5.4 Wrapping Up Multiplication and Division with Multi-Digit Numbers

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.

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

1. \(27 \times 53\)
2. \(518 \times 6\)

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

What is the volume of this rectangular prism? Explain or show your reasoning.

1. Explain or show how the drawing shows \(2 \div 5\).
2. Explain or show how the drawing shows \(\frac{2}{5}\).

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

1. \(100 \times 50\)
2. \(120 \times 50\)
3. \(127 \times 50\)

Complete the diagrams and use each of them to find \(253 \times 31\).

How are the strategies the same? How are they different?

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

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

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

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

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.

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

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.

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?

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?

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.

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.

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

   

   

2. Can you use partial quotients to find \(636 \div 12\) in a different way?

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

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

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.

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 \(7,\!448 \div 76\).

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?

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?

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.

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.

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.