ALEX Learning Activity

  

Using Number Lines to Model Real-World Problems

A Learning Activity is a strategy a teacher chooses to actively engage students in learning a concept or skill using a digital tool/resource.

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  This learning activity provided by:  
Author: Kimberly Dockery
System:Alabama Department of Education
School:Alabama Department of Education
  General Activity Information  
Activity ID: 2704
Title:
Using Number Lines to Model Real-World Problems
Digital Tool/Resource:
Math Learning Digital Number Line
Web Address – URL:
Overview:

In this activity, students will compute real-world problems with rational numbers while using a digital number line. Students are provided a sample problem to work through to become familiar with the digital number line. Since problems can be solved using multiple methods, students are asked to provide a number sentence to represent their number line model as well as the solution to the problem. Through the online digital tool, students can also share a link to their work with their teacher or classmates. This provides a great opportunity for students to investigate how to solve problems using multiple methods.

Using Number Lines to Model Real-World Problems Student Response Page
  Associated Standards and Objectives  
Content Standard(s):
Digital Literacy and Computer Science
DLIT (2018)
Grade: 7
30) Apply the problem-solving process to solve real-world problems.

Unpacked Content
Evidence Of Student Attainment:
Students will:
  • define a problem.
  • analyze the problem.
  • seek multiple solutions to the problem.
  • evaluate possible solutions to the problem.
  • select a best solution to the problem.
  • seek feedback on the solution.
Teacher Vocabulary:
  • problem-solving process
Knowledge:
Students know:
  • the steps to the problem-solving process.
Skills:
Students are able to:
  • select and dissect a problem.
  • seek solutions.
  • select a best alternative.
Understanding:
Students understand that:
  • often there are multiple solutions to real
  • world problems.
Mathematics
MA2019 (2019)
Grade: 7
4. Apply and extend knowledge of operations of whole numbers, fractions, and decimals to add, subtract, multiply, and divide rational numbers including integers, signed fractions, and decimals.

a. Identify and explain situations where the sum of opposite quantities is 0 and opposite quantities are defined as additive inverses.

b. Interpret the sum of two or more rational numbers, by using a number line and in real-world contexts.

c. Explain subtraction of rational numbers as addition of additive inverses.

d. Use a number line to demonstrate 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.

e. Extend strategies of multiplication to rational numbers to develop rules for multiplying signed numbers, showing that the properties of the operations are preserved.

f. Divide integers and explain that division by zero is undefined. Interpret the quotient of integers (with a non-zero divisor) as a rational number.

g. Convert a rational number to a decimal using long division, explaining that the decimal form of a rational number terminates or eventually repeats.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Explain situations where opposite quantities combine to make zero, known as additive inverses.
  • Apply their knowledge of addition and subtraction of rational numbers to describe real-world contexts.
  • Add and subtract rational numbers using number lines to show connection to distance
  • Explain the connection between subtraction and addition of additive inverses.
  • Model multiplication and division of rational numbers (number horizontal and vertical number lines, integer chips, bar models).
  • Use properties of operations to multiply signed numbers.
  • Convert rational numbers to a decimal using long division and determine if the result is terminating or repeating.
Teacher Vocabulary:
  • Integers
  • Rational numbers
  • Additive inverses
  • opposite quantities
  • Absolute value
  • Terminating decimals
  • Repeating decimals
Knowledge:
Students know:
  • a number and its opposite have a sum of 0.
  • A number and its opposite are called additive inverses.
  • Strategies for adding and subtracting two or more numbers.
  • Absolute value represents distance on a number line, therefore it is always non-negative.
  • Strategies for multiplying signed numbers.
  • Every quotient of integers (with non-zero divisor) is a rational number.
  • If p and q are integers, then -(p/q) = (-p)/q = p/(-q).
  • The decimal form of a rational number terminates or eventually repeats.
Skills:
Students are able to:
  • add rational numbers.
  • Subtract rational numbers.
  • Represent addition and subtraction on a number line diagram.
  • Describe situations in which opposite quantities combine to make 0.
  • Find the opposite of a number.
  • Interpret sums of rational numbers by describing real-world contexts.
  • Show that the distance between two rational numbers on the number line is the absolute value of their difference.
  • Use absolute value in real-world contexts involving distances.
  • Multiply and divide rational numbers.
  • Convert a rational number to a decimal using long division.
Understanding:
Students understand that:
  • finding sums and differences of rational numbers (negative and positive) involves determining direction and distance on the number line.
  • Subtraction of rational numbers is the same as adding the additive inverse, p - q = p + (-q).
  • If a factor is multiplied by a number greater than one, the answer is larger than that factor.
  • If a factor is multiplied by a number between 0 and 1, the answer is smaller than that factor.
  • Multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers.
  • Integers can be divided, provided that the divisor is not zero.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.7.4.1: Define rational numbers, horizontal, and vertical.
M.7.4.2: Recall how to extend a horizontal number line.
M.7.4.3: Recall how to extend a vertical number line.
M.7.4.4: Demonstrate addition and subtraction of whole numbers using a horizontal or vertical number line.
M.7.4.5: Give examples of rational numbers.
M.7.4.6: Define absolute value and additive inverse.
M.7.4.7: Explain that the sum of a number and its opposite is zero.
M.7.4.8: Locate positive, negative, and zero numbers on a number line.
M.7.4.9: Recall properties of addition and subtraction.
M.7.4.10: Model addition and subtraction using manipulatives.
M.7.4.11: Show addition and subtraction of 2 or more rational numbers using a number line within real-world context.
M.7.4.12: Define absolute value and additive inverse.
M.7.4.13: Show subtraction as the additive inverse.
M.7.4.14: Give examples of the opposite of a given number.
M.7.4.15: Show addition and subtraction using a number line.
M.7.4.16: Discuss various strategies for solving real-world and mathematical problems.
M.7.4.17: Identify properties of operations for addition and subtraction.
M.7.4.18: Recall the steps for solving addition and subtraction of rational numbers.
M.7.4.19: Identify the difference between two rational numbers on a number line.
M.7.4.20: Recall the steps of solving multiplication of rational numbers.
M.7.4.21: Identify the pattern for multiplying signed numbers.
M.7.4.22: Recall the steps of solving division of rational numbers.
M.7.4.23: Explain that dividing a rational number zero is undefined.
M.7.4.24: Recall that a fraction can be written as a division problem.
M.7.4.25: Recall the steps to divide two rational numbers.
M.7.4.26: Identify whether a decimal is terminating or repeating.

Prior Knowledge Skills:
  • Define parentheses, braces, and brackets.
  • Recall addition and subtraction of fractions as joining and separating parts referring to the same whole.
  • Identify two fractions as equivalent (equal) if they are the same size or the same point on a number line.
  • Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.
  • Generate equivalent fractions.
  • Show on a number line that numbers that are equal distance from 0 and on opposite sides of 0 have opposite signs.
  • Define rational number.
  • Arrange integers and /or rational numbers on a horizontal or vertical number line.
  • Locate the position of integers and/or rational numbers on a horizontal or vertical number line.
  • Identify a rational number as a point on the number line.
Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.7.4 Add and subtract integers up to 15.

Mathematics
MA2019 (2019)
Grade: 7
Accelerated
8. Apply and extend knowledge of operations of whole numbers, fractions, and decimals to add, subtract, multiply, and divide rational numbers including integers, signed fractions, and decimals.

a. Identify and explain situations where the sum of opposite quantities is 0 and opposite quantities are defined as additive inverses. 

b. Interpret the sum of two or more rational numbers, by using a number line and in real-world contexts.

c. Explain subtraction of rational numbers as addition of additive inverses.

d. Use a number line to demonstrate 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.

e. Extend strategies of multiplication to rational numbers to develop rules for multiplying signed numbers, showing that the properties of the operations are preserved.

f. Divide integers and explain that division by zero is undefined. Interpret the quotient of integers (with a non-zero divisor) as a rational number.

g. Convert a rational number to a decimal using long division, explaining that the decimal form of a rational number terminates or eventually repeats. [Grade 7, 4]
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Apply their knowledge of addition and subtraction of rational numbers to describe real-world contexts.
  • Use physical and visual models to add and subtract integers.
  • Add and subtract rational numbers.
  • Model multiplication and division of rational numbers.
  • Apply the distributive property to rational numbers.
  • Convert rational numbers to a decimal using long division to determine if the result is terminating or repeating.
Teacher Vocabulary:
  • Integers
  • Rational numbers
  • Additive inverses
  • opposite quantities
  • Absolute value
  • Terminating decimals
  • Repeating decimals
Knowledge:
Students know:
  • a number and its opposite have a sum of 0.
  • A number and its opposite are called additive inverses.
  • properties of operations.
  • Absolute value represents distance on a number line, therefore it is always non-negative.
  • Every quotient of integers (with non-zero divisor) is a rational number.
  • If p and q are integers, then -(p/q) = (-p)/q = p/(-q).
  • The decimal form of a rational number terminates in 0s or eventually repeats.
Skills:
Students are able to:
  • add rational numbers.
  • Subtract rational numbers.
  • Represent addition and subtraction on a number line diagram.
  • Describe situations in which opposite quantities combine to make 0.
  • Find the opposite of a number.
  • Interpret sums of rational numbers by describing real-world contexts.
  • Show that the distance between two rational numbers on the number line is the absolute value of their difference.
  • Use absolute value in real-world contexts involving distances.
  • Multiply and divide rational numbers.
  • Convert a rational number to a decimal using long division.
Understanding:
Students understand that:
  • p + q is the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative.
  • Subtraction of rational numbers is the same as adding the additive inverse, p - q = p + (-q).
  • If a factor is multiplied by a number greater than one, the answer is larger than that factor.
  • If a factor is multiplied by a number between 0 and 1, the answer is smaller than that factor.
  • 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.
  • Integers can be divided, provided that the divisor is not zero.
Diverse Learning Needs:
Learning Objectives:

The student will solve real-world problems involving rational numbers.

The student will use a digital number line to solve real-world problems.
  Strategies, Preparations and Variations  
Phase:
During/Explore/Explain
Activity:

Students will explore using a digital number line to solve real-world problems that include rational numbers. The digital number line offers many settings that include negative values, decimals, integers, and fractions. Students will work through an example problem with detailed instructions to become familiar with how to change the settings on the number line. Students are then provided five additional problems to complete. Optional number line settings are included for each problem. Have each student complete the first problem independently. At this point, the teacher should conduct a Think-Pair-Share with the class. Each student should have a partner. Ask the students to work with their partners to compare their two number lines to determine if the models and number sentences are the same or different. Allow each group to share their results and reasoning with the rest of the class. Students can then answer the remaining questions using their digital number line. Since the digital tool can create a link to share their number lines, the students can easily send this information to the teacher. For an extension or additional practice, students can be asked to write their own real-world problems and then create a model for their problem. The problems could also be exchanged with other classmates.
Assessment Strategies:

Have students complete an exit slip explaining if they found the digital number line helpful while solving the word problems.

Through the link provided from the student, evalute the student's digital number line used to solve the real-world problem.
Advanced Preparation:

Visit the Digital Number Line Tool to become familiar with how to use the tool. 

Students will need online access to the digital number line.

Copy the “Using Number Lines to Model Real-World Problems” student response page.
Variation Tips (optional):

During the Think-Pair-Share the teacher can display the number line models the students designed and allow them to explain the different ways they built their number lines. 

The teacher can also allow students to write their own word problems with given parameters and allow students to exchange problems to solve.

Notes or Recommendations (optional):

If students do not have access to the digital tool, the teacher can provide paper copies of number lines and allow the students to build their own.
  Keywords and Search Tags  
Keywords and Search Tags: Decimals, Fractions, Model, Negative, Number Line, Positive, Rational Number, Real World, RealWorld, Word Problem, Word Problems