End-of-Unit Assessment

1.

Select all the expressions that equal $6 \times 10^5$ .

A.

$(3 \times 10^1)(2 \times 10^{5})$

B.

$600,000,000 \times 10^{\text-3}$

C.

$\dfrac{1.2 \times 10^7}{2 \times 10^2}$

D.

600,000

E.

$60^5$

Answer:

B, D

Teaching Notes

Students who select choice A may think that when multiplying powers of 10, the exponents are multiplied together. Students who do not select choice B did not rewrite 600,000,000 as $ or may not understand the rules for integer exponents. Students who select choice C may have written the expression as $0.6\times10^5$ but did not recognize that this is equal to $6\times10^4$ . Students who do not select choice D may think the exponent corresponds to the total number of digits that a number has. Students who select choice E made an error in the order of operations, multiplying the two terms before raising 10 to a power.

2.

Select all the expressions that equal $7^8$ .

A.

$7^{\text-2} \boldcdot 7^{10}$

B.

$\left(7^3\right)^5$

C.

$\dfrac {(7^3)^4}{7^{\,\text-4}}$

D.

$\left(7^4\right)^{\text-2}$

E.

$\dfrac {7^6}{7^{\,\text-2}}$

Answer:

A, E

Teaching Notes

Students who select choice B may have added the exponents instead of multiplying them. Students who select choice C may have subtracted incorrectly. Students who select choice D may have multiplied incorrectly.

3.

About $3.2 \times 10^8$ people live in the United States. About $3.9 \times 10^7$ people live in Canada, and about $1.1 \times 10^8$ people live in Mexico. About how many people live in the three countries altogether?

A.

$4.69 \times 10^7$

B.

$4.69 \times 10^8$

C.

$8.2 \times 10^7$

D.

$8.2 \times 10^8$

Answer:

B

Teaching Notes

Students who select choices C or D may have added 3.2, 3.9, and 1.1 and did not attend to the place value denoted by the powers of 10. Students who select choice A may think that the answer must be in terms of $10^7$ .

4.

What number is represented by point $P$ ?

Answer:

$4.2 \times 10^6$ , or 4,200,000 (or equivalent). Point $P$ lies between $4 \times 10^6$ and $5 \times 10^6$ . On the zoomed-in number line, point $P$ lies on the second of ten tick marks, so it represents $4.2 \times 10^6$ .

Teaching Notes

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

5.

A honey bee’s wings beat about 680,000 times per hour when flying. The honey bee can also spend up to 12 hours per day flying around to collect pollen and nectar. Estimate how many times a honey bee will beat its wings in one day of work. Express your answer using scientific notation.

Answer:

An estimate between $7\times10^6$ and $8.4\times10^6$ is reasonable. (Multiply $(6.8\times10^5)\boldcdot(12)=81.6\times10^5$ . Then express as $8.16\times10^6$ in scientific notation. Students may also use $7\times10^5$ as an estimate for 680,000, or 10 hours as an estimate for 12 hours.)

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.

$\large5^\boxed{\phantom{3}}\boldcdot 5^3 = 5^ \boxed{\phantom{3}}$
$\large\dfrac{5^ \boxed{\phantom{3}}}{5^ \boxed{\phantom{3}}} = 5^0$
$\large(5^\boxed{\phantom{3}})^\boxed{\phantom{3}} = 5^{\text-20}$

Answer:

Answers vary. Sample response: $5^{\text-1}\boldcdot 5^3 = 5^2$
Answers vary. Sample response: $\frac{5^{\text-7}}{5^{\text-7}} = 5^0$
Answers vary. Sample response: $(5^{\text-4})^5 = 5^{\text-20}$

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 $(a^n)^m=a^{m\boldcdot n}$ .

7.

Here are the approximate populations of three cities in the United States, expressed in scientific notation: San Jose, CA: $1.1 \times 10^6$ ; Honolulu, HI: $3.5 \times 10^5$ ; Atlanta, GA: $4.8 \times 10^5$ .

Lin says that more people live in San Jose than in Atlanta and Honolulu combined. Do you agree with Lin? Explain or show your reasoning.
About how many times more people live in San Jose than in Atlanta?

Answer:

Yes, I agree with Lin. Sample reasoning: The population of Atlanta and Honolulu combined is $(4.8 \times 10^5) + (3.5 \times 10^5) = 8.3 \times 10^5$ , which is smaller than $1.1 \times 10^6$ .
About 2 times (or 2.3 times) as many people live in San Jose than live in Atlanta. The population of Atlanta is a little less than 500,000 and the population of San Jose is a little more than 1,000,000 which is about twice, or a little more than twice that of Atlanta.

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; minor calculation error when finding sum of Atlanta's and Honolulu's 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 adding the exponents of the powers of 10 when adding expressions with the same place value; 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 do not agree with Lin may not recognize that the population of San Jose is at a different order of magnitude, or may have incorrectly added the populations of Honolulu and Atlanta.

Students who do not correctly find how many times more people live in San Jose than live in Atlanta may not be applying the exponent rules for division correctly.