In this lesson, students will create and solve word problems involving money (addition, subtraction, multiplication, and division of decimals), and develop a script for a short scene and perform it using the elements of performance. Media resources and teacher materials are included.

This QuickTime movie explains a solution to a word problem using subtraction. The teacher and student discuss how to work through it together to find a solution. As you watch and listen to them interact it helps clarify the thinking behind applying this concept. [5:54]

Working With Rational Numbers

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Compare and order positive and negative numbers and place them on a number line.
Understand the concepts of opposites absolute value.

Lesson Flow

The unit begins with students using a balloon model to informally explore adding and subtracting integers. With the model, adding or removing heat represents adding or subtracting positive integers, and adding or removing weight represents adding or subtracting negative integers.

Students then move from the balloon model to a number line model for adding and subtracting integers, eventually extending the addition and subtraction rules from integers to all rational numbers. Number lines and multiplication patterns are used to find products of rational numbers. The relationship between multiplication and division is used to understand how to divide rational numbers. Properties of addition are briefly reviewed, then used to prove rules for addition, subtraction, multiplication, and division.

This unit includes problems with real-world contexts, formative assessment lessons, and Gallery problems.

Students use the Hot Air Balloon simulation to model integer subtraction. They then move to modeling subtraction on a number line. They use patterns in their work and their answers to write a rule for subtracting integers.Key ConceptsThis lesson introduces the number line model for subtracting integers. To subtract on a number line, start at 0. Move to the location of the first number (the minuend). Then, move in the negative direction (down or left) to subtract a positive integer or in the positive direction (up or right) to subtract a negative integer. In other words, to subtract a number, move in the opposite direction than you would if you were adding it.The Hot Air Balloon simulation can help students see why subtracting a number is the same as adding the opposite:Subtracting a positive number means removing heat from air, which causes the balloon to go down, in the negative direction.Subtracting a negative number means removing weight, which causes the balloon to go up, in the positive direction.The rule for integer subtraction (which extends to addition of rational numbers) is easiest to state in terms of addition: to subtract a number, add its opposite. For example, 5 – 2 = 5 + (–2) = 3 and 5 – (–2) = 5 + 2 = 7.Goals and Learning ObjectivesModel integer subtraction on a number line.Write a rule for subtracting integers.

Students critique and improve their work on the Self Check, then work on more addition and subtraction problems.Students solve problems that require them to apply their knowledge of adding and subtracting positive and negative numbers.Key ConceptsTo solve the problems in this lesson, students use their knowledge of addition and subtraction with positive and negative numbers.Goals and Learning ObjectivesUse knowledge of addition and subtraction with positive and negative numbers to write problems that meet given criteria.Assess and critique methods for subtracting negative numbers.Find values of variables that satisfy given inequalities.

Students use the distributive property to rewrite and solve multiplication problems. Then they apply addition and multiplication properties to simplify numerical expressions.Key ConceptsThe distributive property is stated in terms of addition: a(b + c) = ab + ac, for all numbers a, b, and c. However, it can be extended to subtraction as well: a(b − c) = ab − ac, for all numbers a, b, and c. Here is a proof. (We have combined some steps.)a(b − c)Original expression= a(b + (−c))Subtracting is adding the opposite.= a(b) + a(−c)Apply the distributive property.= ab + a(−1 ⋅ c)Apply the property of multiplication by −1.= ab + −1(ac)Apply the associative and commutative properties of multiplication.= ab + −(ac)Apply the property of multiplication by −1.= ab − acAdd the opposite is subtracting.We can use the distributive property to make some multiplication problems easier to solve. For example, by rewriting $1.85 as $2.00 − $0.15 and applying the distributive property, we can change 6($1.85) to a problem that is easy to solve mentally.6($1.85)=6($2−$0.15)=6($2) − 6($0.15)=$12 − $0.90=$11.10One common error students make when simplifying expressions is to simply remove the parentheses when a sum or difference is subtracted. For example, students may rewrite 10 − (6 + 9) as 10 − 6 + 9. In fact, 10 − (6 + 9) = 10 − 6 − 9. To see why, remember that that subtraction is equivalent to adding the opposite, 10 − (6 + 9) = 10 + [−(6 + 9)]. Applying the property of multiplication by −1, this is 10 + (−1)(6 + 9). Using the distributive property, we get 10 + (−6) + (−9) = 10 − 6 − 9.Goals and Learning ObjectivesApply addition and multiplication properties to simplify numerical expressions.

Students explore what happens to a hot air balloon when they add or remove units of weight or heat. This activity is an informal exploration of addition and subtraction with positive and negative integers.Key ConceptsThis lesson introduces a balloon simulation for adding and subtracting integers. Positive integers are represented by adding units of heat to air and negative integers are represented by adding units of weight. The balloon is pictured next to a vertical number line. The balloon rises one unit for each unit of heat added or each unit of weight removed. The balloon falls one unit for each unit of weight added or each unit of heat removed from the air.Mathematically, adding 1 to a number and subtracting −1 from a number are equivalent and increase the number by 1. Adding −1 to a number and subtracting 1 from a number are equivalent and decrease the number by 1. Addition and subtraction with positive and negative numbers are explored formally in the next several lessons.Goals and Learning ObjectivesExplore the effects of adding or subtracting positive and negative numbers.

Students use number lines to solve addition and subtraction problems involving positive and negative fractions and decimals. They then verify that the same rules they found for integers apply to fractions and decimals as well. Finally, they solve some real-world problems.Key ConceptsThe first four lessons of this unit focused on adding and subtracting integers. Using only integers made it easier for students to create models and visualize the addition and subtraction process. In this lesson, those concepts are extended to positive and negative fractions and decimals. Students will see that the number line model and rules work for these numbers as well.Note that rational number will be formally defined in Lesson 15.Goals and Learning ObjectivesExtend models and rules for adding and subtracting integers to positive and negative fractions and decimals.Solve real-world problems involving addition and subtraction of positive and negative fractions and decimals.

If you are looking for a website that explains and defines subtraction with decimals you have arrived at an excellent resource. There are even practice problems at the very end of the lesson.

This website explains and defines basic subtraction. There are practice problems at the very end of this lesson.

This lesson teaches students how to use colored blocks to model word problems. There are several sets of problems each for addition / subtraction, multiplication / division, and ratio word problems.

Download these free worksheets to sharpen your word problem skills. Sheets focus on addition, multiplication, subtraction, division, and other math topics.

All the activities in this lesson are addition and subtraction based. It is not designed to introduce addition and subtraction, rather, to supplement and enrich lessons already being taught. This lesson is not designed to be completed in one sitting. It may be done throughout an entire addition and subtraction unit. These activities may be used as starter activities when introducing new math concepts, particularly those that relate to addition and subtraction.

The online virtual self-checking resource has students practice subtracting two digit numbers with borrowing. Solutions are included.

Math variety of concepts, this colorful website is sure to give your students the chance show what they know. Concepts covered include rounding, division with decimals, order of operations, ratio, percent, multiplication, and division of fractions, simplifying fractions, and more. Not an interactive site, but great practice activities.

This lesson explains how to add and subtract matrices when it is possible to do so, noting matrix addition is commutative and associative. [2:59]

In this Cyberchase video segment, the CyberSquad helps resolve a dispute about the squares on the Gollywood Walk of Fame.

Demonstrates how to find missing numbers in addition and subtraction equations within 100. [5:44]

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