✏️Exploring Integers and Absolute Value

Lesson plan

Objectives

• Students will be able to define an integer and identify positive, negative, and zero on a number line.
• Students will be able to compare and order integers using appropriate inequality symbols (<, >, =).
• Students will be able to define absolute value and calculate the absolute value of any given integer.
• Students will be able to apply their understanding of integers and absolute value to solve real-world problems.

Materials

• Whiteboard or projector
• Markers or pens
• Individual number lines (printable or drawn)
• Integer chips or counters (optional, for visual learners)
• Student worksheet
• Exit tickets
• Quiz copies

Warm-up

Begin by asking students to think about situations where they might encounter numbers less than zero. Prompt them with examples like temperature below freezing, elevations below sea level, or debts. Ask them to share their ideas and discuss how these numbers are different from the positive numbers they usually work with. This sets the stage for introducing negative numbers and their real-world context.

Direct instruction

1. **Introduction to Integers:** Define integers as the set of whole numbers and their opposites. Explain that integers include positive numbers (1, 2, 3...), negative numbers (...-3, -2, -1), and zero. Emphasize that zero is neither positive nor negative.
2. **Number Line Representation:** Draw a large number line on the board. Demonstrate how to plot positive integers to the right of zero and negative integers to the left. Explain that numbers increase in value as you move to the right and decrease as you move to the left.
3. **Comparing Integers:** Introduce the inequality symbols: less than (<), greater than (>), and equal to (=). Use the number line to visually compare integers. For example, show that -3 is less than 1 because -3 is to the left of 1 on the number line. Work through several examples like comparing -5 and -2, or 0 and -4.
4. **Real-World Applications of Integers:** Discuss practical scenarios where integers are used. Examples: temperature (20 degrees above zero vs. 5 degrees below zero), elevation (200 feet above sea level vs. 50 feet below sea level), finances (earning 10 vs. owing 5). Ask students to generate their own examples.
5. **Introduction to Absolute Value:** Define absolute value as the distance of a number from zero on the number line. Emphasize that distance is always a positive quantity. Introduce the notation: |x|.
6. **Calculating Absolute Value:** Provide examples: |5| = 5 (5 units from zero), |-5| = 5 (also 5 units from zero). Explain that the absolute value of zero is zero: |0| = 0. Discuss how absolute value always results in a non-negative number, regardless of whether the original number was positive or negative.
7. **Common Misconceptions:** Address the misconception that 'absolute value just means making a number positive.' Clarify that it means the *distance* from zero, which happens to be positive. For instance, |-7| is 7 because -7 is 7 units away from zero, not just because the negative sign is removed.

Guided practice

The teacher will lead the class through a series of practice problems, encouraging student participation and discussion. First, we'll practice comparing integers. 'Let's compare -8 and -3. On our number line, which number is further to the left?' (Expected answer: -8). 'So, -8 is less than -3, we write -8 < -3.' Next, we'll practice finding absolute values. 'What is the absolute value of 6?' (Expected answer: 6). 'Why?' (Because 6 is 6 units away from zero). 'Now, what is the absolute value of -10?' (Expected answer: 10). 'Why?' (Because -10 is 10 units away from zero). The teacher will then present a real-world problem: 'A submarine is at -250 feet relative to sea level. What is its depth?' Students will identify that depth refers to the absolute value of its position, which is 250 feet. Students will work on individual number lines to plot and compare integers as the teacher guides them through additional examples.

Independent practice

Students will work independently on the 'Integers and Absolute Value Practice' worksheet. The worksheet includes problems on identifying integers, plotting them on a number line, comparing integers using <, >, or =, ordering integers from least to greatest, and calculating absolute values for various integers. The teacher will circulate to provide support and answer questions, ensuring students are applying the concepts correctly.

Closure

To conclude the lesson, gather students' attention and review the main concepts. Ask students to share one new thing they learned about integers or absolute value. Distribute exit tickets with the prompt: '1. What is an integer? Give an example of a positive and a negative integer. 2. Explain what absolute value means in your own words. Give an example.' Collect the exit tickets to gauge understanding and inform future instruction.

Assessment

Student mastery will be measured through several methods: observation during guided and independent practice, completion and accuracy of the 'Integers and Absolute Value Practice' worksheet, student responses on the exit ticket, and performance on the 'Integers and Absolute Value Check-up' quiz.

Differentiation

For struggling learners, provide pre-drawn number lines with key integers labeled. Allow the use of integer chips or counters to visualize positive and negative values. Pair students with stronger peers for collaborative problem-solving. For advanced learners, introduce multi-step problems involving integers (e.g., finding the difference between two temperatures, one positive and one negative) or challenge them to create their own real-world integer scenarios and absolute value problems.

Integers and Absolute Value Practice

Read each question carefully and show your work where indicated. Use a number line to help you visualize if needed.

1. 1. Identify all the integers in the following list: 4, -7, 0.5, 0, -12, 3/4, 25.
2. 2. Plot the following integers on a number line: -6, 2, -1, 5, 0.
3. 3. Compare the following integers using <, >, or =: a) -5 ____ 3 b) 0 ____ -9 c) -10 ____ -1 d) 7 ____ | -7 |
4. 4. Order the following integers from least to greatest: -8, 2, 0, -1, 5, -3.
5. 5. Find the absolute value of each number: a) | 15 | b) | -20 | c) | 0 | d) | -99 |
6. 6. The temperature in Anchorage, Alaska, was -10 degrees Fahrenheit. The temperature in Miami, Florida, was 75 degrees Fahrenheit. Which city had a colder temperature?
7. 7. A diver is 30 feet below sea level. Write this depth as an integer. What is the absolute value of this depth?
8. 8. If you owe $25, is that represented by a positive or negative integer? What is the absolute value of the amount you owe?
9. 9. A mountain peak has an elevation of 1,200 feet above sea level. A valley has an elevation of 300 feet below sea level. Write both elevations as integers. Which elevation is greater?
10. 10. Which number has a greater absolute value: -18 or 12? Explain your reasoning.

Integers and Absolute Value Check-up

1. 1. Which of the following is NOT an integer?
   • -5
   • 0
   • 3/4
   • 100
   Answer: 3/4
2. 2. On a number line, which integer is furthest to the left?
   • -2
   • 5
   • -7
   • 0
   Answer: -7
3. 3. Compare: -15 ____ -10
   • <
   • >
   • =
   Answer: <
4. 4. What is the absolute value of -23?
   • -23
   • 23
   • 0
   • 1/23
   Answer: 23
5. 5. Which situation best represents the integer -50?
   • Earning $50
   • A temperature of 50 degrees above zero
   • A debt of $50
   • Climbing 50 feet up a mountain
   Answer: A debt of $50
6. 6. Order the following integers from greatest to least: -4, 3, 0, -7, 1.
   • -7, -4, 0, 1, 3
   • 3, 1, 0, -4, -7
   • 0, 1, 3, -4, -7
   • 3, 0, 1, -4, -7
   Answer: 3, 1, 0, -4, -7
7. 7. If |x| = 8, what are the possible values for x?
   • 8 only
   • -8 only
   • 8 and -8
   • 0
   Answer: 8 and -8
8. 8. A fish is swimming 15 feet below the surface of the water. Which integer represents the fish's position relative to the surface?
   • 15
   • -15
   • 0
   • 30
   Answer: -15

Integers and Absolute Value at Home

Dear Parents/Guardians, This week in math, your student learned about integers and absolute value. Integers are whole numbers, their opposites (negative numbers), and zero. We discussed how to represent and compare these numbers, especially using a number line. We also explored absolute value, which is the distance of a number from zero, always resulting in a non-negative value. These concepts are fundamental for understanding more complex number systems and are used in everyday situations like tracking temperatures, elevations, and finances. Please encourage your student to complete the tasks below to reinforce their learning. A brief discussion about where you might encounter integers in real life would also be very beneficial!

• 1. Review your notes from today's lesson on integers and absolute value. Make sure you understand the definitions and examples.
• 2. Draw a number line from -10 to 10. Plot and label the following integers: -9, -3, 0, 4, 7.
• 3. Write down five real-world examples where you would use negative numbers (e.g., temperature below zero, debt).
• 4. Find the absolute value of each number: | 25 |, | -14 |, | -3.5 | (challenge!), | 0 |.
• 5. Compare the following using <, >, or =: -6 and -2; 10 and | -10 |; -1 and 0; | -5 | and 4.
• 6. Explain to a family member what absolute value means using a real-world example (e.g., distance traveled).
• 7. Challenge: A city's elevation is 50 feet below sea level, and another city's elevation is 120 feet above sea level. What is the difference in their elevations? (Hint: Think about absolute value and distance on a number line.)

Vocabulary

Integer · noun

A whole number (not a fraction or decimal) that can be positive, negative, or zero.

"The numbers -3, 0, and 7 are all integers."

Positive Integer · noun

An integer greater than zero.

"The temperature rose to a positive integer of 15 degrees Celsius."

Negative Integer · noun

An integer less than zero.

"If you owe money, your bank balance might show a negative integer."

Zero · noun

The integer that is neither positive nor negative; it represents no quantity or a neutral point.

"The score was zero to zero at halftime."

Number Line · noun

A line on which numbers are marked at intervals, used to visualize and compare numbers.

"We used a number line to see that -5 is less than -2."

Absolute Value · noun

The distance of a number from zero on the number line. It is always a non-negative value.

"The absolute value of -10 is 10, because -10 is 10 units away from zero."

Opposite · noun

Two numbers that are the same distance from zero on a number line but on opposite sides (e.g., 5 and -5).

"The opposite of 7 is -7."

Compare · verb

To examine two or more numbers to see how they are similar, different, or to determine which is greater or lesser.

"We need to compare the two temperatures to see which one is colder."

Order · verb

To arrange numbers in a specific sequence, typically from least to greatest or greatest to least.

"The task was to order the integers from smallest to largest."

Less Than (<) · prepositional phrase

A mathematical symbol indicating that the first number is smaller than the second number.

"Five is less than ten, written as 5 < 10."

Greater Than (>) · prepositional phrase

A mathematical symbol indicating that the first number is larger than the second number.

"Eight is greater than three, written as 8 > 3."

Equation · noun

A mathematical statement that shows two expressions are equal.

"The equation x + 2 = 5 tells us that x must be 3."

Activities

• Number Line Walk · 10 minutes

  Draw a large number line on the classroom floor using tape. Call out an integer (e.g., '-3'). Students will walk to that position on the number line. Then, call out another integer and ask students to determine if they need to walk left or right and how many steps to get to the new number. This kinesthetic activity helps students visualize integer positions and movement.

• Integer Card Sort · 10 minutes

  Provide small groups with a set of cards, each containing an integer (e.g., -10, 5, 0, -2, 8). Students must sort these cards into categories: 'Positive Integers', 'Negative Integers', and 'Zero'. Then, challenge them to order the cards from least to greatest within their groups. This reinforces identification and comparison skills.

• Absolute Value Match-Up · 10 minutes

  Prepare two sets of cards: one with integers (e.g., -7, 12, -3, 0) and another with their corresponding absolute values (e.g., 7, 12, 3, 0). Students work in pairs or small groups to match each integer card with its correct absolute value card. This game helps students quickly recall and apply the concept of absolute value.

• Real-World Integer Scenarios · 15 minutes

  Divide students into small groups. Give each group a few real-world scenarios (e.g., 'a submarine descends 100 feet', 'you deposited $50 into your account', 'the temperature dropped 5 degrees'). Each group must represent the scenario with an integer and then find its absolute value, explaining what the absolute value represents in that context. Groups then share their scenarios and solutions with the class.