DETAILED LESSON PLAN: TREASURE HUNT · MidSchoolMath Lesson Plan: Treasure Hunt 5 Lesson Plan Day...

MidSchoolMath Lesson Plan: Treasure Hunt 1

8.NS.A.2The Number System

The best way to visualize and use irrational numbers is to approximate them with a decimal. In Treasure Hunt, Caterina Fenn hires Tiberius Picklebottom to help her find her grandfather’s hidden treasure. Based on a map and a few calculations, Tiberius determines the distance to the treasure is √2 miles. They must approximate this distance in order to use the theodolite (the measuring device), and want to be as accurate as possible to minimize their digging efforts.

DETAILED LESSON PLAN: TREASURE HUNTHow far away is the final map point?

Prerequisite Standards• 7.NS.A.2

Lesson Plan OverviewLesson Length: 4 Days

English Language Learners: Provide sentence frames/stems for use during student discussion. These can help to promote the accurate use of vocabulary and increase the volume of discourse. Language solidifies concept development. “I started by thinking about _____” . . . “ _____ is too high.” . . . “_____ is too low.” . . . “______ is closer than _______.” ”I used the square root of ____ and the square root of ____ to start my approximation.”

Standards for Mathematical PracticeMP3: Construct viable arguments and critique the reasoning of others.On Day 1, during the Data & Computation phase, students are working to analyze each other’s work in order to determine the best approximation for √2. The same is happening as they are working through the Clicker Quiz and Practice Printable on Day 2. During this time you may want to put a critique protocol in place to facilitate student discourse.

•

Cluster Connection

Direct Connection: In Treasure Hunt, students find that the distance toward a treasure leaves them with an irrational number. They are asked to devise a way to approximate the value of √2. Students find that they can use rational square roots to begin approximating usable values for irrational numbers.Cross-Cluster Connection: This activity connects 8.NS to 8.G as students will likely approximate irrational numbers when they utilize the Pythagorean Theorem.

•

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Cluster Heading: Know that there are numbers that are not rational, and approximate them by rational numbers.

VocabularySquare Root: A quantity that, when multiplied by itself, gives the square.Perfect Square: A number that is produced when multiplying an integer by itself twice.Rational Number: A number that can be written as a quotient of two integers.Irrational Number: A number that cannot be written as a quotient of two integers.

••••

Common MisconceptionsWhen students find approximations for irrational numbers (√2 ≈ 1.41) they may use them interchangeably. Reinforce the idea that the decimal approximations of these numbers are not exact values.Students may not yet know that the square roots of perfect squares are rational, and that all other square roots are irrational. This knowledge will help them internalize why we use the perfect squares to help us approximate irrational square roots.

•

•

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.

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8.NS.A.2The Number System

KevinSimpson

GladysGraham

MeganLeBleu

Practice Printable

Instruction at a Glance Clicker Quiz

Full-sized Answer Key available in printed Teacher’s Guide

Gladys: Students will lean heavily on their knowledge of perfect square roots in order to approximate irrational numbers. Consider giving a brief review.

Kevin: To maximize student thinking and to help conceptualize what it means to be irrational, have students approximate irrational numbers without a calculator. Encourage the use of the number line to aid students in their thinking.

Megan: Consider giving small groups various pairs of numbers to compare (√14 and 4, 3π and 9, 2√5 and 5, etc). Ask groups to decide which of the two numbers in each pair is greater and to create as good of an argument as possible to present to the class. No calculators will best serve student thinking.

TREASURE HUNTWhat is the distance to the treasure?

Date PeriodName

MidSchoolMath Treasure Hunt 1 of 2

8.NS.A.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. π 2 ). 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.

About this standard

Because of his success with Caterina Fenn, Tiberius Picklebottom expects he may be hired to assist other treasure hunters on their search for valuable items.

He is using his newfound knowledge about √2 to prepare for other strange distances he may encounter on his other adventures. He has started an entry in his diary to keep this helpful information.

Help Tiberius complete his diary entry.

APPLYING THE STANDARD


Date PeriodName

How might this standard appear on a test?

Find two consecutive integers to complete each inequality.

Compare each value using <, > or =.

Order each list of values from least to greatest.

10) Place each value on the number line, and label it with the appropriate letter.

HOW FAR FROM HOME PLATE TO SECOND BASE?

The catcher at home plate often needs to make a quick throw to second base. According to the measurements shown on the baseball diamond, the exact distance is √16,200 feet. How far is that in decimal form? (Round to the nearest hundredth.)

1) ? < √13 < ?

5.3 5 25

√25, 3√3 ,941 2

3 , √10π,,

3 and 4 8 and 9 13 and 14

< > >

,

√16,200 feet ≈ 127.28 feet

4) √3π2 5) √22 4 5

6 6) -√29 -5 25

8) 5.35 25 ,

38

√25,3√3

-3

,7) 941 2

3 , √10, π,

, , , -√9

2) ? < √78 < ? 3) ? < √181 < ?

a) π3 c) 2√5b) √35 d) √10 e) -√6 f) -√1 g) -2√2

-2√3

9) 23

38- -3, , ,-√9 78

25-

7825

-

Check out my worked example #5


8.NS.A.2

MaterialsTreasure Hunt Immersion videoChart paper/Interactive whiteboard

••

Procedure1. Play the Immersion video to the whole class.2. Restate the questions and keep them visible: How far away is the final map point?3. Use the first part of the Low Guess, High Guess, Best Guess protocol. After 1-2 minutes, ask students to share out their guesses and an explanation if they have one. Record all answers on the board without passing judgment. Then ask students what they know and what they need to find out in order to assign a measurement to √2.

MaterialsCopies of Treasure Hunt Data Artifact, one per student •

Procedure

Low Guess, High Guess, Best GuessStart with a math problem presented to students, whole-class (≈ 1-2 min). Ask students to write down a guess that is too low, a guess that is too high and a guess that is best (≈ 1-2 min). Ask students to share what informed their thinking for their initial best guess (≈ 2 min).

1. Distribute the Data Artifact to each student.2. Invite students to work individually or with a partner to arrive at a solution.3. Observe students at work (avoid confirmation of the solution). Look for opportunities to clarify vocabulary, identify student strategies or work samples to be shared, and ask students questions to clarify their thinking.

• What do you know about square roots? • What are other square roots that are more familiar? • Are your “too high/too low” guesses rational numbers? How might they help you to estimate? • If ___ was too high and ____ was too low, which one is closest? • If ___ was too high and ____ was too low, how can you get closer?

Lesson Plan Day 1

Allow 20 minutes

Data & Computation 2

The Math SimulatorTM

Allow 12 minutes

Immersion1

As a warm-up, tell students to log into their account and access Test Trainer Pro. Specify the domain in which you would like students to work (preferably a different one than the prior day) and also the length of time you wish students to work (not a number of items). It is important to remind students to work out the math using paper and pencil when necessary and to look at their feedback.

Test Trainer ProAllow 7 to 10 minutes


Lesson Plan Day 1, cont'd.

MaterialsTreasure Hunt Resolution video•

• Do you have to stop at the tenths place? • How can you be strategic about what number you try next? 4. After determining students are nearing ‘sufficient’ progress, either consider using an additional teaching protocol, or ask students to provide a thumbs up/middle/down to indicate readiness to see the Resolution video. You may grant students additional work time, if necessary.

Procedure1. Play Resolution video to the whole class, and have the students mentally compare their solutions as they watch.2. After the video, prompt students with the following questions:

• What did you do that was the same? • What was different? • What strategy do you think was more efficient? Why It is not always necessary for students to respond. The questions can simply be used to cue thinking prior to instruction.

8.NS.A.2

Here are examples of statements you might make to the class:

In Treasure Hunt, we helped Professor Picklebottom and Cat determine the distance to the treasure by approximating √2. We found that √2 does not have an exact decimal equivalent, so we estimated its value.Numbers like √2 are called irrational numbers because they have an infinite decimal expansion that does not repeat. We can use rational numbers (those that have a finite decimal expansion) to approximate the value of irrational numbers.

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Allow 14 minutes

Resolution3

Teacher InstructionAllow 6 to 8 minutes

Answer:The final map point and treasure is approximately 1.41 miles away!

Let’s make a list of consecutive perfect squares as square roots, starting with √1. [√1, √4, √9, √16, √25, √36, √49, √64 √81, √100, √121, √144, √169,√196, √225, √256, √289, etc… ]How can we use what we know to approximate √7? • Because 7 is between 4 and 9, both perfect squares, we know √7 must be between 2 and 3. √4 < √7 < √9 2 < √7 < 3 • Remember, we're looking for a number that when multiplied by itself is as close to 7 as possible. And since 7 is closer to 9 than 4, let’s try a number closer to 3, like 2.5. 2.5 • 2.5 = 6.25 • That is too low. Let’s try something just a little higher: 2.7 • 2.7 = 7.29 • That is too high. Let's try a number a bit lower: 2.6 • 2.6 = 6.76

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•


Lesson Plan Day 1, cont'd. 8.NS.A.2

• We now know that √7 has to be between 2.6 and 2.7.• So let's try going to the hundredths place to get an approximation that is even closer. We'll keep testing numbers until we find one whose square is extremely close to 7. 2.61 • 2.61 = 6.8121 7 – 6.8121 = 0.1879 2.65 • 2.65 = 7.0225 7.0225 – 7 = 0.0225 This one is closest to 7. 2.64 • 2.64 = 6.9696 7 – 6.9696 = 0.0304 • 2.65 is the closest approximation of √7.

We can use approximations to compare irrational numbers, like these: -2√28 and -(3π). • First, let’s get an approximation for -2√28. We start by finding perfect squares close to 28. √25 < √28 < √36 5 < √28 < 6 • 28 is closer to 25 than 36, so let’s try squaring numbers close to 5 that get us near 28. 5.1 • 5.1 = 26.01 28 – 26.01 = 1.99 5.3 • 5.3 = 28.09 28.09 – 28 = 0.09 • 5.3 is very close! Let’s try one more, a little bit lower, to see if we can get closer. 5.29 • 5.29 = 27.9841 28 – 27.9841 = 0.0159 • 27.9841 is closer to 28 than 28.09, but just barely! • Let’s use 5.29 as our approximation for √28. • But now we have to multiply that by -2: -2 • 5.29 = -10.58 • The other value we needed to approximate was -(3π). • Not all irrational numbers are in the form of square roots. π is one of them. • We use a very familiar approximation for π: 3.14. -(3∙π) ≈ -(3 • 3.14) = -9.42 • -10.58 is less than -9.42, so -2√28 < -(3π).

• We can use rational numbers to approximate the values of irrational numbers.

•


MaterialsCopies of Treasure Hunt Practice Printable, 1 per student•

Practice Printable/Exit TicketAllow 16 minutes

8.NS.A.2Lesson Plan Day 2



Procedure1. Assign the Simulation Trainer to all students.2. Tell students to navigate to the Simulation Trainer assignment.3. Have students work individually to start.4. Consider using varied protocols that include peer teaching.5. Use Progress Monitoring on the Teacher Dashboard to determine which students are having difficulty. Provide individual help when necessary.

Materials• Treasure Hunt Simulation Trainer• Student Devices• Paper and Pencil• Student Headphones

Allow 25 to 35 minutes

The Math SimulatorTM

Simulation Trainer


8.NS.A.2


About this standard

Date PeriodName





3. Sort the Practice Printables based on student self-assessment and professional teacher judgment of accuracy of the response. This sorting can be used for grouping students for differentiation of instruction the following day.

Procedure

1. Distribute copies of the Practice Printable. Have students work through the front page.

2. Exit Ticket: Ask students to rate their personal understanding of the problem on a scale of 1 to 3.

1 = I need more help 2 = I need more time, yet mostly understand 3 = I’ve got this! Have students put the number on their Practice Printable and turn them in.


Differentiation PlanRemediation

Meet with students who were unsuccessful on the Exit Ticket in a small group. Consider using whiteboards to

work through problems on the Practice Printable together.

Practice

Students who completed the Exit Ticket but need more practice should spend the class period completing the

Practice Printable. Encourage them to confirm strategies and solutions with each other. Any additional time

remaining should be spent getting started on the Student Reflection.

Enrichment

Students who demonstrated confident mastery on the Exit Ticket can finish the Practice Printable and spend the

remaining time getting started on the Student Reflection or completing one or more of the following:

• Have students form two teams and utilize the Lawyer Up! protocol.

• Give students this problem (no calculators):

Place the following numbers in order from least to greatest:

-3√42, 12π2

5 , -√398, 3√217

2 , 34 , √600

Practice PrintableAllow 26 minutes

MaterialsTreasure Hunt Practice Printable•

8.NS.A.2

Lawyer Up! Divide students into two “attorney teams”. Tell each attorney team to complete the problem and prepare a defense for their “case” using the given problem. They will present an argument to convince the other side that their solution is the most accurate (≈ 8 min). Each attorney and/or team is given one minute to present their view, alternating sides (≈ 4 min). Together, the attorneys must decide which case is more defendable (≈ 1 min). Tally results of each group to determine which case wins (≈ 1-2 min).

Lesson Plan Day 3



Procedure1. Group students according to the prior day's Exit Ticket evaluation.2. Redistribute previous copies of the Practice Printable and implement the Differentiation Plan as they finish the Practice Printable. Encourage them to show their work.3. Collect completed Practice Printable.

Don't forget to assign my worked example!


Lesson Plan Day 3, cont'd.

Date PeriodName

My visual representation and title are attractive

and relevant to the story. I put extra

effort into my work.

My math drawing shows complete

understanding of the math concept.

I clearly described how the math is used in the story and used

appropriate math vocabulary.

My multiple choice question is original and directly relates

to the math concept. I have the correct

answer.

My multiple choice question directly

relates to the math concept. I have the

correct answer.

My visual representation and

title are mostly attractive and

relevant to the story. I put good effort into

my work.

My visual representation and title are somewhat

attractive. I only put a little bit of effort

into my work.


title aren’t attractive. I put very minimal


I did not make it clear how the math is used

in the story and/or didn’t use math

vocabulary.

My multiple choice question doesn’t use

the math concept.

I didn’t include a visual representation

or title.

I didn’t include a mathematical

drawing.

I didn’t explain how the math is used in

the story.

I didn’t include a multiple choice

question.

Visual Representation

Exceeds Expectations

4Meets

Expectations

3Nearing

Expectations

2Does Not Meet Expectations

1Incomplete

0

MathematicalDrawing

Math-StoryNarrative

Multiple Choice Question

MidSchoolMath

STUDENT REFLECTION This will help you with in-depth understanding.

Rubric:

Instructions:1. Draw a visual representation of the story. Include a title.

2. Create a mathematical drawing (tables, graphs, equations, figures, etc.) that shows the math used in the story.

3. Write a brief narrative that describes how the math was used in the story.

4. Using an example from your life, create a multiple choice question using the math concept. Circle the correct answer.

5. Use the rubric to grade yourself by circling how you did in each category. Turn this in with your Student Reflection.

Examples:

Student Reflection

I described how the math is used in the

story and used some math vocabulary.

I tried to describe how the math is used

in the story and/or used limited math

vocabulary.

My math drawing shows basic


My math drawing shows limited


My math drawing shows little to no



relates to the math concept, but I have

the incorrect answer.

8.NS.A.2Student Reflection

Allow 20 minutes

Materials• Copies of Student Reflection rubric, 1 per student• White Paper• Colored Pencils

Procedure1. Available in the Student Reflection lesson on Teacher Dashboard, print and distribute the rubric. Discuss requirements with students.2. Distribute white paper and colored pencils to students.3. Have students begin the Student Reflection by sketching a draft. They will have additional time the following day to complete it.


8.NS.A.2Lesson Plan Day 4



Clicker QuizAllow 30 minutes

MaterialsTreasure Hunt Clicker Quiz Student DevicesPaper and Pencil

••

•

1. Ask students to log into their account and access Virtual Clicker. 2. Open the Clicker Quiz, whole class. 3. Prompt students to enter the quiz code on their device. 4. Launch quiz. 5. For each question: a. Show question and give students time to work. Consider using various protocols (i.e., students work individually, work with a partner, or maybe they have to agree with an entire table). b. Click “Vote,” and students will have 10 seconds to enter a response. c. Analyze class distribution. Decide whether more teaching is necessary, either a mini-lesson from you or by having students share strategies. d. Click “>” to advance to the next question. e. You may either then “Skip” the question or repeat steps a through e.

Procedure


Procedure1. Students continue working and complete their Student Reflections.2. Consider having a Gallery Walk when they are complete, using the I Wonder…, I Notice… protocol and sticky notes.

Gallery Walk (16-20 min)Display student work (such as Student Reflections) on classroom walls. Assign groups with tasks focused on

specific details (such as identifying different ways to solve a problem) and/or larger patterns (such as general

misconceptions). Tell groups to walk around, complete their task (≈ 8-10 min), then prepare and report brief

remarks to the class with their broader “a-ha” and “why” understandings (≈ 8-10 min).

I Wonder . . . , I Notice . . . (8-10 min)Following a completed whole-class assignment, set ground rules for peer critique, including being thoughtful,

specific, helpful and to join in (≈ 1 min)! Choose a student to be “the originator” who is tasked to explain his or

her approach and solution to a problem (≈ 2 min), while other students listen only. Then ask other students to

ask “the originator” clarifying questions or comments that start with ‘I wonder’ and ‘I notice’ (≈ 5-6 min).

Date PeriodName

My visual representation and title are attractive

and relevant to the story. I put extra


My math drawing shows complete


I clearly described how the math is used in the story and used

appropriate math vocabulary.

My multiple choice question is original and directly relates

to the math concept. I have the correct

answer.


relates to the math concept. I have the

correct answer.


title are mostly attractive and

relevant to the story. I put good effort into

my work.

My visual representation and title are somewhat

attractive. I only put a little bit of effort

into my work.


title aren’t attractive. I put very minimal


I did not make it clear how the math is used

in the story and/or didn’t use math

vocabulary.

My multiple choice question doesn’t use

the math concept.

I didn’t include a visual representation

or title.

I didn’t include a mathematical

drawing.

I didn’t explain how the math is used in

the story.

I didn’t include a multiple choice

question.

Visual Representation

Exceeds Expectations

4Meets

Expectations

3Nearing

Expectations

2Does Not Meet Expectations

1Incomplete

0

MathematicalDrawing

Math-StoryNarrative

Multiple Choice Question

MidSchoolMath

STUDENT REFLECTION This will help you with in-depth understanding.

Rubric:

Instructions:1. Draw a visual representation of the story. Include a title.

2. Create a mathematical drawing (tables, graphs, equations, figures, etc.) that shows the math used in the story.

3. Write a brief narrative that describes how the math was used in the story.

4. Using an example from your life, create a multiple choice question using the math concept. Circle the correct answer.

5. Use the rubric to grade yourself by circling how you did in each category. Turn this in with your Student Reflection.

Examples:

Student Reflection

I described how the math is used in the

story and used some math vocabulary.

I tried to describe how the math is used

in the story and/or used limited math

vocabulary.

My math drawing shows basic


My math drawing shows limited


My math drawing shows little to no



relates to the math concept, but I have

the incorrect answer.

Student ReflectionAllow 20 minutes

• Student Reflections from Day 3• White Paper• Colored Pencils• Sticky Notes

Materials

Lesson Plan Day 4, cont'd. 8.NS.A.2

TREASURE HUNTArtifact 1


8.NS.A.2


About this standard

Date PeriodName





APPLYING THE STANDARD


Date PeriodName

How might this standard appear on a test?

Find two consecutive integers to complete each inequality.

Compare each value using <, > or =.

Order each list of values from least to greatest.

10) Place each value on the number line, and label it with the appropriate letter.

HOW FAR FROM HOME PLATE TO SECOND BASE?

The catcher at home plate often needs to make a quick throw to second base. According to the measurements shown on the baseball diamond, the exact distance is √16,200 feet. How far is that in decimal form? (Round to the nearest hundredth.)

1) ? < √13 < ? 2) ? < √78 < ? 3) ? < √181 < ?

4) √3π2 5) √22 4 5

6 6) -√29 -5 25

8) 9)5.35 25 , 2

338

7825

√25,3√3 - -3,

a) π3 c) 2√5b) √35 d) √10 e) -√6 f) -√1 g) -2√2

7) 941 2

3 , √10, π, , , ,-√9 -

Check out my worked example #5

DETAILED LESSON PLAN: TREASURE HUNT · MidSchoolMath Lesson Plan: Treasure Hunt 5 Lesson Plan Day...

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