Remaining groups present their unit projects and students discuss teacher and peer feedback.Key ConceptsStudents should demonstrate their understanding of the unit concepts.Goals and Learning ObjectivesPresent projects and demonstrate an understanding of the unit concepts.Provide feedback for others' presentations.Review the concepts from the unit.Review presentation feedback and reflect.

Students will form groups for the unit project, decide on a topic, and write up a project proposal. Students will also complete a Self Check that will be discussed in the next lesson.Key ConceptsStudents will apply what they have learned in the unit so far to determine a project. They will also apply their learning to complete a Self Check problem.Goals and Learning ObjectivesDecide on a project topic and group.Write a project proposal.

Students begin to develop intuition about appropriate sample size by conducting an experiment. They compare different sample sizes and whether increasing the sample size improves the results.Key ConceptsSampling is a way to discover unknown characteristics about a population. The size of the sample is important in determining the accuracy of the results. Ratio and proportion are used to compare the sample to the population.Goals and Learning ObjectivesStudents will use sampling to determine the number of different color marbles in a jar.Students will explore sample size compared to population size.

Students are introduced to the concept of sampling as a method of determining characteristics of a population. They consider how a sample can be random or biased, and think of methods for randomly sampling a population to ensure that it is representative.The idea of sampling is connected to probability; a relatively small set of data (a random sample/number of trials) can be used to generalize about a population (or determine probability). A larger sample (more trials) will give more confidence in the conclusions, but how large of a sample is needed?Students also discuss what random means and how to generate a random sample. Random samples are compared to biased samples and give insight into how statistics can be misleading (intentionally or otherwise).Key ConceptsRandom samples are related to probability. In probability, the number of trials is a sample used to generalize about the probability of an event. The results in probability are random if we are looking at equally likely outcomes. If a data sample is not random, the conclusions about the population will not reflect it.Terminology introduced in this lesson:population: the entire set of objects that can be considered when asking a statistical questionsample: a subset of a population; can be random, where each object in the population is equally likely to be in the sample, or biased, where not every object in the population is equally likely to be in the sampleGoals and Learning ObjectivesIntroduce sampling as a method to generalize about a population.Discuss the concept of a random sample versus a biased sample.Determine methods to generate random samples.Understand that biased samples are sometimes used to mislead.SWD: Some students with disabilities will benefit from a preview of the goals in this lesson. Students can highlight the critical features and/or concepts and will help them to pay close attention to salient information.

Students critique and improve their work on the Self Check, then work on additional problems.Key ConceptsStudents apply what they have learned to date to solve the problems in this lesson.Goals and Learning ObjectivesApply knowledge of probability to solve problems.Determine theoretical probability.Predict expected results.

Students critique and improve their work on the Self Check, then work on additional problems. Students revise the Self Check problem from the previous lesson and discuss their strategies.Key ConceptsStudents apply what they have learned to date to solve the problems in this lesson.Goals and Learning ObjectivesApply knowledge of sampling and data analysis to solve problems.Determine a random, representative sample that is nonbiased and of adequate sample size.Generalize about a population based on sampling.Compare data sets.

Students will extend their understanding of probability by continuing to conduct experiments with outcomes that do not have a theoretical probability. They will make predictions on the number of outcomes from a series of trials, and compare their predictions with the experimental probability calculated from an experiment.Key ConceptsStudents continue to investigate the Law of Large Numbers.Goals and Learning ObjectivesDeepen understanding of experimental probability.Use proportions to predict results for a number of trials and to calculate experimental probability.Understand that some events do not have theoretical probability.Understand that there are often many factors involved in determining probability (e.g., human error, randomness).

Students estimate the length of 50 seconds by starting an unseen timer and stopping it when they think 50 seconds has elapsed. The third attempt is recorded and compiled into a data set, which students then compare to the third attempt from the previous lesson when they estimated the length of 20 seconds. Students analyze the data to make conclusions about how well seventh grade students can estimate lengths of time.Students repeat the timing activity for 50 seconds, but only the third trial is recorded. The task today is to compare this set of data with the third trial for 20 seconds. Students will need to deal with the difference in the spread of data, as well as how to compare the data sets. Students will be reminded of Mean Absolute Deviation (MAD), which will be a useful tool in this situation.Key ConceptsStudents apply the tools learned in Unit 6.8:Measures of center and spreadMean absolute deviation (MAD)Goals and Learning ObjectivesApply knowledge of statistics to compare different sets of data.Use measures of center and spread to analyze data.

Students begin to formalize their understanding of probability. They are introduced to the concept of probability as a measure of likelihood and how to calculate probability as a ratio. The terms discussed (impossible, certain, etc.) in Lesson 1 are given numerical values.Key ConceptsStudents will think of probability as a ratio; it can be written as a fraction, decimal, or a percent ranging from 0 to 1.Students will think about ratio and proportion to predict results.Goals and Learning ObjectivesDefine probability as a measure of likelihood and the ratio of favorable outcomes to the total number of outcomes for an event.Predict results based on theoretical probability using ratio and proportion.

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.

Gallery OverviewAllow students who have a clear understanding of the content thus far in the unit to work on Gallery problems of their choosing. You can then use this time to provide additional help to students who need review of the unit’s concepts or to assist students who may have fallen behind on work.Gallery DescriptionsTemperature ChangesStudents solve a puzzle by using clues about the temperatures and temperature changes between several cities.Time ZonesStudents use integers to solve problems about times in different world time zones.Build ExpressionsPairs of students play a game in which they use cards to build two expressions that are as close in value as possible.Hexagon PuzzleStudents assemble triangular puzzle pieces by matching the problems and answers on their sides. When the puzzle is complete, the pieces will form a large hexagon.Equivalent ExpressionsStudents sort expressions into groups that have the same value.Are They Equivalent?Students decide when given expressions will have the same value as a − b.Graphical Addition and SubtractionThe locations of a and b on a number line are shown, and students must graph −a, −b, a − b, b − a, a + b,and −a − b.p and nStudents decide whether statements about a positive number p and a negative number n are true for all, some, or no values of p and n.

Gallery OverviewAllow students who have a clear understanding of the content in the unit to work on Gallery problems of their choosing. You can then use this time to provide additional help to students who need review of the unit’s concepts or to assist students who may have fallen behind on work.Problem DescriptionsMultiplication WebsStudents fill in the blanks to create expressions equal to the number in the center of the web.Number TreesStudents complete number tree puzzles.Are They Equivalent?Students decide when given expressions will have the same value as ab.Squaring and CubingStudents find solutions to simple equations and inequalities involving squares and cubes.Transforming TrianglesStudents investigate how a triangle changes when they multiply the coordinates of its vertices by positive and negative numbers.True or False?Students determine whether given statements about positive and negative numbers are true or false.Saving MoneyStudents use positive and negative numbers to make sense of changes to Lucy’s savings account.Altitude and TemperatureStudents explore how the air temperature changes as the height of an airplane changes.

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 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.