Comparing a Number and a Product

Task

Decide which number is greater without multiplying.

1. $817$ or $235\times 817$


2. $\displaystyle 99$ or $\displaystyle \frac{1}{4} \times 99$


3. $\displaystyle \frac{51}{100}$ or $\displaystyle \frac{51}{100} \times 301$


4. $\displaystyle \frac{13}{90}$ or $\displaystyle \frac23\times \frac{13}{90}$


5. $\displaystyle \frac{101}{102}$ or $\displaystyle \frac{101}{102} \times \frac{101}{102}$


6. $\displaystyle \frac{99}{5}$ or $\displaystyle \frac{99}{5} \times \frac{1}{2}$


7. $\displaystyle \frac{8}{21} \times 40$ or $\displaystyle \frac{28}{21} \times 40$


8. $\displaystyle \frac{8}{3} \times \frac57$ or $\displaystyle \frac{8}{3} \times \frac{9}{4}$

IM Commentary

The purpose of this task is for students to compare a number and its product with other numbers that are greater than and less than one. As written, this task could be used in a summative assessment context, but it might be more useful in an instructional setting where students are asked to explain their answers either to a partner or in a whole class discussion.

Solution

1. $235\times 817$


2. $\displaystyle 99$


3. $\displaystyle \frac{51}{100} \times 301$


4. $\displaystyle \frac{13}{90}$


5. $\displaystyle \frac{101}{102}$


6. $\displaystyle \frac{99}{5}$


7. $\displaystyle \frac{28}{21} \times 40$


8. $\displaystyle \frac{8}{3} \times \frac{9}{4}$