Unit 5 Arithmetic In Base Ten — Unit Plan

Title	Assessment
Lesson 1 Using Decimals in a Shopping Context	How Did You Compute with Decimals? Reflect on how you made calculations when planning a menu. How did you add dollar amounts that were not whole numbers? Use the numbers $5.89 and $1.45 to show or explain your strategy. How did you multiply dollar amounts that were not whole numbers? Suppose you are computing the cost of 4 pounds of beef at $5.89 per pound. Use this example to explain or show your strategy. Show Solution Sample response: I would add the dollars and cents separately, and then combine the sums at the end. $5 + 1$ is 6 and $89 + 45$ is 134, so it’s $6 plus $1.34, which is $7.34. Sample response: I would round the $5.89 to $5.90 to make it easier to multiply. Then, I would find 4 times $5, which is $20, and 4 times $0.90, which is $3.60. The two products added together is $23.60. The exact cost would be 4 cents less than $23.60, because $5.89 is 1 cent less than $5.90, and 4 times 1 cent is 4 cents. So, the total cost would be $23.56.
Lesson 2 Using Diagrams to Represent Addition and Subtraction	Why or Why Not? Does adding 0.025 and 0.17 give a sum of 0.042? Explain or show your reasoning. If you choose to use a diagram, you can use the following representations of base-ten units. 1 large square labeled "one." 1 medium rectangle labeled “zero point one, or tenth.” 1 medium square labeled “ 0 point 0 one, or hundredth.” 1 tiny rectangle labeled “0 point 0 0 1, or thousandth.” 1 tiny square labeled “0 point 0 0 0 1, or ten-thousandth.” Show Solution No. Sample reasoning: The number 0.17 is greater than 0.042, so 0.042 cannot be the sum of 0.17 and another decimal. A diagram showing 1 medium rectangle (1 tenth), 9 medium squares (9 hundredths), and 5 small rectangles (5 thousandths). $0.025 + 0.17 = 0.02 + 0.005 + 0.1 + 0.07 = 0.125 + 0.07 = 0.195$ . Calculation with numbers should show the decimal points lining up and a sum of 0.195.
Lesson 3 Adding and Subtracting Decimals with Few Non-Zero Digits	Calculate the Difference Find the value of each expression and show your reasoning. $1.56 + 0.083$ $0.2 - 0.05$ Show Solution 1.643. Sample reasoning: Six hundredths and 8 hundredths make 14 hundredths, or 1 tenth and 4 hundredths. The sum has 1 one, 6 tenths, 4 hundredths, and 3 thousandths. 0.15. Sample reasoning:
Lesson 4 Adding and Subtracting Decimals with Many Non-Zero Digits	How Much Farther? A runner has run 1.192 kilometers of a 10-kilometer race. How much farther does she need to run to finish the race? Show your reasoning. Show Solution 8.808 kilometers. Sample reasoning: $9.999 - 1.192 = 8.807$ . Adding 0.001 to 8.807 gives 8.808.
Section A Check Section A Checkpoint	Problem 1 Which calculation shows a correct way to find $31.076 + 4.85$ ? A. B. C. D. Show Solution Problem 2 a. b. Show Solution 98.963 2.958

Lesson 6 Methods for Multiplying Decimals	A Product of Two Decimals Explain or show how you would find the value of $(1.35) \boldcdot (4.2)$ if you know that $135 \boldcdot 42 = 5{,}670$ . Show Solution Sample responses: $135 = (1.35) \boldcdot 100$ and $42 = (4.2) \boldcdot 10$ , so $135 \boldcdot 42$ is $100 \boldcdot 10$ , or 1,000, times $(1.35) \boldcdot (4.2)$ . This means $(1.35) \boldcdot (4.2)$ is $5,670 \div 1,000$ , which is 5.67. $(1.35) \boldcdot (4.2) = \frac{135}{100} \boldcdot \frac{42}{10} = \frac{135 \boldcdot 42}{10 \boldcdot 100} = \frac{5,670}{1,000}$ , which is 5.67.
Lesson 7 Using Diagrams to Represent Multiplication	Find the Product Find the value of $(4.2) \boldcdot (1.6)$ by drawing an area diagram or using another method. Show your reasoning. Show Solution 6.72. Sample reasoning: The sum of the areas of the sub-rectangles is $4 + 0.2 + 2.4 + 0.12 = 6.72$ .
Section B Check Section B Checkpoint	Problem 1 A rectangular wall is 7.2 meters long and is 3.8 meters in height. What is its area in square meters? Show your reasoning. Show Solution 27.36 square meters. Sample reasoning: $(7.2) \boldcdot (3.8) = \frac{72}{10} \boldcdot \frac{38}{10} = \frac{2,736}{100} = 27.36$ Problem 2 Find the product of 64 and 9. Explain how you can use the value of $64 \boldcdot 9$ to find the value of $(6.4) \boldcdot (0.009)$ . Show Solution 576 Sample responses: There is 1 decimal place in 6.4 and 3 decimal places in 0.009, so the product will have 4 decimal places. I can move the digits in 576 to the right 4 places to get 0.0576. 64 is $10 \boldcdot (6.4)$ , and 9 is $1,000 \boldcdot (0.009)$ , so the product of 64 and 9 is 10,000 times the product of 6.4 and 0.009. Dividing 576 by 10,000 gives 0.0576.

Lesson 9 Using Base-Ten Diagrams to Divide	Putting 33 into 4 Groups To find $33 \div 4$ , Clare drew a diagram and thought about how to put the tens and ones into 4 equal-size groups. There aren’t enough tens or ones to put into 4 groups. What can Clare do to find the quotient? Explain or show your reasoning. What is the value of $33 \div 4$ ? Show Solution Sample response: The 3 tens can be decomposed into 30 ones, making a total of 33 ones. Of these, 32 ones can be distributed into 4 groups, 8 ones in each. There is 1 one left. This can be decomposed into 10 tenths and distributed into 4 groups, 2 tenths in each group. There are 2 tenths left. These can be decomposed into 20 hundredths and then distributed into 4 groups, 5 hundredths in each group. 8.25
Lesson 10 Using Partial Quotients	Dividing by 11 Calculate $4,235 \div 11$ using any method. Show Solution 385. Sample reasoning:
Lesson 11 Using Long Division	Dividing by 5 Use long division to find the value of $1{,}875 \div 5$ . Then check your answer by multiplying it by 5. Show Solution 375.
Lesson 12 Dividing Numbers that Result in a Decimal	Calculating Quotients Use long division to find each quotient. Show your computation, and write your answer as a decimal. $43.5 \div 5$ $7 \div 8$ Show Solution 14.5 0.875
Lesson 13 Dividing a Decimal by a Decimal	The Quotient of Two Decimals Write two division expressions that have the same value as $36.8 \div 2.3$ . Find the value of $36.8 \div 2.3$ . Show your reasoning. Show Solution Sample responses: $3.68 \div 0.23$ and $368 \div 23$ . 16. Sample reasoning:
Section C Check Section C Checkpoint	Problem 1 Use long division to find the value of $78.9 \div 2$ . Show Solution 39.45 Problem 2 Write a division expression that has the same value as $1.2 \div 0.75$ and can be used to find the quotient. Find the value of $1.2 \div 0.75$ . Show your reasoning. Show Solution Sample responses: $120 \div 75$ or $1,200 \div 750$ 1.6 (or equivalent). Sample reasoning: $120 \div 75 = \frac{120}{75} = 1\frac{45}{75}$ , which is equal to $1\frac {9}{15}$ or $1\frac{3}{5}$ .

Lesson 14 Solving Problems Involving Decimals	Ribbon for Sharing Jada and Han are sharing a piece of ribbon that is 1.905 meters long for a craft project. Jada cuts 0.82 meter from one end of the ribbon and Han cuts 0.175 meter from the other end. Afterward, they split the ribbon that is left into equal-size pieces that are 0.13-meter long each. How many pieces will they have? Show your reasoning. Show Solution 7 pieces. Sample reasoning: Jada and Han cut a total of $0.82 + 0.175$ , or 0.995 meter, from the two ends. This leaves $1.905-0.995$ , or 0.91 meter. Dividing 0.91 by 0.13 gives 7.
Lesson 15 Making and Measuring Boxes	No cool-down
Unit 5 Assessment End-of-Unit Assessment	Problem 1 A woodworker wants to cut a board that is 8.225 feet long into 5 equal-length pieces. How long will each piece be? A. 0.1645 foot B. 1.645 feet C. 4.1125 feet D. 41.125 feet Show Solution 1.645 feet Problem 2 Select all of the expressions that have the same value as $892 \div 8$ . A. $8920 \div 80$ B. $894 \div 10$ C. $89.2 \div 0.08$ D. $8.92 \div 0.8$ E. $0.892 \div 0.008$ Show Solution A, E Problem 3 Which value is closest to the quotient of $4,367 \div 0.004$ ? A. 1,000 B. 10,000 C. 100,000 D. 1,000,000 Show Solution 1,000,000 Problem 4 One way to convert a measurement from inches to centimeters is to multiply the number of inches by 2.54. How many centimeters are there in $\frac{1}{4}$ inch? Show Solution 0.635 centimeters (or equivalent) Problem 5 Find each quotient using long division. $2,247 \div 7$ $676 \div 13$ Show Solution 321 52 Minimal Tier 1 response: Work is complete and correct. Sample: See calculations. Tier 2 response: Work shows general conceptual understanding and mastery, with some errors. Acceptable errors: A small mistake early on results in different calculations (with no other errors). Sample errors: One or two arithmetic mistakes, such as subtracting incorrectly; one or two mistakes in the algorithm such as bringing down the wrong number, provided there is evidence elsewhere in the work that the student knows how to do this; use of the method of partial quotients rather than long division. Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Sample errors: Correct answers found by a different method, such as guess-and-check multiplication; consistent mistakes with the algorithm; work is only partially completed. Problem 6 A sign in an art store gives these options: 12 poster boards for $29 24 poster boards for $56 50 poster boards for $129 Find each unit price to the nearest cent. Show your reasoning. Which option gives the lowest unit price? Show Solution The unit prices are $2.42, $2.33, and $2.58. Reasoning varies, but most students will use long division. 24 poster boards for $56 has the lowest unit price. Minimal Tier 1 response: Work is complete and correct. Sample: (Accompanied by work showing long division or other calculation methods.) The unit prices are $2.42, $2.33, and $2.58. 24 poster boards for $56 is the best deal. Tier 2 response: Work shows general conceptual understanding and mastery, with some errors. Acceptable errors: Incorrect selection of lowest unit rate comes from errors in calculation of unit rates. Sample errors: One or two errors in long division; incorrect selection of the best deal despite having calculated all three unit rates correctly. Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Sample errors: Calculation of unit rates involves a conceptual error, such as dividing the number of poster boards by the price; three or more errors in long division; correct selection of the best deal without calculation of each unit rate. Problem 7 A stack of 500 pieces of paper is 1.875 inches tall. Diego guesses that each piece of paper is 0.015 inch thick. Explain how you know that Diego’s answer is not correct. Compute the thickness of each piece of paper. Show your reasoning. Show Solution If each piece of paper were 0.015 inch thick, the stack would be 7.5 inches high: $500 \boldcdot (0.015) = 7.5$ . 0.00375 inch, because $1.875 \div 500 = 0.00375$ . Minimal Tier 1 response: Work is complete and correct, with complete explanation or justification. Sample: This can’t be right, because $500 \boldcdot (0.015)$ is 7.5. $1.875 \div 500$ is $(1.875) \boldcdot 2 \div 1000$ , which is $3.75 \div 1000$ . The digits of 3.75 need to move 3 places to the right, so each piece is 0.00375 inch thick. Tier 2 response: Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification. Sample errors: Arithmetic error in multiplication or division; result off by one decimal place but otherwise correct. Tier 3 response: Work shows a developing but incomplete conceptual understanding, with significant errors. Sample errors: Invalid or omitted work on one of the two problem parts; major error in division; division result off by more than one decimal place; incorrect choice of operation. 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.