A Problem Solving Approach To Mathematics For Elementary School Teachers (13th Edition)
13th Edition
ISBN: 9780135183885
Author: Rick Billstein, Shlomo Libeskind, Johnny Lott, Barbara Boschmans
Publisher: PEARSON
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Textbook Question
Chapter 3.5A, Problem 6A
Show that, in general, each of the following is false if
a.
b.
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7. [10 marks]
Let G
=
(V,E) be a 3-connected graph. We prove that for every x, y, z Є V, there is a
cycle in G on which x, y, and z all lie.
(a) First prove that there are two internally disjoint xy-paths Po and P₁.
(b) If z is on either Po or P₁, then combining Po and P₁ produces a cycle on which
x, y, and z all lie. So assume that z is not on Po and not on P₁. Now prove that
there are three paths Qo, Q1, and Q2 such that:
⚫each Qi starts at z;
• each Qi ends at a vertex w; that is on Po or on P₁, where wo, w₁, and w₂ are
distinct;
the paths Qo, Q1, Q2 are disjoint from each other (except at the start vertex
2) and are disjoint from the paths Po and P₁ (except at the end vertices wo,
W1, and w₂).
(c) Use paths Po, P₁, Qo, Q1, and Q2 to prove that there is a cycle on which x, y, and
z all lie. (To do this, notice that two of the w; must be on the same Pj.)
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Let T be a tree with n ≥ 2 vertices and leaves. Let BL(T) denote the block graph of
T.
(a) How many vertices does BL(T) have?
(b) How many edges does BL(T) have?
Prove that your answers are correct.
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Find both a matching of maximum size and a vertex cover of minimum size in
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Chapter 3 Solutions
A Problem Solving Approach To Mathematics For Elementary School Teachers (13th Edition)
Ch. 3.1 - Mathematical Connections Explain why a number and...Ch. 3.1 - If a numeration system has place value, what does...Ch. 3.1 - Explain the difference between zero and nothingCh. 3.1 - What are the major drawbacks to each of the...Ch. 3.1 - Large numbers in the United States are written...Ch. 3.1 - In the Roman numeral system, explain a when you...Ch. 3.1 - Mathematical Connections Research an abacus and...Ch. 3.1 - Prob. 8MCCh. 3.1 - Mathematical Connections Research Adrian Treffers...Ch. 3.1 - Mathematical Connections Joanne asks if the year...
Ch. 3.1 - Betty asks how the two 3s in the numeral 32,356...Ch. 3.1 - A student claims that the Roman system is a...Ch. 3.1 - When using Roman numerals, a student asks whether...Ch. 3.1 - The School Book Page on page 88 shows place value...Ch. 3.1 - Which of the following numbers is represented by...Ch. 3.1 - Which number does the diagram above represent? a...Ch. 3.1 - Write a three-digit number using the digits 2,4...Ch. 3.1 - There are 153,685 people living in a city. What is...Ch. 3.1A - For each of the following pairs of numerals, tell...Ch. 3.1A - A cornerstone representing when a building was...Ch. 3.1A - Write each of the following numerals in Roman...Ch. 3.1A - Indicate which digit in 234,518 has place value....Ch. 3.1A - For each of the following base-ten numerals, give...Ch. 3.1A - Rewrite each of the following as base-ten numeral....Ch. 3.1A - Write the following numerals in words....Ch. 3.1A - How many digits are there in the Hindu-Arabic form...Ch. 3.1A - A certain three-digit natural number has the...Ch. 3.1A - In the counting frame, the value of each dot is...Ch. 3.1A - Write the base four numeral for the following...Ch. 3.1A - Prob. 13ACh. 3.1A - How many different digits are needed for the...Ch. 3.1A - Write 2032four in expanded notation and then...Ch. 3.1A - Determine the greatest three-digit number in each...Ch. 3.1A - Find the numeral preceding and succeeding each of...Ch. 3.1A - What, if anything, is wrong with the following...Ch. 3.1A - Draw base-five blocks to represent 231five.Ch. 3.1A - Without converting to base-ten, tell which is the...Ch. 3.1A - Recall that with base-ten blocks: 1long=10units,...Ch. 3.1A - Convert each of the following base-ten numerals to...Ch. 3.1A - Write each of the following numerals in base ten....Ch. 3.1A - Represent abcdsix, a base-six numeral, in expanded...Ch. 3.1A - In what base does 12b+26b=41b?Ch. 3.1A - You are asked to distribute 900 in prize money....Ch. 3.1A - For each of the following, find b if possible. If...Ch. 3.1A - Write the following numerals in the indicated base...Ch. 3.1A - Complete the following table. Base Ten Base Two...Ch. 3.1B - Prob. 1ACh. 3.1B - Prob. 2ACh. 3.1B - On the United States one-dollar bill, the number...Ch. 3.1B - Write each of the following numerals in Roman...Ch. 3.1B - Indicate which digit in 2,345,925 has place value....Ch. 3.1B - For each of the following base-ten numerals, give...Ch. 3.1B - Rewrite each of the following as a base-ten...Ch. 3.1B - Write the following in words. a. 253,243,810 b....Ch. 3.1B - How many digits are there in the Hindu-Arabic form...Ch. 3.1B - A two-digit number has the property that the units...Ch. 3.1B - Prob. 11ACh. 3.1B - Assessment Write the base-three numeral for the...Ch. 3.1B - Write the first 10 counting numerals for each of...Ch. 3.1B - How many different digits are used in base...Ch. 3.1B - Write 2022three in expanded form and then convert...Ch. 3.1B - Determine the greatest three-digit number in each...Ch. 3.1B - Find the numeral preceding and succeeding each of...Ch. 3.1B - What, if anything, is wrong with the following? a....Ch. 3.1B - Draw base-two blocks to represent 1001two.Ch. 3.1B - Prob. 20ACh. 3.1B - Assessment What is the fewest number of multibase...Ch. 3.1B - Convert each of the following base-ten numerals to...Ch. 3.1B - Write each of the following numerals in base ten....Ch. 3.1B - Represent abcdeight as a base-ten numeral in...Ch. 3.1B - In what base does 54b+45b=132b?Ch. 3.1B - Who Wants the Money, a game show, distributes...Ch. 3.1B - For each of the following numerals, find digit b,...Ch. 3.1B - Using only the number keys on a calculator, fill...Ch. 3.1B - Assessment Complete the following table. Base Ten...Ch. 3.1B - Use fair trading to represent the base-three...Ch. 3.2 - In a survey of 52 students, 22 said they were...Ch. 3.2 - To find 9+7, a student says she thinks of 9+7 as...Ch. 3.2 - Explain in two different ways why if ab and c is a...Ch. 3.2 - Prob. 4MCCh. 3.2 - Describe any model not in this text that you could...Ch. 3.2 - a. Create a word problem in which the set model...Ch. 3.2 - Discuss the merit of the following expanded...Ch. 3.2 - Is the front-end estimate for addition before...Ch. 3.2 - In the new textbooks, there is an emphasis on...Ch. 3.2 - Search for or develop an algorithm for...Ch. 3.2 - Give several examples from real-world situations...Ch. 3.2 - a. Give a numerical example of when front-end...Ch. 3.2 - Zeke claims that he can get the same answer to the...Ch. 3.2 - A child is asked to compute 7+2+3+8+11 and writes...Ch. 3.2 - When Richie was asked to round 2568 to the nearest...Ch. 3.2 - When Ashley was asked to round 345 to the nearest...Ch. 3.2 - Making a ten with ten frames is also helpful in...Ch. 3.2 - The associative, commutative, and identity...Ch. 3.2 - Read the compensation algorithm for addition on...Ch. 3.2 - Page 115 shows adding from left and breaking up...Ch. 3.2 - Prob. 21MCCh. 3.2 - Prob. 22MCCh. 3.2 - Prob. 23MCCh. 3.2 - Prob. 24MCCh. 3.2 - Prob. 25MCCh. 3.2 - Prob. 1NAEPCh. 3.2 - Prob. 2NAEPCh. 3.2 - Prob. 3NAEPCh. 3.2 - Prob. 4NAEPCh. 3.2A - Prob. 1ACh. 3.2A - If n(A)=3, n(B)=5 and n(AB)=6, what do you know...Ch. 3.2A - Give an example to show why, in the definition of...Ch. 3.2A - If n(A)=3, n(AB)=6, answer the following...Ch. 3.2A - Use a number line to model 5+7.Ch. 3.2A - A student uses the counting on strategy for adding...Ch. 3.2A - Use doubles+1 to solve 7+8.Ch. 3.2A - Use ten frames to make a ten. Then complete the...Ch. 3.2A - Prob. 9ACh. 3.2A - Which of the following sets are closed under...Ch. 3.2A - Each of the following equations illustrates an...Ch. 3.2A - Each of the following illustrates a property of...Ch. 3.2A - Show that each of the following is true. Give a...Ch. 3.2A - Find the solution for x in the set W for each of...Ch. 3.2A - Prob. 15ACh. 3.2A - Make a drawing similar to the one in Figure 15 to...Ch. 3.2A - Janet worked her addition problems by placing the...Ch. 3.2A - Use the lattice algorithm to perform each of the...Ch. 3.2A - Use the column-addition algorithm to compute...Ch. 3.2A - Use the opposite-change algorithm to compute...Ch. 3.2A - Perform each of the following addition problems in...Ch. 3.2A - Determine what is wrong with the following...Ch. 3.2A - Use breaking up and bridging to compute each of...Ch. 3.2A - Use trading off to compute each of the following...Ch. 3.2A - Prob. 25ACh. 3.2A - Estimate each answer by rounding. a. 878+2340 b....Ch. 3.2A - Use front-end estimation with adjustment to...Ch. 3.2A - Use the range strategy to estimate each of the...Ch. 3.2A - Suppose you had a balance of 3287 in your checking...Ch. 3.2A - Prob. 31ACh. 3.2A - 32. Complete the following magic square using the...Ch. 3.2A - If A, B, and C each stand for a different single...Ch. 3.2A - A special domino set contains all number pairs...Ch. 3.2A - Find the missing digits in each of the following...Ch. 3.2A - Place the digits 7, 6, 8, 3, 5, and 2 in the boxes...Ch. 3.2A - In the following problem, the sum is correct but...Ch. 3.2A - The following is a supermagic square....Ch. 3.2A - a. Place the numbers 1 through 9 in the following...Ch. 3.2A - Use scratch addition to perform the following...Ch. 3.2B - Prob. 1ACh. 3.2B - Prob. 2ACh. 3.2B - If n(B)=4, and n(AB)=6, answer the following. a....Ch. 3.2B - Use a number line to model 3+9.Ch. 3.2B - A student uses the counting on strategy for adding...Ch. 3.2B - Use the doubles plus strategy for adding the...Ch. 3.2B - Explain whether the following sets are closed...Ch. 3.2B - Prob. 8ACh. 3.2B - Use ten frames to make a ten. Then complete the...Ch. 3.2B - Each of the following equations is an example of...Ch. 3.2B - Each of the following illustrates a properties of...Ch. 3.2B - Prob. 12ACh. 3.2B - Prob. 13ACh. 3.2B - Prob. 14ACh. 3.2B - Make an appropriate drawing like the one in figure...Ch. 3.2B - Hector computed his addition problems by placing...Ch. 3.2B - Use the lattice algorithm to perform each of the...Ch. 3.2B - Use the column-addition algorithm to compute...Ch. 3.2B - Use the opposite-change algorithm to compute...Ch. 3.2B - Perform each of the following addition problems in...Ch. 3.2B - Determine what is wrong with the following...Ch. 3.2B - Prob. 22ACh. 3.2B - Estimate each answer by rounding. a. 937+28 b....Ch. 3.2B - Use front-end estimation with adjustment to...Ch. 3.2B - a. Would the clustering strategy of estimation be...Ch. 3.2B - Use the range strategy to estimate each of the...Ch. 3.2B - The attendance at a Country Fair for six days...Ch. 3.2B - Complete the following magic square using the...Ch. 3.2B - If A,B,C, and D each represent a different single...Ch. 3.2B - George is cooking a meal for Thanksgiving. He can...Ch. 3.2B - Use scratch addition to perform the following...Ch. 3.2B - What is the greatest possible addend that can be...Ch. 3.2B - In the following problem, the sum is correct but...Ch. 3.2B - The following is a super magic square taken from...Ch. 3.2B - a. Place the numbers 24 through 32 in the...Ch. 3.3 - When subtraction and addition appear in an...Ch. 3.3 - Explain whether it is important for elementary...Ch. 3.3 - Prob. 3MCCh. 3.3 - Explain how the model shown can be used to...Ch. 3.3 - Prob. 5MCCh. 3.3 - Prob. 6MCCh. 3.3 - Prob. 7MCCh. 3.3 - Prob. 8MCCh. 3.3 - Prob. 9MCCh. 3.3 - Prob. 10MCCh. 3.3 - Prob. 11MCCh. 3.3 - A student says asks why we use subtraction to...Ch. 3.3 - Prob. 13MCCh. 3.3 - Prob. 14MCCh. 3.3 - A first grader claims that 24=2 because you just...Ch. 3.3 - Ashton claims that if he has 8 cookies and eats 3...Ch. 3.3 - Jordan and Maksim started with the same number of...Ch. 3.3 - Prob. 18MCCh. 3.3 - To find 6819, Joe began by finding 61 and then 98...Ch. 3.3 - Mairead subtracted 415212 by writing 42=2 and...Ch. 3.3 - Olinda found 51849=469. She was not sure she was...Ch. 3.3 - Pauli subtracted 549 from 5004 as follows. How...Ch. 3.3 - Molly computed 26148 by first subtracting 50 from...Ch. 3.3 - Prob. 24MCCh. 3.3 - Prob. 25MCCh. 3.3 - Prob. 26MCCh. 3.3 - Prob. 27MCCh. 3.3 - Prob. 28MCCh. 3.3 - The trading-off strategy for mentally subtracting...Ch. 3.3 - Prob. 30MCCh. 3.3 - Prob. 31MCCh. 3.3 - Prob. 32MCCh. 3.3 - Prob. 33MCCh. 3.3 - 8=21 What number should be put in the box to make...Ch. 3.3 - Prob. 2NAEPCh. 3.3 - Prob. 3NAEPCh. 3.3 - Prob. 4NAEPCh. 3.3 - Prob. 5NAEPCh. 3.3A - Illustrate 85=3 using each of the following...Ch. 3.3A - Write a word problem that corresponds to each of...Ch. 3.3A - a. One fact in a fact family is 6+5=11.What are...Ch. 3.3A - What conditions, if any, must be placed on a and b...Ch. 3.3A - Rewrite each of the following subtraction problems...Ch. 3.3A - a. Solve for x in each of the following equations....Ch. 3.3A - Kelsey has a marbles, Gena has b marbles, and Noah...Ch. 3.3A - Analyze the following computations. Explain what...Ch. 3.3A - a. At a volleyball game, the players stood in a...Ch. 3.3A - Toms diet allows only 1500 calories per day. For...Ch. 3.3A - Use the equal-additions approach to compute each...Ch. 3.3A - Compute each of the following using the counting...Ch. 3.3A - Compute 146-85 using the trades-first algorithm.Ch. 3.3A - Place the digits 7, 6, 8, 3, 5 and 2 in the boxes...Ch. 3.3A - Construct an addition table for base eight and...Ch. 3.3A - Find the number to put in the blank to make each...Ch. 3.3A - Perform the following operation...Ch. 3.3A - Prob. 20ACh. 3.3A - Compute each of the following mentally and...Ch. 3.3A - Use breaking and bridging to compute 99732...Ch. 3.3B - Illustrate 73=4 using each of the following...Ch. 3.3B - Write a word problem that corresponds to each of...Ch. 3.3B - a. One fact in a fact family is 127=5. what are...Ch. 3.3B - Prob. 4ACh. 3.3B - Prob. 5ACh. 3.3B - Find the solution in the set W for each of the...Ch. 3.3B - Ross has 11pencils. Kersten has 5pencils. Which...Ch. 3.3B - Analyze the following computations. Explain what...Ch. 3.3B - If, ab,cb,dc,andced, order the letters from the...Ch. 3.3B - Find the missing digits in each of the following...Ch. 3.3B - Use the equal-additions approach to compute each...Ch. 3.3B - Compute each of the following using the...Ch. 3.3B - Make a table similar to the one in Figure 23 to...Ch. 3.3B - Compute 12858 using trades-first algorithm.Ch. 3.3B - Construct an additional table for base six, and...Ch. 3.3B - Perform each of the following operations using the...Ch. 3.3B - Find the number to put in the blank to make each...Ch. 3.3B - Perform each of the following operations (2c=...Ch. 3.3B - The Hawks played the Elks in a basketball game....Ch. 3.3B - Compute each of the following mentally. a....Ch. 3.3B - Compute each of the following mentally and...Ch. 3.3B - Use trading-off to compute each of the following...Ch. 3.3B - In each of the following, answer the question...Ch. 3.4 - MATHEMATICAL CONNECTIONS Why is an odd number...Ch. 3.4 - MATHEMATICAL CONNECTIONS Can 0 be the identity for...Ch. 3.4 - MATHEMATICAL CONNECTIONS Suppose you forgot the...Ch. 3.4 - MATHEMATICAL CONNECTIONS Is xx ever equal to x?...Ch. 3.4 - MATHEMATICAL CONNECTIONS Describe all pairs of...Ch. 3.4 - MATHEMATICAL CONNECTIONS Explain how the...Ch. 3.4 - MATHEMATICAL CONNECTIONS Describe a real-life...Ch. 3.4 - MATHEMATICAL CONNECTIONS How would you explain to...Ch. 3.4 - MATHEMATICAL CONNECTIONS What happens when you...Ch. 3.4 - If a student presented a new algorithm for...Ch. 3.4 - MATHEMATICAL CONNECTIONS Dina calculated each of...Ch. 3.4 - Prob. 12MCCh. 3.4 - Amy claims that no finite set of whole numbers is...Ch. 3.4 - Prob. 14MCCh. 3.4 - Prob. 15MCCh. 3.4 - A student performs the following computation....Ch. 3.4 - Prob. 17MCCh. 3.4 - The School Book Page on Page 141 shows both the...Ch. 3.4 - MATHEMATICAL CONNECTIONS An example of using the...Ch. 3.4 - Prob. 20MCCh. 3.4 - Prob. 21MCCh. 3.4 - Prob. 22MCCh. 3.4 - Prob. 23MCCh. 3.4 - Prob. 1NAEPCh. 3.4 - Prob. 2NAEPCh. 3.4 - Prob. 3NAEPCh. 3.4 - Prob. 4NAEPCh. 3.4 - Prob. 5NAEPCh. 3.4A - Show that 34=12 using each of the following...Ch. 3.4A - If A={a,b} and B={x,y,z}, find each of the...Ch. 3.4A - What multiplication is suggested by the following...Ch. 3.4A - For each of the following, find, if possible, a...Ch. 3.4A - A sporting goods store has designs for 6 shirts, 4...Ch. 3.4A - Prob. 6ACh. 3.4A - Prob. 7ACh. 3.4A - a. If 5 is removed from the set of whole numbers,...Ch. 3.4A - Rename each of the following using distributive...Ch. 3.4A - Using the distributive property of multiplication...Ch. 3.4A - Which property is illustrated in each of the...Ch. 3.4A - The product 614 can be found by thinking of the...Ch. 3.4A - Use the distributive property of multiplication...Ch. 3.4A - Show that (a+b)2=a2+2ab+b2 using a. the...Ch. 3.4A - If a and b are whole numbers with ab, use the...Ch. 3.4A - Use the property (a+b)2=a2+2ab+b2 to compute the...Ch. 3.4A - Factor each of the following expressions. a. xyy2...Ch. 3.4A - Simplify each of the following multiplications...Ch. 3.4A - a. Which is greater, 280+280 or 2100? Why? b....Ch. 3.4A - Illustrate how to use base-ten blocks to compute...Ch. 3.4A - Perform the following multiplications using the...Ch. 3.4A - Use lattice multiplication to compute...Ch. 3.4A - Find the least values of a and b such that...Ch. 3.4A - a. Use the distributive property of multiplication...Ch. 3.4A - Discuss possible error patterns in the following...Ch. 3.4A - Give reasons for each of the following steps....Ch. 3.4A - Place the digits 7, 6, 8, and 3 in the boxes to...Ch. 3.4A - Assessment Answer the following questions based on...Ch. 3.4A - Dan has 4520 pennies in three boxes. He says that...Ch. 3.4A - Assessment Fill in the missing numbers in each of...Ch. 3.4A - A car trip book 8 hrs of driving at an average of...Ch. 3.4A - A theater has 38 rows with 23 seats in each row....Ch. 3.4A - Without computing, tell which of the following...Ch. 3.4A - Use your calculator to calculate...Ch. 3.4A - A student uses front-end estimation to estimate...Ch. 3.4B - Prob. 1ACh. 3.4B - ASSESSMENT If A={a,b,c,d} and B={x,y,z}, find each...Ch. 3.4B - ASSESSMENT What multiplication is suggested by the...Ch. 3.4B - ASSESSMENT For each of the following, find, if...Ch. 3.4B - ASSESSMENT A new car comes in 5 exterior colors...Ch. 3.4B - ASSESSMENT A local coffee shop has 5 syrup...Ch. 3.4B - ASSESSMENT Determine if the following sets are...Ch. 3.4B - Prob. 8ACh. 3.4B - ASSESSMENT Rename each of the following using the...Ch. 3.4B - Prob. 10ACh. 3.4B - ASSESSMENT Which property of whole numbers is...Ch. 3.4B - The product 524 can be found by thinking of the...Ch. 3.4B - Prob. 13ACh. 3.4B - ASSESSMENT Show that if bc, then a(bc)=abac, using...Ch. 3.4B - ASSESSMENT Use the formula (a+b)(ab)=a2b2 to...Ch. 3.4B - ASSESSMENT Factor each of the following: a. xyyb....Ch. 3.4B - ASSESSMENT Simplify each of the following...Ch. 3.4B - ASSESSMENT a. Which is greater, l 220+220 or 221?...Ch. 3.4B - ASSESSMENT Illustrate how to use base-ten blocks...Ch. 3.4B - The following model illustrates 1312. a. Explain...Ch. 3.4B - Assessment Perform the following multiplications...Ch. 3.4B - Assessment Use lattice multiplication to compute...Ch. 3.4B - ASSESSMENT Find the least values of a and b such...Ch. 3.4B - ASSESSMENT a. Use a distributive property of...Ch. 3.4B - ASSESSMENT Discuss possible error patterns in each...Ch. 3.4B - ASSESSMENT Give reasons for each of the following...Ch. 3.4B - ASSESSMENT Answer the following questions based on...Ch. 3.4B - ASSESSMENT Debbie has 340 dimes in three boxes....Ch. 3.4B - ASSESSMENT Fill in the missing numbers in the...Ch. 3.4B - ASSESSMENT A car trip took 6hours at an average of...Ch. 3.4B - ASSESSMENT About 3540 calories must be burned to...Ch. 3.4B - Without computing, tell which of the following...Ch. 3.4B - Use your calculator to multiply several two-digit...Ch. 3.4B - A student users front-end estimation to estimation...Ch. 3.5 - Is xx always equal to 1? Explain your answer.Ch. 3.5 - Is the answer to 203(2+4) equal to 38,18,2, or 10?...Ch. 3.5 - Prob. 3MCCh. 3.5 - Pick a number Double it. Multiply the result by 3....Ch. 3.5 - Prob. 5MCCh. 3.5 - Prob. 6MCCh. 3.5 - For the given computation, answer the following...Ch. 3.5 - Prob. 8MCCh. 3.5 - Suppose a student argued that 00=1 because every...Ch. 3.5 - A student claims that for all whole numbers...Ch. 3.5 - Prob. 11MCCh. 3.5 - Prob. 12MCCh. 3.5 - A student divides as follows. How do you help?Ch. 3.5 - A student asks how you can find the quotient and...Ch. 3.5 - A student claim that to divide a number with the...Ch. 3.5 - a. A student notices that 39+41=40+40 and wonders...Ch. 3.5 - What errors do you think the student is making in...Ch. 3.5 - Complete the Convince Me on the School Book Page...Ch. 3.5 - Prob. 19MCCh. 3.5 - The School Book Page on page 171 shows the partial...Ch. 3.5 - Prob. 21MCCh. 3.5 - Prob. 22MCCh. 3.5 - At the beginning of a trip, the mileage odometer...Ch. 3.5 - Prob. 1NAEPCh. 3.5 - Prob. 2NAEPCh. 3.5 - Ms. Kim has 45 stickers that she wants to give out...Ch. 3.5 - Which expression has the least value? a. 2+7+0+4...Ch. 3.5 - Prob. 5NAEPCh. 3.5A - Rewrite each of the following division problems as...Ch. 3.5A - For each of the following equation, find whole...Ch. 3.5A - If /108/a=b then find 108/b.Ch. 3.5A - Write the complete fact family for 72/8=9.Ch. 3.5A - Perform each of the following divisions using both...Ch. 3.5A - Show that, in general, each of the following is...Ch. 3.5A - Suppose all of the operations result in whole...Ch. 3.5A - Is it possible to find a whole number less than...Ch. 3.5A - Place the digits 4,5,7, and 3 in the boxes ____ to...Ch. 3.5A - Using a calculator, Ralph multiplied by 10 when he...Ch. 3.5A - Use base ten block to illustrate 1325.Ch. 3.5A - Use the partial quotients algorithm to computer...Ch. 3.5A - Students were divided into 10 teams with 12 on...Ch. 3.5A - Prob. 16ACh. 3.5A - Assessment Deborah Lynn purchases a 50,000 life...Ch. 3.5A - a. Find all whole numbers that leave have 3 upon...Ch. 3.5A - Place parentheses, if needed, to make each of the...Ch. 3.5A - In each of the following, tell what computation...Ch. 3.5A - Think of a number. Multiply it by 5.Add 5. Divide...Ch. 3.5A - In each of the following, determine if the...Ch. 3.5B - Rewrite each of the following division problems as...Ch. 3.5B - For each of the following equations, find whole...Ch. 3.5B - If 64a=b, then find 64b.Ch. 3.5B - Show that, in general, each of the following is...Ch. 3.5B - Suppose all operations result in whole numbers....Ch. 3.5B - Jonah has a large collection of marbles. He...Ch. 3.5B - Place the digits 7, 6, 8, and 3 in the boxes to...Ch. 3.5B - Using a calculator, Jody multiplied by 5 when she...Ch. 3.5B - Use base-ten blocks to illustrate 2568.Ch. 3.5B - Use the partial-quotients algorithm to compute...Ch. 3.5B - Prob. 14ACh. 3.5B - Students were divided into eight teams with nine...Ch. 3.5B - Find all whole numbers that leave remainder 3 upon...Ch. 3.5B - Marry Kaye purchased a 30000 life insurance policy...Ch. 3.5B - a. Find all the whole numbers that have remainder...Ch. 3.5B - Place parentheses, if needed, to make each of the...Ch. 3.5B - In each of the following, tell what computation...Ch. 3.5B - Think of a number. Multiply it by 2. Add 2. Divide...Ch. 3.5B - In each of the following, determine if the...Ch. 3.CR - Prob. 1CRCh. 3.CR - Prob. 2CRCh. 3.CR - Simplify each of the following products, if...Ch. 3.CR - Write the base-three numeral for the base-three...Ch. 3.CR - What is the fewest number of base-three blocks...Ch. 3.CR - Draw multi base blocks to represent. a. 123four b....Ch. 3.CR - Prob. 7CRCh. 3.CR - Prob. 8CRCh. 3.CR - Write 128 in each of the following bases. a. 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Trade base-ten blocks to write 3...Ch. 3 - Now Try this 2 A different method of converting...Ch. 3 - Now Try this 3 If the sets in the preceding...Ch. 3 - Now Try this 4 A common error is that students...Ch. 3 - Prob. 5NTCh. 3 - Prob. 6NTCh. 3 - Prob. 7NTCh. 3 - Prob. 8NTCh. 3 - Prob. 9NTCh. 3 - Prob. 10NTCh. 3 - Prob. 11NTCh. 3 - Now Try this 12 Find a solution for each of the...Ch. 3 - Prob. 13NTCh. 3 - NOW TRY THIS a. Build an addition table for base...Ch. 3 - Now Try this 15 Perform each of the following...Ch. 3 - Now Try this 16 Estimate the following mentally,...Ch. 3 - Prob. 17NTCh. 3 - Prob. 18NTCh. 3 - Prob. 19NTCh. 3 - Prob. 20NT
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- 5. [10 marks] Let G = (V,E) be a graph, and let X C V be a set of vertices. Prove that if |S||N(S)\X for every SCX, then G contains a matching M that matches every vertex of X (i.e., such that every x X is an end of an edge in M).arrow_forwardQ/show that 2" +4 has a removable discontinuity at Z=2i Z(≥2-21)arrow_forwardRefer to page 100 for problems on graph theory and linear algebra. Instructions: • Analyze the adjacency matrix of a given graph to find its eigenvalues and eigenvectors. • Interpret the eigenvalues in the context of graph properties like connectivity or clustering. Discuss applications of spectral graph theory in network analysis. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440 AZF/view?usp=sharing]arrow_forward
- Refer to page 110 for problems on optimization. Instructions: Given a loss function, analyze its critical points to identify minima and maxima. • Discuss the role of gradient descent in finding the optimal solution. . Compare convex and non-convex functions and their implications for optimization. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440 AZF/view?usp=sharing]arrow_forwardRefer to page 140 for problems on infinite sets. Instructions: • Compare the cardinalities of given sets and classify them as finite, countable, or uncountable. • Prove or disprove the equivalence of two sets using bijections. • Discuss the implications of Cantor's theorem on real-world computation. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing]arrow_forwardRefer to page 120 for problems on numerical computation. Instructions: • Analyze the sources of error in a given numerical method (e.g., round-off, truncation). • Compute the error bounds for approximating the solution of an equation. • Discuss strategies to minimize error in iterative methods like Newton-Raphson. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440 AZF/view?usp=sharing]arrow_forward
- Refer to page 145 for problems on constrained optimization. Instructions: • Solve an optimization problem with constraints using the method of Lagrange multipliers. • • Interpret the significance of the Lagrange multipliers in the given context. Discuss the applications of this method in machine learning or operations research. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440 AZF/view?usp=sharing]arrow_forwardOnly 100% sure experts solve it correct complete solutions okarrow_forwardGive an example of a graph with at least 3 vertices that has exactly 2 automorphisms(one of which is necessarily the identity automorphism). Prove that your example iscorrect.arrow_forward
- 3. [10 marks] Let Go (Vo, Eo) and G₁ = (V1, E1) be two graphs that ⚫ have at least 2 vertices each, ⚫are disjoint (i.e., Von V₁ = 0), ⚫ and are both Eulerian. Consider connecting Go and G₁ by adding a set of new edges F, where each new edge has one end in Vo and the other end in V₁. (a) Is it possible to add a set of edges F of the form (x, y) with x € Vo and y = V₁ so that the resulting graph (VUV₁, Eo UE₁ UF) is Eulerian? (b) If so, what is the size of the smallest possible F? Prove that your answers are correct.arrow_forwardLet T be a tree. Prove that if T has a vertex of degree k, then T has at least k leaves.arrow_forwardHomework Let X1, X2, Xn be a random sample from f(x;0) where f(x; 0) = (-), 0 < x < ∞,0 € R Using Basu's theorem, show that Y = min{X} and Z =Σ(XY) are indep. -arrow_forward
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