Dividing Whole Numbers by Decimals 1

OPEN-ENDED

You will need• materials for

modelling decimals (e.g., Hundredths Grids (BLM 8) or base ten blocks)

Pathway 1Dividing Whole Numbers by Decimals

Takuma is using cans that hold 0.❚❚ L to pour water into a container that holds ❚ L.

• Decide how many litres the cans and the container can hold. Record the numbers below.

• Estimate the number of cans needed to fill the container. Explain how you estimated.

• How many full and partial cans (as a decimal) are needed to fill the container? Show your work and check that your answer makes sense.

Each can holds 0. L. The container holds L.

estimated number of cans: about ________ cans

number of full and partial cans: ________ cans

• You can model the division of a whole number by a decimal using a hundredths grid, base ten blocks (where the large cube is the whole), or a number line.

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Copyright © 2012 by Nelson Education Ltd.56

Leaps and BoundsDividing Whole Numbers by Decimals, Pathway 1

2 5 3

12

I thought there would be 4 sets of 0.25 L in each 1 L.

12

Since 0.25 L is 1/4 L, you need to use 4 cans to fill 1 L (I actually poured to make sure and I was right). So for 3 L, you need 3 x 4 = 12. To check, I know that 12 x 0.25 = 3.00, which is 3.

• Repeat 2 more times. Use different digits each time.








Leaps and Bounds Dividing Whole Numbers by Decimals, Pathway 1

3 6 3

9

Since 0.36 is close to 1/3, it would take about 3 sets of 0.36 L to fill each litre, and that's 9.

8.3

There were 8 sets of 0.36 plus 12 squares out of 36 squares, or 1/3 or 0.3 of another can in 3 whole grids. That's 8.3 altogether. To check, I multiplied 8 x 0.36 = 2.88. Another third of 0.36 would be 0.12 and 2.88 + 0.12 = 3.00.

1 5 2

12

Since 6 x 0.15 = 0.9, which is almost 1 L, 12 x 0.15 would be almost 2 L.

13.3

To check: 10 x 0.15 = 1.5, and 3 x 0.15 = 0.45 which is almost 0.5, and that's about 13 altogether.

0 0.1 1.0 1.1 2.0 2.1

GUIDEDPathway 1

You will need• Fraction Strips

(BLM 10)• Hundredths Grids

(BLM 8)• base ten blocks• play coins

Shaher created a footprint that was 0.09 m long. She wanted to know how many footprints would fit along a 5 m wall. She knew she needed to divide 5 by 0.09.

You can use various strategies to divide 5 by 0.09.

• You should estimate first to make sure your answer makes sense.

0.09 is almost 1 tenth and 5 is 50 tenths.50 tenths 4 1 tenth is 50, so about 50 footprints will fit.

• You can divide 5 by 0.09 by renaming both measurements so you can divide by a whole number instead of a decimal.

5 m 5 500 cm and 0.09 m 5 9 cm. So, dividing 500 by 9 is the same as dividing 5 by 0.09.

9 3 55 5 495, so 500 4 9 5 55 R5.A remainder of 5, when you divide by 9, is 59.

This fraction strip model shows that 59 is about 12, or 0.5.

So 5 m 4 0.09 m is about 55.5 footprints.

• You can divide 5 by 0.09 by modelling with hundredths grids.

Each grid represents 1 whole metre, so 5 grids represent 5 m.You can shade sets of 9 small squares, or 9 hundredths (0.09), and then count the number of sets of 9 in the 5 grids.

11 11 11 11 11

That’s 55 sets of 0.09 with 0.05 left over. 0.05 of a 0.09 cm footprint is about half.

That’s 55 footprints and about half a footprint, or 55.5.

19

12

12

59

19

19

19

19

19

19

19

19

Dividing Whole Numbers by Decimals

• You can divide when you want to know how many times one thing fits into another.

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Dividing Whole Numbers by Decimals, Pathway 1 Leaps and Bounds

• About how many 0.10 m footprints would fit along the wall? Why does your answer make sense?

Try These

1. How does each model show the calculation?

a) 3 4 0.5 5 6

b) 2 4 0.4 5 5

c) 2 4 0.2 5 10

d) 4 4 0.4 5 10

4.00 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6

• Base ten blocks can be used to model decimals. If the large cube is the whole, or 1, then a flat is 0.1, a rod is 0.01, and a small cube is 0.001.

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Dividing Whole Numbers by Decimals, Pathway 1Leaps and Bounds

50; e.g., There are 10 sets of 0.10 m in 1 m since 0.10 is 1/10 of a whole so there must be 50 sets in 5 m.

e.g., There are 6 sets of 5 tenths in 3 wholes.

e.g., There are 5 sets of 0.4 in 2 wholes.

e.g., A dime is 0.1 of a dollar and 2 dimes are 0.2 of a dollar, so this

shows how many sets of 2 dimes are in 2 dollars.

e.g., There are 10 jumps of 0.4 in 4. Each jump is a group of 0.4.

2. Explain or show how to model each calculation. Write the quotient.

a) 2 4 0.25 5 ________ c) 1 4 0.22 5 ________

b) 3 4 0.6 5 ________ d) 4 4 0.7 5 ________

3. Estimate each quotient. Explain how you could check your estimate using a model.

a) 1 4 0.33 is about ________.

b) 2 4 0.41 is about ________.

c) 3 4 0.8 is about ________.

d) 5 4 0.7 is about ________.


Dividing Whole Numbers by Decimals, Pathway 1 Leaps and Bounds

8 4.5

5 5.75

e.g., There are 4 quarters in a dollar, so there are 8 quarters in $2.

e.g., There are about 4 sets of 22 squares, with 12 squares left over in 1 whole hundredths grid. Since 12 is about half of 22, that's about 4.5 sets.

e.g., There are 5 jumps of 6 tenths in 3 wholes.

e.g., There are 5 sets of 7 columns or 7 tenths with 5 tenths left over in 4 whole hundredths grids. Since 5 is about 3/4 of 7, that's about 5.75 sets.

3

5

4

7

e.g., I'd show that there were 3 groups of 33 squares in a hundredths grid.

e.g., I'd make 5 jumps of 0.4 from 0 to 2 on a number line.

e.g., I'd make 4 jumps of 0.8 from 0 to just past 3 on a number line.

e.g., I'd show that there were 7 sets of 7 flats in 5 large cubes.

4. Circle the greater quotient. Explain how you know it is greater.

a) 4 4 0.8 or 4 4 0.7

b) 5 4 0.23 or 4 4 0.23

5. A stick is 0.3 m long. How many sticks tall is someone with a height of 2 m? Show your thinking.

6. You divide a whole number by a decimal, and the quotient is about 7. What numbers might you have divided?

________ 4 ________ is about 7.

7. Do you agree with Aaron? Show or explain why.

8. Use the fact that 6 4 0.4 5 15 to help you calculate each quotient. Explain your reasoning.

a) 12 4 0.4 5 ________

b) 6 4 0.8 5 ________

9. Why might you rename 4 as 40 tenths to calculate 4 4 0.8?

2 4 0.4 5 4 4 0.8

It is useful to think of division as how many of one thing fits into another, when dividing by decimals or by fractions.

FYI


Dividing Whole Numbers by Decimals, Pathway 1Leaps and Bounds

e.g., The group size (0.7) is smaller so there are more groups.

e.g., You can make more groups of 0.23 using 5 wholes than using 4 wholes.

e.g., (about) 6.7 2 m divided by 0.3 m is like 200 cm divided by 30 cm. There are 6 groups of 30 in 200 and 20 left over. That's 6 20/30 or about 6.7.

2 0.3

Agree e.g., 2 ÷ 0.4 is the number of sets of 4 tenths in 2 wholes; if you put the 2 wholes, one right under the other, it's the same as figuring out the number of 8 tenths in 4 wholes:

30

7.5

e.g., If you have twice as much to start with, twice as many groups of 0.4 will fit.

e.g., If you are fitting pieces twice as big into the same whole, only half as many fit.

e.g., You can divide 40 tenths by 8 tenths, which is just 40 ÷ 8.

OPEN-ENDED


modelling decimals (e.g., Tenths Grids (BLM 9), Hundredths Grids (BLM 8), Thousandths Grids (BLM 7), base ten blocks, or money)

Pathway 2Dividing Decimals by Whole Numbers

• To write a decimal to the nearest hundredth, decide which number in the form ❚.❚❚ the decimal is closest to.

e.g., 5.128 is close to 5.13.

• One way to check a quotient is by multiplying. e.g., If 12 4 3 5 4, then 4 3 3 5 12.

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The area of a city needs to be divided into equal-sized zones for garbage and recycling collection.

Part A

• Decide on the area of the city. Record the area in the blanks below.

• Decide on a number of zones between 3 and 9.

• Estimate the area of each zone.

• Determine the actual area of each zone to the nearest hundredth and show your work. Then check your answer.

area of city: . km2 number of zones: ________

estimated area of each zone: about ________ km2

actual area of each zone: ________ km2


Leaps and BoundsDividing Decimals by Whole Numbers, Pathway 2

7 8 2 6

13

13.0378.2 = 78 + 0.2 78 ÷ 6 = 13 0.2 = 0.20 = 20 hundredths 20 hundredths ÷ 6 is 3 hundredths with a bit left over. So 78.2 km2 ÷ 6 is 13.03 km2. Check: 6 x 13.03 = 78.18, which is 78.2 to the nearest tenth.

Part B

• Repeat Part A, but use an area that is not too close in size and express it as a decimal hundredth. Use a different number of zones.




Part C

• Repeat Part A, but use an area that is not too close in size and express it as a decimal thousandth. Use a different number of zones. Determine the actual area to the nearest thousandth.





Leaps and Bounds Dividing Decimals by Whole Numbers, Pathway 2

4 6 2 4 6

9

9.25

To share 1.24 among 5 zones, each zone gets 2 columns (0.4) from the first grid and 5 squares (0.05) of the second grid. That's 0.25 for each zone with 1 square left over. So, 46.24 km2 ÷ 5 = 9.25. Check: 5 x 9.25 = 46.25, which is close to 45.24.

2 9 1 2 4 4

7

7.281 Share 1.124 among 4 zones: Trade the large cube for 10 flats and then share 11 flats. Each zone gets 2 flats and there are 3 flats left to share. Trade the 3 flats for 30 rods and then share the 32 rods. Each zone gets 8 rods. Share the 4 small cubes. Each zone gets 1 small cube. That's 2 flats, 8 rods, and 1 small cube, in each zone, which is 0.281. So, 29.124 km2 ÷ 4 = 7.281 km2. Check: 4 x 7.281 = 29.124.

29.124 = 28 + 1.124 28 ÷ 4 = 7 1.124 ÷ 4:

GUIDEDPathway 2


modelling decimals (e.g., Tenths Grids (BLM 9), Hundredths Grids (BLM 8), Thousandths Grids (BLM 7), base ten blocks, or play money)

Rihanna needs to divide a 4.12 m2 piece of paper into 6 equal sections for her group to share. How big will each section be?

You can use different strategies to divide 4.12 m2 by 6 to solve the problem.

• If you estimate the quotient of 4.12 4 6 first, you can use the estimate to decide if your actual answer is reasonable.

4.12 4 6 is more than 3 4 6 5 0.5.

So each student will get a section that is a bit bigger than 0.5 m2.

• To determine the actual quotient of 4.12 4 6, you can use hundredths grids to model 4.12 and share it equally among 6 students.

There are 412 squares, or 412 hundredths, to share.

412 4 6 5 68 R4, which means there are 68 hundredths (0.68) in each share and 4 squares, or 4 hundredths, left over.

If the 4 leftover squares, or hundredths, are shared among the 6 students, each gets almost 1 more square or hundredth, which is about 0.09. So 4.12 4 6 is about 0.69.

Each person in the group will get about 0.69 m2 of paper.

Ryan had a 9.346 m long strip of paper to share equally among 8 students. How long will each student’s share be?

• You can estimate each student’s share.

9.346 4 8 is more than 8 4 8 5 1, so each student will get a strip that is a bit longer than 1 m.

Dividing Decimals by Whole Numbers

• When you divide, you share an amount equally into equal groups, except for any “remainder,” or leftover amount.

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• To determine the actual quotient of 9.346 4 8, you can rename the decimal as a whole number, and then rename the quotient.

Since 1 m 5 1000 mm, then 9.346 m 5 9346 mm.

9346 4 8 5 1168 R2A remainder of 2 when dividing by 8 is not even half.

Since 9346 4 8 is about 1168, then 9.346 4 8 is about 1.168.

Each student will get a strip of about 1.168 m.

Try These

1. Explain how each model shows each calculation.

a) 3.66 4 3

b) 3.48 4 8

c) 1.236 4 6

2. Estimate each quotient.

a) 5.132 4 4 is about ________.

b) 13.458 4 8 is about ________.

c) 9.12 4 5 is about ________.

d) 11.04 4 3 is about ________.


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e.g., It shows 3 wholes (large cubes), 6 tenths (flats), and 6 hundredths (rods),

which is 3.66, shared into 3 equal groups.

e.g., It shows 32 tenths (columns), or 3.2,

shared 8 ways, and 24 hundredths (squares), or

0.24 shared 8 ways, with 4 hundredths left over.

e.g., It shows 12 tenths (flats) and 36 thousandths (small cubes), which is 1 whole,

2 tenths, 3 hundredths, 6 thousandths, or

1.236, shared into 6 equal groups.

1.2

1.5

2

3.7

3. Explain or show how to model each calculation. Write the quotient.

a) 5.2 4 4 5 ________ b) 3.72 4 6 5 ________

4. Each part below is about a different way to calculate 3.8 4 4.

a) How do you know 3.8 4 4 is a bit less than 1?

b) 3.8 4 4 5 38 tenths 4 4, and 38 tenths 4 4 5 9 12 tenths.

How do you know the quotient of 3.8 4 4 is 0.95?

c) Why might you think of 3.8 as $3.80 to help you solve 3.8 4 4?

5. Calculate. Show your thinking.

a) 5.13 4 3 5 ________ c) 10.48 4 4 5 ________

b) 10.52 4 2 5 ________ d) 9.48 4 6 5 ________



1.3

e.g., Model 5.2 using hundredths grids:

Share the shaded part into 4 equal groups so each group gets 1 whole grid and 3 columns of the last two grids. That's 1.3 each.

0.62

e.g.,

e.g., 3.8 is a little less than 4 and 4÷ 4 = 1.

e.g., 9 tenths = 0.9; 1/10 = 10/100, so 1/2 of 1/10 = 5/100.

e.g., I'd share $3.80 between 2 people and then, for 4 people, each would get half of that. $3.80 ÷ 2 = $1.90 and $1.90 ÷ 2 = $0.95.

1.71 2.62

5.26 1.58

e.g., 10 divided by 2 is 5. 52 hundredths divided by 2 is 26 hundredths. 5 and 26 hundredths is 5.26.

e.g., e.g.,

e.g.,

6. Marco calculated 3.123 4 2 by calculating 3123 4 2. What does he need to remember to do? Why?

7. A car drove 228.3 km in 3 h. How far did it go each hour, on average?

________________

8. A decimal divided by a whole number is 3.41. What numbers could have been divided? Write 2 possible solutions.

________ 4 ________ 5 3.41

________ 4 ________ 5 3.41

9. If 4.26 L of juice is shared among 6 people, each gets 0.71 L.Use 4.26 4 6 5 0.71 to help you figure out each share below. Explain your reasoning.

a) 10.26 L 4 6 5 ________ b) 4.26 L 4 3 5 ________

10. Create a problem that you could solve by dividing 5.2 by 4.

11. Use the digits 2, 3, 4, 8, and 9 in the blanks.

a) Create the greatest possible quotient.

. 4 5 ________

b) Create the least possible quotient.

. 4 5 ________

When you are dividing by a whole number, it makes sense to think of division as sharing.

FYI



e.g., The answer is in thousandths. When he divided 3123 ÷ 2 he was dividing 3123 thousandths by 2.

76.1 km/h

6.82 2

10.23 3

1.71 L 1.42 L

e.g., I'd add 1 to 0.71 since there is an extra 6 L for the 6 to share.

e.g., I'd double 0.71 since half as many people are sharing.

e.g., If 5.2 m of ribbon is cut into 4 equal pieces, how long is each piece?

9 8 4 3 2 4.9215

2 3 4 8 9 0.261

OPEN-ENDED


modelling decimals (e.g., Tenths Grids (BLM 9), Hundredths Grids (BLM 8), Thousandths Grids (BLM 7), base ten blocks, or money)

Pathway 3Multiplying with Decimals

Estelle bought between 3 and 9 packages of meat that all had the same mass.

Part A

• Decide on a number of packages of meat.

• Decide on a mass less than 5 kg in the form ❚.❚ kg for each package. Record the mass below. Do not use a 0 in the blanks.

• Figure out the total mass. Show your work.

• Estimate to check your answer. Show how you estimated.

• Repeat with another number of packages and mass.

number of packages: ______

mass of each: . kg

total mass: ___________ kg

estimate: about ___________ kg


mass of each: . kg



• You can multiply to calculate the total amount when all the groups are equal.

• You can model multiplication using grids, base ten blocks, or money.

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Leaps and BoundsMultiplying with Decimals, Pathway 3

9 3

4 3 1 9

38.7 5.7 Multiply 10 x 4.3 = 43. Then subtract 4.3. 43 - 4.3 = 38.7

3 x 1.9 = 3 x 1 + 3 x 0.9 = 3 + 3 x 9 tenths = 3 + 27 tenths = 3 + 2.7

6 36 9 x 4 = 36, so 38.7 makes sense 3 x 2 = 6, so 5.7 makes sense

Part B

• Repeat Part A, but use a different number of packages and a mass in the form ❚.❚❚ that is less than 5 kg.


mass of each: . kg




mass of each: . kg



Part C

• Repeat Part A, but use a different number of packages and a mass in the form ❚.❚❚❚ that is less than 5 kg.


mass of each: . kg




mass of each: . kg




Leaps and Bounds Multiplying with Decimals, Pathway 3

5

2 1 7 1 2 4

7

85 8.68 5 x 2.17 = 5 x 2 + 5 x 0.17 = 10 + 5 x 0.17 For 5 x 0.17, I shaded 5 groups of 0.17 on a grid, which is 85 squares or 85 hundredths, or 0.85. So 5 x 2.17 = 10 + 0.85 = 10.85

10

I thought about money and multiplied 7 x $1.25 = $8.75 and then subtracted 7 cents to get $8.68.

8.75 7 x 1.25 = 8.75, so 8.68 makes sense.5 x 2 = 10, so 10.85 makes sense.

8 4

2 1 3 5 4 0 2 8

17.080 4.028 I thought of 8 x 2.135 as 8 sets of 2 large cubes (2), 1 flat (0.1), 3 rods (0.03) and 5 small cubes (0.005). 8 sets of 2 = 16 8 sets of 0.1 = 0.8 8 sets of 0.03 = 0.24 8 sets of 0.005 = 0.040 16 + 0.6 + 0.24 + 0.404 = 17.080 kg

4 groups of 1 (4) and 4 groups of 7 thousandths (28 thousandths) is 4.028.

more than 16 more than 1 8 x 2 = 16, so 17.080 makes sense. 4 x 1 = 4, so 4.028 makes sense.

GUIDED


modelling decimals (e.g., Tenths Grids (BLM 9), Hundredths Grids (BLM 8), Thousandths Grids (BLM 7), base ten blocks, or play money)

Pathway 3

Vincent used a trundle wheel to measure the distance from home to school. It was 2255 m, which is 2.255 km. How far does he ride his bike each day if he bikes to and from school twice a day?

There are several strategies you can use to multiply 4 3 2.255.

• You can estimate.

4 3 2.255 is a bit more than 4 3 2.So the distance is about 8 km.

• You can model 4 3 2.255 using base ten blocks.

2.255 is 8 large cubes, 8 flats, 20 rods, and 20 small cubes.

You can trade blocks, starting from the thousandths (small cubes), to figure out the product of 4 3 2.255.

4 3 2.255 5 9.02

So the distance is 9.02 km.

• You can multiply 4 3 2.255 by renaming the decimal so you can work with a whole number instead.

2.255 5 2255 thousandths4 3 2255 thousandths 5 9020 thousandths, or 9.0204 3 2.255 5 9.02

So Vincent rides 9.02 km each day.

Multiplying with Decimals


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• This strategy is like ignoring the decimal point, multiplying the decimal like a whole number, and then estimating to decide where to put the decimal point.

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• You can multiply 4 3 2.255 in parts.

2.255 5 2 ones 1 2 tenths 1 5 hundredths 1 5 thousands

4 3 2.255 5 8 ones 1 8 tenths 1 20 hundredths 1 20 thousandths

You can trade as you would with base ten blocks:

Ones Tenths Hundredths Thousandths

8 8 20 20

9 2

8 ones 1 8 tenths 1 20 hundredths 1 20 thousandths 5 9 ones 1 2 hundredths 5 9.02

4 3 2.255 km 5 9.02 km

Try These

1. Explain how each model shows each calculation.

a) 2 3 0.23

b) 3 3 2.135

2. Explain or show how to model each calculation. Write the product.

a) 5 3 1.4 5 ________________ b) 6 3 2.103 5 ________________



e.g., It's 2 sets of 23 hundredths. e.g., It's 3 sets of 2.135.

e.g., 5 whole tenths grids (5 x 1) and 2 other grids with 4 columns shaded 5 times (5 x 0.4) is 7 whole grids, or 7.

7 12.618

e.g., 6 sets of 2 large cubes, 1 flat and 3 small cubes is 12 large cubes, 6 flats and 18 small cubes altogether, or 12.618.

3. Estimate each product.

a) 5 3 3.105 is about ________.

b) 9 3 2.412 is about ________.

c) 6 3 2.046 is about ________.

d) 7 3 8.23 is about ________.

4. Salman figured out 7 3 1.361 by multiplying 7 3 1361. What does he have to remember to do? Why?

5. Circle the greater product in each pair. How do you know it is greater? If the products are equal, circle both and explain how you know they are equal.

a) 4 3 0.823 or 4 3 0.83

b) 4 3 2.28 or 8 3 1.14

6. Gurmeet laid six 0.235 m wooden strips end to end. What is the total length? Show your thinking.

7. Suppose you had $8.24 in your left pocket and 3 times as much in your right pocket. How much is in your right pocket? Show your thinking.



15 20

12 58

e.g., That 1361 is 1361 thousandths, or 1.361.

e.g., There are the same number of groups but more in each.

e.g., If one factor is twice as much and the other is half as much, the product is the same

1.416 m e.g., 0.235 m x 6 = 235 mm x 6 = 1416 mm = 1.416 m

$24.72 e.g., 3 x (8 loonies + 2 dimes + 4 pennies) = 24 loonies + 6 dimes + 12 pennies That's $24.72.

8. If you were to drive an average of 78.2 km/h for 8 h, how far would you get? Show your work.

9. Create a problem that you could solve by multiplying 6 3 1.4.

10. a) Do you agree with Lim’s strategy? Why?

b) Why would that be a good strategy for multiplying?

11. Use the digits 2, 3, 4, 8, and 9 in the blanks.

a) Create the greatest possible product.

. 3 5 ________________

b) Create the least possible product.

. 3 5 ________________

12. Complete this sentence:

You usually multiply when

To multiply 5 3 4.248, I would multiply 10 3 2.124 instead.

• Km/h is a short form for kilometres per hour. e.g., 78.2 km/h means you drive 78.2 km in 1 hour.

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Multiplication is always about the total amount of some equal groups, no matter what types of numbers you are multiplying—decimals, fractions, or whole numbers.

FYI



625.6 km

8 x 78.2 = 8 x 78 + 8 x 0.2 = 624 + 1.6 = 625.6

e.g., Chantal bought 6 packages of fish. Each package is 1.4 kg. What is

the total mass?

Yes, e.g., If you had 10 groups of 2.124 and you paired up the groups and you'd have 5 groups of 4.248, so it's the same product.

e.g., You can multiply by 10 in your head.

8 4 3 2 9 75.888

3 4 8 9 2 6.978

you have a lot of equal groups.

OPEN-ENDED


modelling decimals (e.g., Place Value Charts (to Thousandths) (BLM 6), Thousandths Grids (BLM 7), or base ten blocks)

Pathway 4

Ingrid and Javier used different tools to measure the length and width of a rectangular room. Ingrid measured the width to the nearest thousandth of a metre (the nearest millimetre). Javier’s tool was less precise, so he measured the length only to the nearest hundredth of a metre (the nearest centimetre).

0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7

Part A

• Choose a width close to 7.5 m in the form 7.❚❚❚ m.

• Choose a length a bit longer than 13 m in the form 13.❚❚ m.

• What is the perimeter of the room? Show your work.

• How much longer is the length than the width? Show your work.

width: 7. m length: 13. m

What is the perimeter? ________ m

How much longer is the length than the width? ________ m

Adding and Subtracting Decimals

• The perimeter of a rectangle is the sum of the 2 lengths and 2 widths.

RRRememberRRRRRRRRRRRRRRRRRRRRRRReeeeeeeeeeeeeeeeeeeeemmmmmmmmmmmmmmmmmmmmmeeeeeeeeeeeeeeeeeeeeeemmmmmmmmmmmmmmmmmmmmmmmbbbbbbbbbbbbbbbbbbbbbbeeeeeeeeeeeeeeeeeeeeeeerrrrrrrrrrrrrrr


Leaps and BoundsAdding and Subtracting Decimals, Pathway 4

4 9 2 0 9

41.164

e.g., To add 13.09 + 7.492 + 13.09 + 7.492, I recorded the length and width, each twice, in a place value chart and then added each place value. Then I traded when there was 10 or more of any place value.

5.598

e.g., To subtract 13.09 - 7.492, I traded the 1 ten, 3 ones, and 9 hundredths of the length for 12 ones, 9 tenths, 18 hundredths and 10 thousandths. Then I compared it to 7 ones, 4 tenths, 9 hundredths and 2 thousandths.

Part B

• Repeat Part A twice using different widths and lengths.








Leaps and Bounds Adding and Subtracting Decimals, Pathway 4

5 0 2 1 0

41.204

To add 13.10 + 13.10 + 7.502 + 7.502, I added the whole numbers and got 40. I added the tenths and got 12 tenths. I added the hundredths and thousands and got 0.004 m. Since 12 tenths is 1 whole and 2 tenths, the perimeter is 41.204 m.

5.598 To calculate 13.10 - 7.502, I added 0.008 to 7.502 to get to 7.51, and then 0.49 to get to 8 and then 5.1 to get to 13.1. I put together 5.59 + 0.008 to get 5.598 m.

4 9 5 0 5

41.09 To add 13.05 + 13.05 + 7.495 + 7.495, I added the whole number parts to get 40 and used a thousandths grid to add the decimals. 0.05 + 0.05 is 1 column, or 0.1. 0.495 + 0.495 is 98 squares and 2 half squares, which is 99 full squares, or 0.99. That's 40 + 1 whole grid + 9 squares = 41.09 m.

5.555 To calculate 13.05 - 7.495, I thought of 13.05 as 12 + 1.05 and 7.495 as 7 + 0.495 and subtracted 7 from 12 to get 5 for the whole number part. To calculate 1.05 - 0.495, I thought of 1.05 as 1 large cube and 5 rods, which I traded for 9 flats, 14 rods and 10 small cubes. I compared that to 4 flats, 9 rods and 5 small cubes, and the difference was 5 flats, 5 rods, and 5 small cubes, or 0.555. That was a difference of 5.555 altogether.

GUIDEDPathway 4


modelling decimals (e.g., Place Value Charts (to Thousandths) (BLM 6), Thousandths Grids (BLM 7), or base ten blocks)

Jennifer’s family is taking a trip from Surrey, British Columbia, to Laval, Québec, and then to Halifax, Nova Scotia.

Laval

Surrey Halifax

from Surrey to Laval 4518.169 kmfrom Laval to Halifax 1249.74 kmfrom Laval to Fredericton 821.098 km

Fredericton

Adding Decimals

To figure out how far they will drive from Surrey to Halifax, you can add 4518.169 1 1249.74.

• You can estimate the sum.

4518.169 1 1249.74 is about 4500 1 1250 5 5750 km.

• You can add 4518.169 1 1249.74 by adding the whole number parts and then using a grid to add the decimal parts.

Add the whole numbers: 4518 1 1249 5 5767

Add the decimals, 0.169 1 0.74, using a thousandths grid:

0.169 is 1 tenth 1 6 hundredths 1 9 thousandths, which is 1 column 1 6 squares 1 9 small rectangles.

0.74 is 7 tenths 1 4 hundredths, which is 7 columns 1 4 squares.

Altogether, that’s 8 columns 1 10 squares 1 9 small rectangles, or 9 columns 1 9 small rectangles, which is 9 tenths 1 9 thousandths, or 0.909.

Combine the whole number and decimal sums: 5767 1 0.909 5 5767.909 km

0.169 0.74

Adding and Subtracting Decimals



• You can add 4518.169 1 1249.74 by adding in steps.

Add the whole numbers: 4518 1 1249 5 5767

Think of 0.169 in parts: 0.169 5 0.1 1 0.06 1 0.009

Add 0.169 1 0.74 in steps: 0.74 1 0.1 5 0.84 0.84 1 0.06 5 0.90 0.90 1 0.009 5 0.909

Add the whole number and decimal: 5767 1 0.909 5 5767.909 km

Subtracting Decimals

If Jennifer’s family went to Fredericton instead of Halifax, the distance would be less. To figure out how much less, you can calculate 1249.74 2 821.098.

• You can estimate the difference.

1249.74 2 821.098 is about 1200 2 800 5 400.

• You can calculate 1249.74 2 821.098 by adding up from 821.098 to 1249.74 in steps.

821.098821.1

0.002 0.9 78 349.74

822 900 1249.74

To add 349.74 1 78 1 0.09 1 0.002, you can add 349 1 78 5 427 and then add 0.74 1 0.9 1 0.002 5 0.092. That’s 428.642 altogether.

The distance to Fredericton would be 428.642 km less.

• You can also calculate 1249.74 2 821.098 using whole numbers, by renaming the measurement units used.

1249.74 km 2 821.098 km 5 1 249 740 m 2 821 098 m5 428 642 m5 428.642 km

• You can also subtract using a place value chart and regrouping.

Thousands Hundreds Tens Ones Tenths Hundredths Thousandths

1–

12 2

842

91

67

0

349

1008

4 2 8 6 4 2

Without a place value chart, the subtraction looks like this:

The distance would be 428.642 km less.

12 6 3 10

12 49.7402 8 21.098

4 28.642



Try These

1. Estimate each sum.

a) 42.8 1 37.192 is about ____________.

b) 510.8 1 62.49 is about ____________.

c) 311.4 1 47.922 is about ____________.

d) 511.8 1 1204.38 is about ____________.

2. Estimate each difference.

a) 42.8 2 37.192 is about ____________.

b) 510.8 2 62.49 is about ____________.

c) 311.4 2 47.922 is about ____________.

d) 1204.38 2 511.8 is about ____________.

3. Why can you ignore the decimal parts of the numbers to estimate 14.298 1 27.123, but not to estimate 0.312 1 0.947?

4. Sketch a model or describe a strategy to calculate.

a) 4.126 1 3.8 5 ________ b) 15.1 2 3.872 5 ________

5. How could you have predicted the thousandths digit in each answer in Question 4?



80

573

60

1710

6

450

250

700

e.g., 14.298 and 27.123 are close to 14 and 27 and, even if you did include the decimals, it wouldn't even be another 0.5 over 41. If you

ignore the decimals for 0.312 + 0.947, the estimated sum of 0 would be way off from the actual sum, which is greater than 1.

7.926

e.g.,

11.228

e.g., I'd add up from 3.872 to 15.1. Add 0.008 to get to 3.88; then 0.12 to get to 4; then 11.1 to get to 15.1. Altogether that's 11.228.

a) e.g., If you have a thousandths digit in one number, like the 6 in 4.126, and not in the other, the thousandths digit in the sum is 6.

b) e.g., If you add up from 2 thousandths to 0 thousandths, you add 8 thousandths (then there's be 10 thousandths to trade for 1 hundredth).

6. Use the Parts of a Year chart to help you.

a) What part of a year is 37 days? Show your work. Explain why your answer makes sense.

b) What part of a year is 29 days? How do you know?

7. a) Why might someone think it is easy to add 4.23 1 0.006?

b) Why might someone think it is easy to subtract 4.236 2 3.0?

8. Write 2 possible solutions for each situation.

a) The sum of 2 decimals is 5.123.

________________ 1 ________________ 5 5.123

________________ 1 ________________ 5 5.123

b) The difference between 2 decimals is 3.923.

________________ 2 ________________ 5 3.923

________________ 2 ________________ 5 3.923

9. a) Use the digits 1 to 9 to create the greatest possible difference.

. 2 . 5 ______________

b) Use the digits 1 to 9 to create the least possible difference.

. 2 . 5 ______________

It’s easy to add 4.23 1 0.006 and easy to subtract 4.236 2 3.0.

Adding and subtracting are closely related. You sometimes use one operation to help you with the other. For example, to add 2.34 1 0.99, you can add too much, 2.34 1 1, and then subtract 0.01 to compensate.

FYI

Parts of a Year

Estimated part of a

year

1 day 0.003

1 week 0.019

1 month(30 days) 0.08



0.099; e.g., Add the decimal for 30 days, 0.08, and the decimal for 1 week (0.019). 0.08 + 0.019 = 0.099

0.077; e.g., The decimal for 30 days, 0.08, subtract the decimal for 1 day, 0.003, is 0.077.

e.g., You add 6 thousandths to 4.23 by putting a 6 in the thousandths place of 4.23.

5.12 0.003 5.1 0.23

4.0 0.077 8.1 4.177

9 8 7 6 1 2 3 4 5 86.415

2 3 4 5 1 9 8 7 6 3.574

e.g., You subtract 3 from 4 and don't change the decimal.

Documents

Dividing Whole Numbers by Decimals 1