Grade 8 - Lesson Title: Real Numbers
Unit 1: Real Numbers (Lesson 1 of 3) Time Frame: 1-2 weeks
Essential Question: Why are quantities represented in multiple ways?
Targeted Content Standard(s): / Student Friendly Learning Targets§ 8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
§ 8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of 2, show that 2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. / I can…
· Convert a repeating decimal into a rational number. (8.NS.1)
· Convert a fraction into a repeating decimal. (8.NS.1)
· Identify rational and irrational numbers. (8.NS.1)
· Find rational approximations of irrational numbers. (8.NS.2)
· Use rational approximations of irrational numbers to compare the size of irrational numbers. (8.NS.2)
· Locate rational approximations of irrational numbers approximately on a number line. (8.NS.2)
· Estimate the value of expressions involving irrational numbers. (8.NS.2)
Targeted Mathematical Practice(s):
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for an express regularity in repeated reasoning
Supporting Content Standard(s): (optional)
Purpose of Lesson:
The purpose of this lesson is for students to learn the definition of irrational numbers and to be able to differentiate between rational and irrational numbers. Students also use rational approximations to compare numbers, locate irrational numbers on a number line, and estimate the value of expressions involving irrational numbers.
Explanation of Rigor: (Fill in those that are appropriate.)
Conceptual:
Students understand that every number has an infinite decimal expansion that either repeats the digit zero (terminates) or another digit pattern. Students explore the definition of rational and irrational numbers. (8.NS.1) / Procedural:
Students use rational approximations of irrational numbers to compare numbers, locate irrational numbers on a number line, and estimate the value of expressions involving irrational numbers. (8.NS.2) / Application:
There are no application level targets in this lesson.
Vocabulary:
Radical, Irrational number, Rational number, Square root, Perfect square
Evidence of Learning (Assessment):
Pre-Assessment: Real Number Test (same as summative)Prior Knowledge Pre-Test, Real Number Pre-Test
Formative Assessment(s): Irrational Estimation Card Game, Irrational Estimation Dice Game, Scavenger Hunt, Number Line Code
Summative Assessment: Real Number Post-Test
Self-Assessment: Built into lessonRational/Irrational Identification
Lesson Procedures:
Segment 1 – Converting Decimals to Fractions (8.NS.1) Created by ISBE
Approximate Time Frame:
90 minutes / Lesson Format:
Whole Group
Small Group
Independent
Modeled
Guided
Collaborative
Assessment / Resources:
One note card per student with a fraction on one side. Suggested fractions would be single digit numerators and denominators.
Focus:
The focus of this segment is conceptual understanding of repeating decimals in both fraction and decimal form and converting them. / Modalities Represented:
Concrete/Manipulative
Picture/Graph
Table/Chart
Symbolic
Oral/Written Language
Real-Life Situation
Math Practice Look For(s):
MP.7: Students make use of the structure of decimal numbers by utilizing the place value regardless of the length of the decimal.
MP.3: The work presented is as if from another student. The students critique the reasoning of this work. / Differentiation for Remediation:
Differentiation for English Language Learners:
Differentiation for Enrichment:
Convert decimals with three or more repeating digits into a fraction.
Potential Pitfall(s):
When converting repeating single digit decimals to fractions, students may struggle with how to multiply a repeating decimal by 10. Students may also struggle with translating this process to two repeating digits. They may continue to multiply by 10 instead of by 100 if they don’t understand why they multiplied by 10 earlier. / Independent Practice (Homework):
Exercises, such as convert 0.8, 0.76, or 1.3 to fractions.
Steps:
1. The Repeating Decimal Challenge: (5 min) Tell students that you think any number is really a repeating decimal. Ask students to give you numbers and give them the decimal expansion of that number. Some students may know that numbers like π, e, or most square roots don’t have a repeating digit or digits pattern. If they think of these numbers, they have won the challenge.
a. Ideally, students will begin by giving you whole numbers and decimals, such as three, seven, a million, two point five (yes, it’s two and five tenths), etc. These we have traditionally called terminating decimals but are actually repeating decimals since they repeat the digit zero.
b. Next, lead students to suggesting fractions which give you a decimal expansion that repeats forever. (12=0.50, 38=0.3750, 13=0.3, 211=0.18) This should lead to a discussion of how to convert fractions to decimals. / Teacher Notes/Reflections:
2. Fraction/Decimal Sort: (10 min) Give each student a card with a fraction on it. Ask them to convert the fraction to a decimal without a calculator and move to the left side of the room if their decimal repeats and the right side of the room if their decimal does not repeat.
a. Students should all end up on the left side of the room since all the decimals repeat the digit zero if nothing else. Discuss with them why this happens. / Teacher Notes/Reflections:
The decimal expansion of the listed fraction could be listed on the back if you so desire. Then the students would be trying to discover how to get that decimal expansion.
Potential Pitfall: Students may not realize that a terminating decimal actually is a repeating decimal since it repeats the digit zero.
3. Terminating Decimals to Fractions: (5 min) Since we know how to turn fractions into decimals, we should be able to turn decimals into fractions. Ask students how this is accomplished by starting with examples like 0.2 or 0.35. They may be familiar with the place value method whereby 0.2 is two tenths. We can write that as the fraction 210 which in turn can be simplified to 15. Have students give a couple of examples of how to do this in a whole group discussion. / Teacher Notes/Reflections:
Inquiry Prompt: Would the place value method work for a decimal like 0.2?
4. A Crazy Method: (10-15 min) Walk students through the following work for converting 0.1 to a fraction. Split students into groups of two to three and have them discuss whether or not they believe this method actually works or not:
If x=0.1, then 10x=1.
This means that 10x-x=1-0.1.
That simplifies to 9x=0.9.
Solving for x gives us 9x9=0.99 or x=0.11*1010=110.
This means that x=0.1=110.
a. After students have had time to discuss in groups, have each group share out what they think and why. / Teacher Notes/Reflections:
Observations: Students should be discussing what is happening at each step. The key question to ask is why each step works. If x=0.1, why does that mean that 10x=1? Students should see that both sides of the equation were multiplied by 10. More importantly, why did the person who did this work choose to multiply by 10 in the first place? Students should eventually see that this method works.
5. Expanding on Crazy: (10-15 min) What if we used this same method on a repeating decimal like 0.2? Ask students to try to duplicate the same method of turning decimals into fractions with the repeating decimal 0.2. Split students back into groups to do this.
a. Have students share out their results which should be 29.
b. Have students complete a few more conversions in their groups such as 0.4, 0.26, and 0.15. Circulate and continue to guide students. / Teacher Notes/Reflections:
6. One More Time: (10-15 min) What should we do when two digits repeat such as with the number 0.23 instead of only one digit? Follow the same process of a whole group discussion, small group exploration, and then whole group sharing. / Teacher Notes/Reflections:
7. Assessment: (5-10 min)
a. Formative Assessment: Before closing out this portion of the lesson, ask students to describe the process of turning a repeating decimal into a fraction by listing the steps in the process on a note card or sheet of paper. This should be turned in to the teacher for analysis.
b. Self-Assessment: After the formative assessment, ask students to turn the decimal 0.35 into a fraction. Tell them they should get the answer of 1645. If they get the conversion incorrect on the first try, they should try to correct their work and write out in words what error they may have made during their first attempt. Students should hold on to this write-up for themselves. / Teacher Notes/Reflections:
8. Independent Practice: (5-10 min) Give students some independent practice problems such as 0.8, 0.76, or 1.3. Notice the repeating decimal that is greater than one. / Teacher Notes/Reflections:
Segment 2 – Identifying Rational and Irrational Numbers (8.NS.1) Adapted from:
http://alex.state.al.us/lesson_view.php?id=24079
http://www.regentsprep.org/Regents/math/ALGEBRA/AOP1/Lrat.htm
http://www.quia.com/pop/37541.html
Approximate Time Frame:
50 minutes / Lesson Format:
Whole Group
Small Group
Independent
Modeled
Guided
Collaborative
Assessment / Resources:
Irrational Number Power Point.
Rational and Irrational Sort. Print the 32 rational and irrational number cards. You will need one for each student.
Rational and Irrational Identification Self-Assessment
Focus:
The focus of this segment is conceptual understanding of rational and irrational numbers and being able to accurately identify them. / Modalities Represented:
Concrete/Manipulative
Picture/Graph
Table/Chart
Symbolic
Oral/Written Language
Real-Life Situation
Math Practice Look For(s):
MP.4: Students model rational approximations of irrational numbers on the number line. / Differentiation for Remediation:
Differentiation for English Language Learners:
Differentiation for Enrichment:
Potential Pitfall(s):
Students may think that irrational means “goes forever”. You’ll need to remind them that all numbers “go forever” as we saw in the previous section of the lesson. / Independent Practice (Homework):
Steps:
1. Opening: (5 min) Refer to the opening of the previous concept and challenge students (if they did not come up with any from the previous day) to find a decimal that does NOT repeat. Ex: π, 2, 0.12122122212… / Teacher Notes/Reflections:
Inquiry Prompt: Can you think of a number that has a pattern but does not repeat?
2. Exploration: (10-15 min) Show the Irrational Numbers Power Point. The purpose of this is for students to understand the definition and concept of an irrational number. They should be able to differentiate between them fluently.
a. Using a few examples, give the class some rational and irrational numbers and have them hold up 1 finger for irrational and 2 fingers for rational (or any variation of student monitoring you chose). When the majority of students understand the difference between irrational and rational numbers, continue with the number activity. / Teacher Notes/Reflections:
While the definition of a rational number has traditionally been about the ratio of integers, the focus now is more on the idea that a rational number is any number whose decimal expansion eventually repeats a pattern. Notice that a terminating decimal such as 2.5 repeats the digit zero beginning in the hundredths place. This means an irrational number does not eventually repeat a pattern.
3. Card Sort: (5 min) Give every student a number card making sure that there are at least two irrational number cards handed out. Ask all the students holding rational numbers to move to one side of the room, and all the students holding irrational numbers to move to the opposite side of the room. Students should feel free to discuss with someone nearby what type of number they have as long as they can explain why they think so. After students have chosen a side of the room, make sure they are correct. Discuss any misconceptions. / Teacher Notes/Reflections:
Materials: Print the Rational and Irrational Sort cards so that you have enough for each student.
4. The Number Line: (10-15 min) Ask students holding integers to go to line up in numerical order creating a human number line.
a. Next ask each remaining student to place themselves in their appropriate location on the human number line. Students should feel free to discuss their position with each other.
b. Students with irrational number cards may have a difficult time placing themselves. Discuss with the class why this is and ask if there is another way to represent these numbers?
c. This will lead to the need to approximate irrational numbers with rational numbers which is discussed in the next segment of the lesson. Ask students to determine what integers the square root is between and have the students with irrational numbers to stand between those integers. / Teacher Notes/Reflections:
This connects heavily to the essential question as we need to represent the fractions as decimals so that we can more easily compare the numbers.
Potential Pitfall: Students may have trouble placing the fractions since they are not in decimal form. Connect this to the previous learning about converting fractions to decimals.
5. Self-Assessment: Go to this link and students can answer the questions on the computer where it will tell them the correct answer if they’re wrong. You can print out the worksheet to make copies for students using the “print” link on the right side, but make sure to include the answers if you want it to be a self-assessment.
http://www.quia.com/pop/37541.html
(If you have no internet access, you may use the Rational and Irrational Identification Self-Assessment worksheet provided.)
a. Once students have completed the self-assessment, have them write down any particular type of number that they struggle with identifying as rational or irrational. As independent practice, have students write down three numbers of the type they struggle with and identify them as rational or irrational. For example, a student may struggle with remembering that terminating decimals are rational since they don’t see the repeating zero. That student should write three terminating decimals and identify them as rational. / Teacher Notes/Reflections: