Check Your Readiness

1.

This diagram shows 5 small squares and 1 rectangle. The rectangle is composed of 10 small squares.

Andre says this diagram can represent 1,500. What does a small square represent for Andre?
Name a number greater than 1,500 that this diagram can also represent.
Clare says this diagram can represent 1.5. What does a small square represent for Clare?
Name a number less than 1.5 that this diagram can also represent.

Answer:

100
Sample responses: 15,000; 150,000; 3000
0.1
Sample responses: 0.15; 0.015; 0.0015; 0.75

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 150 to 1,500, 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 2 small squares and 3 rectangles. Each rectangle is composed of 10 small squares.

Priya says that this diagram can represent 320. Do you agree with Priya? 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 $10 \boldcdot 10 = 100$ . The 3 rectangles represent $3 \boldcdot 100 = 300$ , and the whole diagram represents $300 + 10 +10 = 320$ .
The value selected should show understanding that the rectangles of 10 small squares are each 10 times as much as 1 small individual square. Sample responses:
1. 3,200, where the small square represents 100.
2. 32,000, 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 are 3 groups of 10 squares and 2 individual squares, so the 3 needs to go immediately to the left of the 2.

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

3.

Here are some fractions:

$\displaystyle \frac{820}{100} \hspace{0.3in} \frac{820}{1000} \hspace{0.3in} \frac{82}{1000} \hspace{0.3in} \frac{820}{10} \hspace{0.3in} \frac{82}{10} \hspace{0.3in} \frac{82}{100}$

Select all the fractions that are equal to 8.2.
Select all the fractions that are equal to 0.082.

Answer:

$\frac{82}{10}$ , $\frac{820}{100}$
$\frac{82}{1000}$

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.

Lin had $24.16 and spent $8.50. How much did she have left?
Elena had $6.75 and Tyler had $7.85. How much did they have altogether?
A job paid $65.40. If 3 friends shared the work and the pay equally, how much should each person get?
Diego saved $36.50 each week for 7 weeks. What was the total amount he saved?

Answer:

$15.66
$14.60
$21.80
$255.50

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 $(3,902) \boldcdot 85$ .

Answer:

331,670

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.

Kiran has 3 containers of blocks with the same number in each container. If there are 852 blocks altogether, how many blocks are in each container? Explain your reasoning.

Answer:

There are 284 building blocks in each container. Sample reasoning: If Kiran puts 250 blocks in each container, the total would be 750 blocks. If he puts 30 more blocks in each container, which makes 280 blocks per container, the total would be 840. The actual total is 12 more than 840, so there are 4 more blocks in each container or 284 blocks.

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 50 units long, select all the ways to find the length of Rectangle B.

A.

Multiply 50 by 10.

B.

Divide 50 by 10.

C.

Multiply 50 by $\frac{1}{10}$ .

D.

Divide 50 by $\frac{1}{10}$ .

E.

Multiply 10 by 50.

F.

Divide 10 by 50.

Answer: B, C

Teaching Notes

Students who select choice B only may not understand that multiplying by $\frac{1}{10}$ is equivalent to dividing by 10. Students who select choice D 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.