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Mathematics 8: Unit 1 1 Teacher’s Guide

UNIT 1 TEACHER’S GUIDE

Unit 1 Introduction

Folder

The student may need direction on making a folder to hold their work for this unit. The student may be using an electronic folder, a physical folder, or a combination of both depending on teacher preference.

If the teacher decides to give a unit test, there is one already prepared in the Black Line Masters of the MathLinks 8 Teacher’s Resource from McGraw-Hill Ryerson.

Some of the Learn Alberta multimedia pieces in the lessons have written resources for extra practice or reinforcement.

For Lesson 4, the resource can be found at

http://www.learnalberta.ca/content/mejhm/html/object_interactives/square_roots/flashHelp/pdf/SquareRootsPrintActivity.doc

Note that question 7.b. and 7.d. should perhaps be removed and/or postponed until after integers have been discussed in Unit 2.

For Lesson 5, the resource can be found at

http://www.learnalberta.ca/content/mejhm/html/video_interactives/squareroots/printActivities.doc

Note that question 6 refers to the Video Interactive applet screen, and content from there is needed to complete that question.

There are extra-practice exercises for each test section already prepared in the Black Line Masters of the MathLinks 8 Teacher’s Resource from McGraw-Hill Ryerson. You may choose to use parts of them as needed for some students or save them for review at the end of the unit.

Discuss and Share

Before the student begins this course, you may want to set up an online-course discussion area to accommodate the social dimension of student learning. Making a table formatted to show the title or subject of the discussions, and the lesson the discussions appear in, will help you in your preparations.


Parts of the Lessons

Get Focused

This is the first major part of lesson that introduces the lesson and focuses the unit inquiry to level of lesson. This section includes lesson introduction, objectives, list of materials needed, critical question, and list of assessments.

Explore

A major part of lesson that presents a somewhat open-ended problem during which time the student (and partners) struggle with concepts with minimal direct teaching. The student constructs his or her own meaning in an environment of social interaction.

Connect

Third major part of lesson in which understandings developed in the Explore part are verified, consolidated, extended, and applied.

Going Beyond

The Going Beyond provides the student a choice of challenging and enriching his/her knowledge and skills beyond the learning objectives of the lesson.

Lesson Summary

Fourth major part of lesson in which the content and concepts developed in the lesson are summarized and in which the answer to the essential question posed in the Get Focused part is answered.

Read

This is a part of the lesson that directs student to an external text-based medium. In Math 8, this will consist only of readings from the authorized textbook.

Self-Check

A non-graded question with the answer provided in the lesson that is checked by the student. The answer will appear in a drop-down box within the lesson that is accessed by choosing the “Compare your answers” link.

Try This

A question or task that is to be completed without the provision of a model answer; that is, the answer to this question is contained only in the teacher guide.

The Try This questions or tasks may benefit from teacher marking and used for formative assessment. The teacher and the student should arrange prior to beginning the lesson to determine which Try This questions will be graded for assessment.


My Guide

This serves as guidance during a task by providing a hint or some questions to consider (along with answers). This will appear in a drop-down box within the lesson and can be accessed when/if student chooses.

Discuss and Share

This directs students to exchange ideas and knowledge with others and to think about, and respond to, what has been presented in this exchange. Typically, the student will be instructed to submit one or more Try This questions in a lesson to other students for input and comparison.

It may involve the submission of evidence of such interaction for teacher marking.

Extra Practice

This is a set of optional questions or tasks to reinforce lesson concepts and skills.

Assignment

This presents a set of questions that are to be completed by the student and submitted for marks. It appears in the lesson as a Word document for the student to complete.

Watch and Listen

This heading directs the student to multimedia content that may be passive and interactive (e.g., podcasts, videos, or interactive Flash activities).

About Group Work

In the beginning lessons of the course, the student is directed to ask his or her teacher about working with a partner. Expect the student to ask you about this. You may have designed unique arrangements for the student to do group work.

In subsequent lessons, the student will be directed to work with a partner, if possible. But the understanding can be that group work will be according to your directions.

About the Math 8 Folder

The student’s work placed in the Math 8 folder serves as a student portfolio. The student may need directions for using the Math 8 folder. The folder should be stored in an accessible location that you choose. The folder may be a physical folder or an electronic one.

You may ask the student to notify you whenever he or she adds work to this folder. Then you can check the submission. The submission can be used to inform you as to student progress. You may find evidence of misconceptions that indicate some scaffolding is appropriate. Alternately, you may decide to mark the student’s work as part of the course assessment.


About the Unit 1 Summary

In each unit summary the student is directed to check with the teacher about a unit test. Chapter tests are available from the folder “Chapter BLMs” on the MathLinks 8 Teacher’s Guide CD. You may decide to use a chapter test, use another assessment instrument you have developed for a unit test, or not administer a test at the end of each unit.

It may be helpful for you to indicate your plan for unit tests to the student.

Lesson 1: Square Numbers and Area Models

TT 1.

1. The rectangles have areas of 15 cm2, 16 cm2, 9 cm2, 12 cm2, and 36 cm2.

2. a. Squares can be made from the tiles used in the 16 cm2, 9 cm2, and 36 cm2 rectangles.

b. The side lengths are 4 cm, 3 cm, and 6 cm respectively.

c. Each square’s area is the side length multiplied by itself.

3. a. Examples of perfect squares would include 4, 64, and 81. Examples of non-perfect squares would include 5, 12, and 75.

b. 4 = 2 × 2, 64 = 2 × 2 × 2 × 2, and 81 = 3 × 3 × 3 × 3

5 = 5, 12 = 2 × 2 × 3, and 75 = 3 × 5 × 5

c. In 4, 2 occurs two times.

In 64, 2 occurs four times.

In 81, 3 occurs four times.

In 5, 5 occurs once.

In 12, 2 occurs once, and 3 occurs two times.

In 75, 3 occurs once, and 5 occurs two times.

The comparisons will vary depending on which numbers were chosen. 4. a. The prime factors of perfect squares each occur an even number of times.

b. At least one of the prime factors of non-perfect squares occurs an odd number of times.


5. Answers will vary; sample answers are given.

a. Use the tiles needed to cover the area to try to make a square. If you succeed, the number is a perfect square.

b. You can use the number of prime factors to see if a given number is a perfect square. If all prime factors occur and even number of times, the number is a perfect square.

TT 2. It is not possible to make square shapes on the grid paper for numbers 12 and 15.

TT 3. It is possible to make square shapes on the grid paper for numbers 9, 16, and 36.

TT 4. Multiplying the side length by itself gives back the number you started with.

TT 5. Students should relate a square shape and its side length to a square number. Answers will vary:

• A student could have given 49 tiles as the number. With a side length of 7, you would use 7 times 7 tiles. So you would use 49 tiles.

• A student could have given 64 tiles as a possibility. There is a number, 8, that can be multiplied by itself to make 64. So a square shape with a side length of 8 would be put together with 64 tiles.

TT 6. Yes, because 1 = 1 × 1, so 1 is a product of the whole number 1 multiplied by itself. (The student may also have given the following answer: Yes, because the unit tile by itself can make up a square shape on a grid.)

TT 7.

a. The student may have provided either of the following answers:

• Place the right number of unit tiles on the grid to represent the area and arrange the tiles into a square shape. Count the number of squares along a side to get the side length.

• Find a factor that you multiply by itself to get the number for the area. That factor is the side length.

b. The student may have provided either of the following answers:

• The perimeter is 4 times the side length. • Multiply this factor by 4 to get the perimeter of the area.


Unit 1: Lesson 1 Question Set

1.

The number 12 is not a perfect number.

None of the possible rectangles with an area of 12 unit squares are square, so 12 is not a perfect square.(4 marks)


2.

It is possible to draw a square having an area of 49 square units on the grid. The number 49 is a perfect square.(3 marks)


3.

A square with an area of 169 tiles has 13 tiles along each edge.(3 marks)

4. The numeral 196 is a square number. 196 = 14 × 14

Alternately, a student may draw a square measuring 14 units by 14 units to show that 196 is a square number.(2 marks)


5. a. The number 64 is the largest perfect square less than 65.

So, the largest area of the square kennel is 64 m2.(2 marks)

b.

(1 mark)

c. The side length of the square kennel is 8 m.

The length of fence along the perimeter of the kennel is 32 m.(2 marks)

Lesson 2: Squares and Square Roots

TT 1. The list of factors can tell you if a number is a prime or a composite and whether it is a perfect square or not.

TT 2. A number that only has two factors is a prime number.

TT 3. A number that has an odd number of factors is a square number (or a perfect square).

TT 4. A number that has any even number of factors is a number that is not a square number (or not a perfect square).

TT 5. The following are inverse operations:

• multiplication and division• addition and subtraction

TT 6. Factors come in pairs, but in only one of the pairs does a factor occur twice. So only this one factor can be multiplied by itself to produce the perfect square. Therefore, this one factor is the only factor that is the square root of the perfect square.



1. 144 ÷ 12 = 12 (2 marks)

2. 576 ÷ 24 = 24 (2 marks)

3. The square of is 16. That is because = 4 and 42 = 16. Squaring and taking the square root are inverse operations.

16 square units

4

4

(2 marks)

4. The square root of 1212 is 121. That’s because taking the square root undoes the operation of squaring. (2 marks)

5. C (2 marks)

6. The factors are 1, 2, 4, 7, 14, 28, 49, 98, and 196.

There are nine factors. The middle factor is the square root. Therefore, the square root of 196 is 14.(3 marks)


7.

A square with an area of 361 tiles has a side length of 19 units.

So the square root of 361 is 19.(3 marks)

8. number = 48

Factor Pairs1 482 243 164 126 88 612 424 216 348 1

The number 48 is not a perfect square because none of the pairs have a repeated factor.(3 marks)


9. a. The patio is square so 9801 must be a perfect square. Perfect squares have an odd number of factors.

or

The number 9801 has 15 factors:

1, 3, 9, 11, 27, 33, 81, 99, 121, 297, 363, 891, 1089, 3267, 9801

This is an odd number of factors.(1 mark)

b. The number 9801 is a square number. A number that has an odd number of factors is a square number.

or

The patio is square so 9801 must be a square number.(1 mark)

c. The middle factor in the list of ascending factors is 99. This is the square root of 9801.

So, 99 × 99 = 9801.

There must be 99 tiles along one side of the patio.(1 mark)

Lesson 3: Measuring Line Segments

TT 1. From the diagram, you can see that the side of the square is the hypotenuse of a right triangle with side lengths of 2 and 5.

The square has an area of 29 square units.

TT 2.

Square Area Side Lengthsquare A 9 unit squares 3 unitssquare B 29 unit squares unitssquare C 49 unit squares 7 units


TT 3. The side length of a square is the square root of the area of that square.

TT 4. Either strategy may be preferred; the important thing is that the student put the reason for the preference into words.

TT 5. No, not all squares have an area equal to a perfect square number. Squares drawn on a grid that do not have their sides drawn on the grid lines may not have an area equal to a perfect square number. For example, a diamond-shaped square with 6 units across and 6 units high has an area of 18 square units. The number 18 is not a perfect square.

width = 6 units

height = 6 units

TT 6. When the area of the square drawn on the line segment is not a perfect square, the length of the line segment is not a whole number.


1.

Its side length is 11 cm.(2 marks)

2.

Its side length is cm.(2 marks)


3.

The area of the square is 115 m2.(2 marks)

4. a. There are several possible solutions. Following are two of the possible solutions:

Similar coloured triangles add together to make a rectangle.

area of centre square: 7 units × 7 units = 49 square unitsarea of red area rectangle: 5 units × 12 units = 60 square unitsarea of yellow area rectangle: 5 units × 12 units = 60 square unitstotal area of copied large square: = 169 square units


The area of the enclosing square is 172 square units, which equals 289 square units.

There are four congruent rectangles outside of the angled square and within the enclosing square. Each of these triangles has an area of 30 square units. The combined area of these 4 triangles is 120 square units. The area of the angled square is equal to 289 square units minus 120 square units, which equals 169 square units.

So the area of the copied square is 169 square units(6 marks)

b.

The side length of the copied square is 13 units.(2 marks)

Lesson 4: Estimating Square Roots

TT 1.

1. Any number between 36 and 49 would be acceptable, but one nearer 42 seems likely to be the choice students would make.


2. The smallest mat has a side length of

The largest mat has a side length of

3.

4. The middle mat would likely have a side length of around 6.5 m.

5. a. This will vary with the estimates other students have made.

b.

c. Student answers will vary. A student might, for example, say that you choose a number part way between the bracketing perfect squares. That is, if you need an estimate for the square root of 79, you’d look at the square roots of 64 and 81 to give guidance.

TT 2.

0 1 2 3 4 5 6

2 1.4 11 3.3 24 4.95 2.2 18 4.2

TT 3. Various strategies could be used. Some sample strategies follow:

• Using the tool “Explore Square Roots,” the student may have dragged the red circle to create a square having an area equal to the number on the square root sign. The square root value could have been estimated from the side length of the square.

• The student could have tried to draw on grid paper a square having an area equal to the number on the square root sign. The square root value could have been estimated from the side length of the square.

• The student could have considered the nearest lower and higher exact square root to the given square roots as benchmarks and used these to estimate and place the given square root.

TT 4. You could multiply an estimate by itself—that is, square it—and see how close the product is to the number under the square root sign. Or you could use the square root button on the calculator.



1. a. 36 < 38 < 496 < < 7

is between the whole numbers 6 and 7(2 marks)

b.25 < 35 < 365 < < 6

is between the whole numbers 5 and 6(2 marks)

c. 16 < 21 < 254 < < 5The square root is between the whole numbers 4 and 5.(2 marks)

2.1 < 2 < 41 < < 2

So placed between 1 and 2 is correct.

18 > 16> 4

So should be placed to the right of 4 (rather than to the right of 3).

36 < 38 < 496 < < 7

So placing between 6 and 7 and closer than 7 is correct. (5 marks)


3. a. The side length of the square is m.

80 < 90 < 100

So 9 < < 10

Guess and check:

The side length of the square flower bed is approximately 9.49 m.(5 marks)

b. The side length of the square is m.

The side length of the square flower bed is approximately 9.486832981 m.(2 marks)

Lesson 5: The Pythagorean Theorem

Watch and Listen

The “Math Continuum” program “Pythagoras Theorem” had a few errors. The errors may have been corrected by the time the student encounters them. If the errors are not yet corrected, be aware of them in order to help your students past them.

• For A6 B3 Scene 4, the student needs to be able to rotate the ruler or be told to estimate the length of the hypotenuse and the horizontal side instead of measuring them.

• For A6 B3 Scene 5 last frame, the "length" of the hypotenuse should be asked for, instead of the "square" of the hypotenuse (twice). This matches the correct answer keyed in.

• In A6 B6 Scene 2, d2 should be d2.


TT 1. Questions 1, 2, and 3 have to be assumed to have been done. Questions 4, 5, 6, and 7 are shown in the following table.

Side Side LengthAngle

Opposite the Side

Area of Square

Right Triangle?

(Yes or No)

Triangle 1a 6 37 36

Yesb 8 53 64c 10 90 100

Triangle 2a 5 29 25

Nob 7 41 49c 10 110 100

Triangle 3a 5 23 25

Yesb 12 57 144c 13 90 169

TT 2. The hypotenuse must always be c.

TT 3. No, the Pythagorean theorem does not work for triangles other than right triangles.

TT 4. We can add the side lengths without squaring them first, but the number we get will not work in the Pythagorean relationship and will not help.

TT 5. Answers may vary. Sample: The area of the square on the longest side of a right triangle equals the number you get when you add the areas of the squares on the other two sides.


1. a. Triangle H is a right triangle. (1 mark)

b. If the theorem of Pythagoras holds for the triangle, then it is a right triangle. The sum of the area of the squares on the two shorter sides must be equal to the area of the square on the hypotenuse if it is a right triangle.

15 cm2 + 35 cm2 = 50 cm2

(2 marks)


2. a. Triangle A is not a right triangle. The square of the two shorter sides is not equal to the square of the hypotenuse.

22 + 42 ≠ 52, 4 + 16 ≠ 25(2 marks)

b. Triangle B is a right triangle. The square of the two shorter sides is equal to the square of the hypotenuse.

82 + 152 = 172, 64 + 225 = 169(2 marks)

c. Triangle C is not a right triangle. The square of the two shorter sides is not equal to the square of the hypotenuse.

102 + 122 ≠ 152, 100 + 144 ≠ 225(2 marks)

d. Triangle B is a right triangle. The square of the two shorter sides is equal to the square of the hypotenuse.

102 + 242 = 262, 100 + 576 = 676(2 marks)

3. a. The area of square Q is 25 square units. (1 mark)

b. The area of square R is 34 square units. (2 marks)

c. The area of square R was predicted using the Pythagorean relationship (or the Pythagorean theorem). (1 mark)

4. a. B (1 mark)

b. According to the theorem of Pythagoras,

a2 + b2 = c2

42 + b2 = 192

16 + b2 = 361b2 = 361 − 16b2 = 345 cm2

The area of square M is 345 cm2.

The student’s answer need not be as complete as this for full marks, but communication of thought and following conventions is essential.

Note: If part 4.a. is answered wrong, part marks may still be given in 4.b. so that a student is not penalized twice for the same mistake.(3 marks)


Lesson 6: Exploring the Pythagorean Theorem

Watch and Listen

The “Math Continuum” program “Pythagoras Theorem” had a few errors. The errors may have been corrected by the time the student encounters them. If the errors are not yet corrected, be aware of them in order to help the student past them.

• For A6 B3 Scene 4, students need to be able to rotate the ruler or be told to estimate the length of the hypotenuse and the horizontal side instead of measuring them.

• For A6 B3 Scene 5 last frame, the "length" of the hypotenuse should be asked for, instead of the "square" of the hypotenuse (twice). This matches the correct answer keyed in.

• In A6 B6 Scene 2, d2 should be d2.

TT 1.

1. Answers may vary. A sample answer follows:

2. Answers may vary. A sample answer follows:

You can measure the hypotenuse of the triangle directly, or you can use the Pythagorean relationship.


3. a. A sample answer follows:

If the triangle is very large, it might not be possible to measure the hypotenuse directly. If the length of the legs is known, you could use the Pythagorean relationship to find the length of the hypotenuse.

b. 38.2 m

TT 2. a2 + b2 = c2

TT 3. Take the square root of the area of the unknown square.

TT 4. Add when the unknown side is the hypotenuse, and subtract when the unknown side is one of the legs of the triangle.

SC 2. This is information that may help the student after he or she has completed SC 2. Here are a few simple Pythagorean triples:

3, 4, 55, 12, 138, 15, 177, 24, 2512, 35, 37

Any of these triples multiplied by an integer will also be a Pythagorean triple.



1.

The length of the hypotenuse would be 17 m.(4 marks)

2. a.

The height of the truss is 120 cm.(4 marks)

b. Yes, the measurements form a Pythagorean triplet because all three of the numbers are whole numbers.(2 marks)

3. a.

The side length of the lawn area is 50 m.(2 marks)


b.

The length of the diagonal sidewalk would be 70.7 m.(4 marks)

c. No, the measurements do not form a Pythagorean triplet because all three of the numbers are not whole numbers. (2 marks)

Lesson 7: Applying the Pythagorean Theorem

TT 1.

1. You would expect the students to do a calculation similar to the following.

2. a. Answers will vary. The student will likely say something like without corner or straight.

b. The student will likely do a calculation like the following.

400 m + 600 m – 721.1 m = 278.9 m

3. Answers will vary. The student will likely mention the houses and other buildings that get in the way of a direct measurement.


TT 2.

The ship travelled 22.9 km north.

TT 3. The distance north is a leg of the right triangle, so subtraction is needed.

or

The unknown variable in the equation is not isolated, so subtraction is needed.

TT 4. The navigator could use GPS (global positioning satellite) information, or he could use radar if the coastline was visible on his screen.


1.

The bottom of the diagonal brace should be approximately 2.73 m from the wall. (4 marks)


2.

25 km

95 km

?

The plane has travelled approximately 98.2 km from the take-off point.(4 marks)

3.

The height of the stairs is 5.7 m.(4 marks)


4.

The screen is approximately 43-cm wide.(4 marks)

5.

The length of each side of the roof truss will be approximately 2.46 m.(5 marks)

Unit 1 Summary

TT 1. One way of marking the project would be to use the Master 1 Project Rubric found in the MHR Teacher’s Resource in the Chapter 3 Closer file on page 145. It is useful for marking tasks such as the “Challenge in Real Life” and explains how to identify the level of specific answers for this project. If you are going to use this rubric, it is advisable to share it with the students well ahead of time.

1. 5 steps × 18 cm/step = 90 cm

The total rise of the staircase is 90 cm.

2. 5 steps × 24 cm/step = 120 cm

The total run of the staircase is 120 cm.

3.

The length of wood needed for the stringer is 150 cm.


4. a. Designs will vary. Here is a sample:

40cm

58cmstringer26cm

13cm

13cm

26cm

b. total rise: 2 steps × 13 cm/step = 26 cm

total run: 2 steps × 26 cm/step = 52 cm

Length of stringer:

The length of the stringer for the stool is 58 cm.

Unit Review

There are extra-practise exercises for each test section already prepared in the Black Line Masters of the MathLinks 8 Teacher’s Resource from McGraw-Hill Ryerson. You may choose to use them for individual strengthening if required for some students at the end of the unit.

There is a BLM Chapter 3 Test on MathLinks 8 Teacher’s Guide CD and on the MathLinks 8 Instructor Edition website which you may decide to use as a unit assessment.

Are You Ready?

TT 2. When the student is finished, he or she is instructed to submit the answers to the teacher to correct. The mark will not be used for report cards. It is just to confirm how well the student is doing at this point. The answers to the “Practice 3 Test” on pages 114 and 115 of the student text are found in the MHR MathLinks 8 Teacher’s Resource on page 138.

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