English Language Arts 10 – 2
MATHEMATICS 10C
REAL NUMBERS
High School collaborative venture with
Harry Ainlay, Jasper Place, McNally, Queen Elizabeth,
Ross Sheppard and Victoria Schools
[pic]
Harry Ainlay: David Cunningham, Christine Dibben
Jasper Place: Linda Aschenbrenner, Shelaine Kozakavich, Nic Ryan
Ross Sheppard: Tim Gartke, Jeremy Klassen, Don Symes
Victoria: Kevin Bissoon, Elisha Pinter
Facilitator: John Scammell (Consulting Services)
Editor: Rosalie Mazurok (Contracted)
2009 - 2010
TABLE OF CONTENTS
| | |
|STAGE 1 DESIRED RESULTS |PAGE |
| | |
|Big Idea |4 |
| | |
|Enduring Understandings |4 |
| | |
|Essential Questions |5 |
|Knowledge |6 |
| | |
|Skills |7 |
| | |
|Stage 2 ASSESSMENT EVIDENCE | |
| | |
|Teacher Notes For Transfer Tasks |8 |
| | |
|Transfer Task | |
|A Radical Board Game |9 |
|Teacher Notes for A Radical Board Game and Rubric |10 - 12 |
|Transfer Task |13 - 14 |
|Rubric | |
| | |
|The Golden Ratio in a Face |15 |
|Teacher Notes for The Golden Ratio in a Face and Rubric |16 - 22 |
|Transfer Task |23 - 24 |
|Rubric |25 - 32 |
|Possible Solution | |
| | |
|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 |
[pic] 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.
[pic] 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.
[pic] 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?
****
[pic] Knowledge:
|Enduring |Specific |Knowledge that applies to this Enduring Understanding |
|Understanding |Outcomes | |
| | | |
|Students will understand… | |Students will know… |
| | | |
|numeracy as it relates to real | |that numbers can be ordered. |
|numbers. |*AN 2 |that numbers can be approximated. |
| | |the relationship between sets of numbers (union of sets). |
|that the set of real numbers is | |that the number line is infinitely continuous. |
|continuous and is made up of | | |
|rational and | | |
|irrational numbers. | | |
| | | |
|Students will understand… | |Students will know… |
| |*AN 1 | |
|that there are various ways of |*AN 2 |that numbers can be approximated. |
|representing numbers including |*AN 3 |what an exponent is. |
|exponents, fractions, and radicals. | |what integral and rational exponents mean. |
| | |the exponent laws. |
| | |components of radicals. |
*
[pic] Skills:
|Enduring |Specific |Skills that apply to this Enduring Understanding |
|Understanding |Outcomes | |
| | | |
|Students will understand… | |Students will be able to… |
| | | |
|numeracy as it relates to real | |sort real numbers into categories. |
|numbers. | |approximate irrational numbers. |
| | |order real numbers. |
|that the set of real numbers is |*AN 1 |apply exponent laws. |
|continuous and is made up of |*AN 2 |solve problems involving real numbers. |
|rational and |*AN 3 |solve problems involving real numbers. |
|irrational numbers. | |express radicals in mixed and entire forms and convert between forms. |
| | |express a number as a product of its prime factors. |
| | | |
|Students will understand… | |Students will be able to… |
| | | |
|that there are various ways of | |sort real numbers into categories. |
|representing numbers including | |approximate irrational numbers. |
|exponents, fractions, and radicals. |*AN 1 |order real numbers. |
| |*AN 2 |apply exponent laws. |
| |*AN 3 |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. |
| | | |
| | | |
*
| |
|STAGE 2 Assessment Evidence |
1 Desired Results Desired Results
[pic] 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.
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.
[pic] A Radical Board Game - Student Assessment Task
Task: 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 |Completed |
|Zero Exponent Law |[pic] |[pic] | |
|Product of Powers |[pic] |[pic] | |
|Quotient of Powers |[pic] |[pic] | |
|Power of a Power |[pic] |[pic] | |
|Power of a Product |[pic] |[pic] | |
|Power of a Quotient |[pic] |[pic] | |
|Negative Exponent |[pic] |[pic] | |
|Rational Exponents |[pic] |[pic] | |
|Converting from Mixed to Entire |A radical with a coefficient of 1 |[pic] | |
|Converting from Entire to Mixed |Product of a rational number and a|[pic] | |
| |radical | | |
|Number Line |Placing radicals in order on a | [pic] [pic] [pic] [pic] | |
| |number line |[pic] [pic]| |
|Rational and Irrational Numbers |Real Numbers |Rational: [pic] | |
| | |Irrational: [pic] | |
[pic] Assessment
Mathematics 10C
Real Numbers
Rubric
| | | | | | |
|Level |Excellent |Proficient |Adequate |Limited* |Insufficient / Blank* |
| |4 |3 |2 |1 | |
|Criteria | | | | | |
|Performs Calculations |Performs precise and |Performs focused and |Performs appropriate |Performs superficial |No score is awarded |
| |explicit |accurate |and generally accurate |and irrelevant |because there is no |
| |calculations. |calculations. |calculations. |calculations. |evidence of student |
| | | | | |performance. |
|Presents Data |Presentation of data is |Presentation of data |Presentation of data is|Presentation of data is|No data is presented. |
| |insightful and astute. |is logical and |simplistic and |vague and inaccurate. | |
| | |credible. |plausible. | | |
|Explains Choice |Shows a solution for the|Shows a solution for |Shows a solution for |Shows a solution for |No explanation is |
| |problem; provides an |the problem; provides|the problem; provides |the problem; provides |provided. |
| |insightful explanation. |a logical |explanations that are |explanations that are | |
| | |explanation. |complete but vague. |incomplete or | |
| | | | |confusing. | |
|Communicates findings |Develops a compelling |Develops a convincing|Develops a predictable |Develops an unclear |No findings are |
| |and precise presentation|and logical |presentation that |presentation with |communicated. |
| |that fully considers |presentation that |partially considers |little consideration of| |
| |purpose and audience; |mostly considers |purpose and audience; |purpose and audience; | |
| |uses appropriate |purpose and audience;|uses some appropriate |uses inappropriate | |
| |mathematical vocabulary,|uses appropriate |mathematical |mathematical | |
| |notation and symbolism. |mathematical |vocabulary, notation |vocabulary, notation | |
| | |vocabulary, notation |and symbolism. |and symbolism. | |
| | |and symbolism. | | | |
[pic]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:
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.
[pic] The Golden Ratio in a Face - Student Assessment Task
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
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
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.
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[pic]represented the Golden 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, [pic], the continued fraction looks like...
The nested ratio for [pic] is...
[pic]
Use the pattern for the continued fraction and the nested radical to determine the value of [pic] 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?
[pic] Assessment
Mathematics 10C
Real Numbers
Rubric
| | | | | | |
|Level |Excellent |Proficient |Adequate |Limited* |Insufficient / Blank* |
| |4 |3 |2 |1 | |
|Criteria | | | | | |
|Performs Calculations |Performs precise and |Performs focused and |Performs appropriate |Performs superficial |No score is awarded |
| |explicit |accurate |and generally accurate |and irrelevant |because there is no |
| |calculations. |calculations. |calculations. |calculations. |evidence of student |
| | | | | |performance. |
|Presents Data |Presentation of data is |Presentation of data |Presentation of data is|Presentation of data is|No data is presented. |
| |insightful and astute. |is logical and |simplistic and |vague and inaccurate. | |
| | |credible. |plausible. | | |
|Explains Choice |Shows a solution for the|Shows a solution for |Shows a solution for |Shows a solution for |No explanation is |
| |problem; provides an |the problem; provides|the problem; provides |the problem; provides |provided. |
| |insightful explanation. |a logical |explanations that are |explanations that are | |
| | |explanation. |complete but vague. |incomplete or | |
| | | | |confusing. | |
|Communicates findings |Develops a compelling |Develops a convincing|Develops a predictable |Develops an unclear |No findings are |
| |and precise presentation|and logical |presentation that |presentation with |communicated. |
| |that fully considers |presentation that |partially considers |little consideration of| |
| |purpose and audience; |mostly considers |purpose and audience; |purpose and audience; | |
| |uses appropriate |purpose and audience;|uses some appropriate |uses inappropriate | |
| |mathematical vocabulary,|uses appropriate |mathematical |mathematical | |
| |notation and symbolism. |mathematical |vocabulary, notation |vocabulary, notation | |
| | |vocabulary, notation |and symbolism. |and symbolism. | |
| | |and symbolism. | | | |
[pic]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
[pic]
Studies suggest that Shania Twain may have one of the most beautifully proportioned faces.
a = Top-of-head to chin = 9.1 cm
b = Top-of-head to pupil = 4.5 cm
c = Pupil to nose tip = 1.6 cm
d = Pupil to lip = 2.9 cm
e = Width of nose = 1.5 cm
f = Outside distance between eyes = 4.2 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 = 5.3 cm
c = Pupil to nose tip = 1.7 cm
d = Pupil to lip = 2.8 cm
e = Width of nose = 1.2 cm
f = Outside distance between eyes = 3.6 cm
Place the measurements for Shania Twain and Johnny Depp here, and calculate the ratios.
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.
[pic]
a = Top-of-head to chin = 7.8 cm
b = Top-of-head to pupil = 3.2 cm
c = Pupil to nose tip = 1.5 cm
d = Pupil to lip = 2.7 cm
e = Width of nose = 1.7 cm
f = Outside distance between eyes = 3.5 cm
g = Width of head = 5.4 cm
h = Hairline to pupil = 2.7 cm
i = Nose tip to chin = 3.1 cm
j = Lips to chin = 1.9 cm
k = Length of lips = 2.1 cm
I = Nose tip to lips = 1.2 cm
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.
| |
|STAGE 3 Learning Plans |
Lesson 1
Factors and Multiples
|[pic]STAGE 1 |
| |
|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: |ESSENTIAL QUESTIONS: |
| | |
|The students will understand… |When and why should we use exact values? |
| | |
|numeracy as it relates to real numbers. |What is a real number? |
|that the set of real numbers is continuous and is made up of rational|What are the different ways of representing real numbers? |
|and irrational numbers. |How can real numbers be classified? |
|there are various ways of representing numbers including exponents, |What strategies can you use to order real numbers appropriately? |
|fractions, and radicals. |. |
| | |
|KNOWLEDGE: |SKILLS: |
| | |
|Students will know… |Students will be able to… |
| | |
|that numbers can be ordered. |express a number as a product of its prime factors. |
|that numbers can be approximated. | |
[pic]Lesson Summary
Students will review whole number factors and multiples. (Note: these were last addressed in grade 6.)
[pic] 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.
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.
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).
[pic] Going Beyond
Strong students should be given a larger set of numbers (find GCF and LCM of 3 or 4 numbers)
[pic]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
[pic]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
Lesson 2
Square Roots and Cube Roots
|[pic]STAGE 1 |
| |
|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: |ESSENTIAL QUESTIONS: |
| | |
|The students will understand… |When and why should we use exact values? |
| | |
|numeracy as it relates to real numbers. |What is a real number? |
|that the set of real numbers is continuous and is made up of rational|What are the different ways of representing real numbers? |
|and irrational numbers. |How can real numbers be classified? |
|that there are various ways of representing numbers including |What strategies can you use to order real numbers appropriately? |
|exponents, fractions, and radicals. |. |
| | |
|KNOWLEDGE: |SKILLS: |
| | |
|Students will know… |Students will be able to… |
| | |
|components of radicals. |solve problems involving real numbers. |
| |Identify square roots, cube roots, perfect squares and perfect cubes.|
| | |
[pic]Lesson Summary
Students will identify perfect squares and perfect cubes, and then determine square roots and cube roots.
[pic] 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 4
o 6
o 8
o 9
o 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 12
o 18
o 25
o 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: [pic]
[pic]
[pic]
Check for Understanding
Ask students to identify perfect squares and square roots of larger numbers without the use of algebra tiles.
Practise New Learning
Assign selected exercises from text.
[pic]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 tiles
snap cubes
interactive whiteboard or a projector
[pic]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
|[pic]STAGE 1 |
| |
|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: |ESSENTIAL QUESTIONS: |
| | |
|The students will understand… |When and why should we use exact values? |
| | |
|numeracy as it relates to real numbers. |What is a real number? |
|that the set of real numbers is continuous and is made up of rational|What are the different ways of representing real numbers? |
|and irrational numbers. |How can real numbers be classified? |
|that there are various ways of representing numbers including |What strategies can you use to order real numbers appropriately? |
|exponents, fractions, and radicals. | |
| |What is the meaning of continuous? |
| |When are there gaps in a number line? |
| |How many numbers are there? |
| |. |
| | |
|KNOWLEDGE: |SKILLS: |
| | |
|Students will know… |Students will be able to… |
| | |
|that numbers can be ordered. |approximate irrational numbers. |
|that numbers can be approximated. |express radicals in mixed and entire forms and convert between forms.|
|components of radicals. |express a number as a product of its prime factors. |
| | |
[pic]Lesson Summary
Students will estimate and order radicals (using a number line).
[pic] Lesson Plan
Activate Prior Knowledge/Experience
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.
[pic]
Present new learning
Discussion of what is happening between the values above.
What methods can we use to figure out where [pic] would go?
Try to place on a number line: [pic]
Practice new learning
Assign selected exercises from text.
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).
[pic]Resources
Math 10 (McGraw Hill: sec 4.4)
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 4.1)
Lesson 4
Working with Radicals
|[pic]STAGE 1 |
| |
|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: |ESSENTIAL QUESTIONS: |
| | |
|The students will understand… |When and why should we use exact values? |
| | |
|numeracy as it relates to real numbers. |What is a real number? |
|that the set of real numbers is continuous and is made up of rational|What are the different ways of representing real numbers? |
|and irrational numbers. |How can real numbers be classified? |
|that there are various ways of representing numbers including |What strategies can you use to order real numbers appropriately? |
|exponents, fractions, and radicals. |. |
| | |
|KNOWLEDGE: |SKILLS: |
| | |
|Students will know… |Students will be able to… |
| | |
|components of radicals. |express radicals in mixed and entire forms and convert between forms.|
| |express a number as a product of its prime factors. |
| | |
[pic]Lesson Summary
Students will learn to convert between mixed and entire radicals.
[pic] Lesson Plan
Activate Prior Knowledge/Experience
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.
Present New Learning
Expand the definition of a radical to include numbers other than 2 or 3 as your index.
Estimate the value of[pic]. Estimate the value of[pic]. What do you notice? Why is that?
Method 1
Prime factorization
[pic]
Method 2
What are the factors of 18?
[pic]
[pic]
[pic]
Discuss that the second option includes [pic] .
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.
Practise new learning
Assign selected exercises from text.
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.
[pic] Going Beyond
Try simplifying radicals with indexes equal to 4 and 5. Discuss [pic]
[pic]Resources
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 4.3)
Math 10 (McGraw Hill: sec 4.4)
[pic]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 is
under the radical
Lesson 5
Rational Exponents
|[pic]STAGE 1 |
| |
|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: |ESSENTIAL QUESTIONS: |
| | |
|The students will understand… |When and why should we use exact values? |
| | |
|numeracy as it relates to real numbers. |What is a real number? |
|that the set of real numbers is continuous and is made up of rational|What are the different ways of representing real numbers? |
|and irrational numbers. |How can real numbers be classified? |
|that there are various ways of representing numbers including |What strategies can you use to order real numbers appropriately? |
|exponents, fractions, and radicals. |. |
| | |
|KNOWLEDGE: |SKILLS: |
| | |
|Students will know… |Students will be able to… |
| | |
|what an exponent is. |apply exponent laws. |
|what integral and rational exponents mean. |solve problems involving real numbers. |
|the exponent laws. | |
[pic]Lesson Summary
Students will review the exponent laws from math 9 and expand their knowledge set to include rational exponents.
[pic] 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
Diagnostic Assessment of Prior Knowledge
Discuss results of worksheet as a class to ensure everyone came to the same conclusions.
Present New Learning
Use the exponent laws to have students discover the value of [pic] and [pic].
Example #1
a) [pic]
What must [pic] be equal to so that the number multiplied by itself is 4?
b) [pic]
What must [pic] be equal to so that the number multiplied by itself is 9?
What do you think the value of [pic] is? Check your answer with a graphing calculator.
Example #2
a) [pic]
What must [pic] be equal to so that the number multiplied by itself three times is -8?
b) [pic]
What must [pic] be equal to so that the number multiplied by itself three times is 27?
What do you think the value of [pic] 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) [pic] (b) [pic] (c) [pic] (d) [pic]
(e) [pic] (f) [pic] (g) [pic] (h) [pic]
Provide example with variable bases.
Practise New Learning
Assign selected exercises from text.
[pic]Resources
Math 10 (McGraw Hill: sec 4.3)
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 4.4)
Lesson 6
Negative Exponents
|[pic]STAGE 1 |
| |
|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: |ESSENTIAL QUESTIONS: |
| | |
|The students will understand… |When and why should we use exact values? |
| | |
|numeracy as it relates to real numbers. |What is a real number? |
|that the set of real numbers is continuous and is made up of rational|What are the different ways of representing real numbers? |
|and irrational numbers. |How can real numbers be classified? |
|that there are various ways of representing numbers including |What strategies can you use to order real numbers appropriately? |
|exponents, fractions, and radicals. |. |
| | |
|KNOWLEDGE: |SKILLS: |
| | |
|Students will know… |Students will be able to… |
| | |
|what an exponent is. |apply exponent laws. |
|what integral and rational exponents mean. |solve problems involving real numbers. |
|the exponent laws. | |
[pic]Lesson Summary
Students will be able to simplify expressions with negative exponents using reciprocals.
[pic] Lesson Plan
Activate Prior Knowledge/Experience
Review dividing fractions (multiplying by reciprocal of the divisor).
Diagnostic Assessment of Prior Knowledge
Quick check from previous lesson prior to starting (provide students with power expressions with rational exponents).
Present new learning
Investigate.
Given [pic]
• Simplify using factorization:
[pic]
• Simplify using exponent laws.
[pic]
• Can we say that[pic]?
Try simplifying the following using both methods above:
(a) [pic] (b) [pic] (c) [pic]
Use the pattern you see above to evaluate the following:
a) [pic] (b) [pic] (c) [pic]
(d) [pic] (e) [pic] (f) [pic]
Express the rule(s) you used as a general statement.
Provide examples with variable bases.
Practise New Learning
Assign selected exercises from text.
[pic]Resources
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 4.5)
Math 10 (McGraw Hill: sec 4.2)
Lesson 7
Irrational Numbers – Classifying and Ordering
|[pic]STAGE 1 |
| |
|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: |ESSENTIAL QUESTIONS: |
| | |
|The students will understand… |When and why should we use exact values? |
| | |
|numeracy as it relates to real numbers. |What is a real number? |
|that the set of real numbers is continuous and is made up of rational|What are the different ways of representing real numbers? |
|and irrational numbers. |How can real numbers be classified? |
|there are various ways of representing numbers including exponents, |What strategies can you use to order real numbers appropriately? |
|fractions, and radicals. | |
| |What is the meaning of continuous? |
| |When are there gaps in a number line? |
| |How many numbers are there? |
| |. |
| | |
|KNOWLEDGE: |SKILLS: |
| | |
|Students will know… |Students will be able to… |
| | |
|that numbers can be ordered. |sort real numbers into categories. |
|that numbers can be approximated. |approximate irrational numbers. |
|the relationship between sets of numbers (union of sets). |order real numbers. |
|that the number line is infinitely continuous. | |
|what an exponent is. | |
[pic]Lesson Summary
Students will practise classifying and ordering rational and irrational numbers presented in different forms.
[pic] Lesson Plan
Activate Prior Knowledge / Experience
Review of ordering numbers on a number line.
Present New Learning
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 |
| | |
|[pic] [pic] [pic] [pic] [pic] |[pic] [pic] [pic] [pic] |
| | |
|[pic] [pic] [pic] |[pic] [pic] |
| | |
Define real numbers, integers, whole numbers and natural numbers.
Example
Given the following numbers:
[pic], [pic], [pic], [pic], [pic].
• 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 [pic] and [pic] as the extreme values. The students need to find 7 different numbers with values between those extremes and place them on their number line.
Practise new learning
Assign selected exercises from text.
[pic]Resources
Math 10 (McGraw Hill: sec 4.4)
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 4.2)
Lesson 8
Working with Exponent Laws
|[pic]STAGE 1 |
| |
|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: |ESSENTIAL QUESTIONS: |
| | |
|The students will understand… |When and why should we use exact values? |
| | |
|numeracy as it relates to real numbers. |What is a real number? |
|that the set of real numbers is continuous and is made up of rational|What are the different ways of representing real numbers? |
|and irrational numbers. |How can real numbers be classified? |
|that there are various ways of representing numbers including |What strategies can you use to order real numbers appropriately? |
|exponents, fractions, and radicals. |. |
| | |
|KNOWLEDGE: |SKILLS: |
| | |
|Students will know… |Students will be able to… |
| | |
|the exponent laws. |apply exponent laws. |
[pic]Lesson Summary
Students will use their knowledge of exponent laws to apply an appropriate and efficient strategy to simplify a variety of expressions.
[pic] Lesson Plan
Activate Prior Knowledge/Experience
Review adding and subtracting fractions.
Present new learning
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:
[pic] [pic] [pic] [pic] [pic]
[pic] [pic] [pic] [pic]
Check for understanding
Provide students with further examples to try individually, and then go over.
Practice new learning
Assign selected exercises from text.
[pic]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 |
|[pic] |[pic] |[pic] | |
| |=[pic] |or |[pic] |
| |=[pic] |[pic] | |
|[pic] | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
|[pic] | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| |[pic] | | |
| | | | |
| | | | |
| | | | |
| | | | |
Laws of Exponents Extended
Complete the following table (some have been completed for you)
|Exponential form |Expanded Form |Simplified Form (Single |Numerical Value |
| | |base) | |
|[pic] |[pic] | |[pic] |
|[pic] | | | |
|[pic] | | | |
|[pic] | | | |
|[pic] |[pic] | | |
|[pic] | | | |
|[pic] | | | |
Laws of Exponents Review
Complete the following table (some have been completed for you)
|Exponential form |Expanded Form |Simplified Form (Single |Numerical Value |
| | |base) | |
|[pic] |[pic] | |[pic] |
|[pic] | | | |
| |[pic] | | |
|[pic] |[pic] |[pic] |[pic] |
|[pic] | | | |
|[pic] | | | |
|[pic] |[pic] |[pic] |[pic] |
|[pic] | | | |
| |[pic] | | |
|[pic] |[pic] |[pic] |[pic] |
|[pic] | | | |
|[pic] | | | |
|[pic] |[pic] |[pic] |[pic] |
|[pic] | | | |
|[pic] |[pic] |[pic] |[pic] |
|[pic] | | | |
ACKNOWLEDGEMENTS
Pages 15 – 22 and 31-32
Wahl, Mark, A Golden Ratio Activity,
Pictures or Digital Images
Page 11
1.
2.
3.
Pages 17, 26
Pages 18, 27
Pages 19, 28
Page 30
Photograph supplied by Jeremy Klassen
-----------------------
[pic] 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, [pic]means divide measurement [pic] by measurement[pic]; round your answers to 3 decimal places.
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.
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[pic], whose value of that is [pic]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
• 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.
[pic]
[pic]
[pic]
Lips
Chin
Lips
Nose Tip
Pupils
Hairline
Top of the Head
Eye
Nose
Eye
[pic]
[pic]
[pic]
0
1
Implementation note:
Teachers need to consider what performances and products will reveal evidence of understanding?
What other evidence will be collected to reflect
the desired results?
[pic]
[pic]
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
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
Shania
Johnny
Average ratio = __________
Average ratio = __________
[pic]
[pic]
[pic]
0
1
Average ratio = __________
Your name here: ____________________
[pic]
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 [pic]means divide measurement [pic] by measurement[pic]; round your answers to 3 decimal places.
[pic]
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.
*AN = Algebra and Number
*AN = Algebra and Number
Head Width
Implementation note:
Students must be given the transfer task & rubric at
the beginning of the unit. They need to know how
they will be assessed and what they are working
toward.
[pic]
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (). Some terms can be found in more than one division. Some terms have animations to illustrate meanings.
[pic]
[pic]
Implementation note:
Ask students to consider one of the essential questions every lesson or two.
Has their thinking changed or evolved?
Implementation note:
Post the BIG IDEA in a prominent
place in your classroom and refer to it often.
Implementation note:
Teachers need to continually ask
themselves, if their students are acquiring the knowledge and skills needed for the unit.
[pic]
Eye
Chin
Lips
Nose Tip
Pupils
Hairline
Top of the Head
Lips
Nose
Eye
Head Width
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
Top of the head
Hairline
Pupils
Tip of the Nose
Lips
Chin
Mouth
Nose
Eye
Eye
Head width
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
Average ratio = __________
Average ratio = __________
Johnny Depp
Shania Twain
9.9
g = Width of head = 5.6 cm
h = Hairline to pupil = 2.9 cm
i = Nose tip to chin = 2.9 cm
j = Lips to chin = 1.7 cm
k = Length of lips = 2.2 cm
I = Nose tip to lips = 1.1 cm
Implementation note:
Teachers need to consider what performances and products will reveal evidence of understanding?
What other evidence will be collected to reflect
the desired results?
Eye
g = Width of head = 6.2 cm
h = Hairline to pupil = 3.1 cm
i = Nose tip to chin = 3.0 cm
j = Lips to chin = 1.6 cm
k = Length of lips = 2.9 cm
I = Nose tip to lips = 1.4 cm
Eye
Chin
Lips
Nose Tip
Pupils
Hairline
Top of the Head
Lips
Nose
Eye
Head Width
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
Average ratio = __________
3.
[pic]
[pic]
Average ratio = __________
Your name here: _Myself__
[pic]
[pic]
2.
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, [pic], the continued fraction looks like...
[pic]
The nested ratio for [pic] is...
[pic]
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 – 1.468
Shania – 1.590
Johnny – 1.597
Phi – 1.618
Statue – 1.641
Now with reference to Phi I will calculate the percentage error from the exact value for everyone.
Me: [pic]
Shania: [pic]
Johnny: [pic]
Statue: [pic]
1.
Chin
Lips
Nose Tip
Pupils
Hairline
Top of the Head
Lips
Nose
Eye
Head Width
................
................
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