5.4 Wrapping Up Multiplication and Division with Multi-Digit Numbers

Unit Goals

  • Students use the standard algorithm to multiply multi-digit whole numbers. They divide whole numbers up to four-digits by two-digits divisors using strategies based on place value and properties of operations.

Section A Goals

  • Multiply multi-digit whole numbers using the standard algorithm.

Section B Goals

  • Divide multi-digit whole numbers using strategies based on place value, properties of operations, and the relationship between multiplication and division.

Section C Goals

  • Multiply and divide to solve real-world and mathematical problems involving area and volume.
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Section A: Multi-digit Multiplication Using the Standard Algorithm

Problem 1

Pre-unit

Practicing Standards:  4.NBT.A.1

Han says that the value of the 7 in 735,208 is 10 times the value of the 7 in 137,342. Do you agree with Han? Explain or show your reasoning.

Solution

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Problem 2

Pre-unit

Practicing Standards:  4.NBT.B.5

Find the value of each product. Explain or show your reasoning.
  1. \(27 \times 53\)
  2. \(518 \times 6\)

Solution

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Problem 3

Pre-unit

Practicing Standards:  4.NBT.B.6

Find the value of \(7,\!518 \div 6\). Explain or show your reasoning.

Solution

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Problem 4

Pre-unit

Practicing Standards:  5.MD.C.5

What is the volume of this rectangular prism? Explain or show your reasoning.
Rectangular prism. 40 by sixty by eighty centimeters. 

Solution

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Problem 5

Pre-unit

Practicing Standards:  5.NF.B.3

2 diagrams of equal length. 5 equal parts, 1 part shaded. Total length, 1.
  1. Explain or show how the drawing shows \(2 \div 5\).
  2. Explain or show how the drawing shows \(\frac{2}{5}\).

Solution

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Problem 6

Find the value of each product. Explain or show your reasoning.
  1. \(100 \times 50\)
  2. \(120 \times 50\)
  3. \(127 \times 50\)

Solution

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Problem 7

Complete the diagrams and use each of them to find \(253 \times 31\).
ADiagram, rectangle partitioned vertically and horizontally into 6 rectangles. 
BDiagram, rectangle partitioned horizontally into 2 rectangles. Top rectangle, vertical side, 30, horizontal side, two hundred fifty three. Bottom rectangle, vertical side, 1.
How are the strategies the same? How are they different?

Solution

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Problem 8

Find \(315 \times 43\) using partial products.

Solution

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Problem 9

Use the standard algorithm to find the value of \(16,\!452 \times 6\).

Solution

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Problem 10

Find the value of \(322 \times 41\) using the standard algorithm.

Solution

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Problem 11

Find the value of \(562 \times 34\) using the standard algorithm.

Solution

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Problem 12

Andre is playing Greatest Product. He says the greatest product it's possible to make in the game is \(987 \times 65\). Do you agree with Andre? Explain or show your reasoning.

Solution

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Problem 13

Using the digits 1, 2, 3, 4, and 5 make a product that is close to 8,000.

Solution

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Problem 14

The recommended side lengths for a birdhouse for a yellow-bellied sapsucker are 13 cm by 13 cm for the floor and a height of 31 to 38 cm. What are the smallest and largest volumes for these birdhouses? Explain or show your reasoning.

Solution

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Problem 15

Exploration

Jada remembers that the partial products algorithm can go from left to right or from right to left. She wonders if the standard algorithm can also go in either direction.

  1. Calculate \(418 \times 53\) using partial products right to left and left to right.
  2. Calculate \(418 \times 53\) with the standard algorithm. What happens if you try to make the calculation from left to right?

Solution

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Problem 16

Exploration

Clare has a strategy for multiplying a number by 99. To find \(648 \times 99\) she calculates \(648 \times 100\) and then subtracts \(648\).

  1. Use Clare's strategy to calculate \(648 \times 99\).
  2. Use the standard algorithm to calculate \(648 \times 99\).
  3. Which strategy did you prefer? Why?

Solution

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Section B: Multi-digit Division Using Partial Quotients

Problem 1

  1. 480 dancers make groups of 15. How many groups are there? Explain or show your reasoning.
  2. 480 dancers make groups of 30. How many groups are there? Explain or show your reasoning.

Solution

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Problem 2

  1. Explain why \(256 \div 4\) is equivalent to \((200 \div 4) + (40 \div 4) + (16 \div 4)\).
  2. What is the value of \(256 \div 4\)? Explain your reasoning.

Solution

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Problem 3

Use partial quotients to find the value of \(243 \div 9\)

divide. 9, long division symbol with two hundred forty three inside

Solution

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Problem 4

  1. Use partial quotients to find the quotient \(636 \div 12\).

    divide. 12, long division symbol with six hundred thirty six inside
  2. Can you use partial quotients to find \(636 \div 12\) in a different way?

Solution

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Problem 5

Find \(4,\!250 \div 34\) using partial quotients. Explain your calculations.

Solution

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Problem 6

The area of a rectangular field is 8,320 square yards. The width is 65 yards. How long is the field? Explain your reasoning.

Solution

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Problem 7

Exploration

  1. Andre made a noodle that was 102 feet long. The noodle broke into two pieces. One piece was 2 times as long as the other. How long were the two noodles? Explain your reasoning.
  2. Priya made a noodle that was 456 feet long. The noodle broke into two pieces. One piece was 5 times as long as the other. How long were the two noodles? Explain your reasoning.

Solution

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Problem 8

Exploration

Lin is calculating \(6,\!596 \div 68\). She calculates \(6,\!800 - 6,\!596\) and notices that it is \(3 \times 68\). Lin concludes that \(6,\!596 \div 68 = 97\).

  1. Explain Lin's reasoning.
  2. Use Lin's method to calculate \(7,\!448 \div 76\).

Solution

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Section C: Let’s Put it to Work

Problem 1

There are 418 students at a school. Han estimates that they each drink about 5 glasses of water each day.
  1. About how many glasses of water do all of the students together drink in one day?
  2. About how many glasses of water do all of the students together drink in one week?
  3. About how many glasses of water do all of the students together drink in one month?
  4. About how long do you think it would take all of the students to drink 1,000,000 glasses of water?

Solution

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Problem 2

Colorado is 610 kilometers long and 450 kilometers wide.

  1. Recall that New Mexico is about 596 km long and 552 km wide. Do you think the area of Colorado is greater than or less than the area of New Mexico? Explain your reasoning.
  2. What is the area of Colorado?

Solution

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Problem 3

A large cargo ship can carry between 10,000 and 15,000 shipping containers. About how many cargo ships would be needed to carry the 210,000 containers of plastic the United States ships for recycling each year? Explain or show your reasoning.

Solution

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Problem 4

Exploration

Han reads that the length of Florida is 721 kilometers and its width is 582 kilometers. Han says "Florida is bigger than New Mexico!" Do you agree with Han? Explain or show your reasoning.

Solution

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Problem 5

Exploration

The Pentagon has 5 floors and the Empire State Building has 102 floors. Noah says that the Empire State Building is bigger. Do you agree with Noah? Investigate and justify your answer.

Solution

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