This lesson unit is intended to help teachers assess how well students are able to interpret percent increase and decrease, and in particular, to identify and help students who have the following difficulties: translating between percents, decimals, and fractions; representing percent increase and decrease as multiplication; and recognizing the relationship between increases and decreases.
Four full-year digital course, built from the ground up and fully-aligned to the Common Core State Standards, for 7th grade Mathematics. Created using research-based approaches to teaching and learning, the Open Access Common Core Course for Mathematics is designed with student-centered learning in mind, including activities for students to develop valuable 21st century skills and academic mindset.
Working With Rational Numbers
Type of Unit: Concept
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.
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 interactive to model integer addition. They then move to modeling addition on horizontal number lines. They look for patterns in their work and their answers to understand general addition methods.Key ConceptsTo add two numbers on a number line, start at 0. Move to the first addend. Then, move in the positive direction (up or right) to add a positive integer or in the negative direction (down or left) to add a negative integer.Here is −6 + 4 on a number line: The rule for integer addition (which extends to addition of rational numbers) is easiest to state if it is broken into two cases:If both addends have the same sign, add their absolute values and give the result the same sign as the addends. For example, to find −5 + (−9), first find |−5| +|−9| = 14. Because both addends are negative the result is negative. So, −5 + (−9) = −14.If the addends have different signs, subtract the lesser absolute value from the greater absolute value. Give the answer the same sign as the addend with the greater absolute value. For example, to find 5 + (−9), find |−9| − |5| = 9 − 5 = 4. Because −9 has the greater absolute value, the result is negative. So, 5 + (−9) = −4.Goals and Learning ObjectivesModel integer addition on a number line.Learn general methods for adding integers.
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 review the properties of addition and write an example for each. Then they apply the properties to simplify numerical expressions.Key ConceptsThe properties of addition:Commutative property of addition: Changing the order of addends does not change the sum. For any numbers a and b, a + b = b + a.Associative property of addition: Changing the grouping of addends does not change the sum. For any numbers a, b, and c, (a + b) + c = a + (b + c).Additive identity property of 0: The sum of 0 and any number is that number. For any number a, a + 0 = 0 + a = a.Existence of additive inverses: The sum of any number and its additive inverse (opposite) is 0. For any number a, a + (−a) = (−a) + a = 0.These properties allow us to manipulate expressions to make them easier to work with. For example, the associative property of addition tells us that we can regroup the expression (311+49)+59 as 311+(49 +59), making it much easier to simplify.Students must be careful to apply the commutative and associative properties only to addition expressions. For example, we cannot switch the −7 and 8 in the expression −7 − 8 to get 8 − (−7). However, if we rewrite −7 − 8 as the addition expression −7 + (−8), we can swap the addends to get −8 + (−7).Goals and Learning ObjectivesUnderstand the properties of addition.Apply the properties of addition to simplify numerical expressions.
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 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 find the distance between points on a number line by counting and by using subtraction. They then use subtraction to find differences in temperatures.Students discover that the distance between any two points on the number line is the absolute value of their difference, and apply this idea to solve problems.Key ConceptsStudents know from earlier grades that the distance between two positive numbers on the number line can be found by subtracting the lesser number from the greater number. For example, the distance between 5 and 11 is 11 – 5, or 6. We can also state the rule for finding distance as “The distance between two positive numbers is the absolute value of their difference.” With this version of the rule, we don’t have to consider which number is greater; the result is the same either way. Using the example of 5 and 11, the distance is |11 – 5| or |5 – 11|, both of which are equal to 6.This idea extends to the entire number line, including numbers to the left of 0. That is, the distance between any two numbers is the absolute value of their difference. For example, the distance between –5 and 3 is |–5 – 3| = |–8| = 8 or |3 – (–5)| = |8| = 8, and the distance between –12 and –7 is |–12 – (–7)| = |–5| = 5 or |–7 – (–12)| = |5| = 5.Goals and Learning ObjectivesUnderstand the relationship between the distance between two points on the number line and the difference in the coordinates of those points.Find distances in real-life situations.
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.
Students explore the concepts of positive and negative numbers through Newton's Third Law and rocket thrust, and other examples.
This lesson unit is intended to help teachers assess how well students are able to understand and use directed numbers in context. It is intended to help identify and aid students who have difficulties in ordering, comparing, adding, and subtracting positive and negative integers. Particular attention is paid to the use of negative numbers on number lines to explore the structures: starting temperature + change in temperature = final temperature final temperature Đ change in temperature = starting temperature final temperature Đ starting temperature = change in temperature.