Common Core State Standards Math Grade 7,The Number System Cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers Standard: Understand that 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. Interpret products of rational numbers by describing real-world contexts. Lesson Overview Students learn the definition of rational number, and they write rational numbers as ratios of integers and as repeating or terminating decimals.

Study guide Algebra I Sem 1 1.1.1 Study: Rational and Irrational Numbers Name: Date: The questions below will help you keep track of key concepts from this lesson's study activity. Use the study page numbers listed to help you fill in the blanks or solve the problems. Snellville Middle School Math Website. Home 6th Grade 7th Grade 8th Grade Algebra 1 C.A.T. 7th Grade Unit 1: Rational Numbers. Achievement Level Descriptors. The achievement level descriptors (ALD) for this unit explains at what level you can be successful on any assessment. Study Guide Questions.

Key Concepts Students 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 p q, 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 = 36 1 − 1.2 = − 12 10 5% = 5 100 − 1 2 = − 1 2 Any rational number can also be written as a decimal that terminates or that repeats forever in a regular pattern. For example, 3 5 = 0.6 and 7 11 = 0.63636363 Repeating decimals are often written with a bar over the digits that repeat.

For example, 0.63636363 can be written as 0. 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.

Following are some rational numbers written as ratios of integers. Students will study irrational numbers in Grade 8. Lesson Guide. Answers will vary. A rational number is a number that can be written as a fraction, a / b, where a and b are integers. The answer to this lies in the fact that 3.14 is actually π rounded to two. Our experts can answer your tough homework and study questions. Unit 1 Concept Map Unit 1 Study Guide Unit 1 Vocabulary Absolute Value Absolute Value1. Decimals-Terminating and Repeating Integer Word Problems - Answers Integer Word. Subtract rational numbers using an algorithm.

Goals and Learning Objectives. Understand 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. I am a hero english cbr. Task 1: Fractions to Decimals. Lesson Guide Students should use a calculator to change the fractions to decimals and discuss the results with a partner before discussing the results as a class. Mathematics Explain to students that the decimals in the second column repeat forever.

The calculator shows as many digits as fit in the display and rounds the last digit. So, although the display shows 11 12 as 0., the 7 is the result of rounding. The pattern of 6s actually repeats forever. Tell students that one way to show that a decimal continues forever is to use an ellipsis. Write 11 12 as 0.9166666 to demonstrate.

Rational Numbers Read and Discuss These fractions are examples of rational numbers. A rational number is a number that can be written in the form p q, where p and q are integers and q is not 0. That is, a rational number is a number that can be written as a ratio of integers. All the integers, fractions, decimals, and percents you have worked with are rational numbers. All rational numbers can be written as decimals that either terminate (end), like those in the first column, or repeat forever in a pattern, like those in the second column.

Task 3: Math Mission. Lesson Guide Have students work individually or with a partner on the problems. Select both correct and incorrect answers for all the problems to be presented during Ways of Thinking.

Mathematical Practices Mathematical Practice 2: Reason abstractly and quantitatively. Students must work with rational numbers both in a general sense and with specific examples. Mathematical Practice 7: Look for and make use of structure. Students must discern and describe a pattern in the decimal forms in order to write numbers as ratios. Answers Answers will vary.

Possible answers:. 7 10 or 70 100. − 6 1 or 6 − 1. 15 100 or 3 20. − 4 11 or 4 − 11. − 43 10. Interventions Student has difficulty showing that − p q = − p q = p − q.

Try to first show that − p q = − p q, and then show that p − q = − p q. How can you use the multiplication property of −1 to rewrite the numerator of − p q?. Now what can you do?

Remember you are trying to get to − p q. Answers. Answers will vary. Possible answer: − 3 4 = − ( 3 ÷ 4 ) = − 0.75 − 3 4 = − 3 ÷ 4 = − 0.75 3 − 4 = 3 ÷ ( − 4 ) = − 0.75 All three expressions equal −0.75, so they are all equal to each other. Answers will vary.

Possible answer: − p q = − 1 ⋅ p q = − 1 ⋅ p q = − p q p − q = p − 1 ⋅ q = 1 − 1 ⋅ p q = − 1 ⋅ p q = − p q Since both − p q and p − q equal − p q, they are also equal to each other, so − p q = p − q = − p q. Mathematics common error Watch for students who incorrectly place the bar in a repeating decimal. The bar should only be over the digits that repeat and over the smallest possible set of digits. So, for example, 11 12 = 0.916666666 and is written as 0.91 6 ¯, not as 0.

916 ¯ or 0.91 66 ¯. Interventions Student incorrectly uses bar notation to show a repeating decimal. Did you draw the bar only over the digits that repeat?.

What is the smallest set of digits that repeats?. Did you include too many digits under your bar?

ELL: During class discussions, make sure you provide wait time (5–10 seconds) and acknowledge student responses, both verbally and with gestures. 6 ¯. 0.91 6 ¯.

0. 27 ¯ Challenge Problem Answer Let x = 0. Then: 10 x = 7. −( x = 0.) 9 x = 7 So, x = 7 9. Repeating Decimals To indicate that a decimal repeats forever in a specific pattern, you write a bar over the repeating digits.

For example, 6 11 = 0.5. You can write this decimal as 0. The use of a line to show the repeating digits in a decimal is called bar notation. Enable jit debugging windows vista. Write each of these fractions in decimal form using bar notation. 1 3.

2 3. 11 12. 3 11 Challenge Problem It is easy to change a terminating decimal to a fraction; for example, 0.09 = 9 100 and 3.2 = 32 10. Changing a repeating decimal to a fraction is trickier. The steps that follow describe a method for changing the repeating decimal 0.

12 ¯ to a fraction. Write the decimal to show the repeating pattern: For 0. 12 ¯, you write 0. Let x equal the repeating decimal: x = 0. Multiply both sides of the preceding equation by whatever power of 10 (10, 100, 1000, and so on) moves “one set” of the repeating digits to the left of the decimal point. One set of repeating digits in 0. Is “12.” To move this set to the left of the decimal point, you need to multiply x by 100: 100 x = 212.

Subtract x from the equation in the previous step: 100 x = 212 − x = 0. 99 x = 12. Solve x = 12 99 = 4 33. Use this method to change 0. 7 ¯ to a fraction. Task 7: Make Connections.

Lesson Guide Have students present their solutions as their classmates listen and ask questions. Questions to pose for discussion:. How would you define rational number?. What types of numbers are rational numbers?. What do you know about the decimal form of a rational number?. How can you show that a decimal goes on forever in a repeating pattern?

Students may wonder if there are numbers that are not rational. If so, tell them that there are and that such numbers are called irrational numbers. Irrational numbers have decimal forms that go on forever but do not have any sequence of digits that repeats forever. An example of an irrational number that students may have heard of is π. Have students press the π key on their calculator to see the first several digits of π.

Explain that the calculator could show any number of digits but there would never be any sequence of digits that repeats forever. A Possible Summary A rational number is a number that can be written as a ratio of integers. This means it can be written as p q where p and q are integers and q is not equal to 0. The integers, fractions, decimals, and percents we have worked with are all rational numbers. For example, the integer −3 is rational because it can be written as − 3 1. The decimal 0.35 is rational because it can be written as 35 100. And 12% is rational because it can be written as 12 100.

Decimal equivalents of rational numbers either terminate or go on forever in a repeating sequence. For example, 1 4 = 0.25 and 1 3 = 0.333333 We can use a bar to show the repeating digits in a repeating decimal. For example, 1 3 = 0.333333 = 0.

3 ¯ SWD: Refer to the Hint questions as a checklist of the information they need to include in their summaries. Lesson Guide This task allows you to assess students’ work and determine what difficulties they are having. The results of the Self Check will help you determine which students should work on the Gallery problems and which students would benefit from review before the assessment.

Have students work on the Self Check individually. Assessment Have students submit their work to you. Make notes about what their work reveals about their current levels of understanding and their different problem-solving approaches. Do not score students’ work. Share with each student the most appropriate Interventions to guide their thought process.

Also note students with a particular issue so that you can work with them in the Putting It Together lesson that follows. Interventions Student does not know the rules for determining the sign of a sum or product. What did you learn in prior lessons that would be useful for this problem?. Is the product of a positive number and a negative number positive or negative? What about the sign of a product of two negative numbers?.

Is the quotient of a positive number and a negative number positive or negative? What about the sign of a quotient of two negative numbers? Student has difficulty getting started. Will the value of x that makes the statement true be positive or negative?.

Try guessing a number for x and testing to see if it makes the statement true. If it doesn’t, how can you use the result to make another guess?. If the inequality were an equal sign, what value of x would make the equation true? Now test values greater than this value and less than this value. Student works inefficiently.

What do you need to consider to make an “educated” first guess?. How can you keep track of your guesses and the results?

Can you make a table?. Can your answer and work from one part help you find an answer to other parts?

3.1 Classifying Rational Numbers Answers

Answers Answers will vary. The answer for each could be any number that satisfies the given inequality. Possible answers:. x = −10 or any other number x such that x 3. x = − 1 10 or any other number x such that − 1 4 6.