SESSION: 30 PAGE: 1 1/26/04 8:34 LOGIN IS-joer PATH: @sunultra1/raid/CLS_tpc/GRP_va_sprg03/JOB_03-olptg3/DIV_g3mth-1 VIRGINIA STANDARDS OF LEARNING ASSESSMENTS Spring 2003 Released Test GRADE 3 MATHEMATICS Property of the Virginia Department of Education 䉷 2003 by the Commonwealth of Virginia Department of Education, James Monroe Building, 101 N. 14th Street, Richmond, Virginia, 23219. All rights reserved. Except as permitted by law, this material may not be reproduced or used in any form or by any means, electronic or mechanical, including photocopying or recording, or by any information storage or retrieval system, without written permission from the copyright owner. Commonwealth of Virginia public school educators may photocopy or print any portion of these Released Tests for educational purposes without requesting permission. All others should direct their requests to the Commonwealth of Virginia Department of Education at (804) 225-2102, Division of Assessment and Reporting. SESSION: 26 PAGE: 2 5/8/103 15:47 LOGIN IS-debbie PATH: @sunultra1/raid/CLS_tpc/GRP_virginia/JOB_03-olptg3/DIV_g3mth-1 Mathematics DIRECTIONS 1 Brooke guessed that there are 605,788 buttons in the jar below. Read and solve each question. M3010514 墍 SAMPLE C ArtCodes 18 11 23 35 M3010514.AR1 uess G ? ? How y?? Man David Greg Keiko Betsy Who is holding a card with an even number on it? A B C D Which of the following shows 605,788 written in words? David Greg Keiko 墍 Betsy 2003 Commonwealth of Virginia Department of Education 2 A Six hundred, seventy-eight B Six hundred five, seven hundred eighty-eight C Six hundred five thousand, seventy-eight D Six hundred five thousand, seven hundred eighty-eight 墍 SESSION: 26 PAGE: 3 5/8/103 15:47 LOGIN IS-debbie PATH: @sunultra1/raid/CLS_tpc/GRP_virginia/JOB_03-olptg3/DIV_g3mth-1 4 Sara can use the fact 12 ⴑ 6 ⴔ 6 to help solve a related problem. Which of the following could be the M3040507 problem she is trying to solve? 墍 2 M2050201 墌 C F ⫹ 6 ⫽ 12 墍 G ⫻6⫽6 H ⫺6⫽0 J ⫼ 6 ⫽ 12 C ArtCodes M2050201.AR1 Each of these bags has exactly 5 buttons in it. How many buttons are there in all? F G H J 17 80 85 墍 90 5 The figure shown below is shaded to represent a fraction. M3060502 墍 C Which of the following is shaded to ArtCodes represent a fraction with a value M3060502.AR1 EQUAL TO the one above? 3 Greg read about a 365,970 mile space flight. What is the value of the 3 in 365,970? M3010502 墌 C L A B C D 300 3,000 30,000 300,000 墍 A B C D 2003 Commonwealth of Virginia Department of Education 3 墍 ArtCodes M3060502.AR2 M3060502.AR3 M3060502.AR4 M3060502.AR5 SESSION: 26 PAGE: 4 5/8/103 15:47 LOGIN IS-debbie PATH: @sunultra1/raid/CLS_tpc/GRP_virginia/JOB_03-olptg3/DIV_g3mth-1 6 7 Virginia covers one hundred two thousand, five hundred fifty-eight square kilometers of land. Which shows this number? M2030206 墌 C M3010507 墍 C A ArtCodes B M2030206.AR1 C D 1,258 12,558 102,558 墍 1,200,558 8 A clown walks on stilts that are 147 centimeters tall. Which figure is on the 15th space from START? M3020507 墍 C F ArtCodes M3020507.AR1 ArtCodes M2030206.AR2 M2030206.AR4 M2030206.AR3 M2030206.AR5 G 墍 H J What is 147 rounded to the nearest hundred? F G H J 2003 Commonwealth of Virginia Department of Education 4 250 200 150 100 墍 SESSION: 26 PAGE: 5 5/8/103 15:47 LOGIN IS-debbie PATH: @sunultra1/raid/CLS_tpc/GRP_virginia/JOB_03-olptg3/DIV_g3mth-1 9 Which figure shows 1 shaded? 10 10 This is a whole. M2040307 M3070512 墌 墍 C C L A ArtCodes ArtCodes M3070512.AR1 M2040307.AR1 M2040307.AR2 M2040307.AR3 M2040307.AR4 Which tells the exact number of pizzas shown in the picture below? B C ArtCodes 墍 M3070512.AR2 D F G H J 2003 Commonwealth of Virginia Department of Education 5 5.4 4.5 墍 0.54 0.45 SESSION: 26 PAGE: 6 5/8/103 15:47 LOGIN IS-debbie PATH: @sunultra1/raid/CLS_tpc/GRP_virginia/JOB_03-olptg3/DIV_g3mth-1 11 12 M2050300 M3050402 墌 墍 C L C ArtCodes ArtCodes M2050300.AR1 M3050402.AR1 What fraction of the candles on the cake is lit? Tracy bought these pencils. Each pencil cost 2 cents. How much did the pencils cost all together? A B C D 5¢ 15¢ 30¢ 墍 45¢ 2003 Commonwealth of Virginia Department of Education 6 F 4 墍 7 G 4 3 H 3 7 J 1 4 SESSION: 26 PAGE: 7 5/8/103 15:47 LOGIN IS-debbie PATH: @sunultra1/raid/CLS_tpc/GRP_virginia/JOB_03-olptg3/DIV_g3mth-1 13 The table below shows the length of some of the longest rivers in the world. M3030502 14 Diane has the money shown below. 墌 K BER T Y LI 19 9 9 LI 19 9 9 BER T Y BER T Y LI IN GOD WE TRUST LI 19 9 9 BER T Y BER T Y LI IN GOD WE TRUST LI BER T Y LI 19 9 9 BER T Y LI BER T Y BER T Y LI 墍 F BER T Y E D W T RU S GO Y BE ER RTT LL II B BE ER RTT LL II B BE ER RTT LL II B 1999 19 9 9 T Y 1999 IN 7 LI E D W T RU S GO IN IN IN IN IN BE ER RTT LL II B BE ER RTT LL II B BE ER RTT LL II B BE ER RTT LL II B 19 9 9 1 9 99 R TY BE 1999 BER T Y LIBERTY 1999 T S 19 9 9 Y LI IN GOD WE TRUST D WE T R U GO T S 19 9 9 BER T Y T S LI IN GOD WE TRUST T S BER T Y T S LI 1999 D WE T R U GO LIBERTY 1999 1999 IN GOD WE TRUST T S BER T Y IN GOD WE TRUST LIBERTY 1999 BER T Y IN GOD WE TRUST D WE T R U GO D WE T R U GO LIBERTY 1999 LI 1999 LIBERTY 1999 1999 BER T Y IN GOD WE TRUST T S LIBERTY LIBERTY 1999 LI 1999 T S 1999 Y BER T Y D WE T R U GO 1999 D WE T R U GO LIBERTY 1999 IN GOD WE TRUST 1999 D WE T R U GO Y T S LI LIBERTY D WE T R U GO LIBERTY 1999 BER T Y IN GOD WE TRUST 1999 D WE T R U GO T S T S T S T S T S IN D WE T R U GO LIBERTY D WE T R U GO LI T 1999 1999 LI IN GOD WE TRUST 1999 D WE T R U GO LIBERTY BER T Y IN GOD WE TRUST 1999 1999 BER T Y LI T S D WE T R U GO LIBERTY 1999 BER T Y IN GOD WE TRUST 1999 D WE T R U GO IN GOD WE TRUST LI IN GOD WE TRUST 1999 LI 1999 1999 R TY BE IN GOD WE TRUST 1999 1999 BER T Y IN BER T Y LI Y LIBERTY 1999 IN GOD WE TRUST BER T Y LI BER T Y LI IN GOD WE TRUST LIBERTY J LIBERTY 1999 IN LI BER T Y LI 1999 LI M3130509.AR2 M3130509.AR3 M3130509.AR4 M3130509.AR5 D WE T R U GO IN BER T Y 1999 Y ArtCodes D WE T R U GO LIBERTY 1999 1999 LIBERTY D WE T R U GO LIBERTY 1999 IN GOD WE TRUST D WE T R U GO IN LIBERTY 1999 1999 H D WE T R U GO 1999 T S LI IN GOD WE TRUST LI D WE T R U GO 1999 T S BER T Y IN GOD WE TRUST G 1999 IN GOD WE TRUST T S LI IN GOD WE TRUST T S 1999 1999 LI T IN GOD WE TRUST IN 1999 Y BE ER RTT LL II B IN GOD WE TRUST IN IN GOD WE TRUST 1999 Mississippi River Nile River 墍 Yangtze River Yenisei River 2003 Commonwealth of Virginia Department of Education BER T Y IN GOD WE TRUST 19 9 9 T S D BER T Y LI IN GOD WE TRUST 1999 R TY BE C 1999 IN LI IN GOD WE TRUST IN GOD WE TRUST B 1999 1999 Shelly has a group of coins with the same total value as Diane’s money. Which could be Shelly’s money? The Amazon River is 4,000 miles long. This is LESS THAN the length of which river? A BE ER RTT LL II B 3,442 IN GOD WE TRUST 1999 Y IN Yenisei IN GOD WE TRUST 1999 Y IN 4,000 IN GOD WE TRUST M3130509.AR1 BER T Y IN Amazon LI BE ER RTT LL II B 3,964 BER T Y E D W T RU S GO Yangtze LI IN 4,145 BER T Y IN Nile LI IN 2,340 IN M3030502.AR1 Mississippi ArtCodes O N EODNOELDLOALRL A R IN ArtCodes Length in Miles C ONEONE K B 53451446 B B 53451446 B Name of River 墍 B 53451446 B B 53451446 B River Lengths IN C L M3130509 THE UNITED STATES OF AMERICA THE UNITED STATES OF AMERICA 1999 1999 SESSION: 26 PAGE: 8 5/8/103 15:48 LOGIN IS-debbie PATH: @sunultra1/raid/CLS_tpc/GRP_virginia/JOB_03-olptg3/DIV_g3mth-1 15 A school library had 2,089 books. It received 853 more books during the year. How many books did the M3080513 墌 library have then? 18 This is a whole cake. C A B C D 1,236 2,832 2,836 2,942 墍 Mark's part of the cake is shaded below. M3110512 墍 C 42 ⴓ 7 ⴔ 16 F M3090516 He gave these 5 pieces away. 墌 G C H J 5 6墍 7 8 Cathy bought 25 packages of balloons. Each package has 4 balloons. How many balloons did she buy in all? 墌 C L Which of the following number sentences can Ben use to solve this problem? A 25 ⫻ 4 ⫽ B 25 ⫹ 4 ⫽ C 25 ⫺ 4 ⫽ D 25 ⫼ 4 ⫽ M3110512.AR1 M3110512.AR2 M3110512.AR3 What fraction of the whole cake did Mark have left? 17 The word problem below is part of Ben’s homework. M3100522 ArtCodes 墍 2003 Commonwealth of Virginia Department of Education 8 F 2 4 G 2 6 H 2 7 J 2 墍 8 SESSION: 26 PAGE: 9 5/8/103 15:47 LOGIN IS-debbie PATH: @sunultra1/raid/CLS_tpc/GRP_virginia/JOB_03-olptg3/DIV_g3mth-1 19 21 At the craft fair, Tom sold toy trains like the one shown below. 56.4 ⴑ 32.7 M3120118 M3100509 墌 墍 C C A B C D 14.7 23.7 墍 24.3 24.7 $28 ArtCodes M3100509.AR1 He sold each train for $28. How much did Tom make by selling 4 trains? 20 A M3110504 墌 B C L C D ArtCodes $24 $32 $112 墍 $832 M3110504.AR1 M3110504.AR2 22 This is one. This is one tenth. M3120301 墍 C F 1 3 G 3 8 H 2 3 J What is M3120301.AR1 – F G H 5 墍 8 2003 Commonwealth of Virginia Department of Education ArtCodes J 9 2.4 墍 2.0 1.6 0.1 ? SESSION: 26 PAGE: 10 5/8/103 15:47 LOGIN IS-debbie PATH: @sunultra1/raid/CLS_tpc/GRP_virginia/JOB_03-olptg3/DIV_g3mth-1 23 Paul wanted to buy a shirt that cost $15. He had $7. How much more did Paul need to buy the shirt? M3080107 墌 C A B C D 25 Use your centimeter ruler to help you answer this question. Which is CLOSEST to the length of this tennis racket? $22 $12 $8 墍 $3 M314A504 墍 C L ArtCodes M314A504.AR1 7ⴒ9ⴔ 24 F M3090509 墌 G C L H J 16 32 56 63 墍 A B C D 9 centimeters 10 centimeters 墍 11 centimeters 12 centimeters Note that due to varying printer properties, measurement items may not appear in exact proportions. 2003 Commonwealth of Virginia Department of Education 10 SESSION: 26 PAGE: 11 5/8/103 15:47 LOGIN IS-debbie PATH: @sunultra1/raid/CLS_tpc/GRP_virginia/JOB_03-olptg3/DIV_g3mth-1 26 Each small square on the grid is 1 square unit. 27 Martin drew the picture shown below. M2130203 M1150602 墌 墍 C L C ArtCodes ArtCodes M2130203.AR1 M1150602.AR1 Which of the following is NOT true about the boy in the picture? How many square units are needed to make the shaded figure shown on the grid? F G H J A B 5 7 10 墍 14 C D He He He He is is is is far from the tree. 墍 close to the tree. beside the tree. near the tree. 28 Linda and Tom have been friends for 1 year. How many months make 1 year? M3160514 F G H J 2003 Commonwealth of Virginia Department of Education 11 6 12 墍 18 24 墍 C SESSION: 26 PAGE: 12 5/8/103 15:47 LOGIN IS-debbie PATH: @sunultra1/raid/CLS_tpc/GRP_virginia/JOB_03-olptg3/DIV_g3mth-1 29 Timmy saw this picture of an old television. M3190601 墌 31 In which of the figures below is it possible to draw a line of symmetry? C M3200509 墍 C L A ArtCodes M3190601.AR1 ArtCodes M3200509.AR1 M3200509.AR2 M3200509.AR3 M3200509.AR4 Which part of the television is the BEST model of an angle? B A ArtCodes 墍 B M3190601.AR2 M3190601.AR3 M3190601.AR4 M3190601.AR5 C C D 30 What shape are the faces of a cube? F M3180104 墌 G C H J D Circle Triangle Square 墍 Pentagon 2003 Commonwealth of Virginia Department of Education 12 墍 SESSION: 26 PAGE: 13 5/8/103 15:47 LOGIN IS-debbie PATH: @sunultra1/raid/CLS_tpc/GRP_virginia/JOB_03-olptg3/DIV_g3mth-1 32 Which of the following is the BEST example of a sphere? 33 Which 2 shapes below are congruent? M2180601 M3200513 墌 墍 C ArtCodes L F 墍 P M2180601.AR1 M2180601.AR2 M2180601.AR3 M2180601.AR4 A B C D C M ArtCodes N M3200513.AR1 L and P M and N 墍 N and L P and N G 34 The school principal has a fire drill planned for the time shown on the clock below. M3150506 墍 11 H 12 C L 1 10 2 ArtCodes 3 9 8 4 7 6 5 J Which is closest to the time shown on the clock? F G H J 2003 Commonwealth of Virginia Department of Education 13 12:06 12:29 墍 1:06 1:29 M3150506.AR1 SESSION: 26 PAGE: 14 5/8/103 15:47 LOGIN IS-debbie PATH: @sunultra1/raid/CLS_tpc/GRP_virginia/JOB_03-olptg3/DIV_g3mth-1 35 Which thermometer shows 78ⴗF? M3170114 墌 90° 36 Look at the pairs of shapes. Which is a pair of rectangles? °F M3180106 墍 C C F 80° ArtCodes A M3170114.AR1 M3170114.AR2 M3170114.AR3 M3170114.AR4 ArtCodes 70° M3180106.AR1 M3180106.AR2 M3180106.AR3 M3180106.AR4 60° G 90° °F 80° B H 70° 60° 90° °F J 80° 墍 C 70° 60° 90° °F 80° D 70° 60° 2003 Commonwealth of Virginia Department of Education 14 墍 SESSION: 26 PAGE: 15 5/8/103 15:47 LOGIN IS-debbie PATH: @sunultra1/raid/CLS_tpc/GRP_virginia/JOB_03-olptg3/DIV_g3mth-1 37 The chart below shows the number of some different snacks sold before a movie. M3210607 Snacks Sold 墌 C Snack Number Sold Candy 15 Drink 11 Popcorn 18 38 NOVEMBER Sun Mon Tue Wed Thu 1 2 3 4 7 8 9 10 11 14 15 16 17 18 21 22 23 24 25 28 29 30 Fri 5 12 19 26 Sat 6 13 20 27 M2220212 墍 C ArtCodes M2220212.AR1 ArtCodes M3210607.AR1 Diana’s birthday is on November 23. Hilary’s birthday is six days before that. When is Hilary’s birthday? Which graph below shows the correct number sold for each kind of snack? F G H A ArtCodes Snack Snacks Sold M3210607.AR2 M3210607.AR3 M3210607.AR4 M3210607.AR5 J 墍 0 2 4 6 8 10 12 14 16 18 20 Number Sold B Snack Snacks Sold 0 2 4 6 8 10 12 14 16 18 20 Number Sold C Snack Snacks Sold 0 2 4 6 8 10 12 14 16 18 20 Number Sold D Snack Snacks Sold 0 2 4 6 8 10 12 14 16 18 20 Number Sold 2003 Commonwealth of Virginia Department of Education 15 November November November November 6 17 墍 18 24 SESSION: 26 PAGE: 16 5/8/103 15:47 LOGIN IS-debbie PATH: @sunultra1/raid/CLS_tpc/GRP_virginia/JOB_03-olptg3/DIV_g3mth-1 39 The bar graph below shows the favorite breakfast cereals of some students. M3220530 墌 C L ArtCodes Number of Students M3220530.AR1 18 16 14 12 10 8 6 4 2 0 40 The chart below shows the number of each kind of kite that are mixed up in a box. M3230504 墍 Favorite Breakfast Cereals Kites for Sale Kind of Kite C L Number of Kites ArtCodes 4 M3230504.AR1 7 18 11 Nutty Crispy Fun Yummy Pops Crunch Flakes OÕs Name of Cereal If Cathy picks a kite from the box without looking, what kind is it MOST LIKELY to be? How many MORE students like Nutty Pops than Yummy O’s? A B C D 16 6墍 3 1 F ArtCodes M3230504.AR2 M3230504.AR3 M3230504.AR4 M3230504.AR5 G H J 2003 Commonwealth of Virginia Department of Education 16 墍 SESSION: 26 PAGE: 17 5/8/103 15:47 LOGIN IS-debbie PATH: @sunultra1/raid/CLS_tpc/GRP_virginia/JOB_03-olptg3/DIV_g3mth-1 41 These are the different erasers and pencils Adrian has. M3230520 墌 42 The picture graph below shows the number of sweaters ordered in each size in one week. M3220507 墍 C Sweaters Ordered Number Ordered Size ArtCodes C Small M3230520.AR1 ArtCodes Which of the following shows all the different ways that Adrian can combine 1 eraser and 1 pencil? M3220507.AR1 Medium Large Extra Large Key: A = 3 sweaters. ArtCodes M3230520.AR2 M3230520.AR3 M3230520.AR4 M3230520.AR5 Which size of sweater had exactly 15 sweaters ordered? F G H B J C D 墍 2003 Commonwealth of Virginia Department of Education 17 Small 墍 Medium Large Extra Large SESSION: 26 PAGE: 18 5/8/103 15:47 LOGIN IS-debbie PATH: @sunultra1/raid/CLS_tpc/GRP_virginia/JOB_03-olptg3/DIV_g3mth-1 43 The bar graph below shows the number of miles each member of a club hiked in one week. M3220548 45 Look at the pattern of shapes below. M2250519 墌 C 墍 C Hiking Miles ArtCodes Number of Miles M3220548.AR1 35 30 25 20 15 10 5 0 George Nancy Tony Rose Paula ArtCodes M2250519.AR1 If the pattern continues in the same way, what will the next shape look like? A ArtCodes M2250519.AR2 M2250519.AR3 M2250519.AR4 M2250519.AR5 Name Which is CLOSEST to the number of miles Paula hiked? A B C D B 5 14 20 24 墍 C 44 Look at the pattern of numbers shown below. D 墌 21 M2250525 29 37 45 ? C If the pattern continues in the same way, what will be the next number? F G H J 46 52 53 墍 63 2003 Commonwealth of Virginia Department of Education 18 墍 SESSION: 26 PAGE: 19 5/8/103 15:47 LOGIN IS-debbie PATH: @sunultra1/raid/CLS_tpc/GRP_virginia/JOB_03-olptg3/DIV_g3mth-1 46 The table below shows the number of vans needed for different numbers of students to go on a M3240511 墌 camping trip. C L ArtCodes M3240511.AR1 Number of Vans Number of Students A 1 2 3 4 5 6 G H J B C 6 12 18 24 30 ? If the pattern in the table continues, how many students are able to ride in 6 vans? F 47 Shauna had some toy cars in a bag. She gave 8 of the cars to Lucas. After that, Shauna had 10 toy cars. How many toy cars were in the bag before Shauna gave any to Lucas? 31 36 墍 40 42 D 2 9 12 18 墍 M2260305 墍 C 48 Numbers that go into the machine below are changed to a different number using a rule. 20 In 21 In Out 27 墍 C 23 In Out 28 M3240504 Out 30 ArtCodes M3240504.AR1 Which of the following could be the rule the machine uses? F G H J 2003 Commonwealth of Virginia Department of Education 19 Add Add Add Add 1 3 7墍 10 SESSION: 28 PAGE: 20 5/8/103 15:52 LOGIN IS-debbie PATH: @sunultra1/raid/CLS_tpc/GRP_virginia/JOB_03-olptg3/DIV_g3mth-1 49 Look at the pattern of shapes below. 50 Look at the pattern of leaves below. M3240537 M2250517 墌 墍 C C L ArtCodes ArtCodes M3240537.AR1 A ArtCodes M3240537.AR2 M3240537.AR3 M3240537.AR4 M3240537.AR5 M2250517.AR1 If the pattern continues in the same way, what will be the next shape? If the pattern continues in the same way, what will the next leaf in the pattern look like? 墍 B F C ArtCodes D M2250517.AR2 M2250517.AR3 M2250517.AR4 M2250517.AR5 G H J 2003 Commonwealth of Virginia Department of Education 20 墍 SESSION: 26 PAGE: 21 5/8/103 15:47 LOGIN IS-debbie PATH: @sunultra1/raid/CLS_tpc/GRP_virginia/JOB_03-olptg3/DIV_g3mth-1 Answer Key Test Sequence Correct Answer Reporting Category Reporting Category Description 1 D 006 Number and Number Sense 2 H 006 Number and Number Sense 3 D 006 Number and Number Sense 4 F 006 Number and Number Sense 5 B 006 Number and Number Sense 6 G 006 Number and Number Sense 7 C 006 Number and Number Sense 8 J 006 Number and Number Sense 9 C 006 Number and Number Sense 10 G 006 Number and Number Sense 11 C 006 Number and Number Sense 12 F 006 Number and Number Sense 13 B 006 Number and Number Sense 14 F 007 Computation and Estimation 15 D 007 Computation and Estimation 16 G 007 Computation and Estimation 17 A 007 Computation and Estimation 18 J 007 Computation and Estimation 19 B 007 Computation and Estimation 20 J 007 Computation and Estimation 21 C 007 Computation and Estimation 22 F 007 Computation and Estimation 23 C 007 Computation and Estimation 24 J 007 Computation and Estimation 25 B 008 Measurement and Geometry 26 H 008 Measurement and Geometry 27 A 008 Measurement and Geometry 28 G 008 Measurement and Geometry 29 A 008 Measurement and Geometry 30 H 008 Measurement and Geometry 31 C 008 Measurement and Geometry 32 F 008 Measurement and Geometry 33 B 008 Measurement and Geometry 34 G 008 Measurement and Geometry 35 C 008 Measurement and Geometry 36 H 008 Measurement and Geometry 37 A 009 Probability and Statistics 38 G 009 Probability and Statistics 39 B 009 Probability and Statistics 40 H 009 Probability and Statistics 41 D 009 Probability and Statistics 42 F 009 Probability and Statistics 43 D 009 Probability and Statistics 44 H 010 Patterns, Functions, and Algebra 45 D 010 Patterns, Functions, and Algebra 46 G 010 Patterns, Functions, and Algebra 47 D 010 Patterns, Functions, and Algebra 48 H 010 Patterns, Functions, and Algebra 49 A 010 Patterns, Functions, and Algebra 50 J 010 Patterns, Functions, and Algebra
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