Using Place Value in Comparing Quantities
Using Place Value in Comparing Quantities
Objectives
In this unit students review the concept of whole number place values and learn about decimal place values. The emphasis of these lessons is on developing students’ understanding of the relationship between adjacent place values (whole number and decimal), using powers of ten to multiply or divide, and techniques for comparing and rounding decimals. Students will:
- describe and draw representations of numeric place values.
- explain the relationship between adjacent place values.
- write whole and decimal values in base-ten numeral, word, and expanded forms.
- describe how multiplying and dividing by powers of ten impacts a value.
- represent powers of ten with whole number exponents.
- compare numbers (to the thousandths) using <, >, or =.
- round base-ten decimal numbers to specified place values (from the ones place to the thousandths place).
Essential Questions
- How is mathematics used to quantify, compare, represent, and model numbers?
- How can mathematics support effective communication?
- How are relationships represented mathematically?
- What makes a tool and/or strategy appropriate for a given task?
- How can patterns be used to describe relationships in mathematical situations?
Related Unit and Lesson Plans
Related Materials & Resources
The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.
- Interactive activity allowing students to change digits to different place values (represented by candles and layers of a birthday cake) to discover or reinforce the ratio of
between adjacent place values http://www.mathcats.com/explore/age/placevalueparty.html - Interactive base-ten blocks
http://nlvm.usu.edu/en/nav/frames_asid_152_g_2_t_1.html?from=category_g_2_t_1.html
- Manny’s Rumba
http://www.learningbox.com/Base10/BaseTen.html
- Students group one, tens and hundreds to form three-digit numbers, reinforcing place value concepts
https://www.brainpop.com/math/numbersandoperations/decimals/
- Practice placing decimal values to the left, right, or between two other values
http://www.aaamath.com/cmp-dec-order.htm
- This is a video lesson on dividing and multiplying by powers of ten (about 5 minutes)
http://www.mathplayground.com/howto_dividedecimalspower10.html
- Practice rounding decimals from word form to the nearest hundredth
Formative Assessment
-
View
Multiple-Choice Items:
- Which place value describes the location of the digit 4 in 2,576.049?
A
Tenths
B
Hundreds
C
Hundredths
D
Thousandths
- Which expression is the best representation of the number five hundred sixty-eight and four hundred seventy-three thousandths in expanded form?
A
568 × 1 + 473 × 0.001
B
5 × 100 + 6 × 10 + 8 × 1 + 4 × 0.1 + 7 × 0.01 + 3 × 0.001
C
5 × 100 + 68 + 4 × 100 + 73 × 0.001
D
500 + 60 + 8 + 400 + 70 + 3
- Which statement is not correct?
A
If two identical digits are next to each other in a base-ten number, the digit on the left will have a value 10 times greater than the digit on the right.
B
If two identical digits are next to each other in a base-ten decimal number, the digit on the right will have a value
of the value to its left.C
A digit’s numeric value depends on its placement in a number.
D
If two identical digits are next to each other in a base-ten number, the digits have the same numeric value in both places.
- Simplify 46.5 × 103.
A
0.465
B
4,650
C
46,500
D
465,000
5. Simplify 2.05 ÷ 102.
A
0.0205
B
0.00205
C
205
D
20,500
6. Write the base-ten numeral represented by (5 × 1,000) + (3 × 100) + (9 × 10) + (3 × 1).
A
5,391
B
5,393
C
500,393
D
5,003,009,031
7. Which comparison statement is true?
A 3.5
3.45 B 4.042 
4.41C 9.09
9.9 D 5.672 5.7718. Which rounding statement is false?
A
45.809 rounded to the nearest tenth is 45.9
B
4.5809 rounded to the hundredth is 4.58
C
0.45809 rounded to the nearest one is 0
D
4.5809 rounded to the nearest thousandth is 4.581
9. Estimate the following sum by rounding each value to the nearest whole before adding.
2.45 + 2.819 + 5.3
A
9
B
11
C
10.5
D
10
Multiple-Choice Answer Key:
1. C
2. B
3. D
4. C
5. A
6. B
7. C
8. A
9. D
Short-Answer Items:
- Write the base-ten number 478.932 in both word form and expanded form.
_____________________________________________________________________________________________________
_______________________________________________________
- Explain the patterns of zeros and decimal points in problems where a decimal number is multiplied or divided by a power of ten. Use examples to help you explain both.
_____________________________________________________________________________________________________
_____________________________________________________________________________________________________
_____________________________________________________________________________________________________
_____________________________________________________________________________________________________
- Compare the numbers below using <, >, or =. Write a description of your reasoning to help someone else your age understand how to compare decimal numbers accurately.
2.708
2.78______________________________________________________________________________________________________
______________________________________________________________________________________________________
______________________________________________________________________________________________________
______________________________________________________________________________________________________
Short-Answer Key and Scoring Rubrics:
Use the following rubric for short answers 10–12.
Points
Description
2
- Student’s written explanation or work is complete and detailed.
- Student demonstrates thorough understanding of the concept.
- Student’s answer is correct (and in proper form when applicable).
1
- Student’s written explanation or work is correct but brief or simplistic.
- Student demonstrates partial understanding of the concept.
- Student’s answer is partially correct, a minor error in process is evident, or the answer may not be in proper form.
0
- Student’s written explanation or work is incorrect or missing.
- Student demonstrates no understanding of the concept.
- Student’s answer is incorrect, incomplete, or missing.
- Write the base-ten number 478.932 in both word form and expanded form.
Four hundred seventy-eight and nine hundred thirty-two thousandths
478.932 = 4 × 100 + 7 × 10 + 8 × 1 + 9 × 0.1 + 3 × 0.01 + 2 × 0.001
- Explain the patterns of zeros and decimal points in problems where a decimal number is multiplied or divided by a power of ten. Use examples to help you explain both.
Answers will vary. A sample answer is provided.
Multiplying by a power of ten like 10, 102 or 103, moves the decimal one place to the right for each power of 10. For example, 4.5789 × 103 = 4,578.9 because 103 is like multiplying by 10 three times. The decimal moves right three places, and zeros are used to hold the new place values.
When dividing by a power of 10, the decimal moves left one place for each power of ten. For example 6.78 ÷ 103 = 0.00678 because the decimal moves left 3 places.
- Compare the numbers below using <, >, or =. Write a description of your reasoning to help someone else your age understand how to compare decimal numbers accurately.
2.708
2.78Descriptions may vary. A sample response is provided.
One of the numbers I compared is to the thousandths place and the other is to the hundredths. Comparing them would be easier if they were to the same place value. I added a zero to the end of 2.78 to make it 2.780 since this does not change its meaning but changes the place value it ends in. Then I compared the parts of each number. They both have two whole units so I needed to compare the decimals. The number on the left has 708 parts out of 1,000, while the number on the right has 780 parts out of 1,000. Since 780 is greater than 708, the number on the right is greater.
Performance Assessment:
Wise Shopper
Each student will select one (or more) food products to shop for. Collecting data and making math calculations will help students select the best purchasing option for their school.
Directions to students:
- Select a food product: Choose an item that would likely be served at the concession stand at a school event such as a play or basketball game. You will be in charge of finding the best option for purchasing this item.
- Research: Find at least four purchasing options for your product (different brands, store/prices, package sizes…). Record the product name/brand, location, price, package size, and number of servings per package. Find this information on a shopping trip with a family member, online, in grocery advertisements, or another method. Ask your family or teacher to help decide what the best option for you to gather your data is. Record your data in your product table.
- Calculate the price per serving (or unit price): Use a calculator to divide the price of the first package by the number of servings in the package. Record the result exactly as shown on the calculator. Find an organized way to record this result and the remaining data you collect. Round this value to the nearest thousandth, hundredth, and one. Record each of these calculations. Repeat these steps for each package.
- Compare price-per-serving data: Select one form of the price-per-serving data to compare (original calculation or one of the rounded forms). Use the same form for each product. Explain why you selected this form for your comparison. Rewrite the values in order least to greatest and label which product each value belongs with. Select the purchasing option that is the best deal for your school.
- Calculate the cost to purchase multiple servings: Use the product option selected in step 4. Calculate the cost of purchasing one hundred, one thousand, and ten thousand servings of your product. Show your work.
- Prepare a summary or presentation: Organize and display your work and reasoning. Write neatly and be prepared to share your work with the class.
Product Table for ______________________:
Product Name/Brand
Store Name or Location
Package Price
Package Size
Number of Servings
Package 1
Package 2
Package 3
Package 4
Price per Serving (Unit Price)
Original Calculation
Rounded to Thousandths
Rounded to Hundredths
Rounded to Tenths
Rounded to Ones
Package 1
Package 2
Package 3
Package 4
Performance Assessment Scoring Rubric:
Points
Description
4
- Student collected and recorded complete product data for more than four purchasing options or possibly more than one product was researched.
- Student’s price per serving and multiple serving costs were complete and accurate, with detailed work shown.
- Student’s rounding of the prices per serving was complete and accurate, with detailed explanation or work provided.
- Student demonstrates advanced understanding of the mathematical ideas and processes related to decimal place value and powers of ten.
- Student’s written summary or presentation is neat, thorough, and detailed.
- Student worked beyond the problem requirements, possibly by doing more than required and/or incorporating technology or an exceptional amount of detail.
3
- Student collected and recorded complete product data for four purchasing options for one product.
- Student’s price per serving and multiple serving costs were complete and accurate, with work shown, possibly 1–2 minor errors.
- Rounding was complete with some explanation provided, possibly 1–2 minor errors.
- Student demonstrates solid understanding of the mathematical ideas and processes related to decimal place value and powers of ten.
- Student’s written report or presentation is neat and thorough.
- Student’s work meets all of the problem requirements.
2
- Student collected and recorded product data for multiple purchasing options, but 1–5 cells of data were incomplete.
- Student’s price per serving and multiple serving costs were recorded, little work shown, with 3–4 errors or missing data cells.
- Rounding was partially complete, or complete with 3–5 errors.
- Student demonstrates some understanding of the mathematical ideas and processes related to decimal place value and powers of ten.
- Student’s written report is brief, lacking detail on several components of problem.
- Student’s work partially meets the problem requirements.
1
- Student collected and recorded product data for purchasing options, but 6–15 cells of data were incomplete.
- Student’s price per serving and multiple serving costs were only partially attempted, with little or no work shown, or most were complete, but with more than 4 errors.
- Rounding was only partially attempted, or mostly complete, but with more than 5 errors.
- Student demonstrates substantial lack of understanding of portions of the problem related to decimal place value and powers of ten.
- Student’s written report or presentation is significantly incomplete or has inaccuracies throughout.
- Student’s work does not meet several of the problem requirements.
0
- Student did not record product data or only 1–5 cells were filled.
- Price per serving and multiple serving costs were not attempted by student or 1–3 incorrect values, without work, were filled in.
- Rounding was not attempted by student, or was mostly incomplete with no correct responses.
- Student demonstrates a complete lack of understanding of the concepts of decimal place value and powers of ten.
- Student’s written report or presentation is missing.
- Student’s work does not meet any of the problem requirements.
Final 07/12/2013
