Whether you’re a 5th grade teacher, a homeschool parent, or a parent enriching your child’s education in addition to school, this lesson is the perfect way to start teaching your gifted 5th grade student how to multiply decimals by powers of 10. This lesson includes instructions for assessing your child’s current knowledge of decimals, a game to practice multiplying decimals by powers of 10, and a puzzle to practice multiplying decimals by powers of 10.

**Overview**

Fourth grade students learn to use decimal notation to write fractions with denominators that are multiples of 10. In Fifth grade students learn to perform operations with decimals. This article contains a quick pre-assessment to determine if your fifth grader is ready to multiply numbers containing decimals with powers of 10. It also includes three activities to help fifth graders understand the process of multiplying with decimals.

Students should first learn how to multiply numbers containing decimals with powers of 10 before advancing to multiplying numbers containing decimals with whole numbers.

**Standards Covered**

Standard | Brief Description | Long Description |
---|---|---|

5.NBT.a2 | Understand the place value system. | Explain patterns in the number of zeros of the product when multiplying a number by powers of 10 and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10. |

I divide this standard into four units:

- Multiplying Whole Numbers by Powers of 10
- Dividing Whole Numbers by Powers of 10
- Multiplying Decimals by Powers of 10
- Dividing Decimals by Powers of 10

This lesson focuses on the third unit: Multiplying Decimals by Powers of 10.

**Before You Begin**

Place value is a critical component of arithmetic that most parents overlook. I find that parents often focus on ‘how’ to perform operations without considering the conceptional understanding of numbers or operations. Basic concepts seem so natural to adults who have mastered more complex processes, that they assume that their children also naturally understand them.

Ask your child these questions to assess their readiness:

*“I’m thinking of a number. It has an 8 in the tens place. How much will the value of the 8 change if I move it to the hundreds place?”*

This question requires the child to understand the difference between number and digit. It also requires an understanding of place value.

If your child can’t immediately answer the question, provide it in writing and give your child time to think about it. Remain patient and quiet while your child thinks. Remember you want to see if your child can figure this out on his own, not whether you can guide your child to the correct answer.

Use this chart to assess their response.

If your child responds... | Then... |
---|---|

"I don't understand." | Your child does not understand of the basic vocabulary and concepts of place value. Your child should start by a review of 2nd and 3rd grade place value. |

"720" | Your child used 3rd grade understanding of place value and subtraction to answer this question correctly. Try the next question. |

"Ten times as much." | Your child used 4th grade understanding of place answer this question correctly. Try the next question |

No answer or an incorrect answer | Your child does not understand of the basic vocabulary and concepts of place value. Your child should start by a review of 2nd and 3rd grade place value. |

If your child was not able to correctly answer the first question without coaching, don’t read the rest of this article now. Go back and review 2^{nd} and 3^{rd} grade place value first.

If your child was able to correctly answer the first question, ask this question:

“*I’m thinking of another number. It has a mystery digit in the tens place. How much will the value of the mystery digit change if I move it to the thousands place?”*

This is a similar but much harder question that requires a deeper understanding of place value. Again, don’t provide any hints or guidance.

Us this chart to assess their response.

If your child responds... | Then... |
---|---|

"a hundred times" | Your child used 4th grade understanding of place value to answer this question correctly. Try the next question |

No answer or incorrect answer | Your child should start by a review of 4th grade place value |

If your child did not include a phrase that indicated that the value increases by a factor of 100, stop reading this article. Instead, start by reviewing 4^{th} grade place value with your child.

If your child’s answer indicated that the new value of the digit could by found by multiplying the old value by 100, try this question:

*“I have three number. It has a mystery digit in the hundreds place. How much will the value of the mystery digit change if I move it to the tens place?”*

Not only is this question reversed (seeking lower value rather than a higher value), but it tests your child’s understanding of fractions as division.

Use this chart to assess their response.

If your child responds... | Then... |
---|---|

"one-tenth" | Your child used 4th grade understanding of place value and fractions answer this question correctly. |

"divide by 10" |
Your child used 4th grade understanding of place value to describe how to find the value. Ask your child to complete this sentence:
“If you divide it by ten than the new value would be ______ of the old value.”
-If your child answers “one-tenth” then proceed with this lesson. -If your child can’t fill in the blank, review fractions as division before proceeding with this lesson. |

If your child answered all three of these questions correctly, ask one final question.

Write 10 ^{ 3 } on a piece of paper. Ask your child what it means. If your child does not understand exponents, review 5^{th} grade place value before beginning this lesson.

If your child understands that and answered the three previous questions correctly, he/she is ready for this lesson.

In this lesson, students begin with their prior understanding of adding zeroes when multiplying by powers of 10. Students were first introduced to this concept in 3^{rd} grade. In 4^{th} grade students learned to use a decimal point as an alternative method to write a fraction that has a power of ten in the denominator. Now students are ready to combine these to understand that they are not simply “adding zeros” but moving the digits to the left.

I find that the most difficult concepts for students to grasp are the ones involving things that are implied rather than explicitly written. In this case, when dealing with whole numbers, the decimal point following the last digit is implied, rather than explicitly written. I address this by introducing “the invisible decimal” and encourage students to expose it before multiplying. By doing this, students can see that moving the decimal point has the same impact as adding zeros.

After making the connection between their prior understanding of adding zeros and the new concept of moving the decimal point, I find that most students can easily apply this new understanding to cases where the decimal point is explicitly written.

I find that students motivated to work on self-checking tasks. They like the immediate feedback and find satisfaction in knowing that they got the correct answers. I like tasks that naturally provide students with the realization that they did something wrong and motivates them to find their mistake. In the attached lesson I provide a pentomino puzzle to provide practice. Because I want to focus on the placement of the decimal point, I reuse the same digits in different multiplication expressions.

**Materials**:

- Copy of the game board and number-line questions
- One 10-sided die
- 10-15 game markers per player (ex: checkers pieces, coins)

**Rules**:

- On your turn, roll the die.
- Find the number line that matches the die.
- Identify the missing number on the number line.
- Place one marker on a space with the same fraction.
- You must fill the columns from the bottom of the board.

**Winning**:

- The first player to get four markers in a row (horizontally, vertically, or diagonally) wins!

These twelve self-checking puzzles not only reinforce how to multiply decimals by powers of 10, they also strength the connection between standard notation and powers of ten.

I recommend giving your child this puzzle on a different day than when he/she worked on the previous activities. Like the pentomino puzzle in the lesson, these twelve puzzles reuse the same digits so the student must concentrate on the placement of the decimal. Each puzzle consists of 4 pieces:

- One piece contains the multiplication expression using standard notation.
- One piece contains the multiplication expression using exponents.
- One piece shows the process of moving the decimal.
- One piece contains the product.

**Materials**:

- One copy of the twelve puzzles.

**Rules**:

- Assemble all twelve puzzles before looking at any of the pictures on the back.
- After completing all the puzzles, turn them over one at a time.
- If the pictures match, continue turning over the puzzles.
- If the picture doesn’t match, flip the puzzle back on the side with the numbers and try to correct your mistake by switching pieces with the remaining puzzles.

I recommend saving the puzzle for future review. Your child will probably take a long time to complete the puzzle the first time. Have him/her try it again in a week for review.

**The Next Step**

Make sure your child can easily complete the puzzle and play the game without assistance or struggling to remember the steps before moving forward. After your child has mastered this skill, I would recommend advancing to dividing numbers with decimals by powers of 10.