplify the expression  . Show each step in the process. Explain each step.

    Example 1: −8 × 5 = −40; 4 + (−5) = −1; and −40 ÷ −1 = 40

    Example 2: 4 + (−5) = −1; −8 ÷ −1 = 8; and 8 × 5 = 40

    Example 3: 4 + (−5) = −1; 5 ÷ −1 = −5; and −8 × −5 = 40

    Points

    Description

    2
    • The student correctly simplifies the expression to 40.
    • The student shows all three calculations; all mathematical calculations are accurate; and negative signs are correctly used.
    • Each step includes adequate explanation.

    1
    • The student incorrectly simplifies the expression.
    • The student shows all three calculations; two of the three mathematical calculations are accurate; and negative signs are correctly used.
    • Two of the three steps include adequate explanation.

    0
    • The student incorrectly simplifies the expression.
    • The student fails to show all three calculations; one or none of the three mathematical calculations are accurate; and negative signs are incorrectly used.
    • No part of the explanation is accurate or adequate.

     

    11. Write a short paragraph explaining what kinds of real-life situations and applications negative numbers can represent. Back up your explanation with a specific example showing how negative numbers affect the real-life situation. Answers will vary.

    Points

    Description

    2
    • The student’s answer is complete and accurate.
    • The student’s explanation makes logical sense.
    • Specific examples are present to show understanding of how negative numbers affect real-life situations.

    1
    • The student’s answer is partially correct and accurate.
    • The student’s explanation is partially correct, but logical errors may exist and/or parts are missing. For example, the student may include the addition of a negative number without indicating that the number must be subtracted.
    • Some understanding of how negative numbers affect real-life situations is evident.

    0
    • The student’s answer is missing, incomplete, or inaccurate.
    • No explanation is provided. For example, the student may add negative numbers inappropriately by adding rather than subtracting.
    • The provided answer shows little to no understanding of how negative numbers affect real-life situations.

     

    12. Write a story problem that would represent the following calculation:
    7 + (−10) + 4 + 20 + (−25). Show the solution to your story problem. Answers will vary.

    Points

    Description

    2
    • The student creates a credible story problem for the sum of 7, −10, 4, 20, and −25.
    • The student calculates the correct sum (−4) with all work evident.

    1
    • The student creates a story problem that begins appropriately but has errors or missing parts. For example, the student may represent
      7 + (−10) as indicative of an initial temperature of positive 7 degrees that then drops to negative 10 degrees, rather than the initial temperature of 7 degrees decreasing by 10 degrees.
    • The sum may be incorrect due to a minor calculation error.

    0
    • The student does not provide a story problem, or the provided story problem is illogical or unclear.
    • The solution is not present or the sum is incorrect due to major calculation error(s).

     

    Performance Assessment:

    Be the Club Treasurer! Keeping It Balanced

    Suppose you are the treasurer of a club. The club has a checking account balance of $74. You have paid bills in the amounts of $17, $14, and $25 for supplies. You have collected dues of $6 from 7 different members. You are to deposit the collected dues into the account.

    • Create a table that shows the account transactions described above.
    • In words, how do you find the new balance?
    • If the bank charged a $10 service fee, find the current balance.

    Performance Assessment Scoring Rubric:

    Points

    Description

    4
    • Table shows beginning balance of $74.
    • Table shows deductions of $17, $14, and $25 as debits or subtracted from the beginning balance.
    • Table shows 7 discrete dues entries for $6 totaling $42 as a credit or added to the balance OR $42 associated with 7 members and credited to the balance.
    • Table shows $10 fee as a debit or subtracted from the balance.
    • Table shows current balance of $50.
    • Explanation shows clear understanding of adding and subtracting positive and negative numbers.

    3
    • Table shows beginning balance of $74.
    • Table shows subtraction of 17, 14, and 25.
    • Table shows addition of 42.
    • Table shows subtraction of 10.
    • Table shows incorrect current balance due to minor computation error.
    • Explanation shows some understanding of adding and subtracting positive and negative numbers.

    2
    • Table shows beginning balance of $74.
    • Table shows some of the necessary steps.
    • No more than 2 of the required steps are missing.
    • A few minor computation errors may be present, but no more than one major computation error is present.
    • Explanation shows basic understanding of adding and subtracting positive and negative numbers.

    1
    • Student work is incomplete.
    • Some of the necessary steps are present but major computation errors may be present.
    • Supporting work shown does not necessarily contribute to the solution.
    • Table shows incorrect current balance.
    • Explanation shows little understanding of adding and subtracting positive and negative numbers.

    0
    • Student work has incorrect balance.
    • Supporting work is insufficient to indicate engagement with the concept of adding and subtracting to find the current balance.
    • Explanation shows no understanding of adding and subtracting positive and negative numbers.
Final 07/05/2013
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