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 use number lines to represent products of a negative integer and a positive integer, and they use patterns to understand products of two negative integers. Students write rules for products of integers.Key ConceptsThe product of a negative integer and a positive integer can be interpreted as repeated addition. For example, 4 • (–2) = (–2) + (–2) + (–2) + (–2). On a number line, this can be represented as four arrows of length 2 in a row, starting at 0 and pointing in the negative direction. The last arrow ends at –8, indicating that 4 • (–2) = –8. In general, the product of a negative integer and a positive integer is negative.The product of two negative integers is hard to interpret or visualize. In this lesson, we use patterns to help students see why a negative integer multiplied by a negative integer equals a positive integer. For example, students can compute the products in the pattern below.4 • (–3) = –123 • (–3) = –92 • (–3) = –61 • (–3) = –30 • (–3) = 0They can observe that, as the first factor decreases by 1, the product increases by 3. They can continue this pattern to find these products.–1 • (–3) = 3–2 • (–3) = 6–3 • (–3) = 9In the next lesson, we will prove that the rules for multiplying positive and negative integers extend to all rational numbers, including fractions and decimals.Goals and Learning ObjectivesRepresent multiplication of a negative integer and a positive integer on a number line.Use patterns to understand products of two negative integers.Write rules for multiplying integers.

Students solve division problems by changing them into multiplication problems. They then use the relationship between multiplication and division to determine the sign when dividing positive and negative numbers in general.Key ConceptsThe rules for determining the sign of a quotient are the same as those for a product: If the two numbers have the same sign, the quotient is positive; if they have different signs, the quotient is negative. This can be seen by rewriting a division problem as a multiplication of the inverse.For example, consider the division problem −27 ÷ 9. Here are two ways to use multiplication to determine the sign of the quotient:The quotient is the value of x in the multiplication problem 9 ⋅ x = −27. Because 9 is positive, the value of x must be negative in order to get the negative product.The division −27 ÷ 9 is equivalent to the multiplication −27 ⋅ 19. Because this is the product of a negative number and a positive number, the result must be negative.Goals and Learning ObjectivesUse the relationship between multiplication and division to solve division problems involving positive and negative numbers.Understand how to determine whether a quotient will be positive or negative.

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 use properties of multiplication to prove that the product of any two negative numbers is positive and the product of a positive number and a negative number is negative.Key ConceptsMultiplication properties can be used to develop the rules for multiplying positive and negative numbers.Students are familiar with the properties from earlier grades:Associative property of multiplication: Changing the grouping of factors does not change the product. For any numbers a, b, and c, (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c).Commutative property of multiplication: Changing the order of factors does not change the product. For any numbers a and b, a ⋅ b = b ⋅ a.Multiplicative identity property of 1: The product of 1 and any number is that number. For any number a, a ⋅ 1 = 1 ⋅ a = a.Property of multiplication by 0: The product of 0 and any number is 0. For any number a, a ⋅ 0 = 0 ⋅ a = 0.Property of multiplication by −1: The product of −1 and a number is the opposite of that number. For any number a, (−1) ⋅ a = −a.Existence of multiplicative inverses: Dividing any number by the same number equals 1. Multiplying any number by its multiplicative inverse equals 1. For every number a ≠ 0, a ÷ a = a ⋅ 1a = 1a ⋅ a = 1.Distributive property: Multiplying a number by a sum is the same as multiplying the number by each term and then adding the products. For any numbers a, b, and c, a ⋅ (b + c) = a ⋅ b + a ⋅ c.In this lesson, students will encounter a proof showing that the product of a positive number and a negative number is negative and two different proofs that the product of two negative numbers is positive. Two alternate proofs are as follows.Proof that the product of two negative numbers is positive:Represent the negative numbers as −a and −b, where a and b are positive.(−a) ⋅ (−b)Original expression= ((−1) ⋅ a) ⋅ ((−1) ⋅ b)   Property of multiplication by −1= (−1) ⋅ (a ⋅ (−1)) ⋅ b   Associative property of multiplication= (−1) ⋅ ((−1) ⋅ a) ⋅ b   Commutative property of multiplication= ((−1) ⋅ (−1)) ⋅ (a ⋅ b)   Associative property of multiplication= 1 ⋅ (a ⋅ b)   Property of multiplication by −1= a ⋅ b   Multiplicative identity property of 1Because a and b are positive, a ⋅ b is positive.Proof that the product of a positive number and a negative number is negative:Let a be the positive number. Let −b be the negative number, where b is positive.a ⋅ (−b)Original expression= a ⋅ ((−1) ⋅ b)     Property of multiplication by −1= (a ⋅ (−1)) ⋅ b     Associative property of multiplication= ((−1) ⋅ a) ⋅ b     Commutative property of multiplication= (−1) ⋅ (a ⋅ b)     Associative property of multiplication= −(a ⋅ b)     Property of multiplication by −1Because a and b are positive, a ⋅ b is positive, so −(a ⋅ b) must be negative.Goals and Learning ObjectivesReview properties of multiplication.Explain why the product of two negative numbers is positive and the product of a negative number and a positive number is negative.

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 critique and improve their work on the Self Check. They then extend their knowledge with additional problems.Students solve problems that require them to apply their knowledge of multiplying and dividing positive and negative numbers. Students will then take a quiz.Key ConceptsTo solve the problems in the Self Check, students must apply their knowledge of multiplication and division of positive and negative numbers learned throughout the unit.Goals and Learning ObjectivesUse knowledge of multiplication and division of positive and negative numbers to solve problems.

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 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.

Lesson OverviewStudents learn the definition of rational number, and they write rational numbers as ratios of integers and as repeating or terminating decimals.Key ConceptsStudents have been working with rational numbers throughout this unit, but the term rational number is formally defined in this lesson. A rational number is a number that can be written in the form pq, where p and q are integers. All the integers, fractions, decimals, and percents students have worked with so far in their math classes are rational numbers. Following are some rational numbers written as ratios of integers:36=361−1.2=−12105%=5100 −12=−12Any rational number can also be written as a decimal that terminates or that repeats forever in a regular pattern. For example, 35 = 0.6 and 711 = 0.63636363… Repeating decimals are often written with a bar over the digits that repeat. For example, 0.63636363… can be written as 0.63¯.There are numbers that are irrational. These numbers include π and the square root of any whole number that is not a perfect square, such as 2. The decimal form of an irrational number does not terminate, and the digits do not follow a repeating pattern. Students will study irrational numbers in Grade 8.Goals and Learning ObjectivesUnderstand the definition of rational number.Write rational numbers as ratios of integers.Write rational numbers as terminating or repeating decimals.SWD: Students with disabilities may have difficulty working with decimals and fractions, especially moving between the two. If students demonstrate difficulty to the point of frustration, provide direct instruction on the basics for finding equivalent fractions and decimals.ELL: Target and model key language and vocabulary. Specifically, focus on the term rational, as well as terms such as terminate. As you’re discussing the key points, write the words on the board or on large sheets of paper and explain/demonstrate what the words mean. Since these are important points that students will be using throughout the module, write them on large poster board so that students can use them as a reference. Have students record new terms, definitions, and examples in their Notebook. 

Zooming In On Figures

Unit Overview

Type of Unit: Concept; Project

Length of Unit: 18 days and 5 days for project

Prior Knowledge

Students should be able to:

Find the area of triangles and special quadrilaterals.
Use nets composed of triangles and rectangles in order to find the surface area of solids.
Find the volume of right rectangular prisms.
Solve proportions.

Lesson Flow

After an initial exploratory lesson that gets students thinking in general about geometry and its application in real-world contexts, the unit is divided into two concept development sections: the first focuses on two-dimensional (2-D) figures and measures, and the second looks at three-dimensional (3-D) figures and measures.
The first set of conceptual lessons looks at 2-D figures and area and length calculations. Students explore finding the area of polygons by deconstructing them into known figures. This exploration will lead to looking at regular polygons and deriving a general formula. The general formula for polygons leads to the formula for the area of a circle. Students will also investigate the ratio of circumference to diameter ( pi ). All of this will be applied toward looking at scale and the way that length and area are affected. All the lessons noted above will feature examples of real-world contexts.
The second set of conceptual development lessons focuses on 3-D figures and surface area and volume calculations. Students will revisit nets to arrive at a general formula for finding the surface area of any right prism. Students will extend their knowledge of area of polygons to surface area calculations as well as a general formula for the volume of any right prism. Students will explore the 3-D surface that results from a plane slicing through a rectangular prism or pyramid. Students will also explore 3-D figures composed of cubes, finding the surface area and volume by looking at 3-D views.
The unit ends with a unit examination and project presentations.

Students will resume their project and decide on dimensions for their buildings. They will use scale to calculate the dimensions and areas of their model buildings when full size. Students will also complete a Self Check in preparation for the Putting It Together lesson.Key ConceptsThe first part of the project is essentially a review of the unit so far. Students will find the area of a composite figure—either a polygon that can be broken down into known areas, or a regular polygon. Students will also draw the figure using scale and find actual lengths and areas.GoalsRedraw a scale drawing at a different scale.Find measurements using a scale drawing.Find the area of a composite figure.SWD: Consider what supplementary materials may benefit and support students with disabilities as they work on this project:Vocabulary resource(s) that students can reference as they work:List of formulas, with visual supports if appropriateClass summaries or lesson artifacts that help students to recall and apply newly introduced skillsChecklists of expectations and steps required to promote self-monitoring and engagementModels and examplesStudents with disabilities may take longer to develop a solid understanding of newly introduced skills and concepts. They may continue to require direct instruction and guided practice with the skills and concepts relating to finding area and creating and interpreting scale drawings. Check in with students to assess their understanding of newly introduced concepts and plan review and reinforcement of skills as needed.ELL: As academic vocabulary is reviewed, be sure to repeat it and allow students to repeat after you as needed. Consider writing the words as they are being reviewed. Allow enough time for ELLs to check their dictionaries if they wish.

Students discover the formula for finding the volume of a pyramid and apply the formula to solve problems.Key ConceptsThe volume of a pyramid is one-third the volume of a prism with the same base and height. The shape of the base does not matter (including if it’s a circle), and students will see the same one-third comparison between a cylinder and cone.GoalsUnderstand the formula for the volume of a pyramid.Apply the volume formula to solve problems.

Lesson OverviewStudents will compare the formula for the area of a regular polygon to discover the formula for the area of a circle.Key ConceptsThe area of a regular polygon can be found by multiplying the apothem by half of the perimeter. If a circle is thought of as a regular polygon with many sides, the formula can be applied.For a circle, the apothem is the radius, and p is C.A=a(p2)→A=rC2→A=rπd2→A=rπ2r2→A=rπr=πr2 GoalsDerive the formula for the area of a circle.Apply the formula to find the area of circles.SWD: Consider the prerequisite skills for this lesson: understanding and applying the formula for the area of a regular polygon. Students with disabilities may need direct instruction and guided practice with this skill.Students should understand these domain-specific terms:apothemparallelogramderivationheightapproximate (estimate)scatter plotpiperimetercircumferenceIt may be helpful to preteach these terms to students with disabilities. 

Students will complete the first part of their project, deciding on two right prisms for their models of buildings with polygon bases. They will draw two polygon bases on grid paper and find the areas of the bases.Key ConceptsProjects engage students in the application of mathematics. It is important for students to apply mathematical ways of thinking to solve rich problems. Students are more motivated to understand mathematical concepts if they are engaged in solving a problem of their own choosing.In this lesson, students are challenged to identify an interesting mathematical problem and choose a partner or a group to work collaboratively on solving that problem. Students gain valuable skills in problem solving, reasoning, and communicating mathematical ideas with others.GoalsSelect a project shape.Identify a project idea.Identify a partner or group to work collaboratively with on a math project.SWD: Consider how to group students skills-wise for the project. You may decide to group students heterogeneously to promote peer modeling for struggling students. Or you can group students by similar skill levels to allow for additional support and/or guided practice with the teacher. Or you may decide to create intentional partnerships between strong students and struggling students to promote leadership and peer instruction within the classroom.ELL: In forming groups, be aware of your ELLs and ensure that they have a learning environment where they can be productive. Sometimes, this means pairing them up with English speakers, so they can learn from others’ language skills. Other times, it means pairing them up with students who are at the same level of language skill, so they can take a more active role and work things out together. Other times, it means pairing them up with students whose proficiency level is lower, so they play the role of the supporter. They can also be paired based on their math proficiency, not just their language proficiency.

Students will extend their knowledge of volume to find the volume of right prisms, seeing that the volume is the area of the base multiplied by the height.Key ConceptsVolume is measured in cubic units. The area of the base of a prism indicates how many cubic units are in the first unit “layer” of the prism. Multiplying by height gives the number of layers, and therefore the volume.GoalsFind the volume of right prisms.SWD: Some students with disabilities may have difficulty connecting newly introduced information with previously learned concepts. Consider ways to help students with disabilities to make connections between what they have learned in previous lessons about volume and right prisms and finding the volume of right prisms.Consider the prerequisite skills for this lesson. Students with disabilities may need direct instruction and guided practice with the skills, measurement, and concepts needed for this lesson.Students should understand these domain-specific terms:volumeright (domain-specific)prismcubicIt may be helpful to preteach these terms to students with disabilities.ELL: As new vocabulary is introduced, be sure to repeat it several times and allow students to repeat after you as needed. Write the new words as they are introduced, and allow enough time for ELLs to check their dictionaries or briefly consult with another student who shares the same primary language if they wish.

Students further explore scale, taking a scale drawing floor plan and redrawing it at a different scale.Key ConceptsStudents explore change from one scale to another, focusing on the ratios. Students will draw a scale model of a house.GoalsRedraw a scale drawing at a different scale.Find measurements using a scale drawing.

Students will explore the cross-sections that result when a plane cuts through a rectangular prism or pyramid. Students will also see examples of cross-section cuts in real-world situations.Key ConceptsStudents are very familiar with rectangular prisms, and to a lesser degree, they are familiar with rectangular pyramids. However, students haven’t been exposed to the myriad possibilities for solids that result from planar slices. The purpose of the lesson is for students to explore these possibilities.GoalsIdentify the plane figures that result from a plane cutting through a rectangular prism or pyramid.