Repeated Addition of Fractions 1

OPEN-ENDEDPathway 1

You will need• materials for

modelling fractions (e.g., Fraction Strips (BLM 10), Fraction Circles (BLM 18), pattern blocks, or other commercial materials)

• 2 cm Grid Paper (BLM 11)

• coloured counters

Imagine you have a punch bowl of juice and some glasses.

Part A

• Choose a number of glasses between 5 and 10. Assume each glass holds 1 cup of juice when it’s full.

• Choose what fraction of a cup of juice you want in each glass.

• Estimate how many cups of juice you will need altogether to put that amount into all the glasses.

• Exactly how many cups will you need? Show or explain your answer.

number of glasses: ________ fraction of a cup in each: ________

estimated amount of juice: ________ exact amount: ________

• Repeat 2 more times. Use a different number of glasses and a fraction of a cup with a different denominator each time.





Repeated Addition of Fractions

• You can model fraction addition using pattern blocks, fraction circles, fraction strips, counters on grids, and number lines.

• You can write improper fraction answers as mixed numbers so they are easier to understand.

e.g., 54

5 114

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Copyright © 2012 by Nelson Education Ltd.114

Leaps and BoundsRepeated Addition of Fractions, Pathway 1

5 1/2

3 cups 2 1/2 cups

If each block is 1/2, when you put 5 together, it makes 2 1/2 wholes.

10 3/4

7 cups 7 1/2 cupsTen 3/4 circles make 7 whole circles and 2 fourths, or 7 1/2.

9 3/5

6 cups 5 2/5 cupsNine 3/5s on a number line ends up at 5 2/5.

e.g.,

e.g.,

e.g.,

Part B

• Choose a number of glasses between 5 and 10 to fill to the top using the ladle. Assume each glass holds 1 cup.

• Choose a fraction of a cup that your ladle will hold.

• Estimate how many times you will need to use the ladle to fill all the glasses.

• Exactly how many times will you use the ladle? Show or explain your answer.

number of glasses: ________ The ladle holds ________ of a cup.

estimated number of ladles: ________ exact number: ________

• Repeat 2 more times. Use a different number of glasses and a fraction with a different denominator each time.






Leaps and Bounds Repeated Addition of Fractions, Pathway 1

6 1/2

about 12 12 ladles

If 2 ladles fill 1 glass, 12 ladles fill 6 glasses.

about 12 12 ladles

4 ladles fill 3 glasses, so 3 times that many or 12 will fill 9 glasses.

8

9 3/4

If you use 8 ladles, each 2/3 cup, to pour juice into 8 glasses, there are still 8 thirds left to fill and 8 thirds makes 2 2/3. That's 10 2/3 altogether.

about 12 10 2/3 ladles

2/3

e.g.,

e.g.,

e.g.,

GUIDEDPathway 1





A recipe calls for 23 cup of sugar to make 12 cookies.How many cups of sugar are needed to make 60 cookies?

You can solve the problem using different strategies.

• You can use fraction materials to model the problem.

Count the number of thirds of cups you need to make 60 cookies.

12 24 36 48 60

It takes 10 thirds, or 103 cups of sugar to make 60 cookies.

23

1 23

1 23

1 23

1 23

5 103

5 3 23 5 103 , which is 3 13 cups of sugar.

• You could also solve the problem by reasoning like this:

If 23 of a cup makes 12 cookies, then 13 of a cup makes 6 cookies. 60 cookies is 10 times 6 cookies, so you need 10 sets of 13 cups. That is 10

3 , or 3 13 cups.

Suppose you have exactly 4 cups of sugar. How many batches of 12 cookies can you make?

• You can solve the problem by modelling it.

You can draw 4 wholes (as 4 rectangles) and divide each into thirds. Then circle and count the number of sets of 23 in the 4 wholes.

1 2 3 4 5 6

There are 6 sets of 23 in 4 wholes, so you can make 6 batches of cookies.

Repeated Addition of Fractions

• Some multiplication problems can be solved using repeated addition.

• Some division problems can also be solved using repeated addition.

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Repeated Addition of Fractions, Pathway 1 Leaps and Bounds

• You can think about the rectangle model in 3 ways:

– As division, 4 4 23 5 6, since there are 6 sets of 23 in 4 wholes.

– As multiplication, 6 3 23 5 4, since 6 sets of 23 is 4.

– As repeated addition, 23 1 23 1 23 1 23 1 23 1 23 5 123 ,

since adding 23 six times is 123 , or 4.

Try These

1. What repeated addition does each model show?

a)

b)

c)

2. How do you know that each statement is true?

a) 8 groups of 35 is more than 4.

b) 6 groups of 34 is more than 4.

c) There are more than 5 groups of 25 in 3.

• You can write improper fraction answers as mixed numbers so they are easier to understand.

e.g., 54

5 114

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Repeated Addition of Fractions, Pathway 1Leaps and Bounds

e.g., 2/3 + 2/3 + 2/3 + 2/3 + 2/3 + 2/3

3/5 + 3/5 + 3/5 + 3/5 + 3/5 + 3/5

e.g., 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2

e.g., 8 groups of 3/5 is 24/5

and that's more than 20/5 = 4.

e.g., 6 groups of 3/4 is 18/4 and

that's more than 16/4 = 4.

e.g., 5 groups of 2/5 is

10/5 = 2, so you need more than 5 groups for 3.

3. a) Suppose you need to figure out 5 4 14. Why might it be helpful to think about how many quarters are in $5?

b) What fraction division would be like figuring out how manydimes are in $12?

________________

4. Calculate. Show or explain your thinking.

a) 34

1 34

1 34

1 34

1 34

5 ________ d) 8 3 25

5 ________

b) 38

1 38

1 38

1 38

5 ________ e) 2 4 12

5 ________

c) 3 3 45

5 ________ f) 3 4 34

5 ________


Repeated Addition of Fractions, Pathway 1 Leaps and Bounds

e.g., Quarters are 1/4 of a dollar, so you can use 4 quarters in $1

12 ÷ 1/10

e.g., 5 groups of 3 fourths is 15 fourths.

15/4 16/5

e.g., 8 groups of 2 fifths is 16 fifths or 16/5.

12/8

e.g., 4 groups of 3 eighths is 12 eighths, or 12/8.

4

e.g., There are 4 halves in 2 wholes.

12/5

e.g., 4/5 + 4/5 + 4/5 = 12/5

4

e.g., There are four 3/4s in 3 wholes.

to figure out the number of quarters in $5.

5. Fill in the blank to complete each equation.

a) ________ 3 59

5 459

c) 3 3 ________ 5 123

b) ________ 3 78

5 7 d) 5 3 ________ 5 6

6. a) Calculate.

6 3 56

5 ________ 5 3 45

5 ________ 8 3 38

5 ________

b) What do you notice about the equations in part a)?

7. Calculate. How are the calculations in each pair related?

a) 3 3 45

5 ________ 6 3 45

5 ________

b) 4 3 34

5 ________ 5 3 35

5 ________

c) 3 4 35

5 ________ 3 4 15

5 ________

d) 2 4 23

5 ________ 1 4 13

5 ________

8. Suppose you add 45 repeatedly. How do you know the sum will either be a fraction with a denominator of 5 or a whole number? If you know how

addition, multiplication, and division are related, you can use the operation you are most comfortable with to solve a problem.

FYI


Repeated Addition of Fractions, Pathway 1Leaps and Bounds

9

8

4/3

6/5

5 4 3

e.g., The denominator matches the first number and the product is

the same as the numerator.

e.g., 6 x 4/5 is 2 times as much as 3 x 4/5.

12/5 24/5

15

e.g., 3 ÷ 1/5 is 3 times as much as 3 ÷ 3/5.

3

e.g., They have the same answer.

e.g., They have the same answer.

3 3

3 3

e.g., When you are adding fifths, you always end up with fifths (or 5

as a denominator). If you end up with a numerator that is a multiple of 5, the fraction can be renamed as a whole number.

OPEN-ENDEDPathway 2

You will need• 2 cm Grid Paper

(BLM 11)• coloured counters

Nellie is planting vegetables in rows in her garden.

Part A

• Choose 4 crops for Nellie’s garden. Record the names in the chart below.

• Choose a mixed number between 1 and 5 for the number of rows for each crop. (A mixed number is the sum of a whole number and a proper fraction.) Use a different denominator for each fraction part.

Crops for Nellie’s Garden

Crop A Crop B Crop C Crop D

Name of crop

Number of rows

• Choose 4 of the pairs of crops in the chart below. Determine the total number of rows for both crops. Record your answers in the chart.

• Estimate to see if your answer is reasonable.

• Show or explain your work in the space provided.

Number of Rows of Crops

Crop pairs

Total number of rows Estimate

A and B

A and C

A and D

B and C

B and D

C and D

Adding and Subtracting Mixed Numbers

I could plant lettuce, carrots, peppers, and zucchini.


Leaps and BoundsAdding and Subtracting Mixed Numbers, Pathway 2

tomatoes lettuce cucumbers carrots

2 1/101 3/53 1/42 1/2

65 3/4

4 1/24 3/5

54 17/20

5 1/25 35/100

e.g., A + B: 2 1/2 + 3 1/4 = 5 3/4: 2 1/2 + 3 1/4 = 5 3/4, since 2 + 3 = 5 and 1/2 + 1/4 = 3/4. A + D: 2 1/2 + 2 1/10 = 4 3/5: 2 1/2 + 2 1/10 = 2 + 2 + 1/2 + 1/10 = 4 + 5/10 + 1/10 = 4 6/10 = 4 3/5 B + C: 3 1/4 + 1 3/5 = 4 17/20: 3 1/4 + 1 3/5 = 3 + 1 + 1/4 + 3/5 = 4 + 1/4 + 3/5 I used a grid to add 1/4 + 3/5 = 17/20.

B + D: 3 1/4 + 2 1/10 = 5 35/100: 3 1/4 + 2 1/10 = 5 + 1/4 + 1/10 = 5 + 25/100 + 10/100 = 5 35/100

e.g.,

e.g.,

Part B

• Choose 4 of the pairs of crops in the chart below. Determine how many more rows there are of one than the other in each pair. Record your answers in the chart.

• Estimate to see if your answer is reasonable.

• Show or explain your work in the space below.

Comparing the Number of Rows of Crops

Crop pairs

Difference between rows

Estimate

A and B

A and C

A and D

B and C

B and D

C and D

• Any mixed number can be written as an improper fraction and vice versa.

e.g., 113

5 43

114

5 2 34

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Leaps and Bounds Adding and Subtracting Mixed Numbers, Pathway 2

19/10

1/42/5

1 3/41 13/20

11 3/20

e.g., A and C: 2 1/2 - 1 3/5 = 9/10 2/5 + 1/2 = 4/10 + 5/10 = 9/10

A and D: 2 1/2 - 2 1/10 = 4/10 2 - 2 = 0 and 1/2 = 5/10, so 2 5/10 - 2 1/10 = 4/10

B and D: 3 1/4 - 2 1/10 = 1 3/20: 3 - 2 = 1 and 1/4 > 1/10, so I needed to know 1/4 - 1/10. I filled 1 out of 4 rows (1/4) and then moved counters so I could take away 1 column (1/10). I was left with 6/40 = 3/20 of the grid filled.

B and C: 3 1/4 - 1 3/5 = 1 13/20 2/5 + 1 + 1/4 = 1 + 2/5 + 1/4 = 1 13/20

e.g.,

GUIDEDPathway 2





Kyle was serving lasagna at the community dinner. He served 2 25 meat lasagnas and

123 vegetarian lasagnas.

How much lasagna did he serve altogether?

You can use different strategies to add the mixed numbers 2 25 1 12

3 to figure out how much lasagna Kyle served.

• You can estimate the sum.

2 25 1 123 is between 3 and 5, since

the sum is more than 2 1 1 5 3 and less than 3 1 2 5 5.

• You can add the whole number parts mentally and use a grid model to add the fraction parts.

2 25

1 123

5 3 1 25

1 23

A 3-by-5 grid makes it easy to show both fifths and thirds:

– You can model 25 by placing counters in 2 of the 5 columns.

– You can model 23 by placing counters in 2 of 3 rows.

– You can move double counters to empty squares to model 1 115.

25

1 23

25

1 23

5 11

15

That’s 3 1 1 115, or 4 1

15 altogether.

• You can write each fraction as an improper fraction and then add using equivalent fractions with the same denominator.

2 25

1 123

5 125

1 53

5 3615

1 2515

5 6115

, or 4 1

15

Adding and Subtracting Mixed Numbers

• A mixed number is the sum of a whole number and a proper fraction (less than 1).

e.g., 2 34

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• Any mixed number can be written as an improper fraction and vice versa.

e.g., 113

5 43

114

5 2 34

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Leaps and BoundsAdding and Subtracting Mixed Numbers, Pathway 2

You can use 2 25 2 123 to figure out how much more meat lasagna

than vegetarian lasagna Kyle served.

• You can estimate the difference.

2 25 2 123 is about 1, since 2 2 1 5 1 and 25 and 23 are close.

• You can subtract by adding up on a number line.

31 1 1 2 2 2 2 223

13

13

25

25

15

45

35

13 1

25 5

515 1

615, which is 11

15.

Since 123 1 11

15 5 2 25, then 2 25 2 123 5 11

15.

• You can write each mixed number as an improper fraction and then subtract using equivalent fractions with the same denominator.

Try These

1. What addition does each model show?

a)

b)

42 2 2 3 3 3

1

23

13

23

13

13

c)

2 25

2 123

5 125

2 53

5 3615

2 2515

5 1115


Leaps and Bounds Adding and Subtracting Mixed Numbers, Pathway 2

3 1/3 + 2 1/4

2 2/3 + 1 1/3

1 2/3 + 1 3/4

2. What subtraction does each model show?

a)

b)

42 2 2 3 3 3 3 323

13

13

25

45

15

45

35

c)

3. Calculate. Show your work.

a) 3 12

1 5 18

5 ________ c) 2

35

1 3 18

5 ________

b) 7 13

1 2 35

5 ________

d) 189

1 56

5 ________

Adding and Subtracting Mixed Numbers, Pathway 2 Copyright © 2012 by Nelson Education Ltd.124

Leaps and Bounds

3 2/3 - 1 1/3

3 4/5 - 2 2/3

e.g., 1 4/5 - 2/3

8 5/8

9 14/15

5 29/40

2 13/18

e.g., 3 4/8 + 5 1/8 = 8 5/8

e.g., 7 5 + 2 9 = 9 14 15 15 15

e.g., 2 24/40 + 3 5/40 = 5 29/40

e.g., 1 48 + 45 = 1 93 54 54 54 = 2 39 54 = 2 13 18

Adding and Subtracting Mixed Numbers, Pathway 2

Although you can change fractions to decimals to calculate, knowing different strategies might make it easier to work with the fractions.

FYI

4. Calculate. Show your work.

a) 5 12

2 3 18

5 ________ c) 6 18

2 2 56

5 ________

b) 7 13

2 2 35

5 ________ d) 156

2 89

5 ________

5. Create a problem that could be solved using each calculation.

a) 3 2 123

b) 4 12

1 2 12

6. Circle the 2 mixed numbers that have a sum that is close to 6.

2 23

4 12

3 14

2 13

7. The sum of 2 mixed numbers is 4 13, and they are about 2 apart.What could the numbers be? ________ ________

8. It is often easier to estimate the sums and differences of 2 mixed numbers than of 2 improper fractions. Why?


Leaps and Bounds

2 13/18

4 11/15

3 7/24

17/18

e.g., I had 3 pies and 1 2/3 were eaten. How many pies are left?

e.g., A recipe called for 4 1/2 cups of flour to be

added and then another 2 1/2 cups. How much flour is

needed altogether?

e.g., 3 1/6 1 1/6

e.g., You can sometimes ignore the proper fraction parts and just work

with the whole number parts.

e.g., 5 4/8 - 3 1/8 = 2 3/8 e.g., 6 3/24 - 2 20/24 = 5 27/24 - 2 20/24 = 3 7/24

e.g., 7 5/15 - 2 9/15 = 6 20/15 - 2 9/15 = 4 11/15

e.g., 1 45/54 - 48/54 = 99/54 - 48/54 = 51/54 = 17/18

OPEN-ENDEDPathway 3


modelling fractions (e.g., Fraction Strips (BLM 10), Fraction Circles (BLM 18), or commercial materials)



Each fuel gauge below is designed to show how full a gas tank is. The first fuel gauge shows that its tank is 38 full.

• Shade the other 4 fuel gauges to tell how full each tank might be. Do not shade any to the top.Record each fraction you shaded.

38

____ ____ ____ ____

• Choose 2 of the fractions you shaded. Estimate how much more full one tank is than the other. Explain your estimate.

• Exactly how much more is it? Show or explain what you did.

fractions: ________ ________

estimated difference: about ________

exact difference: ________

FUEL

Subtracting Fractions

• You can model fraction subtraction using fraction circles, fraction strips, counters on grids, and number lines.

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This gauge shows the tank is 38 full.


Leaps and BoundsSubtracting Fractions, Pathway 3

3/4 2/3 1/6 7/10

2/3 3/4

1/10

3/4 is a bit more than 2/3, so it's about 1/10 more.

1/12I lined up fraction strips showing 2/3 and 3/4. A 1/12 strip was needed to fill the gap, so 3/4 - 2/3 = 1/12.

e.g.,

• Repeat for another pair of fractions.

fractions: ________ ________

estimated difference: about ________

exact difference: ________

• Choose one of the 5 fractions with a denominator that is not 4. Show or explain how you would use a model to determine how much gas would be left if you used 14 of a tank.


Leaps and Bounds Subtracting Fractions, Pathway 3

3/8 1/6

1/4

1/6 is close to 1/8 and 3/8 - 1/8 = 2/8 or 1/4.

5/243/8 is 18 out of 48 grid squares and 1/6 is 8 out of 48 grid squares, so 3/8 is 10 more squares. That's 10/48, which is 5/24.

e.g., 2/3 - 1/4 I would fill 2 of 3 rows (2/3) of a 3-by-4 grid with counters. I'd move a counter so that had 1 of 4 columns (1/4) that I could take away. The fraction of the grid still covered with counters is the answer, 5/12.

e.g.,

GUIDEDPathway 3





Liang and Jens agreed to do the same number of volunteer hours. Liang has completed 25 of his hours.

Jens has completed 13 of her hours.How much more has Liang completed than Jens?

You can use various strategies to calculate 25 2 13 to solve the problem.

• You can estimate.25 and 13 are very close, so the difference will be close to 0.

• You can use grids and counters to model 25 2 13.

– Draw two 3-by-5 grids to make it easy to show both thirds and fifths.

– Model 13 by filling 1 of the 3 rows in one grid.

– Model 25 by filling 2 of the 5 columns in the other.

– Compare the number of squares filled in the grids.

You can see that 13 fills 5 of the 15 squares, or 5

15, and 25 fills 6 of the 15 squares, or 615.

Liang has completed 115 more of his hours

than Jens has.

• You can use fraction circles or strips to model 25 2 13.– Model both fractions using fraction strips divided into fifteenths. – Compare to see how much longer 6

15 is than 515.

25

2 13

5 1

15

615

2 5

15 5

115

25

2 13

Subtracting Fractions

• Subtraction can mean different things:– taking away one

amount from another

– how much needs to be added to a number for a given amount

– comparing 2 numbers

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• You can use a common denominator to subtract 25 2 13.

Rename the fractions as equivalent fractions with the same denominator, and then subtract the numerators.

25

213

56

152

515

51

15

Try These

1. What fraction comparison does each model show? Record the number sentence, including the difference.

a)

________________

b)

________________

c)

________________

d)

________________

e)

________________

• One way to figure out a common denominator is to multiply the denominators.

e.g., 34

237

53 3 74 3 7

23 3 47 3 4

52128

21228

• To model a fraction greater than 1, you would need more than 1 grid.

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e.g., 1/2 - 2/5 = 1/10

e.g., 4/3 - 3/4 = 7/12

e.g., 6/7 - 2/3 = 4/21

4/6 - 1/3 = 1/3

e.g., 1 2/4 - 3/6 = 1

2. A bowl of soup was 23 full, and then someone ate 14 of a bowl’s worth.

How much is left? Use a model to calculate 23 2 14. Show or explain your model.

3. Circle an estimate for each calculation.

a) 56

234

less than 12 between 12 and 1 more than 1

b) 56

21

10 less than 12 between 12 and 1 more than 1

c) 43

212


d) 1110

218


4. Calculate. Sketch models to show your thinking.

a) 58

2 28

5 ________ c) 136

2 34

5 ________

b) 118

2 78

5 ________ d) 83

2 75

5 ________



e.g., 5/12 is left. I used a 4-by-3 grid and modelled 2/3 by filling 2 of 3 columns. To be able to take away 1/4, I moved counters to fill 1 of 4 rows (1/4). After I took away 1/4, there was 5 of 12 squares still full. That's 5/12.

3/8 17/12

4/8 19/15

e.g., e.g., 26/12 - 3/4 = 17/12

e.g., e.g.,


a) 78

2 34

5 ________ b) 116

2 107

5 ________

6. Anne-Lise got a new Sudoku puzzle book. She finished 13 of the puzzles on Monday and another 25 on Tuesday. How much more of the puzzle book did she do on Tuesday than on Monday?

7. Create a problem that could be solved using 710 2

23.

8. Suppose you subtracted a fraction that is a bit less than 23 from the fraction shaded in the model on the right.

Estimate the difference. ________________

9. Complete the fractions to make each equation true.

a)

3 1

4 5

1912

b)

3 2

6 5 2

10. Why do you think subtracting fractions with common denominators is easier than when their denominators are different?

Learning how to subtract fractions will help you work with algebraic expressions.

FYI



1/8 17/42

e.g., 7/8 - 3/4 = 1/8

e.g., 2/5 - 1/3 = 6/15 - 5/15 = 1/15 Anne-Lise did 1/15 more on Tuesday than on Monday.

e.g., Jane poured 1 L of juice into 10 equal size cups and gave away 7 of

them. Michel poured 1 L of juice into 3 equal size mugs and gave away

2. How much more of a litre did Jane give away?

e.g., more than 1/6

7 3 11 10

e.g., You can just compare the parts (the numbers in the numerators) and

don't need to see a model of them. If the denominators are different,

you have to figure out what fraction size works with both fractions first.

e.g., 77/42 - 60/42 = 17/42

-

OPEN-ENDEDPathway 4





Stefan runs on the high school track every day. He varies the distance day to day—some days he runs the whole track, some days more, and some days less.

• Choose 5 fractions for how much of the track he runs each day. Use proper and improper fractions between 14 and 32 with 4 or 5 different denominators. Record the fractions in this chart.

Fraction of Track Run by Stefan

Monday Tuesday Wednesday Thursday Friday

• Choose 2 of the days in the chart for Stefan to run.

• Is his total distance more or less than the whole track? How do you know?

• Exactly how much would he run altogether in those 2 days?

• Show or explain how you added.

fractions: ________ ________

Adding Fractions

• A proper fraction is less than 1 (e.g., 56 ).

• An improper fraction is greater than 1, with a numerator greater than the denominator (e.g., 65 ).

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• You can model fraction addition using fraction circles, fraction strips, counters on grids, and number lines.

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Leaps and BoundsAdding Fractions, Pathway 4

1/3 1/2 3/4 2/3 6/5

1/3 2/3

It's 1 track exactly. 1/3 + 2/3 = 1

e.g.,

e.g.,

• Repeat for 3 other pairs of days.

fractions: ________ ________

fractions: ________ ________

fractions: ________ ________


Leaps and Bounds Adding Fractions, Pathway 4

1/3 1/2

It's less than 1 track, since 1/3 < 1/2 and 1/2 + 1/2 = 1. 1/3 + 1/2 = 5/6

3/4 6/5 It's more than 1 track, since 6/5 is more than 1 track. 3/4 + 6/5 = 1 19/20 I used 5-by-4 grids and modelled 3 of 4 columns with Xs (3/4) and 6 of 5 rows (6/5) with Os. Then I moved double counters to empty squares. The result was 1 whole grid plus 19 out of 20 squares.

2/3 3/4

It's more than 1 track, since 1/4 + 3/4 = 1 and 2/3 > 1/4. 2/3 + 3/4 = 1 5/12 2/3 + 3/4

= 8/12 + 9/12 = 17/12 = 1 5/12

e.g.,

e.g.,

e.g.,

GUIDEDPathway 4





Rebecca used 23 of a cup of sugar to bake

a cake and 34 of a cup of sugar for a pie. How much sugar did she use altogether?

You can use various strategies to add 23 and 34 to solve the problem.

• You can estimate the sum of 23 and 34.

Since both 23 and 34 are more than 12, 23 1 34 is more than 1.

• You can use a grid and counter model to add 23 1 34.

– Draw a 3-by-4 grid so you can show both thirds and fourths.

– Represent 23 by filling 2 of the 3 rows.

– Represent 34 by filling 3 of the 4 columns.

– Move any double counters to empty squares.

Since there are 6 double counters and only 1 empty square, another grid is needed for the 5 extra counters.

Since 1 full grid and 5 of 12 squares of a second grid are filled, the sum is 17

12, or 1 512.

Adding Fractions



• You can use a fraction strip model to add 23 1 34.

– Model both fractions using fraction strips and see what their total length is.

– Model the total length using strips divided into twelfths (since 23 5 8

12 and 34 5 912).

23 1 34 5 1 5

12, which is 1712.

• You can use common denominators to add 23 1 34.

Rename the fractions as equivalent fractions with the same denominator, and then add the numerators.

23

134

58

121

912

51712

Rebecca used 1712, or 1 5

12 cups of sugar.

Try These

1. Suppose Rebecca had used 43 cups of sugar for the cake and 34 cup for the pie. Why would her addition model have looked like this?

2. What addition does each grid model show? Include the sum.

a)

b)

• One way to figure out a common denominator is to multiply the denominators.

e.g., 34

137

53 3 74 3 7

13 3 47 3 4

52128

11228

RRRememberRRRRRRRRRRRRRRRRRRRRRRRRRReeeeeeeeeeeeeeeeeeemmmmmmmmmmmmmmmmmmmmmmmeeeeeeeeeeeeeeeeeeemmmmmmmmmmmmmmmmmmmmbbbbbbbbbbbbbbbbbbbbbbbeeeeeeeeeeeeeeeerrrrrrrrrrrrrrrr

____________________

____________________



e.g., The grey counters show 4/3 is 1 whole 3-by-4 grid

plus 1/3 or 1 row of another 3-by-4 grid. The white

counters show 3/4 is 3 of 4 columns of a 3 by 4 grid.

1/4 + 4/12 = 2/3

4/3 + 2/7 = 1 13/21

3. What addition does each fraction strip model show? Include the sum.

a)

________________________

b)

________________________

4. Circle more than 1 or less than 1 to estimate the sum.

a) 56

134

more than 1 less than 1

b) 56

113


c) 23

114


d) 49

138


5. A fraction is added to 35 and the sum is less than 1.List 2 possibilities for the fraction.

________ ________

6. Calculate. Draw models to show your thinking.

a) 38

1 28

5 ________

c) 56

1 34

5 ________

b) 38

1 78

5 ________

d) 23

1 65

5 ________

35

1j

j, 1



1/6 + 2/3 = 5/6 1 2/4 + 3/6 = 2

e.g., 1/10 1/20

5/8

10/8 28/15

19/12

e.g., e.g.,

e.g., e.g.,


a) 38

1 34

5 ________

b) 56

1 87

5 ________

8. The juice in a pitcher that is 78 full of juice is added to another

pitcher that is 34 full. If one of the pitchers is filled to the top,how full will the second pitcher be?

9. Create a problem that could be solved using 512 1

23.

10. Estimate each sum.

a) Add a fraction that is a bit less than 23 to a fraction that is a bit more than 23. about ________

b) Add a fraction that is a bit less than 78 to a fraction that is a bit less than 13.

about ________

c) Add 3 fractions, each a bit less than 12. about ________

11. Why do you think adding fractions with common denominators is easier than when their denominators are different?

Learning how to add fractions will help you work with algebraic expressions.

FYI



9/8, or 1 1/8 28/15 , or 1 13/15

e.g., e.g.,

e.g., 7/8 + 3/4 = 7/8 + 6/8 = 13/8 = 1 5/8 The juice fills 1 5/8 pitchers, so the second pitcher will be 5/8 full.

e.g., In one egg carton, there were 5 eggs. Another carton was 2/3 full.

How many cartons of eggs are there altogether?

1 1/4

3/2 or 1 1/2

e.g., You can just add up the parts (the numbers in the numerators) and

don't need to see a model of them. If the denominators are different,

you have to figure out what fraction size works with both fractions first.

4/3 e.g.,

Repeated Addition of Fractions 1

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