5.5 Place Value Patterns and Decimal Operations

Unit Goals

  • Students build from place value understanding in grade 4 to recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and $\frac{1}{10}$ of what it represents in the place to its left. They use this place value understanding to round, compare, order, add, subtract, multiply, and divide decimals.

Section A Goals

  • Compare, round and order decimals through the thousandths place based on the value of the digits in each place.
  • Read, write, and represent decimals to the thousandths place, including in expanded form.

Section B Goals

  • Add and subtract decimals to the hundredths using strategies based on place value.

Section C Goals

  • Multiply decimals with products resulting in the hundredths using place value reasoning and properties of operations.

Section D Goals

  • Divide decimals with quotients resulting in the hundredths using place value reasoning and properties of operations.
Read More

Section A: Numbers to Thousandths

Problem 1

Pre-unit

Practicing Standards:  5.NF.B.4

Find the value of each expression.

  1. \(\frac{1}{3} \times \frac{1}{10}\)
  2. \(\frac{1}{10} \times \frac{1}{10}\)
  3. \(\frac{1}{10} \times \frac{1}{100}\)

Solution

For access, consult one of our IM Certified Partners.

Problem 2

Pre-unit

Practicing Standards:  5.NF.B.4.b

  1. Write a multiplication equation shown by the shaded region of the diagram.

    Diagram, square. Length and width, 1. Partitioned into 10 rows of 10 of the same size squares. 36 squares shaded. 

  2. What is the value of \(\frac{7}{10} \times \frac{5}{10}\)? Use the grid if it is helpful.

    Diagram, square. Length and width, 1. Partitioned into 10 rows of 10 of the same size squares. No squares shaded. 

Solution

For access, consult one of our IM Certified Partners.

Problem 3

Pre-unit

Practicing Standards:  4.NBT.B.5

Find the value of \(73 \times 28\). Use the diagram if it is helpful.

Diagram, rectangle partitioned vertically and horizontally into 4 rectangles.

Solution

For access, consult one of our IM Certified Partners.

Problem 4

Pre-unit

Practicing Standards:  4.NBT.A.1

  1. What is the value of the 6 in 618,923?
  2. How many times greater is the value of the 6 in 618,923 than the 6 in 27,652?

Solution

For access, consult one of our IM Certified Partners.

Problem 5

Pre-unit

Practicing Standards:  4.NBT.B.6

Find the value of \(3,\!724 \div 7\). Explain or show your reasoning.

Solution

For access, consult one of our IM Certified Partners.

Problem 6

Pre-unit

Practicing Standards:  4.NBT.B.4

Find the value of each sum or difference.


  1. Add. 13 thousand, 8 hundred, 17, minus, 6 thousand, 5 hundred, forty 4, equals.

  2. Subtract. eight thousand, seven hundred, ninety three, minus, four thousand six hundred, seventy five, equals.

Solution

For access, consult one of our IM Certified Partners.

Problem 7

  1. What fraction of the whole square is shaded? Explain or show your reasoning.

    Diagram, square. Length and width, 1. Partitioned into 10 rows of 10 of the same size squares. 1 square shaded.

  2. What fraction of the whole square is shaded? Explain or show your reasoning.

    Diagram, square. Length and width, 1. Partitioned into 10 rows of 10 of the same size squares. Top left square partitioned into 10 rows. 1 row shaded.

Solution

For access, consult one of our IM Certified Partners.

Problem 8

  1. Write a decimal number to represent how much of the square is shaded.

    Diagram, square. Length and width, 1. Partitioned into 10 rows of 10 of the same size squares. 66 squares shaded. 

  2. Shade one hundred fifteen thousandths of the square.

    Diagram, square. Length and width, 1. Partitioned into 10 rows of 10 of the same size squares. No squares shaded. 

Solution

For access, consult one of our IM Certified Partners.

Problem 9

Write the decimal 0.418 as a fraction, in words, and in expanded form.

Solution

For access, consult one of our IM Certified Partners.

Problem 10

  1. A gold nugget weighs 0.265 ounces. Name 2 different sets of 0.1 ounce, 0.01 ounce, and 0.001 ounce weights you can use to balance the nugget.

    Scale, tilted left-side down, right-side up. Left side, gold nuggets. Right side, nothing.
  2. One gold nugget weighs 0.008 ounces. A second gold nugget weighs 0.8 ounces.
    • How many times as much as the first nugget does the second nugget weigh?
    • How many times as much as the second nugget does the first nugget weigh?

Solution

For access, consult one of our IM Certified Partners.

Problem 11

Noah threw the frisbee 4.89 yards. 

  1. Noah threw the frisbee farther than Lin. How far could Lin have thrown the frisbee?
  2. Andre threw the frisbee farther than Noah but less than 4.9 yards. How far could Andre have thrown the frisbee? Explain your reasoning.

Solution

For access, consult one of our IM Certified Partners.

Problem 12

  1. Label the tick marks. Use the number line to explain your reasoning.

    Number line. Eleven evenly spaced tick marks. First tick mark, 65 hundredths. Last tick mark, 66 hundredths.
  2. Which is greater, 0.654 or 0.658? Explain or show your reasoning.

Solution

For access, consult one of our IM Certified Partners.

Problem 13

A \$5 gold coin weighs 8.359 grams.
  1. Locate 8.359 on the number line.
  2. A scale measures to the nearest 0.01 gram. What will the scale show for the weight of the coin? Explain or show your reasoning.

Solution

For access, consult one of our IM Certified Partners.

Problem 14

  1. What is 0.374 rounded to the nearest hundredth? Explain or show your reasoning. Use the number line if it's helpful.
  2. What is 9.893 rounded to the nearest tenth? What about to the nearest hundredth? Draw a number line if it is helpful.

Solution

For access, consult one of our IM Certified Partners.

Problem 15

List the decimals from least to greatest: 6.95, 6.895, 6.598, 6.985, 5.986

Solution

For access, consult one of our IM Certified Partners.

Problem 16

To the nearest hundredth of a mile per hour, a luge rider's top speed was 81.73 mph. What are some possible speeds to the thousandth of a mile per hour? Use the number line if it is helpful.

Number line. Scale, 81 and 72 hundredths to 81 and 74 hundredths, by hundredths.

Solution

For access, consult one of our IM Certified Partners.

Problem 17

Exploration

  1. Jada has 3 doubloons. She knows that two of them have the same weight and one of them is heavier than the other two. Jada also has a balance which she can use to compare the weights of coins. Explain or show how Jada can use the balance to figure out which doubloon is heavier and which two are the same weight.
  2. What if Jada has 5 doubloons and knows that 4 of them have the same weight and one of them is heavier?

Solution

For access, consult one of our IM Certified Partners.

Problem 18

Exploration

There are two packages of ground beef at the store. One package says it has 1 pound of beef. The second package says it has 0.97 pounds of beef. Jada says that the 1 pound package has more beef. Do you agree with Jada? Explain or show your reasoning.

Solution

For access, consult one of our IM Certified Partners.

Section B: Add and Subtract Decimals

Problem 1

Mai and Tyler were playing “Target Number Addition.”

  1. Mai rolled 6 sixes. How close can Mai get to 1 without going over?
  2. Tyler rolled 6 fours. How close can Tyler get to 1 without going over?

Solution

For access, consult one of our IM Certified Partners.

Problem 2

  1. Which whole number is \(3.62 + 1.49\) closest to? Explain or show your reasoning.
  2. Find the value of \(3.62 + 1.49\).

Solution

For access, consult one of our IM Certified Partners.

Problem 3

Find the value of the expression \(215.7 + 64.94\).

Solution

For access, consult one of our IM Certified Partners.

Problem 4

  1. Which whole number is \(9.36 - 6.52\) closest to? Explain or show your reasoning.
  2. Find the value of \(9.36 - 6.52\).

Solution

For access, consult one of our IM Certified Partners.

Problem 5

  1. Here is how Elena found the value of \(15.37 - 8.19\).

    Subtract. 15 and 37 hundredths, minus, 8 and 19 hundredths, equals, 7 and 18 hundredths.
    Explain Elena's calculations and the meaning of the 15 above the 5 and the 17 above the 7 in 15.37.
  2. Use Elena's algorithm to calculate \(52.63 - 17.55\).

Solution

For access, consult one of our IM Certified Partners.

Problem 6

Find the value of each expression.

  1. \(37.06 - 22.57\)
  2. \(555 - 4.44\)

Solution

For access, consult one of our IM Certified Partners.

Problem 7

Exploration

  1. Kiran finds the value of \(35.16 - 18.79\) with these calculations.
    \(18.79 + 0.21 = 19\)
    \(19 + 16.16 = 35.16\)
    \(16.16 + 0.21 = 16.37\).
    Explain why Kiran’s strategy works.
  2. Find the difference \(22.86 - 9.99\) in a way that makes sense to you.

Solution

For access, consult one of our IM Certified Partners.

Problem 8

Exploration

Lin is trying to use the digits 1, 3, 4, 2, 5, and 6 to make 2 two-digit decimals whose sum is equal to 1.

  1. Explain why Lin can not make 1 by adding together 2 two-digit decimal numbers made with these digits.
  2. What is the closest Lin can get to 1? Explain how you know.

Solution

For access, consult one of our IM Certified Partners.

Section C: Multiply Decimals

Problem 1

Diagram, square. Partitioned into 10 rows of 10 of the same size squares. No squares shaded. 
Diagram, square. Partitioned into 10 rows of 10 of the same size squares. No squares shaded. 
  1. Shade \(5 \times 0.07\) on the first diagram.
  2. What is the value of \(5 \times 0.07\)? Explain or show your reasoning.
  3. What is the value of \(5 \times 0.2\)? Use the second diagram if it is helpful.

Solution

For access, consult one of our IM Certified Partners.

Problem 2

  1. Mai says that \(7 \times 0.4\) and \(7 \times 0.04\) both have the same value. She says that they are both 28. Do you agree with Mai? Explain or show your reasoning.
  2. Explain why \(8 \times 0.03 = (8 \times 3) \times 0.01\).

Solution

For access, consult one of our IM Certified Partners.

Problem 3

  1. Explain why each expression is equivalent to \(9 \times 0.45\).

    \((9 \times 0.4) + (9 \times 0.05)\)

    \((9 \times 45) \div 100\)

    \((10 \times 0.45) - (1 \times 0.45)\)

  2. Find the value of \(9 \times 0.45\) using one of the expressions or your own strategy.

Solution

For access, consult one of our IM Certified Partners.

Problem 4

Shade the diagram to represent \(0.7 \times 0.4\).

What is the value of \(0.7 \times 0.4\)?

Diagram, square. Length and width, 1. Partitioned into 10 rows of 10 of the same size squares. No squares shaded.

Solution

For access, consult one of our IM Certified Partners.

Problem 5

  1. Explain or show why \(5.6 \times 3.4 = (56 \times 34) \times 0.01\).
  2. Use this strategy to calculate \(5.6 \times 3.4\).

Solution

For access, consult one of our IM Certified Partners.

Problem 6

Exploration

Here is Diego's strategy to find the value of \(17.5 \times 3.3\). I know \(\frac{175}{10} \times \frac{33}{10} = \frac{175 \times 33}{100}\) so I just find \(175 \times 33\) and then divide by 100.

  1. Explain or show why Diego's method works.
  2. Use Diego's method to find the value of \(17.5 \times 3.3\).

Solution

For access, consult one of our IM Certified Partners.

Problem 7

Exploration

  1. Han says the picture shows \(4 \times 0.5 = 2\). Label the diagram to show Han's thinking.

    2 hundredths grids. All squares shaded in each grid.

  2. Mai says it shows \(10 \times 0.2 = 2\). Label the diagram to show Mai's thinking.

    2 hundredths grids. All squares shaded in each grid.

  3. What other products can the diagram represent? Explain or show your reasoning.

Solution

For access, consult one of our IM Certified Partners.

Section D: Divide Decimals

Problem 1

  1. Find the value of \(1 \div 0.01\). Use the diagram if it is helpful.

    Diagram, square. Partitioned into 10 rows of 10 of the same size squares. No squares shaded.

  2. Jada says that there are 100 hundredths in 1 so \(1 \div 0.01\) is 100. Do you agree with Jada? Show or explain your reasoning.

Solution

For access, consult one of our IM Certified Partners.

Problem 2

  1. Find the value of \(2 \div 0.2\). Use the diagram if it is helpful.

    Two diagrams. Each squares. Each partitioned into 10 rows of 10 of the same size squares. No squares shaded. 
  2. Find the value of \(21 \div 0.2\).

Solution

For access, consult one of our IM Certified Partners.

Problem 3

Here is a diagram.

Two diagrams. Each squares.

  1. Explain or show how the diagram shows \(200 \div 25\). What is the value of the expression?

  2. Explain or show how the diagram shows \(2 \div 0.25\). What is the value of the expression?

Solution

For access, consult one of our IM Certified Partners.

Problem 4

Find the value of each expression. Explain or show your reasoning.

  1. \(0.2 \div 5\). Use the diagram if it is helpful.
    Diagram, square. Length and width, 1. Partitioned into 10 rows of 10 of the same size squares. No squares shaded.
  2. \(6 \div 3\)
  3. \(6 \div 0.3\)

Solution

For access, consult one of our IM Certified Partners.

Problem 5

Find the value of each expression. Explain or show your reasoning.

  1. \(0.5 \div 0.1\)
  2. \(0.5 \div 0.01\)
  3. \(3.5 \div 0.01\)

Solution

For access, consult one of our IM Certified Partners.

Problem 6

Exploration

Noah has a scale that weighs to the nearest ounce. The table shows the weights of different numbers of paper clips in ounces.

paper clips weight
1 0
10 0
20 1
25 1
50 2
100 3

How many ounces do you think each paper clip weighs? Explain or show your reasoning.

Solution

For access, consult one of our IM Certified Partners.

Problem 7

Exploration

The daily recommended allowance of vitamin C for a 5th grader is 0.05 grams.

  1. A vitamin C tablet has 1 gram of vitamin C. How many times the daily recommended allowance of vitamin C is one vitamin C tablet? Use the diagram if it is helpful.

    Diagram, square. Partitioned into 10 rows of 10 of the same size squares. No squares shaded.

  2. A large orange has 0.18 grams of vitamin C. How many times the daily recommended allowance of vitamin C is in a large orange? Use the diagram if it is helpful.

    Diagram, square. Partitioned into 10 rows of 10 of the same size squares. No squares shaded.

Solution

For access, consult one of our IM Certified Partners.