English Language Arts 10 – 2 › 2014 › 01 › … · Web viewcube – the result of a number or term being multiplied by itself twice more. cube root – a number that when

MATHEMATICS 10C

REAL NUMBERS

High School collaborative venture withHarry Ainlay, Jasper Place, McNally, Queen Elizabeth,

Ross Sheppard and Victoria Schools

Harry Ainlay: David Cunningham, Christine DibbenJasper Place: Linda Aschenbrenner, Shelaine Kozakavich, Nic Ryan Ross Sheppard: Tim Gartke, Jeremy Klassen, Don SymesVictoria: Kevin Bissoon, Elisha PinterFacilitator: John Scammell (Consulting Services)Editor: Rosalie Mazurok (Contracted)

2009 - 2010

Mathematics 10C Real Numbers Page 2 out of 66

TABLE OF CONTENTS

STAGE 1 DESIRED RESULTS PAGE

Big Idea

Enduring Understandings

Essential Questions

4

4

5

Knowledge

Skills

6

7

Stage 2 ASSESSMENT EVIDENCE

Teacher Notes For Transfer Tasks 8

Transfer Task A Radical Board Game

Teacher Notes for A Radical Board Game and Rubric Transfer Task

Rubric

The Golden Ratio in a Face Teacher Notes for The Golden Ratio in a Face and Rubric Transfer Task Rubric Possible Solution

910 - 1213 - 14

1516 - 2223 - 2425 - 32

Stage 3 LEARNING PLANS

Lesson #1 Factors and Multiples 33 - 35

Lesson #2 Square Roots and Cube Roots 36 - 39

Lesson #3 Estimating Radicals 40 - 41

Lesson #4 Working with Radicals 42 - 44

Lesson #5 Rational Exponents 45 - 48

Lesson #6 Negative Exponents 49 - 51

Lesson #7 Irrational Numbers – Classifying and Ordering 52 - 54

Lesson #8 Working with Exponent Laws 55 - 56

APPENDIX - Handouts

Real Numbers Unit Handouts 58 - 60


Mathematics 10C Real Numbers

STAGE 1 Desired Results

Big Idea:

Real Numbers provide students with a foundation upon which they build an understanding of the different ways to represent and order real quantities. This understanding will enable students to solve problems related to all disciplines.

Enduring Understandings:

The students will understand…

numeracy as it relates to real numbers.

o that the set of real numbers is continuous and is made up of rational and irrational numbers.

that there are various ways of representing numbers including exponents, fractions, and radicals.


Implementation note:Post the BIG IDEA in a prominentplace in your classroom and refer to it often.

Essential Questions:

When and why should we use exact values?

What is a real number?o What are the different ways of representing real numbers?o How can real numbers be classified?o What strategies can you use to order real numbers appropriately?

What is the meaning of continuous?o When are there gaps in a number line?o How many numbers are there?

****

Knowledge:


Implementation note: Ask students to consider one of the essential questions every lesson or two.Has their thinking changed or evolved?

Enduring Understanding

Specific Outcomes

Knowledge that applies to this Enduring Understanding

Students will understand…


o that the set of real numbers is continuous and is made up of

rational and irrational numbers.

*AN 2

Students will know…

that numbers can be ordered. that numbers can be approximated. the relationship between sets of numbers

(union of sets). that the number line is infinitely continuous.



*AN 1 *AN 2 *AN 3


that numbers can be approximated. what an exponent is. what integral and rational exponents mean. the exponent laws. components of radicals.

*


*AN = Algebra and Number

Skills:

Enduring Understanding

Specific Outcomes

Skills that apply to this Enduring Understanding



o that the set of real numbers is continuous and is made up of

rational and irrational numbers.

*AN 1 *AN 2 *AN 3

Students will be able to…

sort real numbers into categories. approximate irrational numbers. order real numbers. apply exponent laws. solve problems involving real numbers. solve problems involving real numbers. express radicals in mixed and entire forms

and convert between forms. express a number as a product of its prime

factors.



*AN 1 *AN 2 *AN 3


sort real numbers into categories. approximate irrational numbers. order real numbers. apply exponent laws. solve problems involving real numbers. solve problems involving real numbers. express radicals in mixed and entire forms

and convert between forms. express a number as a product of its prime

factors.

*


*AN = Algebra and Number

Implementation note:Teachers need to continually askthemselves, if their students are acquiring the knowledge and skills needed for the unit.

STAGE 2 Assessment Evidence

1 Desired Results Desired Results

A Radical Board Game or The Golden Ratio in a Face

Teacher Notes

There are two transfer tasks to evaluate student understanding of the concepts relating to slope. The teacher (or the student) will select one for completion. Photocopy-ready versions of the two transfer tasks and rubric are included in this section.

Each student will: (A Radical Board Game) demonstrate their understanding of the exponent laws. demonstrate their understanding of vocabulary related to real numbers.

(The Golden Ratio in a Face) demonstrate an appreciation for the beauty of mathematics. create an approximate value for phi and understand that it is a special irrational

number.


Implementation note:Students must be given the transfer task & rubric atthe beginning of the unit. They need to know howthey will be assessed and what they are workingtoward.

Teacher Notes for A Radical Board Game Transfer Task

Board game considerations should be tailored – only share as much as you feel is necessary (differentiated instruction). Teachers should feel free to add any suggestions that may move students along. For example, chance cards could be created that would require students to use (an additional) law or power rule before moving.

Teacher Notes for Rubric

No score is awarded for the Insufficient/Blank column , because there is no evidence of student performance.

Limited is considered a pass. The only failures come from Insufficient/Blank.

When work is judged to be Limited or Insufficient/Blank, the teacher makes decisions about appropriate intervention to help the student improve.


Implementation note:Teachers need to consider what performances and products will reveal evidence of understanding?What other evidence will be collected to reflectthe desired results?

A Radical Board Game - Student Assessment TaskTask: You have been hired by HAZBRO to create an award-winning mathematical board game focusing on exponents and radicals. You are expected to present your game idea and a working prototype to the board of directors. The following concepts need to be included in your design:

Name Definition Example CompletedZero Exponent LawProduct of PowersQuotient of PowersPower of a Power

Power of a Product

Power of a Quotient

Negative Exponent

Rational Exponents

Converting from Mixed to Entire

A radical with a coefficient of 1

Converting from Entire to Mixed

Product of a rational number and a radical

Number Line Placing radicals in order on a number

line

Rational and Irrational Numbers Real Numbers Rational:

Irrational:

Board Game Considerations

You may want to include the following:

Board design (Do you need a game board?)o Example: Locate some cardboard that can be used to form the playing board for

your math board game. You can use whatever you have on hand, as long as one side contains no writing. Use a black marker and a meter stick to mark evenly spaced squares around the perimeter.

Playing cards

Game pieces o Instead of using standard dice, create one where each side’s value is a radical.

You may choose a die with 8, 12 or 20 sides. For Example, if you rolled , whose value of that is 2.236067977… so you would move 2 spaces. Essentially, you would always round to the nearest whole number.

o Moving pieces

Rules

How to start

How to win

1. 2. 3.

How to move or score points

o Example: Roll the dice--the player who rolls the highest roll goes first. Take turns rolling the dice and moving game pieces around the board. Each time you land, your opponent will read a math problem from a card that matches the space you have landed on. If you answer correctly, you get the points assigned to that colour. If you answer incorrectly, you do not get any points. The first player to reach 100 points wins!

Struggling?

Make a long list of math problems and come up with the solutions - every problem you include may represent a game card, or a board space.

Look for inspiration for your game. Feel free to use ideas from other games that you have played in the past (e.g., Monopoly, Snakes and Ladders, Sorry!, Trivial Pursuit, Cranium, etc.).

Still stuck?...

Use colour markers to assign a point value to each square on your math game board. For example, use red to denote spaces that are worth ten points. Use yellow to denote spaces that are worth five points. Try not to assign points to every square, you can add some fun to the board by including lose your turn spaces, roll again spaces, free points spaces, a bet-your-own points space, chance cards etc.

Equally divide the math problems on your list into categories that match the colour point values. Make sure that the most challenging math problems are placed into the category with the most points assigned to it and that all others are grouped accordingly, as well.

Write the math problems onto cards. If you can find colour note cards to match the various point categories, use them. If you cannot, just colour the edges of the note cards with a marker for identification. Group the note cards into piles.

Assessment

Mathematics 10C

Real Numbers

Rubric

Level

CriteriaExcellent

4Proficient

3Adequate

2Limited*

1Insufficient /

Blank*

Performs Calculations

Performs precise and explicitcalculations.

Performs focused and accuratecalculations.

Performs appropriate and generally accuratecalculations.

Performs superficial and irrelevant calculations.

No score is awarded because there is no evidence of student performance.

Presents Data Presentation of data is insightful and astute.

Presentation of data is logical and credible.

Presentation of data is simplistic and plausible.

Presentation of data is vague and inaccurate.

No data is presented.

Explains Choice

Shows a solution for the problem; provides an insightful explanation.

Shows a solution for the problem; provides a logical explanation.

Shows a solution for the problem; provides explanations that are complete but vague.

Shows a solution for the problem; provides explanations that are incomplete or confusing.

No explanation is provided.

Communicates findings

Develops a compelling and precise presentation that fully considers purpose and audience; uses appropriate mathematical vocabulary, notation and symbolism.

Develops a convincing and logical presentation that mostly considers purpose and audience; uses appropriate mathematical vocabulary, notation and symbolism.

Develops a predictable presentation that partially considers purpose and audience; uses some appropriate mathematical vocabulary, notation and symbolism.

Develops an unclear presentation with little consideration of purpose and audience; uses inappropriate mathematical vocabulary, notation and symbolism.

No findings are communicated.

Glossary

accurate – free from errors

astute – shrewd and discerning

appropriate – suitable for the circumstances

compelling – convincing and persuasive

complete – including every necessary part

convincing – impressively clear or definite

credible – believable

explicit – expressing all details in a clear and obvious way

focused – concentrated on a particular thing

incomplete – partial

inaccurate – not correct

inappropriate – not suitable

insightful – a clear perception of something

irrelevant – not relevant or important

logical - based on facts, clear rational thought, and sensible reasoning

precise - detailed and specific

plausible – believable

predictable - happening or turning out in the way that might have been expected

simplistic – lacking detail

superficial - having little significance or substance

unclear – ambiguous or imprecise

vague - not clear in meaning or intention

Teacher Notes for The Golden Ratio in a Face Transfer Task

This task leads students through discovering the golden ratio. Students will use pictures of faces, measure set dimensions and calculate ratios to approximate the golden ratio.

The introduction to this project comes from the following website:

http://www.markwahl.com/index.php?id=22

The calculation of the golden ratio lends itself quite nicely to an excel application. Avoid the implication that beauty can be measured by the proximity of your proportions to the golden ratio. For example, Julia Roberts has a wide mouth and big lips, but these are considered her most beautiful and distinguishing feature.

Teacher Notes for Rubric

No score is awarded for the Insufficient/Blank column , because there is no evidence of student performance.

Limited is considered a pass. The only failures come from Insufficient/Blank.

When work is judged to be Limited or Insufficient/Blank, the teacher makes decisions about appropriate intervention to help the student improve.


Implementation note:Teachers need to consider what performances and products will reveal evidence of understanding?What other evidence will be collected to reflectthe desired results?


The Golden Ratio in a Face - Student Assessment Task

Statues of human bodies that the ancient Greeks considered most “perfect” embodied many Golden Ratios. It turns out that the “perfect” (to ancient Greeks) human face has a whole flock of Golden Ratios as well.

In this task, you will measure distances on the face of a famous Greek statue (despite its broken nose) by using the instructions on this page. Before you start, notice that near the face on the second page are names for either a location on the face or a length between two places on the face. Lines mark those lengths or locations exactly.

Using your cm/mm ruler and the face picture on the next page, find each measurement below to the nearest millimetre that is tenth of a cm or .1cm (___._ cm). Remember, you are measuring the distance or length between the two locations mentioned. You can use the marking lines to place the ruler for your measurements. Complete the table.

a = Top-of-head to chin = ___ . __ cm

b = Top-of-head to pupil = ___ . __ cm

c = Pupil to nose tip = ___ . __ cm

d = Pupil to lip = ___ . __ cm

e = Width of nose = ___ . __ cm

f = Outside distance between eyes = ___ . __ cm

g = Width of head = ___ . __ cm

h = Hairline to pupil = ___ . __ cm

i = Nose tip to chin = ___ . __ cm

j = Lips to chin = ___ . __ cm

k = Length of lips = ___ . __ cm

I = Nose tip to lips = ___ . __ cm

Now use these letters and go on to the next page to compute ratios with them with your

calculator. Remember: the first ratio means divide measurement by measurement ;

round your answers to 3 decimal places.

Studies suggest that Shania Twain may have one of the most beautifully proportioned faces.

a = Top-of-head to chin = _____cm

b = Top-of-head to pupil = _____cm

c = Pupil to nose tip = _____cm

Eye Nose

Lips

Top of the Head

Hairline

Pupils

Nose Tip

Lips

Chin

Eye

Head Width

h = Hairline to pupil = _____cm

i = Nose tip to chin = _____cm

j = Lips to chin = _____cm

k = Length of lips = _____cm

I = Nose tip to lips = _____cm

d = Pupil to lip = _____cm

e = Width of nose = _____cm

f = Outside distance between eyes = _____cm

g = Width of head = _____ cm


How closely proportioned to the golden ratio is Johnny Depp’s face?

a = Top-of-head to chin = _____ cm

b = Top-of-head to pupil = _____ cm

Head WidthEye Nose

Lips

Top of the Head

Hairline

Pupils

Nose Tip

Lips

Chin

Eye

g = Width of head = _____ cm

h = Hairline to pupil = _____ cm

i = Nose tip to chin = _____ cm

j = Lips to chin = _____ cm

k = Length of lips = _____ cm

I = Nose tip to lips = _____ cm

c = Pupil to nose tip = _____ cm

d = Pupil to lip = _____ cm

e = Width of nose = _____ cm

f = Outside distance between eyes = _____ cm


Place the measurements for Shania Twain and Johnny Depp here, and calculate the ratios.

Shania Johnny

Average ratio = __________ Average ratio = __________

Take a photo of yourself straight on or find a straight on headshot from a magazine or the internet. Identify the same ratios to see how closely you (or your chosen picture) match the golden ratio.

What you have is an approximation of the golden ratio. Originally the golden ratio was developed using the following ratio.

The ancient Greeks set up the following ratio, where represented the Golden ratio.

10 x 1 x

Your name here: ____________________

Average ratio = __________

We can approximate many different types of constants with something called nested radicals and continued fractions. Nested radicals are radicals within radicals and continued fractions are fractions within fractions both of which continue without end.For the golden ratio, , the continued fraction looks like...

The nested ratio for is...

Use the pattern for the continued fraction and the nested radical to determine the value of to four decimal places. You will know that you have done it correctly when consecutive terms no longer change the value of the 4th decimal place.How closely do the Greek statue, Shania, Johnny, and yourself match the value of the golden ratio calculated using continued fractions and nested radicals?

Assessment

Mathematics 10C

Real Numbers

Rubric

Level

CriteriaExcellent

4Proficient

3Adequate

2Limited*

1Insufficient /

Blank*

Performs Calculations

Performs precise and explicitcalculations.

Performs focused and accuratecalculations.

Performs appropriate and generally accuratecalculations.

Performs superficial and irrelevant calculations.

No score is awarded because there is no evidence of student performance.

Presents Data Presentation of data is insightful and astute.

Presentation of data is logical and credible.

Presentation of data is simplistic and plausible.

Presentation of data is vague and inaccurate.

No data is presented.

Explains Choice

Shows a solution for the problem; provides an insightful explanation.

Shows a solution for the problem; provides a logical explanation.

Shows a solution for the problem; provides explanations that are complete but vague.

Shows a solution for the problem; provides explanations that are incomplete or confusing.

No explanation is provided.

Communicates findings

Develops a compelling and precise presentation that fully considers purpose and audience; uses appropriate mathematical vocabulary, notation and symbolism.

Develops a convincing and logical presentation that mostly considers purpose and audience; uses appropriate mathematical vocabulary, notation and symbolism.

Develops a predictable presentation that partially considers purpose and audience; uses some appropriate mathematical vocabulary, notation and symbolism.

Develops an unclear presentation with little consideration of purpose and audience; uses inappropriate mathematical vocabulary, notation and symbolism.

No findings are communicated.

Glossary

accurate – free from errors

astute – shrewd and discerning

appropriate – suitable for the circumstances

compelling – convincing and persuasive

complete – including every necessary part

convincing – impressively clear or definite

credible – believable

explicit – expressing all details in a clear and obvious way

focused – concentrated on a particular thing

incomplete – partial

inaccurate – not correct

inappropriate – not suitable

insightful – a clear perception of something

irrelevant – not relevant or important

logical - based on facts, clear rational thought, and sensible reasoning

precise - detailed and specific

plausible – believable

predictable - happening or turning out in the way that might have been expected

simplistic – lacking detail

superficial - having little significance or substance

unclear – ambiguous or imprecise

vague - not clear in meaning or intention


The Golden Ratio in a Face –Possible Solution

Statues of human bodies that the ancient Greeks considered most “perfect” embodied many Golden Ratios. It turns out that the “perfect” (to ancient Greeks) human face has a whole flock of Golden Ratios as well.

In this task, you will measure distances on the face of a famous Greek statue (despite its broken nose) by using the instructions on this page. Before you start, notice that near the face on the second page are names for either a location on the face or a length between two places on the face. Lines mark those lengths or locations exactly.

Using your cm/mm ruler and the face picture on the next page, find each measurement below to the nearest millimetre that is tenth of a cm or .1cm (___._ cm). Remember, you are measuring the distance or length between the two locations mentioned. You can use the marking lines to place the ruler for your measurements. Complete the table.

a = Top-of-head to chin = 7.4 cm

b = Top-of-head to pupil = 3.7 cm

c = Pupil to nose tip = 1.4 cm

d = Pupil to lip = 2.2 cm

e = Width of nose = 1.4 cm

f = Outside distance between eyes = 3.8 cm

g = Width of head = 4.9 cm

h = Hairline to pupil = 2.2 cm

i = Nose tip to chin = 2.3 cm

j = Lips to chin = 1.5 cm

k = Length of lips = 2.3 cm

I = Nose tip to lips = 0.8 cm

Now use these letters and go on to the next page to compute ratios with them with your

calculator. Remember: the first ratio, means divide measurement by measurement ;

round your answers to 3 decimal places.



7.44.93.72.22.31.52.31.41.40.83.82.22.31.4

1.510

1.682

1.533

1.643

1.750

1.727

1.643

1.641

Studies suggest that Shania Twain may have one of the most beautifully proportioned faces.










Head Width

Eye Nose

Lips

Top of the Head

Hairline

Pupils

Nose Tip

Lips

Chin

Eye






How closely proportioned to the golden ratio is Johnny Depp’s face?


Head Width

Eye Nose

Lips

Top of the Head

Hairline

Pupils

Nose Tip

Lips

Chin

Eye

a = Top-of-head to chin = cm













9.9

Place the measurements for Shania Twain and Johnny Depp here, and calculate the ratios.


Shania Twain Johnny Depp

Average ratio = __________ Average ratio = __________

9.16.24.52.93.01.63.01.61.51.44.23.12.91.5

9.95.65.32.82.81.72.91.71.21.13.62.92.21.2

1.468

1.552

1.875

1.875

1.071

1.355

1.933

1.590

1.768

1.893

1.647

1.706

1.091

1.241

1.833

1.597

Take a photo of yourself straight on or find a straight on headshot from a magazine or the internet. Identify the same ratios to see how closely you (or your chosen picture) match the golden ratio.


Top of the headHairline

Pupils

Tip of the Nose

Lips

Chin

Head widthEye EyeNose

Mouth













What you have is an approximation of the golden ratio. Originally the golden ratio was developed using the following ratio.

The ancient Greeks set up the following ratio, where x represented the Golden ratio.


Your name here: _Myself__


7.85.43.22.73.11.93.11.51.71.23.52.72.11.7

1.444

1.185

1.632

2.067

1.417

1.296

1.235

1.468

10x 1 x


We can approximate many different types of constants with something called nested radicals and continued fractions. Nested radicals are radicals within radicals and continued fractions are fractions within fractions both of which continue without end.For the golden ratio, , the continued fraction looks like...

The nested ratio for is...

How closely do the Greek statue, Shania, Johnny, and yourself match the value of the golden ratio calculated using continued fractions and nested radicals? In order we come in at:

Me – 1.468Shania – 1.590Johnny – 1.597Phi – 1.618Statue – 1.641

Now with reference to Phi I will calculate the percentage error from the exact value for everyone.

Me:

Shania:

Johnny:

Statue:

This eventually equals 1.618.

This also eventually equals 1.618.

STAGE 3 Learning Plans

Lesson 1

Factors and Multiples

STAGE 1

BIG IDEA:


ENDURING UNDERSTANDINGS:


numeracy as it relates to real numbers.o that the set of real numbers is

continuous and is made up of rational and irrational numbers.

there are various ways of representing numbers including exponents, fractions, and radicals.

ESSENTIAL QUESTIONS:


What is a real number?o What are the different ways of

representing real numbers?o How can real numbers be

classified?o What strategies can you use to

order real numbers appropriately?.

KNOWLEDGE:


that numbers can be ordered. that numbers can be approximated.

SKILLS:


express a number as a product of its prime factors.


Implementation note:

Each lesson is a conceptual unit and is not intended to be taught on a one lesson per block basis. Each represents a concept to be covered and can take anywhere from part of a class to several classes to complete.

Lesson Summary

Students will review whole number factors and multiples. (Note: these were last addressed in grade 6.)

Lesson Plan

Activate Prior Knowledge

Quick discussion of Multiples vs. Factors.

Play the “Buzz”.

Have students stand in a circle – the teacher says a number, and the students count. When the count gets to a multiple of the starting number the student says “buzz” rather than the number. Students who make a mistake sit down. The winner is the last person standing. For a more advanced game, use two or more numbers as the “buzz” factors.

Factors Activity

Students build rectangles to explore the factors of a given number. This can be done as a class using a projector or an interactive whiteboard or in a computer lab where students work independently.

http://www.shodor.org/interactivate/activities/FactorizeTwo/

Introduce Greatest Common Factor and Least Common Multiple

Provide definitions of Greatest Common Factor (GCF) and Least Common Multiple (LCM).

Provide groups of students with several pairs of numbers and ask them to find both GCF and LCM. Have students describe their strategies on posters and share with the rest of the class.


This lesson is review for students (previously learned in grade 6) and will likely not take an entire class. This topic is also covered in Polynomials, and so may have been previously taught.

http://www.shodor.org/interactivate/activities/FactorizeTwo/

Check for Understanding

Quick Check: put two numbers on the board and ask the students to individually find the GCF and LCM.

Practise new learning

Assign selected exercises from text.

Assess learning

Exit slip at the end of the class.

Give the students sets of numbers and have them find GCF and LCM (this should be completed individually).

Going Beyond

Strong students should be given a larger set of numbers (find GCF and LCM of 3 or 4 numbers)

Resources

Math 10 (McGraw Hill: sec 5.2)Foundations and Pre-calculus Mathematics 10 (Pearson: sec 3.1)

Interactive whiteboard or Projector or class set of computers

Glossary

greatest common factor (GCF) – the largest or most complex factor that a set of terms have in common

least common multiple (LCM) – the smallest or least complex multiple that a set of terms have in common


Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more than one division. Some terms have animations to illustrate meanings.

http://www.learnalberta.ca/content/memg/index.html



Lesson 2

Square Roots and Cube Roots

STAGE 1

BIG IDEA:













KNOWLEDGE:


components of radicals.

SKILLS:


solve problems involving real numbers. Identify square roots, cube roots, perfect

squares and perfect cubes.

Lesson Summary

Students will identify perfect squares and perfect cubes, and then determine square roots and cube roots.


Lesson Plan

Activate Prior Knowledge/Experience

Provide students with a set of 1 unit square algebra tiles.

Ask students to make a rectangle of area 15 and identify the side lengths.

Ask students if we can make a square with area 15.

Ask students to make rectangles with the following areas and identify which can be made into a square.

o 4o 6o 8o 9o 12

Use Perfect Squares and Cubes notebook file (see notebook file in the share site) to model on the board if necessary.

Discuss perfect squares and square roots based on the investigations.

Diagnostic Assessment of Prior Knowledge

Ask students to identify perfect squares and square roots of larger numbers without the use of algebra tiles.

Present New Learning

Provide students with a set of snap cubes.

Ask students to build a 3 dimensional shape of volume 8.

Ask students if they can use the 8 pieces to make a cube. What is the side length of the cube?

Ask students to make rectangular prisms (if possible) with the following volumes and identify which can be made into a cube.

o 12o 18o 25o 27


Discuss perfect cubes and cube roots based on the investigation.

Define a radical (and components of), specifically in terms of perfect squares and perfect cubes.

Make a connection between radicals and prime factorization (link to Lesson #4)

Examples:

Check for Understanding

Ask students to identify perfect squares and square roots of larger numbers without the use of algebra tiles.

Practise New Learning


Resources

Foundations and Pre-calculus Mathematics 10 (Pearson: sec 3.2)Math 10 (McGraw Hill: sec 4.1)

Perfect Squares and Cubes notebook file (in share site)

algebra tilessnap cubesinteractive whiteboard or a projector


Glossary

cube – the result of a number or term being multiplied by itself twice more

cube root – a number that when multiplied by itself twice more will result in the original number

index – the order of root being taken (e.g. 2 is the index for square root and 3 is the index for a cube root)

radical – the radical symbol or the symbol together with the index and the radicand

radicand – the number or expression of which the root is being taken

square – the result of a number or term being multiplied by itself

square root (of a non-negative number) – a non-negative number that when multiplied by itself results in the original number








Lesson 3

Estimating Radicals

STAGE 1

BIG IDEA:












order real numbers appropriately?

What is the meaning of continuous?o When are there gaps in a number

line?o How many numbers are there?

.

KNOWLEDGE:


that numbers can be ordered. that numbers can be approximated. components of radicals.

SKILLS:


approximate irrational numbers. express radicals in mixed and entire forms

and convert between forms. express a number as a product of its

prime factors.

Lesson Summary

Students will estimate and order radicals (using a number line).


Lesson Plan


Review of perfect squares and perfect cubes – given a list of radicals that are perfect squares and cubes and whole numbers, put them in order.

Present new learning

Discussion of what is happening between the values above.

What methods can we use to figure out where would go?

Try to place on a number line:

Practice new learning


Review and consolidate learning

Number line game – each student is given a card with a radical or whole number – they need to put themselves in a line from least to greatest without using calculators (and without talking for an extra challenge).

Resources



Lesson 4

Working with Radicals

STAGE 1

BIG IDEA:













KNOWLEDGE:


components of radicals.

SKILLS:


express radicals in mixed and entire forms and convert between forms.

express a number as a product of its prime factors.

Lesson Summary

Students will learn to convert between mixed and entire radicals.


Lesson Plan


Review prime factorization. Put three or four examples on the board, have students try them.

Review factors of a number from Lesson #1 (Example: 36 = 1x36, 2x18, 3x12, 4x9, 6x6)

Review definition of a radical.


Expand the definition of a radical to include numbers other than 2 or 3 as your index.

Estimate the value of . Estimate the value of . What do you notice? Why is that?

Method 1

Prime factorization

Method 2

What are the factors of 18?

Discuss that the second option includes .

Handout “Simplifying Radicals” Worksheet (see Appendix). Have students complete the table. Then follow with a class discussion.

Expand lesson to include moving from mixed to entire radicals.




Self / Peer Assessed Practice Work

“Matching Game” – in partners, given a set of cards with one radical per card – half are entire radicals, half are mixed radicals – have students match up the cards that are equal. It could be in the form of a jigsaw puzzle.

Going Beyond

Try simplifying radicals with indexes equal to 4 and 5. Discuss

Resources


Glossary

entire radical – an expression where an entire term is under a radical sign or symbol

mixed radical – an expression where part of the term is outside the radical and part isunder the radical


Lesson 5

Rational Exponents

STAGE 1

BIG IDEA:













KNOWLEDGE:


what an exponent is. what integral and rational exponents

mean. the exponent laws.

SKILLS:


apply exponent laws. solve problems involving real numbers.

Lesson Summary

Students will review the exponent laws from math 9 and expand their knowledge set to include rational exponents.


Lesson Plan

Activate Prior Knowledge / Experience

Students have seen the exponent laws in Math 9 but were limited to natural number exponents.

Hand out worksheet ‘Laws of Exponents Review’ (see Appendix). Have students work in group to complete the worksheet, using the examples provided and their past knowledge


Discuss results of worksheet as a class to ensure everyone came to the same conclusions.


Use the exponent laws to have students discover the value of and .

Example #1

(a)

What must be equal to so that the number multiplied by itself is 4?

(b)

What must be equal to so that the number multiplied by itself is 9?

What do you think the value of is? Check your answer with a graphing calculator.


Example #2

(a)

What must be equal to so that the number multiplied by itself three

times is -8?

(b)

What must be equal to so that the number multiplied by itself three

times is 27?

What do you think the value of is? Check your answer with a calculator.

Discuss why cube roots can have negative bases, but square roots cannot when in brackets.

Hand out worksheet – “Laws of Exponents Extended” (see Appendix). Discuss the strategy used and the ‘rule’

Define radicals – including index and radicand.

How could you use the definition to write the following expressions in radical form?

(a) (b) (c) (d)

(e) (f) (g) (h)

Provide example with variable bases.




Resources



Lesson 6

Negative Exponents

STAGE 1

BIG IDEA:













KNOWLEDGE:


what an exponent is. what integral and rational exponents

mean. the exponent laws.

SKILLS:


apply exponent laws. solve problems involving real numbers.

Lesson Summary

Students will be able to simplify expressions with negative exponents using reciprocals.


Lesson Plan


Review dividing fractions (multiplying by reciprocal of the divisor).


Quick check from previous lesson prior to starting (provide students with power expressions with rational exponents).


Investigate.

Given

Simplify using factorization:

Simplify using exponent laws.

Can we say that ?

Try simplifying the following using both methods above:

(a) (b) (c)


Use the pattern you see above to evaluate the following:

(a) (b) (c)

(d) (e) (f)

Express the rule(s) you used as a general statement.

Provide examples with variable bases.



Resources



Lesson 7

Irrational Numbers – Classifying and Ordering

STAGE 1

BIG IDEA:






there are various ways of representing numbers including exponents, fractions, and radicals.






order real numbers appropriately?

What is the meaning of continuous?o When are there gaps in a number

line?o How many numbers are there?

.

KNOWLEDGE:


that numbers can be ordered. that numbers can be approximated. the relationship between sets of numbers

(union of sets). that the number line is infinitely

continuous. what an exponent is.

SKILLS:


sort real numbers into categories. approximate irrational numbers. order real numbers.

Lesson Summary

Students will practise classifying and ordering rational and irrational numbers presented in different forms.


Lesson Plan

Activate Prior Knowledge / Experience

Review of ordering numbers on a number line.


Taken from Pearson Foundations and Pre-Calculus 10

Given a table with a selection of rational and irrational numbers, what generalizations can you make about rational and irrational numbers based on the values provided below.

Rational Irrational Numbers

Define real numbers, integers, whole numbers and natural numbers.

ExampleGiven the following numbers:

, , , , .

Classify each number as being rational or irrational.

Order the numbers from lowest to highest.

Locate them on a number line.


Check for understanding

Have the students create a number line with and as the extreme values. The students need to find 7 different numbers with values between those extremes and place them on their number line.



Resources



Lesson 8

Working with Exponent Laws

STAGE 1

BIG IDEA:













KNOWLEDGE:


the exponent laws.

SKILLS:


apply exponent laws.

Lesson Summary

Students will use their knowledge of exponent laws to apply an appropriate and efficient strategy to simplify a variety of expressions.

Lesson Plan



Review adding and subtracting fractions.


Separate the students into groups. Give each group an example of a multi-step question (see examples below). Each group comes up with an approach to simplify each expression, arriving at the correct answer based on the information from the previous lessons. Once groups have verified that their answer is correct, they will ‘teach’ the example to the class. Questions should be given at varying levels to accommodate student levels.

Simplify each of the following:

Check for understanding

Provide students with further examples to try individually, and then go over.

Practice new learning


Resources

Foundations and Pre-calculus Mathematics 10 (Pearson: sec 4.6)Math 10 (McGraw Hill: sec 4.3 and 4.4)


Appendix

Handouts


Simplifying Radicals

Complete the following table.

Entire Radical Form

Prime Factorization Method Factor Form Mixed Radical Form

==

or


Laws of Exponents Extended

Complete the following table (some have been completed for you)

Exponential form

Expanded Form Simplified Form (Single base)

Numerical Value

Laws of Exponents Review

Complete the following table (some have been completed for you)

Exponential form

Expanded Form Simplified Form (Single base)

Numerical Value

ACKNOWLEDGEMENTS

P ages 15 – 22 and 31-32

Wahl, Mark, A Golden Ratio Activity, http://www.markwahl.com/index.php?id=22

Pictures or Digital Images

Page 11

1. http://www.toolsforeducators.com/boardgames/loop.php

2. http://www.zazzle.com/game_board_poster-228628221447007502

3. http://www.toolsforeducators.com/boardgames/loop.php

Pages 17, 26


Pages 18, 27

http://images.smh.com.au/2009/12/22/994337/shania_twain_420-420x0.jpg

Pages 19, 28

http://www.southshields-sanddancers.co.uk/photos_posters/johnny_depp_calendar_photo.jpg

Page 30

Photograph supplied by Jeremy Klassen


http://www.southshields-sanddancers.co.uk/photos_posters/johnny_depp_calendar_photo.jpg

http://images.smh.com.au/2009/12/22/994337/shania_twain_420-420x0.jpg


http://www.toolsforeducators.com/boardgames/loop.php

http://www.zazzle.com/game_board_poster-228628221447007502

http://www.toolsforeducators.com/boardgames/loop.php


Documents

English Language Arts 10 – 2 › 2014 › 01 › … · Web viewcube – the result of a number or term being multiplied by itself twice more. cube root – a number that when