Check Your Readiness

1.

Select all expressions that are equal to $362 \boldcdot 5.9$ .

A.

$361 \boldcdot 6.9$

B.

$3.62 \boldcdot 5,900$

C.

$36.2 \boldcdot 59$

D.

$36,200 \boldcdot 0.059$

E.

$0.362 \boldcdot 0.0059$

Answer: C, D

Teaching Notes

In this unit, students rewrite multiples of powers of 10 as equivalent expressions. The key idea is that if one factor is multiplied by a power of 10, the other factor must be divided by the same factor of 10 for the value to remain the same.

If most students struggle with this item, use this item or a similar question before the first activity. Prompt students to think about what they know about writing numbers using powers of 10 to make their decisions.

2.

Which is closest to the product $19,898 \boldcdot 0.002$ ?

A.

4,000

B.

400

C.

40

D.

4

Answer:

40

Teaching Notes

In this unit students estimate the quotients of very large numbers. While students could find the answer through long division, the focus here is on using estimation and their knowledge of place value.

If most students struggle with this item, use this item or a similar question before first activity. Prompt students to think about what they know about writing numbers using powers of 10 to make their estimate.

3.

Select all the expressions that equal $2^8 \boldcdot 2^4$ .

A.

$2^4$

B.

$2^{12}$

C.

$4^6$

D.

$2^{32}$

E.

$4^{32}$

Answer:

C, D

Teaching Notes

In this unit, students will examine patterns to develop exponent rules for multiplication. While some students may attempt to calculate the values of each expression in this problem, the key idea is to look at the factors of each expression and what an exponent means.

If most students struggle with this item, revisit it in Activity 3 or include it in the Cool-Down. If most students do well with this item, it may be possible to skip Lesson 2 Activity 3 and to move faster through Lesson 2 in general.

4.

A population of red ants was 640. After a season of heavy rainfall, the ant population decreased by 50%. In the following dry season, the population increased by 5%. What is the ant population after the increase?

Answer:

336 ants. After the decrease, the population was 320 because $(0.5) \boldcdot 640 = 320$ . The increase in population is 5% of 320, which is 16. The ant population after the increase is 336 since $320 +16 = 336$ .

Teaching Notes

In this unit, students will perform decimal arithmetic. This problem assesses their fluency with decimal calculations.

If most students struggle with this item, connect writing numbers as a multiple of a power of 10 with the location of the decimal point in the product of decimals.

5.

Plot and label these numbers on the same number line:

$\text-0.23,\, \text-0.023,\, \text-0.352,\, \text-0.58,\, \text-0.7$

Answer:

Teaching Notes

In this unit, students will plot decimals on a number line labeled using powers of 10. This problem assesses students’ prior understanding of the relative size of decimals.

If most students struggle with this item, revisit it as part of the Warm-up. Show a number line labeled from -1 and 0, and discuss the size of the intervals.

6.

Plot and label these numbers on the same number line:

$2^1,\,2^2,\, \left(\text{-}\frac{1}{2}\right)^2,\, \left(\text{-}\frac{1}{2}\right)^3,\,\left(\text{-}\frac{1}{2}\right)^4$

Answer:

Teaching Notes

In this unit, students investigate patterns resulting from repeated multiplication of integers and fractions. This problem checks student fluency with integer operations, order of operations, and operations with fractions.

If most students struggle with this item, revisit it as part of Activity 3. Show a number line labeled with 0 and 1 and find the halfway point. Continue plotting points that are halfway between the previous point and 0 to represent repeated multiplication by $\frac12$ .

7.

Write three other fractions that are equivalent to $\frac{12}{72}$ . Explain or show your reasoning.

Answer:

Answers vary. Sample response: $\frac{1}{6}$ , $\frac{2}{12}$ , $\frac{3}{18}$ . The fraction $\frac{12}{72}$ can be written as $\frac{12 \boldcdot 1}{12 \boldcdot 6}$ . This equals $\frac16$ because it can be written as $\frac{12}{12} \boldcdot \frac16$ .

Teaching Notes

Some of the work in this unit involves equivalent fractions, notably fractions that are equivalent to integers or reciprocals of integers.

If most students struggle with this item, revisit it after the Warm-up. Ask students to focus on strategies for creating equivalent fractions, such as multiplying or dividing both the numerator and denominator by the same value.