unit c assignment 1 - erlc ongoing professional … · web viewhe multiplied the whole numbers and...

$: Unit C Assignment 1 - ERLC Ongoing Professional … · Web viewHe multiplied the whole numbers and added that product to the product of the fractions, instead of changing the numbers$
Mathematics 8: Unit 3 1 Teacher’s Guide

UNIT 3 TEACHER’S GUIDE

Unit 3 Introduction

You may find that the student needs a refresher on the use of fraction strips to model addition and subtraction. Before embarking on the Explore, you should review this topic.

One way of understanding multiplication by a whole number is by thinking of it as repeated addition. The student may need some review of fraction addition, and the related operation of subtraction, with concrete materials—such as fraction strips—before starting the Explore.

To activate the appropriate prior knowledge, you may present the student with the following example.

As well, you could indicate how pattern blocks (via the multimedia Pattern Blocks) and fraction circles could be used to model the solution.

You may want to use the following as a basis for review:

http://www.learnalberta.ca/content/mejhm/html/object_interactives/fractions/explore_it.html


Lesson 1: Using Models to Multiply Fractions and Whole Numbers

TT 1.

1. a.

With each of the blue rhombuses representing , the sum of fractions can be shown this way.

The four blue rhombuses rearranged make up one hexagon (the same size as a yellow hexagon) and one blue rhombus. The hexagon represents one whole, and the remaining blue rhombus represents . So, together the rearranged rhombuses represent .

The multiplication equation is this:

b. Various answers are possible. The following is a sample answer:

I used the Pattern Blocks because they have a shape—the rhombus—that can be used to represent thirds. I enjoy fitting together shapes. It’s a bit like working out a puzzle. By making a hexagon, I could show the whole number part of the answer. The left over rhombus represented the fractional part of the answer. I also enjoy having blocks in my hand. Being able to handle the blocks makes me feel that fractions are more real. Besides, shading parts of fraction strips or fraction circles is not so appealing to me.

TT 2.

1. a.

b. Answers will vary. Example: A hexagon would be difficult to use because it is difficult to divide into five equal parts.


2. a. Answers will vary. Example: He used 5-by-1 grids because the denominator is 5.

b. Answers will vary. Example: He used four grids because he was multiplying by 4.

c. Answers will vary. Example: Four groups of three counters yields 12 counters, where each counter represents .

3. Yes. Answers will vary. Example:

Unit 1: Lesson 1 Question Set

a.

(2 marks)

b.

(2 marks)

c.

(2 marks)

d.

(2 marks)

2. a.

(2 marks)

b.

(2 marks)


3. a.

(3 marks)

b.

(3 marks)

4. Answers will vary. The following are sample answers.

The surface of a cube is 12 cm2. What is the surface area of one face?

An egg carton (for a dozen eggs) is only full. How many eggs are in the carton?

A crayon has been used so that it is only its original length. The original length of the crayon was 12 cm. What is its length now?

(3 marks)

5. One-half of the squares covering the playing surface are black. (1 mark)


Lesson 2: Dividing a Fraction by a Whole Number

TT 1. Answers will vary. The student should show or describe several models. These models could be based on the use of a number line (divided into sections), rectangles, pattern blocks, fraction strips, or some other concrete materials. The student may explain a preference based on reasons such as ease of use or the liking of 3-D objects (e.g., pattern blocks) that can fitted together. Look for clear communication.

TT 2.

1. No, she needs to divide the strip into ninths.

2. a.

b. No, because the whole would have 12 pieces and would have 9 pieces, and 9 cannot be divided evenly by 6.



1. a. The triangle will represent the quotient.

One rhombus represents because three rhombuses cover the area of the hexagon. Two

rhombuses represent . But two of these shapes cannot be split four ways. So these shapes should be replaced by triangles.

The four triangles can be split 4 ways.

The result of splitting the triangles 4 ways is 1 triangle. So the triangle represents the quotient.(3 marks)

b.

The quotient is represented by the triangle. The triangle represents the fraction .(2 marks)

2 a.

(3 marks)

b.

(3 marks)


3.

On the average, Mike used of a tank each day.(4 marks)

4. a.

(1 mark)

b.

(3 marks)

c. Vancouver has frost on of the days of the year. (1 mark)

Lesson 3: Multiplying Proper Fractions

TT 1. Although slide shows will vary in style and content, the slide show should communicate the following:

6. a. Example: Fold the paper lengthwise into the number of sections indicated by the denominator of the first fraction. Colour the number of sections indicated by the numerator of the first fraction. Then, fold the paper widthwise into the number of sections indicated by the denominator of the second fraction. Colour the number of sections indicated by the numerator of the second fraction a different colour. The number of sections that contain both colours is the numerator of the product. The total number of sections is the denominator. A rule for multiplying two proper fractions is to multiply the numerators and multiply the denominators.

b. Example: I prefer the rule because it is less time consuming.


TT 2.

1. a. Answers will vary. Example:

b. Answers will vary. Example: I chose a diagram because it is like paper folding.

2. a. He did not multiply the denominators.

b. Answers will vary. Example: If he estimated the answer as , he would have noticed that his answer is incorrect.

c.


1. Each rectangle represents . The red and blue colours overlap in only 1 of the 9 equal

rectangles. So the product is .(3 marks)

2.

There are two folds parallel to the length that splits the paper into thirds. Two of these thirds are coloured blue. This coloured area represents the factor .

There are four folds parallel to the width that splits the paper into fifths. Just one of these fifths is coloured yellow. This yellow area represents the factor .

The green area is the overlap of the yellow and blue colours and represents the product. The green area consists of just 2 out of 15 divisions of the paper. So the green represents the value

.

Therefore, the product is .

So, . (7 marks)

3. a.

(2 marks)


b.

(2 marks)

4. a. is close to

is close to 1

The product is closest to .(3 marks)

b. is close to 0

is close to 1

0 × 1 = 0

The product is closest to 0.(3 marks)

5.

Kenji’s height is of his father’s height.(3 marks)

6.

A horse sleeps as much as a two-toed sloth.(3 marks)

Lesson 4: Multiplying Improper Fractions and Mixed Numbers

TT 1. Answers will vary, but look for presentations that convey the following in a clear fashion.

4. Example: To use a diagram, divide a rectangle into sections indicated by each mixed number. Find the area of each section. The sum of the sections is the answer.

TT 2.


1. a. He multiplied the whole numbers and added that product to the product of the fractions, instead of changing the numbers to improper fractions.

b.

2. a. Example: There are four and one in .

b. A rule is to multiply the denominator by the whole number and add the numerator. Place this number over the denominator.

c. A rule is to divide the numerator by the denominator. This quotient is the whole number part of the mixed number. Place the remainder of the quotient over the denominator.

3. a. Yes.

b. She did not need to change each fraction into an equivalent form with a common denominator.


1. a. The diagram represents . (2 marks)

b. Area of A:

1 × 2 = 2 → Area of A is 2 units.

Area of B:

Area of B is of 1 unit.

Area of C:

.

Area of D:

.(4 marks)


c. Area of rectangle:

Area of rectangle is units.(2 marks)

d. The value of the product is . (1 mark)

2. To the nearest whole number, the factors are 3 and 4.

The product of 3 and 4 is 12.(3 marks)

3. To determine the height:

Height:

The height must be m.

To determine the length:

The length of the flag must be m.(8 marks)


Lesson 5: Dividing Fractions and Mixed Numbers

TT 1. The student is to create a division problem and indicate a preferred method for finding the answer to the problem. Answers will vary. The student may show either method—“division using common denominators” or “division using multiplication.” Plausible reasons should be given for a choice. Some students may have discovered that the common denominator method may work better when fractions can easily be converted to a common denominator and will not indicate a blanket preference for just one method.

After the student has completed TT 1, the student is required to post his or her work from TT 1 on the discussion board. Look for some meaningful exchange between students.

Connect

Demonstrating the solution to one or two division problems through an alternate visual model could provide students with improved understanding of division.

Such an alternate visual model is provided by “Operations with Fractions” from the course Toolkit.

In your introduction to this multimedia, you may indicate that this multimedia uses two-dimensional modelling based on an area representation. In the lesson itself, only one-dimensional models based on the length of a rectangle and its subdivisions are presented.

You could demonstrate the solution to the division problem in textbook “Example 1: Divide Using Diagrams” and “Show You Know” question a) on page 224 using “Operations with Fractions.”

Select the “Multiply & Divide” button. In modelling division, the overlapped area represents the dividend and either one of the chosen fractions can represent the divisor. The other chosen fraction becomes the quotient.

Doing division is not straightforward with this multimedia piece. You would expect to enter the dividend and the divisor of a division problem directly and then the quotient would appear. You do enter one fraction to represent the divisor. But then you don’t enter the dividend directly. Instead, through trial and error, you adjust the value of the other fraction in the interface so that the resulting area is equal to the dividend. The fraction entered indirectly through trial and error becomes the quotient of the division problem.

You and the student can make up some more division problems and solve some of them together using “Operations with Fractions.” But note that this multimedia simulation can only be used for a limited number of division problems. For example, in any division problem, the denominators have to be less than or equal to 4 and the divisor and the dividend must be less than or equal to 6. There are other restrictions too. So you may want to check some division problems in advance to make sure they work out.

You may use this multimedia later when helping the student with some questions from “Check Your Understanding” later in the lesson. The multimedia simulation can be used for questions 5.b), 5.c), 6.c), 11, and 15 on pages 227 and 228.


TT 2.

1. a. She multiplied the numerators and the denominators, but did not replace the second fraction with its reciprocal.

b. ,

2. Mike is incorrect. He should have replaced the second fraction with its reciprocal.

3. Answers will vary. Example: There are between four and five s in .


1. a.

(4 marks)


b.

(4 marks)

2. a. This is approximately equal to 4 ÷ 2.

4 ÷ 2 = 2

is approximately equal to 2.(2 marks)

b. The quotient is approximately equal to 12 ÷ 3.

12 ÷ 3 = 4

So is approximately equal to 4.(2 marks)

3. You would multiply the cell entries to decide whether the entry in one cell is the reciprocal of the other. If the product is 1, then the cell entries are reciprocals. Otherwise, they’re not.

Brent’s conclusion that the numbers are reciprocals of each other is correct.

An alternate solution could be based on changing any mixed fractions into improper fractions and then comparing the proper and improper fractions to see if they are the vertical reflection of each other.(5 marks)


4. To determine the fraction of Earth that Canada covers, you could divide by :

Canada covers of Earth’s surface. Lori had remembered correctly.

An alternate acceptable solution that the student may present involves showing that the area of Russia multiplied by the factor equals . (4 marks)

Lesson 6: Applying Fraction Operations

Explore

The calculations in the Explore involve large numbers. You may have the student express population data as a number of thousands; for example, 190-thousand Aboriginal people live in Alberta. This does make the number more manageable for students.

The calculations involve some approximating. You may indicate to the student that it’s the method, and not the exact answer, that is important. Showing the student about rounding may help them.

Some students may benefit from a review of the use of a calculator for multiplication and division of whole numbers. These whole numbers arise when multiplying numerators and denominators. Division is used when evaluating fractions; the vinculum is treated as the division sign.

You may see a need to discuss that sometimes there are brackets around brackets. Then the outside ones are made as square brackets. This background will allow students to write a mathematical expression for the population of Aboriginal people living in urban areas more easily.

Students should be made aware that answers may differ due to various ways to round.

TT 1. The discussion should show students reacting to each other’s ideas, rather than just listing ideas independently of each other.


Here are some ideas for pros (+) and cons (−) you can expect to see reflected in the transcript of the student discussion.

Solving by Steps Solving by Applying a Mathematical Expression

+ Each step is small and easy to understand.

+ If the calculation to a step is unreasonable, you have a signal that you have to correct it before continuing on.

− It’s challenging to write an expression, especially when it gets complicated.

− It’s easy to mix up the order of operations and get the wrong answer.

− You can end up doing a lot of work in evaluating the expression before you see that you are on the wrong track.

+ You can do all your calculations in one series of operations on your calculator without having to write down values along the way.

TT 2.

1. a. She performed the subtraction before multiplying.

b. She followed the order of operations by performing the multiplication before the subtraction.

c. 9. When following the order of operations, you will obtain the correct answer.

2. 8. Check the answer by taking of 8. The result is 6.

3. a. She multiplied by to obtain ; then, she added to obtain her answer of .

b.


1. a.

(3 marks)


b.

(3 marks)

c.

(3 marks)

2.

Vikram earns $141 on his longer work day. Note: Students may also solve this problem by calculating in stages.(8 marks)

3. a.

Emily sold 4 pages of advertisement space.

b.

The revenue made from the advertisement space sold by Emily was $1000.


Unit 3 Summary

TT 1.

a. Of Canada’s ecozones, are marine.

Canada has 5 marine ecozones.

b. Since of Canada’s ecozones are marine, of them are terrestrial.

Canada has 15 terrestrial ecozones.

TT 2.

Each ecozone covers the area of Canada.

TT 3.

The pacific Maritime ecozone “covers” the area of Canada.

TT 4.

The Northern Arctic ecozone covers the area of Canada.

TT 5.


TT 6.

There are 200 mammal species in all of Canada.

200 – 50 = 150

There are 150 mammal species in Canada outside the Taiga Shield.

Students may approach the question this way:

The number of mammal species in the Taiga Shield is .

So of the number of mammal species of Canada are outside of the Taiga Shield.

So there are 3 times as many mammal species outside of the Taiga Shield but within Canada.

50 × 3 = 150

There are 150 mammal species in Canada outside the Taiga Shield.

TT 7. Answers will vary. Example questions are as follows with the solutions provided in parentheses:

How much area of the Boreal Plains is in Alberta? (390 000 km2)

How much area of the Boreal Plains is in the Prairie provinces? (698 000 km2)

How much of the Boreal Plains is in Canada outside of the listed provinces? (none)

What which two provinces contain about ¾ of the Boreal Planes? (Alberta and Saskatchewan)

Alberta contains how many times as much of the Boreal Plains ecozone as British Columbia? (times).

unit c assignment 1 - erlc ongoing professional … · web viewhe multiplied the whole numbers and...

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