Check Your Readiness

1.

This diagram shows 3 small squares and 2 rectangles. Each rectangle is composed of 10 small squares.

Diagram of 3 small squares and two rectangles composed of 10 small squares.

Jada says this diagram can represent 230. What does a small square represent for Jada?
Name a number greater than 230 that this diagram can also represent.
Lin says this diagram can represent 2.3. What does a small square represent for Lin?
Name a number less than 2.3 that this diagram can also represent.

Answer:

10
Sample responses: 2,300; 23,000; 230,000; 460
0.1
Sample responses: 0.23; 0.023; 0.0023; 1.15

Teaching Notes

This problem involves base-ten diagrams, which are used throughout this unit. Students should understand that the unit square can have any value, but each strip of 10 squares will have 10 times this value. They should also understand that when the digits are shifted one place to the left, for example, from 230 to 2,300, each digit has 10 times its previous value.

If most students struggle with this item, do the optional Lesson 2, and the optional Lesson 3 Activity 2.

2.

This diagram shows 4 small squares and 1 rectangle. The rectangle is composed of 10 small squares.

Diagram of 4 small squares and one rectangle composed of 10 small squares.

Andre says that this diagram can represent 140. Do you agree with Andre? Explain your reasoning.
What is another value this diagram could represent? How do you know?

Answer:

Yes. If a small square represents 10, then the rectangle represents $10\boldcdot 10 = 100$ , and the diagram represents 140.
The value selected should show understanding that the rectangle of 10 small squares is 10 times as much as 1 small individual square. Sample responses:
- 1,400, where the small square represents 100.
- 14,000, where the small square represents 1,000.

Teaching Notes

The main idea in this problem is that a digit in the hundreds place has a value that is 10 times that of the same digit in the tens place. (And a digit in the thousands place has a value that is 10 times that of a digit in the hundreds place, and so on). There is 1 group of 10 squares and 4 individual squares, so the 1 needs to go immediately to the left of the 4.

If most students struggle with this item, do the optional Lesson 2, Activity 1.

3.

Here are some fractions:

$\displaystyle \frac{490}{10} \hspace{0.3in} \frac{49}{10} \hspace{0.3in} \frac{490}{100} \hspace{0.3in} \frac{49}{100} \hspace{0.3in} \frac{490}{1000} \hspace{0.3in} \frac{49}{1000}$

Select all the fractions that are equal to 4.9.
Select all the fractions that are equal to 0.049.

Answer:

$\frac{49}{10}, \frac{490}{100}$
$\frac{49}{1000}$

Teaching Notes

In this problem, students reason about the decimal equivalents of fractions whose denominator is a power of 10. Students may use their understanding of place value or divide each numerator by the denominator.

If most students do well with this item, it may be possible to skip Lesson 5, Activity 2.

4.

Noah had $5.25 and Lin had $8.95. How much did they have altogether?
Diego had $21.32 and gave $9.50 to Elena. How much did he have left?
Andre saved $24.50 each week for 5 weeks. What was the total amount Andre saved?
A job paid $85.20. If 4 friends shared the work and the pay equally, how much should each person get?

Answer:

$14.20
$11.82
$122.50
$21.30

Teaching Notes

This problem assesses all four arithmetic operations with decimals: addition, subtraction, multiplication, and division. If most students struggle with this item, connect place-value diagrams to pennies, dimes, and dollars in Lesson 3.

5.

Compute $(4,803) \boldcdot 95$ .

Answer:

456,285

Teaching Notes

Students will need to know how to perform multi-digit multiplication fluently before generalizing this work to decimal multiplication.

If most students struggle with this item, do the optional Lesson 7, Activity 2.

6.

Noah has 4 bags of marbles with the same number in each bag. If there are 536 marbles altogether, how many marbles are in each bag? Explain your reasoning.

Answer:

There are 134 marbles in each bag. Sample reasoning: If Noah puts 100 marbles in each bag, the total would be 400. If Noah puts 25 more marbles in each bag, which makes 125 per bag, the total would be 500. The actual total is 36 more than 500, so each bag has $36 \div 4$ , or 9, more marbles than 125, which is 134

Teaching Notes

This problem assesses students' understanding of and fluency in whole-number division, another prerequisite skill for this unit. Students may explain the problem conceptually, as in the solution, or they may take a more formulaic approach.

If most students struggle with this item, do additional activities involving fair-sharing, including using manipulatives. Focus on dividing large numbers and ways to make the sharing easier, such as giving each person 10 at a time rather than 1 at a time.

7.

Rectangle A is 10 times as long as Rectangle B.

2 tape diagrams, A and B. A, 10 equal parts, total, 80. B, 1 equal part.

If rectangle A is 80 units long, select all the ways find the length of rectangle B.

A.

Multiply 80 by 10.

B.

Multiply 80 by $\frac{1}{10}$ .

C.

Multiply 10 by 80.

D.

Divide 80 by 10.

E.

Divide 10 by 80.

F.

Divide 80 by $\frac{1}{10}$ .

Answer: B, D

Teaching Notes

Students who select choice D only may not understand that multiplying by $\frac{1}{10}$ is equivalent to dividing by 10. Students who select choice F are making a common type of mistake: dividing by $\frac{1}{10}$ when meaning to divide by 10.

If most students struggle with this item, make sure to do Lesson 5, Activity 2. Launch the activity by doing some error analysis on this item.