12.1: Number Talk: Multiplication Strategies
Find the product mentally.
$19\boldcdot 14$
Let’s solve problems about fractional lengths.
Find the product mentally.
$19\boldcdot 14$
It takes exactly 26 paper clips laid end to end to make a length of $17\frac78$ inches.
Estimate the length of each paper clip.
Calculate the length of each paper clip. Show your reasoning.
Lin has a work of art that is $14$ inches by $20$ inches. She wants to frame it with large paper clips laid end to end.
Write a division expression that can help answer each of the following questions. Then answer the question. If you get stuck, draw a diagram.
The photo shows a situation that involves fractions.
Use the photo to help you complete the following statements. Explain or show your reasoning for the second statement.
The length of the long paper roll is about ______ times the length of the short paper roll.
The length of the short paper roll is about ______ times the length of the long paper roll.
If the length of the long paper roll is $11 \frac 14$ inches, what is the length of each short paper roll?
Use the information you have about the paper rolls to write a multiplication equation or a division equation for the question. Note that $11 \frac 14 = \frac{45}{4}$.
Answer the question. If you get stuck, draw a diagram.
Division can help us solve comparison problems in which we find out how many times as large or as small one number is compared to another. Here is an example.
A student is playing two songs for a music recital. The first song is $1\frac12$ minutes long. The second song is $3 \frac34$ minutes long.
$${?} \boldcdot 1\frac12 = 3\frac 34$$ $$3 \frac 34 \div 1\frac 12 = {?}$$
Let’s use the algorithm we learned to calculate the quotient:$$\begin {align} &3 \frac 34 \div 1\frac 12\\[10px] &= \frac {15}{4} \div \frac 32\\[10px] &= \frac {15}{4} \boldcdot \frac 23\\[10px] &=\frac {30}{12}\\[10px]&=\frac {5}{2}\\[10px] \end {align}$$
This means the second song is $2\frac 12$ times as long as the first song.
$${?} \boldcdot 3\frac 34 = 1\frac 12$$ $$1\frac12 \div 3\frac34 = {?}$$
Let’s calculate the quotient:
$$\begin {align} &1\frac 12\div 3 \frac 34\\[10px] &=\frac 32 \div \frac {15}{4}\\[10px] &=\frac 32 \boldcdot \frac {4}{15}\\[10px] &=\frac {12}{30}\\[10px] &=\frac {2}{5} \end {align}$$
The first song is $\frac 25$ as long as the second song.