End-of-Unit Assessment

1.

Select all the expressions that equal $4 \times 10^6$ .

A.

$(2 \times 10^8)(2 \times 10^{\text-2})$

B.

$40 \times 10^5$

C.

$40^6$

D.

400,000

E.

$\dfrac{1.2 \times 10^9}{3 \times 10^2}$

Answer: A, B, E

Teaching Notes

Students who do not select choice A may not understand the rules for integer exponents. Students who do not select choice B may not have recognized that 40 can be rewritten as $4\times10^1$ . Students who select choice C made an error in the order of operations, combining 4 and 10 incorrectly. Students who select choice D are off by a factor of 10 and may think the exponent corresponds to the total number of digits that a number has. Students who do not select choice E may have written the expression as $0.4 \times 10^7$ but did not recognize that this is equal to $4 \times 10^6$ .

2.

Select all the expressions that equal $6^{\text-10}$ .

A.

$6^{\text-5} \boldcdot 6^2$

B.

$\frac 1 {6^{10}}$

C.

$\left(6^{\text-5}\right)^2$

D.

$\dfrac{6^{\text-3}}{6^{7}}$

E.

$\dfrac{6^5 \boldcdot 6^{\text-3}}{6^{\text-8}}$

Answer: B, C, D

Teaching Notes

Students who select choice A may have multiplied the exponents instead of adding them. Students who select choice E may have made an error in subtracting exponents.

3.

About $3.9 \times 10^7$ people live in California. About $1.3 \times 10^6$ people live in Maine. About how many more people live in California than live in Maine?

A.

$2.6 \times 10^6$

B.

$2.6 \times 10^7$

C.

$3.77 \times 10^6$

D.

$3.77 \times 10^7$

Answer:

$3.77 \times 10^7$

Teaching Notes

Students who select choices A or B may have subtracted 1.3 from 3.9 and did not pay attention to the place value denoted by the powers of 10. Students who select choice C may think that the answer must be in terms of $10^6$ .

4.

What number is represented by point $P$ ?

Answer:

$5.7 \times 10^{\text-3}$ or $0.0057$ (or equivalent) (Point $P$ lies between $5 \times 10^{\text-3}$ and $6 \times 10^{\text-3}$ . In the zoomed-in number line, point $P$ lies on the seventh of ten tick marks, so it represents $5.7 \times 10^{\text-3}$ .)

Teaching Notes

Some students may think that the second number line should use smaller powers of 10.

5.

In 2015, there were about 22 million teenagers (aged 13–17) in the United States. They each sent an average of 900 text messages per month. Estimate how many text messages were sent in 2015 by teenagers in the United States in one month. Express your answer using scientific notation.

Answer:

An estimate between $1.8 \times 10^{10}$ and $2.2 \times 10^{10}$ is reasonable. (Multiply $(22 \times 10^6) \boldcdot (9 \times 10^2) = 198 \times 10^8$ . Then express in scientific notation as $1.98 \times 10^{10}$ . Students may also use $20\times10^6$ as an estimate for 22 million, or 1,000 as an estimate for 900 texts.)

Teaching Notes

Since approximate values are used and students are asked to estimate, reasonable student responses may fall within a range.

6.

Place a number in each box so that each equation is true.

$\large 2^{\boxed{\phantom{0}}} \boldcdot 2^{\boxed{\phantom{0}}} = 2^0$
$\large \dfrac{2^3}{2^{\boxed{\phantom{0}}}} = 2^{\boxed{\phantom{0}}}$
$\large 2^{\text -3} \boldcdot {\boxed{\phantom{0}}}^{\text{ -}3} = 10^{\boxed{\phantom{0}}}$

Answer:

Answers vary. Sample response: $2^5 \boldcdot 2^{\text-5} = 2^0$
Answers vary. Sample response: $\frac{2^3}{2^5} = 2^{\text-2}$
$2^{\text-3} \boldcdot 5^{\text-3} = 10^{\text-3}$

Minimal Tier 1 response:

Work is complete and correct.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: One of the three equations produced is incomplete or incorrect.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Two or more of the three equations produced are incomplete or incorrect.

Teaching Notes

Students who have difficulty with part a may need a review of the exponent rule that $a^n\boldcdot a^m=a^{n+m}$ . Students who have difficulty with part b may need a review of the exponent rule that $\frac{a^n}{a^m}=a^{n-m}$ . Students who have difficulty with part c may need a review of the exponent rule that $a^n\boldcdot b^n=(a\boldcdot b)^n$ .

7.

Here are the approximate populations of three countries, expressed in scientific notation: Panama: $4 \times 10^6$ ; Peru: $3.2 \times 10^7$ ; Thailand: $7 \times 10^7$ .

Lin says that more than 20 times as many people live in Thailand than in Panama. Do you agree with Lin? Explain or show your reasoning.
What is the population of all three countries together?

Answer:

I do not agree with Lin. Sample reasoning: If more than 20 times as many people lived in Thailand as in Panama, the population of Thailand would be at least 80 million, but it is only 70 million.
$1.06\times10^8$ people or about 100 million people (or equivalent)

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Acceptable errors: Estimates that are reasonable but do not match solution exactly

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Statement agreeing or disagreeing with Lin is missing, though explanation of reasoning implies agreement; invalid or incorrect explanation for why there aren’t more than 20 times as many people in Thailand; minor calculation error when finding sum of the 3 countries’ populations, but correct interpretation of place value.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: Two or more error types from Tier 2 response; incorrectly stating there are more than 20 times as many people in Thailand; failing to find equivalent expressions with the same place value before adding or subtracting; incorrectly interpreting numbers written in scientific notation.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Two or more error types from Tier 3 response.

Teaching Notes

Students who agree with Lin, or who think that the population of Panama is greater than the population of Peru, may not recognize that Panama’s population is at a different order of magnitude.

Students who do not find the correct total population of the 3 countries may not be correctly rewriting the populations using the same power of 10 or may not be rewriting the populations as a decimal.