ns7-18 fractions

1Number Sense 7-18

Review fractions. Ask students what they already know about fractions. SAMPLE ANSWERS: they have a top number and a bottom number; the bottom number tells how many parts are in a whole; the top number tells how many parts are being considered (e.g., a team that has played 5 games and won 3 of them has won 3/5 of their games). Some students may know that greater denominators decrease the size of the fraction.

Ask students what they can take a fraction of. EXAMPLES: fraction of an hour, fraction of a cup or teaspoon, fraction of an inch, fraction of an area, fraction of a distance, fraction of a group of people, fraction of an angle, fraction of a line, fraction of number of votes, fractions of shoe sizes (half).

Naming fractions of an area. Emphasize the importance of each part being the same size and start with examples where this is the case (see Question 1); if this is not the case, students should further subdivide the area until all parts are the same size (see Question 2).

Review the terms numerator and denominator, and explain what they refer to (top and bottom numbers in a fraction) and what they represent (number of parts considered and number of parts in a whole).

EXTRA PRACTICE for Question 2: What fraction of each figure is shaded? a) b) c) d) e)

ANSWERS: a) 2/16 b) 5/16 c) 1/4 d) 1/16 e) 1/64

What fraction of this shape (see margin) made from pattern blocks is each colour? ANSWERS: Black: 6/20 White: 4/20 Light grey: 6/20 Dark grey: 4/20

NS7-18 FractionsPage xxxx

CuRRICuLuM EXPECTATIoNSOntario: 6m14, review, 7m2, 7m5, 7m6WNCP: review, [ME, CN, R]

VoCAbuLARy fraction numerator denominator

GoalsStudents will understand fractions as equal parts of a whole or of a set.

PRIoR KNoWLEDGE REQuIRED

Understands the concept of area Recognizes contexts that fractions can be used in Understands logical statements using and, or, and not

MATERIALS

modelling clay, resealable bags, straws, paper (see Activity)

2 Teacher’s Guide for Workbook 7.1

Finding the whole from a part. Have students draw a rectangle 6 cm long in their notebooks (on 1 cm grid paper if available). Tell students that this rectangle is only 3/5 of a whole rectangle. ASK: How can you draw the whole rectangle? Write on the board, “3/5 is ______ out of ______ parts,” and have students fill in the blanks. Then ASK: How many more parts do I need to make a whole? (two more parts) How can you find the size of those two parts? (divide the given rectangle into three equal parts; those parts will be the same size as the two more parts) Have students draw the whole rectangle in their notebooks.

EXTRA PRACTICE: A line 6 cm long is 2/3 of a whole line. How long is the whole line?

Taking the fraction of a set. See Questions 8 to 12.

EXTRA PRACTICE for Questions 8-12: 1.

What fraction of the shapes are ANSWERS: ANSWERS: a) shaded? 5/9 h) not white triangles? 8/9 b) circles? 2/9 i) big squares or big circles? 4/9 c) not circles? 7/9 j) squares or triangles? 7/9 d) not squares? 5/9 k) triangles or circles? 5/9 e) shaded squares? 2/9 l) not small triangles? 7/9 f) big squares? 3/9 m) neither small nor a triangle? 4/9 g) white triangles? 1/9

Have students find parts that have the same answer and decide if it is just a coincidence or if there is a reason why the two parts have the same answer. For example, a) and k) have the same answer, but this is just a coincidence. On the other hand, that d) and k) have the same answer can be deduced logically.

2. What word do you get when you combine a) the first 2/3 of sun and the first 1/2 of person? (ANSWER: super) b) the first 1/2 of grease and the first 1/2 of ends? c) the first 1/3 of trance and the last 3/4 of luck? d) the first 1/2 of wood and the last 2/3 of arm?

Try making up your own questions like these.

3. Draw the set: There are 7 triangles and squares; 2/7 of the figures are triangles,

3/7 are shaded, and 2 triangles are shaded.

4. There are 9 circles and triangles. Can you draw a set so that a) 7/9 are circles and 4/9 are striped? (Have volunteers show different possibilities and ask how many are striped circles.) b) 7/9 are circles and 4/9 are triangles?

Discuss what is the same and what is different in both questions.

PRoCESS ASSESSMENT7m2, [R]

3Number Sense 7-18

ANSWER: The numbers are the same, but one set is possible and the other is not. In the impossible case, the same attribute (shape) was used twice; in the other case, two different attributes (shape and pattern) were used. Note that if the fractions used had added to 1, then it would be possible to use the same attribute twice, e.g., 7/9 are circles and 2/9 are triangles.

5. What fraction of the multiples of 2, up to 3000, are also multiples of 5? ANSWER: 1/5

ACTIVITy

Prepare many small, equal-sized balls of modelling clay in different colours (white, red, blue, and yellow). You could use a teaspoon to measure out the modelling clay, to ensure that all balls are the same size. Prepare resealable bags, each with 6 balls of the same colour; make enough so that each student can choose two bags of different colours.

Students work in pairs. Each student secretly chooses three balls and thoroughly mixes the balls together to create a new ball. Students show their partners the new balls. Knowing that three balls were used in total, can the partner guess how many of each colour were used? What fraction of the new ball is each colour? (EXAMPLE: 1/3 red and 2/3 blue) Students should write the correct “recipe” for each ball of modelling clay on a triangular flag (see below), tape it to a straw, and insert the straw into the ball. (You can make 4 flags from a regular sheet of paper.) One possible recipe is shown on the flag in the margin. Repeat, but this time have students choose 4 balls from their remaining 9 balls.

Finally, have students make a ball with the remaining 5 balls but this time have partners predict the fraction of each colour used before they see the ball. Students should be able to logically deduce the correct fraction at this point.

Students should make recipe flags for all the balls they create. Keep them for use in NS7-25.

8m5, [CN] Workbook Question 5

PRoCESS ASSESSMENT

1 3 red, 2

3 blue

Mental math and estimation, Modelling

PRoCESS EXPECTATIoN


Naming mixed numbers. Introduce mixed fractions by drawing the following picture on the board:

Tell students that some friends ordered 3 pizzas with 4 pieces each. The shaded pieces show how much they have eaten. They ate two whole pizzas plus a quarter of another one. Draw the following pictures on the board. a) b)

c)

ASK: How many whole pizzas are shaded? What fraction of the last pizza is shaded? Write the mixed number for the first picture and have volunteers write the mixed numbers for the second and third pictures.

Draw models of several mixed numbers, asking students to name them in their notebooks. Use a variety of shapes for the whole piece, such as rectangles and triangles.

Representing mixed numbers. Write a fraction such as 3 1/4 on the board. Draw a series of circles divided into the same number of parts, as given by the denominator of the fraction (since the denominator of the fraction in this example is 4, each pie has 4 pieces). Ask your students to shade the correct number of pieces in the pies to represent the fraction.

EXAMPLE: 3 1 4

Students should shade the first 3 circles and 1 part of the fourth circle.

Tell your students that you have drawn more circles than they need so they have to know when to stop shading.

NS7-19 Mixed NumbersPage xxxx

CuRRICuLuM EXPECTATIoNSOntario: 7m1, 7m6, reviewWNCP: review, [V]

VoCAbuLARy mixed number

GoalsStudents will recognize and name mixed numbers.


Understands that fractions can name an area Can use pies as models for fractions Can name fractions Can double numbers

7m1, 7m6, [V] Workbook Question 5

PRoCESS ASSESSMENT

5Number Sense 7-19

Have students sketch the pies for given mixed numbers in their notebooks.

EXAMPLES: 124

, 538

, 318

, 223

, 435

.

If students have trouble, let them first practise drawing whole pies divided into halves, thirds, quarters, and so on.

Have students use their sketches to order the fractions from least to greatest.

Extensions1. Teach students how to count forwards by halves, thirds, quarters, and tenths beyond 1.

Ask students to complete the patterns:

a) 1 4 ,

2 4 ,

3 4 , ____, ____, ____, ____, ____

b) 1 3 ,

2 3 , ____, ____, ____, ____, ____, ____

c) 2 2 4 , 2 3

4 , ____, ____, ____, ____, ____

d) 7 10 ,

8 10 , ____, ____, ____, ____, ____

e) 7 9 ,

8 9 , ____, ____, ____, ____, ____

SAMPLE ANSWER: b) 1 2 1 2 1, , 1 , 1 , 1 , 2, 23 3 3 3 3

2. Which model (A or B) represents 2 3/4 ? How do you know? ANSWER: B, because all the wholes are the same size

3. Ask students to order these mixed numbers from least to greatest:

3 1 5 1 5

7 7 7 11 ANSWER: 1 5

7 , 3 1 5 , 7 7

11

ASK: Did you need to look at the fractional parts at all or just the whole numbers? Why? (Just the whole numbers - for example, 1 5/7 is between 1 and 2, so it is less than 3 1/5, which is between 3 and 4)

bonus Order these numbers: 3 1 8 5 2

7 2 1 11 6 1

5 8 5 7 4 3

10

ANSWER: Just put the whole number parts in order:

12

11

138

34

10 257

165

587

Modelling, Representing

PRoCESS EXPECTATIoN

A B


Fractions equivalent to 1. Draw these shapes on the board:

Have students name the fractions shaded. ASK: How many parts are shaded? How many parts are in one whole? Tell students that they are all one whole and write 1 = 4/4 and 1 = 6/6. Then have student volunteers fill in the blanks:

Fractions equivalent to 2. Tell students that sometimes they might have more than one whole—they might have two whole pizzas, for EXAMPLE:

and

ASK: Which number goes on top—the number of parts that are shaded or the number of parts in one whole? Tell students to look at the pictures and say how many parts are in one whole circle and how many parts are shaded. Then write:

and

Ask a volunteer to come and write the number of shaded pieces. ASK: How are the numerator and denominator in these fractions related? (you double the denominator to get the numerator). Have students fill in the missing numbers:

Introduce Improper Fractions. ASK: How are the fractions above different from the fractions we’ve seen so far? Tell them these fractions are called improper fractions because the numerator is larger than the denominator. Challenge students to guess what a fraction is called if its numerator is smaller than its denominator. (A proper fraction)

NS7-20 Improper FractionsPage xxxx

CuRRICuLuM EXPECTATIoNSOntario: 7m1, 7m6, reviewWNCP: review, [V]

VoCAbuLARy mixed number

GoalsStudents will name improper fractions and fractions representing exactly one whole.


Can identify and represent mixed numbers


PRoCESS ASSESSMENT 1 = 9

1 =7

2 = 2

2 = 3

2 = 4

2 =

282 =

76

2 =10 2 =

62

7Number Sense 7-20

Draw on the board:

ASK: How many pieces are shaded? (9) SAy: I want to write a fraction for this picture. Should 9 be the numerator or the denominator? (numerator) How many equal parts are in one whole? (4) Should this be the numerator or the denominator? (denominator) Tell your students that the fraction is written 9/4.

Have volunteers write improper fractions for these pictures.

a) b)

c)

ASK: How many parts are shaded? How many parts are in one whole?

Draw models of several improper fractions, asking students to name them in their notebooks. Use a variety of shapes, such as rectangles and triangles, for the whole.

Modelling improper fractions. Write a fraction such as 15/4 on the board. Draw a series of circles divided into the same number of parts, as given by the denominator of the fraction (since the denominator of the fraction in this example is 4, each pie has 4 pieces). Ask your students to shade the correct number of pieces in the pies to represent the fraction.

EXAMPLE:

15 4

Students should shade the first 3 circles and 3 parts of the fourth circle.

Tell your students that you have drawn more circles than they need so they have to know when to stop shading.

Have students sketch the pies for given improper fractions in their notebooks.

EXAMPLES: 11 4

, 15 8

, 19 8

, 10 3

, 12 5

.

Have students use their sketches to order the fractions from least to greatest.

ASK: Which fractions show more than a whole? How do you know?

13 5

, 2 9

, 14 15

, 8 3

, 12 7

Modelling, Representing

PRoCESS EXPECTATIoN


Review. ASK: What is a mixed number? What is an improper fraction? Have a volunteer write a mixed number for the picture in the margin and explain their answer.

Have another volunteer write an improper fraction for the same picture and explain their answer.

Draw several models of fractions larger than 1 on the board and have students write both the mixed number and the improper fraction. Use several different shapes other than circles.

Draw several more such models on the board and ask students to write an improper fraction if the model contains more than 2 whole shapes, and a mixed number otherwise. This will allow you to see if students know the difference between mixed numbers and improper fractions.

Converting mixed numbers to improper fractions. Tell students to draw models for the following mixed numbers and to write the corresponding improper fraction that is equal to it.

a) 2 3 4 b) 3 1

3 c) 2 1 6 d) 1 5

6 e) 3 2 5 f) 2 7

8

Converting improper fractions to mixed numbers. Tell students to draw models for the following improper fractions and then to write the mixed number that is equal to each one.

a) 13 4

b) 7 3

c) 11 6

d) 19 5

e) 27 8

Relating improper fractions to mixed numbers. SAy: You have written many numbers as both a mixed number and an improper fraction. What is the same in both? What is different? (The denominators are the same because they tell you how many parts are in a whole, but the numerators are different because a mixed number counts the pieces that make up the wholes separately from the pieces that make up only part of a whole; improper fractions count them all together.)

Ask students to solve these problems by sketching their answers on triangular grid paper: a) Which two whole numbers is 23/6 between? b) What fraction of a pie would you have if you took away 1/6 of a pie

NS7-21 Mixed Numbers and Improper FractionsPage xxxx

CuRRICuLuM EXPECTATIoNSOntario: 6m14; 7m1, 7m6, 7m7, reviewWNCP: 6N4, review, [V,C]

VoCAbuLARy mixed number improper fraction remainder

GoalsStudents will relate mixed numbers and improper fractions.


Can identify and represent mixed numbers and improper fractions Can multiply up to 9 × 9Can divide with remainders Understands the relationship between multiplication and division

7m1, 7m6, 7m7, [R, CN, C] Workbook Question 5

PRoCESS ASSESSMENT

Modelling

PRoCESS EXPECTATIoN

9Number Sense 7-21

from 3 pies? Write your answer as both a mixed number and an improper fraction. bonus Write the following fractions in order:

3 1 4

27 4

11 4

7 1 4

36 4

4 3 4

Converting fourths to improper fractions using multiplication. Draw on the board:

SAy: How many parts are in the pie? There are 4 quarters in one pie. How many quarters are in 2 pies? (8)

What operation can we use to tell us the answer? (multiplication) How many quarters are there in 3 pies? (4 × 3 = 12)

ASK: How many quarters are in 3 3/4 pies?

3 3–4 So there are 15 pieces (quarters) altogether. Continue converting mixed numbers to improper fractions with halves and thirds instead of fourths.

Have students write in their notebooks how many halves are in…

a) 2 1 2 b) 5 1

2 c) 11 d) 11 1 2 bonus 49 1

2 84 1 2

Have students write in their notebooks how many thirds are in…

a) 2 1 3 b) 5 1

3 c) 11 d) 11 1 3 bonus 49 2

3 84 2 3

Solving problems in a context. SAy: I have boxes that will hold 4 cans each. What fraction of a box is each can? (one fourth) How many fourths are in 2 wholes? How many cans will 2 boxes hold? How are these questions the same? How are they different?

A box holds 4 cans. How many cans will the following hold?

a) 1 1 4 boxes b) 2 3

4 boxes c) 1 1 4 boxes d) 1 3

4 boxes

To help your students, encourage them to rephrase the question in terms of fourths and wholes. For example, since a can is one fourth of a whole, a) becomes “How many fourths are in 1 1/4 ?”

7m5, [CN] Workbook Question 10

PRoCESS ASSESSMENT

3 extra pieces 12 pieces

(3 × 4)


Next, students will have to rephrase the question in terms of fractions other than fourths, depending on the number of items in each package.

a) A box holds 6 cans. How many cans will 1 5/6 boxes hold? b) A box holds 8 cans. How many cans will 2 3/8 boxes hold?

bonus c) A box holds 326 cans. How many cans will 1 5/326 boxes hold? d) Tennis balls come in cans of 3. How many balls will 7 1/3 cans hold? e) A bottle holds 100 mL of water. How many mL of water will 7 53/100 bottles hold?

Converting mixed numbers to improper fractions. To change 2 3/8 to an improper fraction, start by calculating how many pieces are in the wholes (2 × 8 = 16) and add on the remaining pieces (16 + 3 = 19), so 2 3/8 = 19/8.

Have students write the following mixed numbers as improper fractions and explain how they found the answers.

a) 3 1 7 b) 5 1

6 c) 4 3 9 d) 7 5

8 e) 6 5 6

Writing Improper fractions and mixed numbers from models. Have students write in their notebooks the mixed numbers and improper fractions for several pictures displaying area:

Then provide examples involving length and capacity as well, as shown in Questions 4 and 5 on the worksheet. EXAMPLES:

How long is the line? How many litres are shown?

Converting whole numbers to improper fractions: Have students write each whole number below as an improper fraction with denominator 2 and show their answer with a picture and a multiplication statement:

a) 3 = 6 2 b) 4 c) 2 d) 7 e) 10

3 × 2 = 6

Converting improper fractions to whole numbers: ASK: If I have the improper fraction 10/2, how could I find the number of whole pies it represents? Draw on the board: whole pies = 10

2 pies

1 m1 litre

11Number Sense 7-21

Ask students how many whole pies 10 half-sized pieces would make? Show students that they simply divide 10 by 2 to find the answer: 10 ÷ 2 = 5 whole pies.

Review converting mixed numbers to improper fractions. Have students write the mixed numbers below as improper fractions and show their answer with a picture. Students should also write a statement involving multiplication and addition for the number of half-sized pieces in the pies.

a) 3 1 2 = 7

2 b) 4 1 2 c) 2 1

2

d) 5 1

2 e) 8 1 2

3 × 2 + 1 = 7 halves

There are 3 pies with 2 halves each.

Converting improper fractions to mixed numbers. SAy: If I have the improper fraction 15/2, how can I know how many whole pies there are and how many pieces are left over? I want to divide 15 into sets of size 2 and I want to know how many full sets there are and how many extra pieces, if any. What operation should I use? (division) What is the leftover part called? (the remainder)

Write on the board: 15 ÷ 2 = 7 Remainder 1, 15 2

= 7 1 2 .

15 ÷ 2 = 7 R 1

Draw the following picture and number sentences on the board.

15 4 = 3 3

4

4 × 3 + 3 = 15

15 ÷ 4 = 3 Remainder 3

Have students relate the number sentences to the picture: When we divide 15 into sets of size 4, we get 3 sets and then 3 extra pieces left over. This is the same as dividing pies into fourths and seeing that 15 fourths is the same as 3 whole pies (with 4 pieces each) and then 3 extra pieces.

Repeat for several pictures, having volunteers write the mixed numbers and improper fractions as well as the multiplication and division statements. Then have students do similar problems individually in their notebooks.

Now give students improper fractions and have them draw the picture, write the mixed fraction, and the multiplication and division statements.

Then show students how to change an improper fraction into a mixed fraction:

[V], 7m6 Workbook Question 19

PRoCESS ASSESSMENT

There is one extra half-sized piece.


Reflecting on what makes a problem easy or hard, selecting tools and strategies

PRoCESS EXPECTATIoN

Recall that 2 1 4 = 2 × 4 + 1 quarters = 9 quarters = 9

4 .

Starting with 9 4 , we can find 9 ÷ 4 = 2 Remainder 1

so 9 4 = 2 wholes and 1 more quarter = 2 1

4 .

Have students change several improper fractions into mixed fractions without using pictures.

Tell students that 7/2 pies is the same as 3 whole pies and another half a pie. ASK: Is this the same as 2 whole pies and 3 halves? Do we ever write 2 3/2? ASK: When we find 7 ÷ 2, do we write the answer as 3 Remainder 1 or 2 Remainder 3? Tell your students that as with division, we want to have the fewest number of pieces left over.

Extensions

1. Write the following numbers in order: 3 1 4 ,

27 4 ,

21 4 , 2 1

4 , 36 4 , 4 3

4

2. Investigation If Stick A is a/b of Stick B, what fraction of Stick A is Stick B?

A. Draw Stick A 5 units long and Stick B 3 units long (see margin).

b. What fraction of Stick A is Stick B?

Put Stick B on top of Stick A. If Stick A is the whole, then the

denominator is 5 because Stick A has 5 equal-sized parts. How many of those parts does Stick B take up? (3). So Stick B is 3/5 of Stick A.

C. What fraction of Stick B is Stick A?

If Stick B is the whole, what is the denominator? (3) Why? (because Stick B has 3 equal-sized parts) How many of those equal-sized parts does Stick A take up? (5) So stick A is 5/3 of Stick B.

Have students write their answers as both a mixed number (1 2/3) and an improper fraction (5/3).

D. Repeat the above with several examples. In this way, your students will discover reciprocals. If you know what fraction Stick A is of Stick B, what fraction is Stick B of Stick A? (just turn the improper fraction upside down!)

ASK: Why was it more convenient to use improper fractions instead of mixed numbers?

Stick A

Stick B

13Number Sense 7-22

What can you take a fraction of? For example, you can take a fraction of a length, an area, a capacity or volume, time, or an angle. You can even sometimes take a fraction of a person, if the context is right. You can’t say 3 1/2 people went skating, but you can say that half of a person is covered in paint. In the latter case, you are taking the fraction of a surface area. You can also talk about fractions of people in averages. For example, families in a certain region might have an average of 1 1/2 children. Of course, no family can have exactly 1 1/2 children, but that average means that probably most families have 1 child, some have 2 children, some have 3, and so on. For example, the average of the following ten numbers is 1 1/2: 1, 1, 1, 1, 1, 1, 1, 2, 2, 4. Each data value is a whole number, but the average is a fraction.

Introduce fractions of a number. ASK: Can you take a fraction of a number? What would that mean? What does “half of six” mean? If there are six friends and half of them are girls, how many are girls? If there are six juice boxes, and half of them are of apple juice, how many are of apple juice? If the distance to a friend’s house is 6 km, how far away is the halfway point? If I want to finish a race in six hours, when should I be at the halfway point? Explain that since all of these questions have the same numeric answer, we can say that the number 3 is half of the number 6.

use pictures to show half. Explain that if you want to eat half a pizza, you would divide the pizza into two equal parts and eat one of them:

Similarly, if you wanted to eat half of six cherries, you would divide the six cherries into two equal groups and eat one of the two groups:

There are three cherries in each group, so 3 is half of 6.

Have students draw pictures to show half of these numbers: a) 4 b) 10 c) 16

Challenge students to decide how they would show half of 7. Explain that you would put three in each group and split the one left between the two groups, so you would put 3 1/2 in each group; half of 7 is 3 1/2.

NS7-22 Fractions of Whole NumbersPage 34-35

GoalsStudents will use models to find fractions of whole numbers.


Can name and model fractions Can convert mixed numbers to improper fractions and vice versa

CuRRICuLuM EXPECTATIoNSOntario: 7m1, 7m5, 7m6, 7m7, 7m25 WNCP: essential for 7N3, [V, R, C, N, C]

VoCAbuLARy fraction of a whole number

Modelling

PRoCESS EXPECTATIoN


use pictures to find other fractions of whole numbers. ASK: If I wanted to eat only one third of the cherries, how many groups should I make? (3) Show this:

When you divide six cherries into three equal groups, and take one of those groups, you are taking two of the cherries. So 1/3 of 6 cherries is 2 cherries. Indeed, 1/3 of 6 anythings is 2 anythings.

ASK: What would two thirds be? Explain that you still need to make three groups, but now you take two of the groups instead of just one:

Have students draw pictures to find these fractions of numbers:

a) 23

of 9 b) 35

of 10 c) 45

of 15

d) 34

of 12 e) 23

of 12 f) 56

of 12

ANSWERS: a) 6 b) 6 c) 12 d) 9 e) 8 f) 10

use fractions of numbers to compare and order fractions. Have students look at their answers above. ASK: How can you use your answers to decide what is larger: 2/3 or 3/4? Which two answers did you look at? (from the answers to d) and e), 3/4 of 12 is 9 and 2/3 of 12 is 8, so 3/4 is more than 2/3)

Have students find the following fractions of 20 and then use the answers to write the fractions in order from smallest to largest. a) 3/4 b) 7/10 c) 3/5 d) 4/5

using division to find fractions (with numerator 1) of whole numbers. ASK: How would you find 1/3 of 15 dots? (divide the dots into 3 groups and count how many are in 1 group) Show this using a picture. Ask students to find a multiplication statement and then a division statement that suits the model. (3 groups × 5 dots in each group = 15 dots, so 3 × 5 = 15 or 15 divided into 3 groups gives 5 dots in each group, so 15 ÷ 3 = 5)

Explain that to find 1/3 of 15, students just have to find how many are in one group. ASK: What question is this the answer to? To guide students if necessary, write 15 3 = 5 and have students write the correct operation symbol (÷) in the blank.

Have students write division statements to find these:

a) 16

of 9 b) 15

of 10 c) 14

of 15

d) 14

of 12 e) 12

of 12 f) 13

of 12

Modelling, Connecting

PRoCESS EXPECTATIoN

Using logical reassuring

PRoCESS EXPECTATIoN

15Number Sense 7-22

ANSWERS: a) 9 ÷ 6 = 1.5 b) 10 ÷ 5 = 2 c) 15 ÷ 4 = 3.75 d) 12 ÷ 4 = 3 e) 12 ÷ 2 = 6 f) 12 ÷ 3 = 4

Find any fraction of a whole number using multiplication and division. ASK: If I know 1/3 of 12 is 4, what is 2/3 of 12? Draw a picture to help explain that 2/3 of 12 is twice as many as 1/3 of 12. If 1/3 of 12 is 4, there are 4 in each group, so to find 2/3 of 12, take 2 groups of 4, or 2 × 4 dots. So 2/3 of 12 is 8.

Have students use multiplication and division to find these fractions of numbers:

a) 25

of 20 b) 37

of 14 c) 35

of 15 d) 47

of 35 e) 89

of 36 ANSWERS:

a) 15

of 20 is 20 ÷ 5 = 4, so 25

of 20 is 2 × 4 = 8

b) 17

of 14 is 14 ÷ 7 = 2, so 37

of 14 is 3 × 2 = 6

c) 15

of 15 is 15 ÷ 5 = 3, so 35

of 15 is 3 × 3 = 9

d) 17

of 35 is 35 ÷ 7 = 5, so 47

of 35 is 4 × 5 = 20

e) 19

of 36 is 36 ÷ 9 = 4, so 89

of 36 is 8 × 4 = 32

A small fraction of a large number can still be a large number. Have students place the fraction 1/40 on a number line between 0 and 1.

0 1

Ask students if they think that 1/40 is a small fraction or a large fraction. Then ASK: If you read that 1/40 of the Canadian population had poor access to medical services, do you think that would be a lot of people or only a few? Tell students that Canada has 30 million people. Have students calculate 1/40 of the Canadian population. (750 000) ASK: Is that a lot of people or only a few? ASK: What if only 1 out of 40 people in the world had poor access to clean water? Would that be a lot of people or a little? What if only 1 out of 1000 people in the world had poor access to clean water? Would that be a lot of people or a little? Tell students that there are 6 billion people and have students do the calculations. (1/40 of 6 000 000 000 = 150 000 000 is still a lot of people, even 1/1000 of 6 000 000 000 is 6 000 000 people) Students might be interested in researching what fraction of people actually have no access to clean water. The actual statistic is more like 1 out of 5, but the point to be left with here is that people who want to convince you that a problem isn’t very important might quote what fraction of people rather than the number of people. Students should be aware of this.


Solving word problems. Tell students that each mathematical word in a word problem can be replaced by a symbol. ASK: What number or symbol would you use to replace each of the following: more than [>], is [=], half [1/2], three-quarters [3/4].

Write on the board:

Calli’s age is half of Ron’s age. Ron is twelve years old. How old is Calli?

Teach students to replace each word they do know with a math symbol and what they don’t know with a blank:

= 1 __ 2 of 12

Calli’s age is half of Ron’s age.

Have them do similar problems of this sort.

a) Mark gave away 3 __ 4 of his 12 stamps. How many did he give away?

( = 3 __ 4 of 12)

b) There are 8 shapes. What fraction of the shapes are the 4 squares? ( of 8 = 4)

c) John won three fifths of his five sets of tennis. How many sets did he win? ( =

3 __ 5 of 5)

Then have students change two sentences into one, replacing the underlined words with what they’re referring to:

a) Mark has 12 stamps. He gave away 3 __ 4 of them. (Mark gave away

3 __ 4 of his 12 stamps)

b) A team played 20 games. They won 11 of them.

EXTRA PRACTICE for Questions 7–11.

1. How many hours are in 5/8 of a day? (5/8 of 24 = 15)

2. By weight, about 1/5 of a human bone is water and 1/4 is living tissue. If bone weighs 120 grams, how much of the bone’s weight is water and how much is tissue?

ANSWER: 1/5 of 120 is 24, so the water weighs 24 grams; 1/4 of 120 is 30, so the tissue weighs 30 grams

3. If Ron studied math for 2/5 of an hour and then history for 1/3 of an hour, how long did he study for altogether?

ANSWER: 2/5 of 60 is 24 minutes and 1/3 of 60 is 20 minutes, so altogether he studied for 44 minutes

4. Is 5/8 of one pizza more than, less than, or the same amount as 1/8 of five pizzas?

17Number Sense 7-22

ANSWER: the same, because both of them are five pieces of size 1/8; whether the five pieces are taken all from the same pizza or one each from five different pizzas doesn’t matter.

Extensions1. a) Find:

12

of 2 13

of 3 14

of 4 15

of 5 16

of 6 17

of 7

Do you see a pattern? Predict 1/384 of 384.

ANSWERS: All answers are 1, because each group consists of one (2 ÷ 2, 3 ÷ 3, and so on) object and you are always taking one (the numerator) group.

b) Find:

23

of 3 35

of 5 35

of 9 715

of 15 811

of 11

ANSWERS: 2, 3, 8, 7, and 8. In every case, the answer is the numerator of the fraction because there is one object in each group and the numerator tells how many groups to take.

2. Explain to your students that you can have a fraction of a fraction! Demonstrate finding half of a fraction by dividing it into a top half and a bottom half:

1 __ 2 of

3 __ 5 is

3 ___ 10

Have them find:

a) 1 __ 2 of

2 __ 9

b) 1 __ 2 of

5 __ 7

Then have them draw their own pictures to find:

c) 1 __ 2 of

3 __ 7 d)

1 __ 2 of

2 __ 5 e)

1 __ 2 of

5 __ 6 f)

1 __ 2 of

4 __ 7

bonus Find two different ways of dividing the fraction

4 __ 7 in half.

ANSWER:

1 __ 2 of

4 __ 7 is 4 ___

14

1 __ 2 of

4 __ 7 is 2 __

7

ASK: Are the two fractions 2 __ 7 and

4 ___ 14 equivalent? How do you know?


PRoCESS ASSESSMENT

Looking for a pattern

PRoCESS EXPECTATIoN

Divide in half top to bottom.

Divide the 4 pieces in half from left to right.


Comparing fractions with like denominators. Have students decide which is more and write the appropriate inequality between the numbers:

Repeat with several examples, eventually having students name the fractions as well:4 7

5 7

3 7

2 7

Which is greater:

1 9 or 2

9 ? 1 13 or 2

13 ? 1 74 or 2

74 ? 1 500 or 2

500 ?

2 9 or 5

9 ? 3 17

or 4 17

? 9 17

or

8 17 ? 35

78 or

8 17 ?

1 709 or 2

709 ? 91 1002 or 54

1002 ?

bonus 7 432 24 401 or 869

24 401 ? 52 645 4 567 341 or 54 154

4 567 341 ?

Then ask students to order three fractions with the same denominator (EXAMPLE: 2/7, 5/7, 4/7) from least to greatest, eventually using bigger numerators and denominators. Repeat with lists of four fractions.

bonus 4 21 , 11

21 , 8 21 ,

19 21 ,

6 21 ,

12 21 ,

5 21

Then write on the board: 7 10 , 13

10

Ask students to think of numbers that are between these two numbers.

NS7-23 Comparing Fractions — IntroductionPage xxxx

CuRRICuLuM EXPECTATIoNSOntario: 6m14; 7m1, 7m7, reviewWNCP: 6N4, review, [R, C]

VoCAbuLARy mixed number improper fraction

GoalsStudents will understand that as the numerator increases and the denominator stays the same, the fraction increases; as the numerator stays the same and the denominator increases, the fraction decreases.


Can name fractions Understands that fractions show same-sized parts

MATERIALS

strips of paper (see Activities)

7m7, [C] Workbook Questions 2 and 4

PRoCESS ASSESSMENT

2 5

3 5

19Number Sense 7-23

Allow several students to volunteer answers. Then ask students to write individually in their notebooks at least one fraction between the following pairs.

a) 4 11 9

11 b) 3 12 9

12 c) 4 16 13

16 d) 21 48

25 48 e) 67

131 72 131

bonus 104 18 301

140 18 301

Comparing fractions with like numerators. Draw on the board:

1 2

1 3

1 4

ASK: Which fraction shows the most: 1/2, 1/3 or 1/4? Do you think one fifth of this strip will be more or less than one quarter of it? Will one eighth be more or less than one tenth?

ASK: Is a half always bigger than a quarter. Is half a minute longer or shorter than a quarter of an hour? Is half a centimetre longer or shorter than a quarter of a metre? When is a half always bigger than a quarter? Allow everyone who wishes to attempt to articulate an answer. Summarize by saying: A half of something is always more than a quarter of the same thing. But if we compare different things, a half of something might very well be less than a quarter of something else. When mathematicians say that 1/2 >1/4, they mean that half of something is always more than a quarter of the same thing; it doesn’t matter what you take as your whole, as long as it’s the same whole for both fractions.

Draw the following strips on the board:

ASK: Are the two wholes the same length? (yes) Ask students to name the fractions and then to tell you which is more. (3/4 > 3/8)

ASK: If you cut the same strip into more and more same-sized pieces, what happens to the size of each piece? (the pieces get smaller)

Draw the following picture on the board to help students see the answer:

1 piece = one whole

2 same-sized pieces in one whole



many same-sized pieces in one whole

ASK: Do you think that one third of a pie is more or less pie than one fifth of the same pie? Would you rather have a piece when the pie is cut into 3


pieces or 5 pieces? Which way will you get more? Ask a volunteer to show how we write that mathematically (1/3 > 1/5).

If you get 7 pieces, would you rather the pie be cut into 20 pieces or 30? Which way will get you more pie? How do we write that mathematically? (7/20 > 7/30)

ordering fractions. Ask students to compare pairs of fractions with the same numerator. EXAMPLES: i) 3/5 and 3/7 ii) 6/17 and 6/15 iii) 5/102 and 5/109

Then ask students to order three or more fractions with the same numerator.

EXAMPLES: 7

13 ,

720

, 75

, 7

15Now give students lists of fractions where two have the same numerator and two have the same denominator, and have students order the fractions (construct the list so that this is possible).

EXAMPLE: 3/5, 3/7, 4/5, but not 3/5, 3/7, 4/7. (The second list is not comparable using only the strategies discussed so far because 3/7 is less than both 3/5 and 4/7 and students cannot yet compare 3/5 and 4/7.)

Have students compare and order mixed numbers and improper fractions using this method.

EXAMPLE: 138

, 198

, 257

ANSWER: Changing 3 1/8 to an improper fraction results in 25/8 which is greater than 19/8 but less than 25/7, by the methods of this section.

Finally, have students compare 13/87 and 14/86 by finding a fraction with the same numerator as one and the same denominator as the other: 13/86 or 14/87. For example, 13/86 is more than 13/87 but less than 14/86, which means 13/87 is less than 14/86. Another way to reason about the problem is that 14/86 has more pieces (14 instead of 13) and each piece is bigger (an eighty-sixth instead of an eighty-seventh), so it represents the bigger fraction.

bonus Have students order the following fractions:

21 28

21 22

21 22

8 200

19 105

13 200 19

28 13 105 19

61

SAy: Two fractions have the same numerator and different denominators.

How can you tell which fraction is bigger? Why? The same number of pieces gives more when the pieces are bigger. The numerator tells you the number of pieces, so when the numerator is the same, you just look at the denominator. The bigger the denominator, the more pieces you have to share between and the smaller the portion you get. So bigger denominators give smaller fractions when the numerators are the same.

ASK: If two fractions have the same denominator and different numerators, how can you tell which fraction is bigger? Why? If the denominators are the same, the size of the pieces are the same. So just as 2 pieces of the same

21Number Sense 7-23

size are more than 1 piece of that size, 84 pieces of the same size are more than 76 pieces of that size.

Have students answer these questions in their notebooks. a) Why is 2/5 greater than 2/7?b) Why is it easy to compare 2/5 and 2/12?

Comparing fractions using 1 whole as a benchmark. Write the two fractions 3/4 and 4/5 on the board. ASK: Do these fractions have the same numerator? The same denominator? (no, neither) Explain that we cannot compare these fractions directly using the methods in this section, so we need to draw a picture:

Have a volunteer shade the fraction 3/4 on the first strip and the faction 4/5 on the second strip. ASK: On which strip is a greater area shaded? On which strip is a smaller area unshaded? How many pieces are not shaded? What is the fraction of unshaded pieces in each strip? (1/4 and 1/5) Can you compare these fractions directly? (yes, they have the same numerator). Emphasize that if there is less unshaded, then there is more shaded. So 4/5 is a greater fraction than 3/4 because the unshaded part of 4/5 (i.e., 1/5) is smaller than the unshaded part of 3/4 (i.e., 1/4 ).

Repeat for other such pairs of fractions. (draw the pictures when the numerators and denominators are small.) EXAMPLES:

7 8 and 8

9 , 7 8 and 5

6 , 89 90 and 74

75 , 3 5 and 4

6 , 72 74 and 34

36 , 56 76 and 39

59

Have students write the following fractions in order from least to greatest and explain how they found the order.13

, 12

, 23

, 34

, 18

ANSWER: 18

, 13

, 12

, 23

, 34

Communicating

PRoCESS EXPECTATIoN

ACTIVITIES 1-2

1. Give each student three strips of paper. Ask students to fold one strip into halves, one into quarters, and one into eighths. Use the strips to find a fraction between

a) 3 8 and 5

8 (one answer is 1 2 )

b) 1 4 and 2


c) 5 8 and 7


Keep these strips for the next lesson.

2. Have students fold a strip of paper (the same length as those folded in ACTIVITy 1) into thirds by guessing and checking. Students should number their guesses. Then ASK: Is 1/3 a good answer for any part of Activity 1? How about 2/3 ?

Reflecting on what made the problem easy or hard

PRoCESS ASSESSMENT


Extensions1. If 13/x > 13/47, what can you say about x?

2. Two players play a game with a red die and a blue die. They make a fraction from rolling the dice as follows: the red die gives the numerator and the blue die gives the denominator. Player A wins if the fraction is more than 1/2 and Player B wins if the fraction is less than 1/2.

a) Ask students to give examples of rolls that result in a win for Player A, a win for Player B, and a tie. Alternatively, give students examples of rolls and ask them to identify the winner, or play the game briefly as a class (you could be Player A and the class as a whole could be Player B). You could record possible plays in a table. EXAMPLE:

b) Is the game fair? If not, who is more likely to win? If necessary, remind students that a fair game is one in which the probability of winning is the same for both players. (The game is not fair because Player A is more likely to win: if the red die is 4 or more, Player A surely wins; otherwise, Player A sometimes wins.)

c) Player B thinks they should change the game so that the blue die gives the numerator and the red die gives the denominator. Will this help Player B win more games? (No, the probability of getting a fraction more than 1/2 is exactly the same in both games.)

d) What fraction, instead of 1/2, should the players use to make the game fair? (1; then A wins if the red die is more than the blue die, and B wins if the blue die is more than the red die)

next guess

First guess

Try folding here too short, so try a little further

red blue fraction Winner

4 3 4/3 Player A

1 5 1/5 Player B

3 6 3/6 tie

Probability

CoNNECTIoN

23Number Sense 7-24

Introduce equivalent fractions. Show several pairs of fractions on fraction strips and have students say which is larger. EXAMPLE:

Include many examples where the two fractions are equivalent. Tell your students that when two fractions look different but actually show the same amount, they are called equivalent fractions. Have students find pairs of equivalent fractions from the fraction strips from NS8-17 Activity 1. Tell them that we have seen other examples of equivalent fractions in previous classes and ask if anyone knows where. (There are 2 possible answers here: fractions that represent 1 whole are all equivalent, as are fractions representing 2 wholes, and mixed fractions have an equivalent improper fraction.)

Finding equivalent fractions using fraction strips. Then have students find equivalent fractions by shading the same amount in the second strip as in the first strip and writing the shaded amount as a fraction:

Show students the fraction strip chart below and have volunteers fill in the blank areas.

NS7-24 Equivalent FractionsPage xxxx

CuRRICuLuM EXPECTATIoNSOntario: 6m14; 7m5, 7m6, reviewWCNP: 7N5, [CN, V]

VoCAbuLARy equivalent fractions

GoalsStudents will understand that different fractions can mean the same amount. Students will find equivalent fractions by using pictures.


Can compare fractions Understands that fractions can name an area or a set

MATERIALS

modelling clay fraction strips (from NS7-23 Activity 1)

1 whole

1 2

1 2

1 4

1 4

1 4

1 4

1 5

1 5

1 5

1 5

1 5

1 2 =

4

8 9

3 4


Shade the fraction 1/2 and then ASK: What other fractions can you see that are equivalent to 1/2 ? (2/4 or 4/8 or 5/10). What other fraction from the chart is equivalent to 3/4?(6/8) Repeat with 8/10, 3/5, 1/5 and 4/10. What fractions on the chart are equivalent to 1 whole?

Finding equivalent fractions using grouping. Draw several copies of a square on the board with two thirds, shaded:

ASK: What fraction of each square is shaded? (2/3) Then draw a line to cut one square into 6 equal parts, draw 2 lines to cut another square into 9 equal parts, draw 3 lines to cut a square into 12 equal parts, and draw 4 lines to cut the last square into 15 equal parts.

Have volunteers name the equivalent fractions shown by the pictures ( 2

3 = 4 6 = 6

9 = 8 12 = 10

15 )

Then write on the board:

For each picture, ASK: How many times more shaded pieces are there? How many times more pieces are there altogether? Emphasize that if each piece (shaded or unshaded) is divided into 4, then, in particular, each shaded piece is divided into 4. Hence if the number of pieces in the figure is multiplied by 4, the number of shaded pieces will also be multiplied by 4: that is why you multiply the top and bottom of a fraction by the same number to make an equivalent fraction.

For the pictures below, have students divide each piece into equal parts so that there are a total of 12 pieces. Then have them write the equivalent fractions with the multiplication statements for the numerators and denominators:

ANSWERS: 2

3 4 4 = 812 1

4 3 3 = 312 5

6 2 2 = 10

12

2 × = 4 — — 3 × = 6

2 × = 6 — — 3 × = 9

2 × = 8 — — 3 × = 12

2 × = 10 — — 3 × = 15

× ×

× ×

× ×

25Number Sense 7-24

Grouping small groups into larger groups to make an equivalent fraction. See Question 3.

EXTRA PRACTICE for Question 3: Group the squares to make an equivalent fraction.

ANSWERS:

15 524 8

= 10 516 8

=

Recall the Activity from NS8-8. Have ready 18 small balls of modelling clay, all the same size: 6 white and 12 red. Ask for two volunteers. Have one volunteer take 1 small white ball and 2 small red balls and thoroughly mix the balls together. At the same time, have another volunteer mix 2 small white balls and 4 small red balls. Discuss with the class which ball they think will look more red when finished.

Next ask for three volunteers. Have each of them mix 2 balls together: one volunteer mixes 2 small white balls together and the other two mix 2 small red balls together. In the end, you will have 1 large white ball and 2 large red balls. ASK: Are all these balls the same size? (yes, because the original balls were all the same size and each larger ball consists of two original balls) Will mixing 1 large white ball and 2 large red balls result in the same colour as mixing 1 small white ball and 2 small red balls? (yes, because the fraction of red modelling clay is 2/3 in each) Verify this directly with the class. Will mixing 1 large white ball and 2 large red balls result in the same colour as mixing 2 small white balls and 4 small red balls? (yes) How do you know? (the amount of white modelling clay and red modelling clay is the same in each; mixing 1 large white and 2 large red balls is just like mixing 2 small white and 4 small red balls because that’s how you made the large balls in the first place) ASK: What equivalent fractions does that show? (1/3 = 2/6 and 2/3 = 4/6)

Discuss how grouping the balls of modelling clay is similar to grouping squares in Question 3. In both cases, you are making larger groups from the same number of equal-sized smaller groups.

ACTIVITy

Give your students 10 counters of one colour and 10 counters of a different colour. Ask them to make a model of a fraction that can be described in at least three different ways.

Here are two possible solutions: 1 = 2 = 4 6 = 2 = 12 4 8 12 4 2

7m6, [V] Workbook Questions 5

PRoCESS ASSESSMENT

Connecting, Modelling

PRoCESS EXPECTATIoN


Extensions1. Use a ruler to draw lines of different lengths on the board, and use an X or shading to mark different fractions of the line. Write two fractions for each line. EXAMPLE:

3 cm 30 mm _____ _______ 7 cm 70 mm ASK: Are these fractions equivalent? Students can draw more lines and write the corresponding equivalent fractions using different units, including decimetres.

2. Write as many equivalent fractions as you can for each picture.

3. List 3 fractions between 1/2 and 1. Hint: Change 1/2 to an equivalent fraction with a different denominator (EXAMPLE: 4/8 ) and then increase the numerator or decrease the denominator. Show your answers on a number line (this part is easier if they increase the numerator instead of decrease the denominator).

4. Ask students to use the patterns in the numerators and denominators of the equivalent fractions below to fill in the missing numbers.

a) 3 = 26 = 39 = 4 = 15 b) 5 = 6 10 = 9

= 12 20 = --

c) 4 = 8 = 9 12 = 12

16 = --

5. Ask students if the same patterning method as in Extension 4 will work to find equivalent fractions in these patterns:

a) 2 3 = 6 = 9 = 5 = 6 b) 1

2 = 3 = 5 = 8 = 10

Have students discuss how these questions differ from the ones above. Ensure that students understand that each fraction in a sequence of equivalent fractions is obtained by multiplying the numerator and denominator of a particular fraction by the same number. In these examples, the patterns are only obtained through adding, not multiplying, so the pattern will not produce a sequence of equivalent fractions.

6. Find a fraction equivalent to 2/3 so that: a) its denominator is 3 more than its numerator (6/9) b) its denominator is 5 more than its numerator (10/15) c) its numerator is a multiple of 3 (e.g. 6/9) d) its denominator is a multiple of 5 (e.g. 10/15)

7. Can you find a fraction equivalent to 3/5 whose denominator is 17 more than its numerator? Hint: Start by listing fractions equivalent to 3/5 in an organized way, and check the differences between numerator and denominator. What do you notice? (list the fractions equivalent to 3/5: 3/5, 6/10, 9/15, … These have denominator - numerator = 2, 4, 6, …. So the difference is always a multiple of 2. 17 is not a multiple of 2, so there is no fraction equivalent to 3/5 whose denominator is 17 more than its numerator)

Revisiting conjectures that were true in one context.

PRoCESS EXPECTATIoN

27Number Sense 7-25

NS7-25 Comparing Fractions using Equivalent Fractions and

NS7-26 Problems and PuzzlesPage 25

Finding fractions with the same denominator to compare fractions. Have students write out sequences of equivalent fractions for two fractions and then find one from each list so that both have the same denominator. Discuss how they can use this to order the fractions.

EXTRA PRACTICE for Workbook p.84 Question 2:

34 8 12 16 20 24 28 32

= = = = = = =

57 14 21 28 35 42 49 56

= = = = = = =

Repeat for 34

and 710

and then for 710

and 57

. Finally, have students attempt to find a common denominator for all three of the fractions above (instead of two at a time). Discuss how this is a lot of work; it is easier to compare them two at a time because the common denominators for the pairs are smaller than the common denominator for all three.

Also, notice that the common denominator is a multiple of both the denominators. You simply made a list of all the multiples until you found one in common. ASK: Have you done this before? Does this remind you of anything? (finding the lowest common multiple)

Finding equivalent fractions by multiplying the numerator and denominator by the same number. Demonstrate how to find equivalent fractions with the same denominators for the three pairs above using the LCM of the denominators:34

and 7

10(LCM of 4 and 7 is 28 so find equivalent fractions with that

denominator: 3 7 214 7 28

×=

× and 5 4 20

7 4 28×

=×

)

CuRRICuLuM EXPECTATIoNSOntario: 6m14; 7m1, 7m2, 7m3, 7m5, 7m6, 7m7, reviewWNCP: 7N5, [C, R, ME, V, CN]

VoCAbuLARy lowest common multiple

GoalsStudents will compare fractions with different denominators by changing both fractions to have the same denominator.


Can compare fractions with the same denominator Can change a fraction to an equivalent fraction with a different denominator Can find lowest common multiples (LCMs)

MATERIALS

Balls and flags from Activity in NS8-8

Changing into a known problem

PRoCESS EXPECTATIoN

Connecting

PRoCESS EXPECTATIoN


34

and 710

(LCM of 4 and 10 is 20: 3 5 154 5 20

×=

× and 7 2 14

10 2 20×

=×

)

710

and 57

(LCM of 7 and 10 is 70: 7 7 4910 7 70

×=

× and 5 10 50

7 10 70×

=×

)

Finally, have students find a common denominator for all three fractions and then find the equivalent fractions with that denominator and order the three fractions. Discuss how this is easier than using lists.

EXTRA PRACTICE: Compare these fractions by using the LCM of the denominators.

a) 35

and 58

b) 23

and 34

c) 23

and 811

Comparing methods of solving problems. Now look at Questions 3 and 4 in the workbook. In this case, would it be easier to compare the fractions in Question 3, two at a time, in order to put them in order for Question 4—or would it be easier to find a common denominator for all 8 fractions? Why? What is different about the two situations? (ANSWER: In the extra practice question at the beginning of the lesson, the lowest common multiple of all 3 denominators was quite large, so finding it required a lot of work; also, comparing 2 fractions at a time required only 3 comparisons because there were only 3 fractions. In Question 3, however, there are 8 fractions, so comparing 2 at a time would be a lot of work; also, the lowest common multiple of all 8 denominators is relatively small.)

using a benchmark to compare fractions. Compare these fractions by comparing them all to 1: 1/4, 4/7, 5/8, 7/10. ASK: By how much is each fraction less than 1? (3/4, 3/7, 3/8, and 3/10) Which fraction is closest to 1? How do you know? (7/10 is closest to 1, because 3/10 is the smallest fraction in the list above) Write the fractions in order from smallest to greatest. (They were in order to begin with: 1/4, 4/7, 5/8, 7/10.) Have students decide which is greater between 37/40 and 43/46. Which fraction is closer to 1?

Have students write many equivalent fractions for 3/4:

34 8 12 16 20 24

= = = = =

Then have students order the following fractions by comparing them all to 3/4:

58

, 1116

, 1724

, 78

, 1316

, 1924

(ANSWER: 58

, 1116

, 1724

, 1924

, 1316

, 78

)

bonus for Question 6: How much more than 1/2 is 4/7? (ANSWER: 4/7 = 8/14 is 1/14 more than 1/2)

Where does 4/7 fit in the list in Workbook Question 6 c)? (It is the smallest fraction because it is greater than 1/2 by the least amount.)

Reflecting on what made the problem easy or hard

PRoCESS EXPECTATIoN

7m2, 7m3, 7m7, [R, C] Workbook p.84 Question 3

PRoCESS ASSESSMENT

Selecting tools and strategies

PRoCESS EXPECTATIoN

Using logical reasoning

PRoCESS EXPECTATIoN

29Number Sense 7-25

Have students decide which of these fractions is greater and explain how they know: 101/200 or 151/300?

Comparing mixed numbers and improper fractions using equivalent fractions. You can compare 13/4 and 3 2/7 two ways:

1. Change both to improper fractions (13/4 and 23/7) and then to fractions with the same denominator ( 13 7 91

4 7 28×

=×

and 23 4 927 4 28

×=

× ).

2. Change both to mixed numbers (3 1/4 and 3 2/7) and then compare the fractional parts only.

Have students do both. Do they get the same answer both ways? Which way is easier? Why? (Some students might say that using mixed numbers is easier because the numbers involved in the fractional parts only are smaller compared to the numbers involved in the improper fraction; others might say that using improper fractions is easier because they find it easier to change mixed numbers to improper fractions than to change improper fractions to mixed numbers. Accept all answers.)

EXTRA PRACTICE: Compare the following using equivalent fractions. Decide whether to change to improper fractions or mixed numbers.

a) 185

and 739

185

= 335

, so compare 35

to 79

. These are equivalent to

3 9 27

5 9 45×

=×

and 7 5 359 5 45

×=

×, so 73

9is greater.

Reflecting on other ways to solve the problem, Reflecting on what made the problem easy or hard

PRoCESS EXPECTATIoN

Give students their balls and flags from the Activity in NS7-18. Have students organize themselves into groups with people who chose the same two colours they did. For example, suppose five people chose red and blue as their two colours. Have those students order their balls by colour, from most red to least red, and then order the fractions for red from the recipes in order from greatest to least. Each student in the group should use equivalent fractions with the same denominator to check the ordering of the fractions. Ask the class as a whole why they might expect slight disagreements between the results from using modelling and their individual results from checking mathematically. Which order of fractions do they think will be the correct order? What mistakes may have been made when comparing the balls of modelling clay? (it might be hard to tell which has more red; the colours may not have been mixed thoroughly; the balls might not have exactly the same amount of modelling clay)

ACTIVITy

Modelling, Mental math and estimation

PRoCESS EXPECTATIoN

8m7, [C]

PRoCESS ASSESSMENT


b) 354

and 12121

c) 378

and 22130

is greater

( 12121

is greater) ( 378

is greater)

Extensions1. Is it easier to compare fractions using numerators or denominators?

Write these fractions on the board: 5/43, 2/19

ASK: Would you compare these fractions using common numerators or common denominators? Why? Students may suggest that it will be easier to use common numerators because the numerators are much smaller than the denominators, so the calculations will be easier. Ask students to compare the fractions both ways. You might have half the class do the comparison one way and the other half do it the other way. Encourage students to do only those calculations that are necessary to determine which fraction is greater. Then discuss the results:

To compare the fractions using common denominators, we compare

5 1943 19

××

and 2 4319 43

××

.

To compare them using common numerators, we compare

5 243 2

××

and 2 519 5

××

.

Point out that in the first case, we don’t need to compute the common denominator 43 × 19, we just need to compare the numerators. Since 5 × 19 = 95 and 2 × 43 = 86, 5/43 is greater (when fractions have the same denominator, the one with the larger numerator is greater). Similarly, to compare the fractions using common numerators, we don’t need to compute the common numerator, only the denominators (when fractions have the same numerator, the one with the smaller denominator is greater). But the denominators we need to calculate—5 × 19 and 2 × 43— are the same as the numerators we calculated using the other method!

While it is tempting to think that the comparison using common numerators is easier because the numerators are smaller than the denominators, the computations required are the same in both cases. Students can compare more such pairs of fractions to confirm that this is true in general.

Then ask students to compare fractions with small denominators and large numerators. EXAMPLE: 81/4, 37/2Is one method easier than the other in this case?

2. If 2 67 x

> , what can you say about x? (x is greater than 21)

Reflecting on what makes a problem easy or hard; Reflecting on other ways to solve a problem

PRoCESS EXPECTATIoN

Revisiting conjectures that were true in one context

PRoCESS EXPECTATIoN

31Number Sense 7-27

NS7-27 Adding and Subtracting Fractions and

NS7-28 Lowest Terms

What it means to add fractions. Draw two large circles and divide them into four quarters each, shading them as shown.

Explain that these are two plates with pieces of pizza on each. How much pizza do you have on each plate? Write the fractions beneath the pictures. Tell students that you would like to combine all the pieces onto one plate, so put the + sign between the fractions and ask a volunteer to draw the results on a different plate. How much pizza do you have now?

Draw on the board:

1

4 + 2

4 =

Tell your students that you would like to regroup the shaded pieces so that they fit onto one circle. SAy: I shaded two fourths of one circle and one fourth of another circle. If I move the shaded pieces to one circle, what fraction of that circle will be shaded? How many pieces of the third circle do I need to shade? Tell students that mathematicians call this process adding fractions. Just as we can add whole numbers, we can add fractions too.

Do several examples of this, like 1/5 + 2/5, 1/3 +1/3, never extending past 1 whole circle. ASK: You are adding two fractions. Is the result a fraction too? Does the size of a piece change when we transport pieces from one plate to the other? What part of the fraction reflects the size of the piece—top or bottom, numerator or denominator? When you add fractions, which part stays the same; the top or the bottom, the numerator or the denominator? What does the numerator of a fraction represent? (the number of shaded pieces) How do you find the total number of shaded pieces when you move them to one pizza? What operation do you use?

CuRRICuLuM EXPECTATIoNSOntario: 7m1, 7m6, 7m24WNCP: 7N5, [R, V]

VoCAbuLARy fraction regrouping numerator denominator equivalent fractions lowest common multiple (LCM) lowest common denominator (LCD) lowest terms reduce to lowest terms greatest common factor (GCF)

GoalsStudents will add and subtract fractions with the like and unlike denominators, and will reduce the result to lowest terms.


Can name fractions, including mixed numbers and improper fractions Can add and subtract whole numbers Can find equivalent fractions Can find lowest common multiples Can find greatest common factors

Page 25


Show more examples using pizzas, and then have students add the fractions without pizzas. Assign lots of questions. EXAMPLES: 3/5 + 1/5, 2/7 + 3/7 , 2/11+ 4/11, etc. Enlarge the denominators gradually.

bonus Add: 18 134

+ 45 134

67 1 567 + 78

1 567 67 456 + 49

456

bonus Add three or more fractions:

3 17

+ 1 17

+ 2 17

5 94

+ 4 94

+ 7 94

3 19

+ 2 19

+ 5 19

+ 1 19

+ 3 19

Introduce subtracting fractions. Return to the pizzas and say that now you are taking pieces of pizza away. There was 3/4 of a pizza on a plate. You took away 1/4. Show on a model the one piece you took away:

How much pizza is left? Repeat with several more fractions and pizzas, and repeat the sequence of questions you asked for addition.

EXAMPLES: 78

- 18 , 5

6 - 3

6 , 35

- 25 .

Then have students subtract some fractions without using pizzas. Assign lots of questions; enlarge the denominators gradually.

EXAMPLES: 11 8

- 13 8

, 1056

- 7 56

, 12 803

- 5 803

.

Finally, have students solve problems that involve both addition and subtraction (see Workbook Question 6).

Adding fractions can result in more than one whole. Draw on the board:

34 + 2

4 =

Ask how many parts are shaded in total and how many parts are in one whole circle. Tell your students that, when adding fractions, we like to regroup the pieces so that they all fit onto one circle. ASK: Can we do that in this case? Why not? Tell them that since there are more pieces shaded than in one whole circle, the next best thing we can do is to regroup them so that we fit as many pieces onto the first circle as we can and then we put only the leftover parts onto the second circle.

Draw on the board:

ASK: How many parts are shaded in the first circle and how many more parts do we need to shade in the second circle? Ask a volunteer to shade that many pieces and then tell them that mathematicians write this as:

34 + 24 = 54 = 1 14

Add mixed numbers and improper fractions. EXAMPLES: 54 + 2 34

33Number Sense 7-

The role of zero in adding and subtracting fractions. Have students do these problems:

a) 35

- 35 b) 27 + 07 c) 38 - 08 d) 67 - 0 e) 123 + 0

Point out that any fraction with 0 in the numerator is equal to 0, since 0 divided by any number is 0.

using what you know to add fractions with unlike denominators. Review adding fractions with the same denominator. Then challenge students to find a way to add fractions with different denominators. Discuss the strategy of changing the problem into one they already know how to do. Emphasize that if students know how to add fractions with the same denominator, and can change fractions into ones with the same denominator, then they can add any pair of fractions.

Challenge students to find two fractions with the same denominator so that they can add 1/3 and 3/5:

13 6 9

= = = = = =

35 10

= = = = = =

Explain that since 1 53 15

= and 3 95 15

= , then 1 3 5 9 143 5 15 15 15

+ = + =

Have students use this method to add: a) 2 1

3 5+

b) 1 24 3

+ c) 3 1

4 5+

Have students subtract, using their lists above: a) 2 1

3 5- b) 2 1

3 4-

c) 3 14 5

-

The lowest common denominator (LCD). Explain that by finding the first fraction in each list with the same denominator, students are finding the lowest number that is a multiple of both denominators. Ask students if they know the name for this number. (lowest common multiple) ASK: When else have you used the lowest common multiple of the denominators of fractions? (when comparing fractions, students found two fractions with the same denominator) Tell students that because the lowest common multiples of the denominators are so useful when working with fractions—either comparing or adding them—mathematicians have come up with a name specifically for the lowest common multiple of the denominators. The lowest common denominator of two or more fractions is the lowest common multiple of the denominators.

Adding and subtracting fractions by finding the lowest common denominator. See Questions 1 and 2 on the worksheet.

Changing into a known problem

PRoCESS EXPECTATIoN


Identifying fractions in lowest terms. A fraction is in lowest terms when the only whole number that divides into its numerator and denominator is 1, i.e., 1 is the only factor of both the numerator and denominator. Have volunteers decide if each fraction is in lowest terms: 2/6, 3/5, 1/4, 2/4. Students could do this by listing the factors of each numerator and denominator and seeing if there are any common factors other than 1.

Have students copy in their notebooks only those fractions that are in lowest terms: 3/6 4/7 4/8 4/9 4/10 3/7 2/8 2/9 3/9 bonus 12/50 42/96 36/175

Reducing fractions to lowest terms. Tell students that the process of rewriting a fraction in lowest terms is called reducing the fraction to lowest terms. It is the most convenient form for a fraction to be in because it names the fraction using the smallest possible numbers. Emphasize that “reducing” here does not mean reducing the value of the fraction—the value stays the same; only the numerator and denominator are reduced.

Remind students that to make an equivalent fraction with a greater numerator and denominator, they can multiply both the numerator and denominator by the same number. To make an equivalent fraction with a smaller numerator and denominator, they can divide both the numerator and denominator by the same number. This is called reducing the fraction. Have students reduce the following fractions by dividing both the numerator and denominator by the same number; continue reducing until the fraction is in lowest terms. 700/750 (= 14/15) 336/504 (= 2/3) 378/420 (= 9/10)

Explain that in order to divide both numerator and denominator by the same number, that number has to be a factor of both. So to get the smallest possible numerator and denominator, divide by the greatest common factor of the numerator and denominator.

Have students add and subtract fractions by finding the LCD (lowest common denominator) and then reduce their answer to lowest terms. See Workbook p.90 Question 3. Extension1. Find the next two fractions in the pattern: 1

2, 2

6 , 3

12 , 4

20Two solutions: 1. Consider separately the patterns in the numerators (1, 2, 3, 4, 5, 6, 7,… ) and denominators (2, 6, 12, 20, 30, 42, 56,…). The gaps between terms in the denominators are 4, 6, 8, 10, 12, 14,…, which gives the pattern “start at 4, add 2 each time.”

2. Reduce the fractions to lowest terms: 1/2, 1/3, 1/4, 1/5. The pattern of lowest terms continues as follows: 1/6, 1/7, 1/8,…. But in the actual pattern, the numerators and denominators of the first fraction are multiplied by 1, 2, 3, 4,…, so the pattern continues as follows: 5/30, 6/42, 7/56.

7m1, [R]

PRoCESS ASSESSMENT

35Number Sense 7-29

Add mixed fractions with the same denominator using pictures. Start with problems where the fractional parts add to less than 1, then progress to problems where students will need to simplify their answer.

EXAMPLE: Start with 3 14 18 8

+ :

+ = = 4 15 58 2

=

Progress to 5 74 1

8 8+ :

+ =

= 4 16 68 2

= Subtract mixed fractions with the same denominator using pictures. See Workbook Question 1. Start with subtraction problems where both the whole number part and the numerator of the fractional part of the number being subtracted are smaller than in the number you are subtracting from.

EXAMPLE: 7 35 28 8

- (not 3 75 28 8

- )

Then progress to problems where the fractional part of the number being taken away is bigger than the fractional part of the number it is being subtracted from. EXAMPLE: 5 3/8 – 2 7/8. ASK: How is this problem different from the problems you have seen so far? (the fractional part of the smaller number is bigger) Can you still subtract by using a picture? (Yes, but you would have to cut one of the whole pies in 5 3/8 into 8 pieces so that you look at the fraction as 4 and 11/8 rather than 5 and 3/8:

Now you can take away 2 and 7/8. This leaves you with 2 wholes and 4 eighths.

So 3 7 11 7 4 15 2 4 2 2 28 8 8 8 8 2

- = - = = .

Add mixed numbers with different denominators. Add the whole number parts and the fractional parts separately. If the fractional parts add to more than 1, students will need to simplify. Have students practise this skill as in Question 5 on the worksheet.

NS7-29 Adding and Subtracting Mixed NumbersPages 29–30

CuRRICuLuM EXPECTATIoNSOntario: 7m1, 7m3, 7m6, 7m24WNCP: 7N5, [R, CN, V]

VoCAbuLARy Mixed numbers Improper fractions

GoalsStudents will add and subtract mixed numbers and improper fractions with like denominators.


Can add and subtract proper fractions with like and unlike denominators Can translate between mixed numbers and improper fractions

PRoCESS EXPECTATIoN


Subtract mixed numbers with different denominators. Subtract the whole number parts and the fractional parts separately. If the fractional part of the smaller fraction is bigger than the fractional part of the bigger fraction, students will need to first rewrite the bigger fraction by taking 1 away from the whole number part and adding 1 to the fractional part (resulting in an improper fractional part). See Question 6.

Adding and subtracting mixed numbers by using improper fractions. Since each mixed number can be written as an improper fraction, students can rewrite the mixed numbers as improper fractions and add or subtract as done in NS8-18.

Have students verify that this method gives the same answer as the method outlined in Questions 1 through 6.

Students can then discuss which method they find easier and why. For example, because the numerators of the fractional part of the mixed numbers are smaller than the numerators of the improper fraction, using mixed numbers might be considered easier. However, it is still useful to know both methods as a self-checking mechanism. Also, some students might find using improper fractions easier because then they don’t have to worry about regrouping at the end.

8m7, [C] Workbook Question 29

PRoCESS ASSESSMENT

Reflecting on what made the problem easy or hard, Reflecting on other ways to solve a problem

PRoCESS EXPECTATIoN

8m1, [R] Workbook Questions 8–10

PRoCESS ASSESSMENT

37Number Sense 7-30

EXTRA PRACTICE for Questions 1 and 2: Students can draw their own number lines on grid paper if it helps them.

a)

253

8 b) 174

9

So 28 53

- = _____ So 19 74

- = _____

c) 538

10 d) 7349

41

So 10 – 538

= _____ So 741 349

- = _____

EXTRA PRACTICE for Question 3:

a) 25 37

- b) 310 54

- c)57 28

- d) 537 307

- EXTRA PRACTICE for Question 4:

a) 243

183

So 1 28 43 3

- = _____

b) 354

1104

So 1 310 54 4

- = _____

c) 579

2129

So 2 512 79 9

- = _____

d) 5387

1427

So 1 542 387 7

- = _____

EXTRA PRACTICE for Question 5:

a) 4 55 37 7

- b) 1 310 54 4

- c) 7 56 28 8

- d) 4 537 327 7

-

CuRRICuLuM EXPECTATIoNSOntario: 7m1, 7m2, 7m7, 7m7, 7m24 WNCP: 7N5, [R, C, ME, PS]

GoalStudents will mentally subtract mixed numbers with like denominators.


Can subtract by counting forward Can add and subtract fractions with like denominators

NS7-30 Mental Math and

NS7-31 Problems and PuzzlesPage 31

Mental Math and Estimation

PRoCESS EXPECTATIoN


e) 1 28 33 3

- f) 1 512 88 8

- g) 4 87 29 9

- h) 3 556 497 7

-

Extensions1. Subtract mentally by mentally changing to a common denominator:

a) 1 35 22 4

- b) 1 16 34 2

- c) 5 37 28 4

- d) 1 16 44 8

-

e) 1 15 16 2

- f) 1 15 16 3

- g) 1 25 12 3

- h) 1 19 43 2

-

ANSWERS:

a) 324

b) 324

c) 748

d) 128

e) 4 23 36 3

= f) 536

g) 536

h) 546

DISCuSS: Which questions have the same answer? Can you see why? Parts a) and b) have the same answer because both fractions in b) are 3/4 more than the corresponding fractions in a). Parts f) and g) have the same answer because both fractions in g) are 1/3 more than the corresponding fractions in f).

2. Have students decide which two whole numbers the answer should be between by estimating:

a) 1 18 35 4

- b) 3 28 35 7

- c) 3 38 34 5

-

d) 4 38 35 8

+ e) 1 28 33 5

+

SoLuTIoNS: a) Because 1/4 is slightly more than 1/5, the answer will be slightly less than 8 – 3 = 5. b) Because 2/7 is less than 3/5, the answer will be more than 8 – 3 = 5, but less than 6 because 3/5 is less than 1. c) Because 3/5 is slightly less than 3/4, the answer will be slightly more than 8 – 3 = 5. d) 4/5 + 3/8 is more than 1 because 4/5 is further from 1/2 than 3/8 is, so the answer will be more than 12. e) 1/3 + 2/5 is less than 1/2 + 1/2 = 1, so the answer will be between 11 and 12.

Students should then check their estimates by calculating the answers.

[ME] Workbook p.94 Question 5

7m1, 7m2, [PS] Workbook p.95 Question 1

7m7, [C] Workbook p.96 Question 11

7m3, [R] Workbook p.96 Question 16

PRoCESS ASSESSMENT

[ME]

PRoCESS EXPECTATIoN

ns7-18 fractions

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