Math Concepts I Final

Show how to find all the factors of the following numbers in an efficient manner. Explain why you can stop checking for factors when you do. a. 75

In all parts, we can stop when we would start rewriting multiplication facts. (i.e. 3 ∙ 5 = 5 ∙ 3) a. 75 = 1 ∙ 75 75 = 3 ∙ 25 75 = 5 ∙ 15 Factors: 1, 3, 5, 15, 25, 7

A class must use the Associative Property of Addition to write an expression equivalent to 28 + (14 + 19). One student incorrectly comes up with the expression28 + (19 + 14) What was the student's error?

They used the commutative property

Describe a way that the children in Mrs. Verner's kindergarten class can tell if there are an even number or an odd number of children present in the class without counting

Split the class into two groups or into groups of 2. If there is one person left out, there is an odd number. If each group has an equal amount, there is an even number

Determine which of the following applies to the child's counting strategy. Explain your answer one two three five nine ten seven twenty eight b. There is one-to-one correspondence: YES NO Explain:

Yes, each shell has one number

Write an equivalent expression using the distributive property so that the expression has no parenthesis −(2𝑏 + 𝑐 − 3)

−2𝑏 − 𝑐 + 3

Use the commutative property of multiplication to write an equivalent algebraic expression -9xX

𝑥 ∙ (−9)

A class must use the Associative Property of Addition to write an expression equivalent to 28 + (14 + 19). One student incorrectly comes up with the expression28 + (19 + 14) Write an expression that is equivalent to 28 + (14 + 19) by the Associative Property of Addition

(28+14)+19

Use the commutative property of multiplication to write an equivalent algebraic expression 5(x+5)

(𝑥 + 5) ∙ 5

Use the standard algorithm to solve the following problems 0.086 − 1.26

-1.174

Complete the stated operation −6 − 6

-12

In each of the following, order the decimals from least to greatest -3.824, -3.8024, -3.834, -3.82

-3.834, -3.824, -3.82, -3.8024

Complete the stated operation −12 + 8

-4

Place and label each of these points on the number line:−1, − 4, 5, − (−3), |−4| , − 2.5, 0.5

-4,-2.5,-1,0.5,-(-3),I-4I,5

Complete the stated operation 6 + (−6)

0

Divide 3/8 (Give answer as a decimal)

0.375

Solve the multiplication problem 23 ∙ 45 in three different ways Using the common method/standard algorithm

1035

Show how to find all the factors of the following numbers in an efficient manner. Explain why you can stop checking for factors when you do c.120

120 = 1 ∙ 120 120 = 2 ∙ 60 120 = 3 ∙ 40 120 = 4 ∙ 30 120 = 5 ∙ 24 120 = 6 ∙ 20 120 = 8 ∙ 15 120 = 10 ∙ 12 Factors: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120

Use divisibility tests to determine whether the following numbers are divisible by 2, 3, 4, 5, 6, and/or 10 3,921,579

2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 1920 21 22 23 24 25Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23

Use divisibility tests to determine whether the following numbers are divisible by 2, 3, 4, 5, 6,and/or 10 675

2, 3, 4, 5, 6, 10

Use divisibility tests to determine whether the following numbers are divisible by 2, 3, 4, 5, 6,and/or 10

2, 3, 6

Show how to use the slide method to determine the GCF and LCM of 2440 and 3600

20. One possible slide method: GCF = 10 ∙ 4 = 40 LCM = 10 ∙ 4 ∙ 61 ∙ 90 = 219,600

Write 2535 in three different expanded forms

2000 + 500 + 30 + 5 2 ∙ 1000 + 5 ∙ 100 + 3 ∙ 10 + 5 2 ∙ 10^3 + 5 ∙ 10^2 + 3 ∙ 10 + 5 2 thousands + 5 hundreds + 3 tens + 5 ones

An annual sporting event is held every 7 years. If the event was held in 2003, will the event beheld in 2049

2049-2003=46 is not a multiple of 7, so no, it will not be held in 2049

Convert the mixed number 7 2/3 to an improper fraction

23/3

There are 25 players on a professional baseball team and 15 players on a professional basketball team. What is the smallest number of athletes that can be split evenly into either baseball teams or basketball teams

25: 25, 50, 75, 100 15: 15, 30, 45, 60, 75 We need 75 athletes to make full teams for each sport

Use the standard algorithm to solve the following problems 900 − 637

263

In each of the following, order the decimals from least to greatest 27.49179, 27.49185, 27.492, 27.4918

27.49179, 27.4918, 27.49185, 27.492

Use divisibility tests to determine whether the following numbers are divisible by 2, 3, 4, 5, 6,and/or 10 120

3

Use the divisibility test for 3 to determine whether the following numbers are divisible by 3 321,321,321,321

3+2+1+3+2+1+3+2+1+3+2+1=4(6), yes divisible by 3

Use divisibility tests to determine whether the following numbers are divisible by 2, 3, 4, 5, 6,and/or 10 846

3, 5

Rewrite each of the following numbers as decimals 9 ∙ 10^−1 + 3 ∙ 10^4

30,000.9

Which of the following numbers are divisible by 2? a. 34, 136, 510, 68, 170, 306, 4284, 51, 17, 255, 153

34, 136, 510, 68, 170, 306, 4284

Write the following numbers in fraction notation. Do not simplify 35.2

352/10

Show how to find all the factors of the following numbers in an efficient manner. Explain why you can stop checking for factors when you do b. 36

36 = 1 ∙ 36 36 = 2 ∙ 18 36 = 3 ∙ 12 36 = 4 ∙ 9 36 = 6 ∙ 6 Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36

Given that 405 = 3^4 ∙ 5, find all the factors of 405 and explain how you know you have found them all

3^0 = 1 3^1 = 3 3^2 = 9 3^3 = 27 3^4 = 81 5^1 = 5 3^1 ∙ 5 = 15 3^2 ∙ 5 = 45 3^3 ∙ 5 = 135 3^4 ∙ 5 = 405 We found all of the combinations of the prime factors. Hence, we have found all of the factors of 405

Explain why an even number can always be written in the form 2𝑁 for some whole number 𝑁. How would you write an odd number using this notation?

An even number is always divisible by 2 so every even number can be written as 2𝑁. (Example:16 = 2 ∙ 8). Odd numbers are written as 2𝑁 + 1 or 2𝑁 − 1

Classify the following problems as Area, Array, Multiplicative Comparison, or Ordered Pair. Then, draw a math picture and solve the problem A marching band has four rows with 6 people in each row. How many people are in the band?

Array - There are 4 ∙ 6 = 24 people in the band

Classify the following problems as Add To/Take Away, Compare, or Put Together/TakeApart. Then, draw a strip diagram and solve the problem Perry solved 19 problems and Chris solved 17 problems. How many more problems has Perry solved than Chris?

Compare - Perry solved 2 more problems

David and Ashley want to calculate $3.27-$1.02 by first calculating $3.27-$1=$2.27.David says that they must subtract $0.02 from $2.27, but Ashley says that they must add$0.02 to $2.27 Draw a number line (which need not be perfectly to scale) to help you explainwho is right and why. Do not just say which answer is numerically correct; usethe number line to help you explain why the answer must be correct

David is correct

For the following word problems, write multiplication and division equations that model the problem, using a question mark for the unknown amount. Determine which interpretation of division is involved (how-many-groups or how-many-units-in-1-group)and solve the problem "There are 5235 tennis balls that are to be put into packages of 3. How many packages of balls can be made?"

How-many-groups 5235/3 =? ?∙ 3 = 5235 ? = 1745 packages

For the following word problems, write multiplication and division equations that model the problem, using a question mark for the unknown amount. Determine which interpretation of division is involved (how-many-groups or how-many-units-in-1-group)and solve the problem "If your car used 12 gallons of gasoline to drive 360 miles, then how many miles per gallon did your car get?"

How-many-units-in-1-group 36012 =? 12 ∙? = 360 ? = 30 mpg

Decide if the following numbers are rational or irrational. Explain your reasoning. √7

Irrational, cannot be written as a fraction made up of two integer

Decide if the following numbers are rational or irrational. Explain your reasoning. 1.24262325 ...

Irrational, does not repeat

One day the temperature dropped from −2°F to −14°F. How many degrees did it drop?

It dropped 12 degrees

Classify the following problems as Area, Array, Multiplicative Comparison, or Ordered Pair. Then, draw a math picture and solve the problem Robert and Samuel each have a trading card collection. Robert has 3 times as many cards as Samuel. Samuel has 8 cards. How many cards does Robert have?

Multiplicative Comparison - Robert has 3 ∙ 8 = 24 cards

Make three different factor trees for 360. Then, write the prime factorization for 360 using exponents to write repeated factors. Does it matter which factor tree you used to write the prime factorization

One possible example of a factor tree for 360. 360 = 2^3 ∙ 3^2 ∙ 5 It does not matter what factor tree you end up with. They will all give the exact same prime factorization.

Classify the following problems as Add To/Take Away, Compare, or Put Together/TakeApart. Then, draw a strip diagram and solve the problem Benny has 41 CDs. 23 of his CDs are hip hop and the rest are rap. How many rapCDs does Benny have

Put Together/Take Apart - He has 18 rap CD

Decide if the following numbers are rational or irrational. Explain your reasoning. 4/9

Rational, can be written as a fraction made up of two integers

In Lisa's closet, 4/8 of the space is used for sweaters and 1/8 of the space is used for shirts. Lisa says that her sweaters and shirts take up 5/16 of the space in her closet. Is Lisa correct? If not, explain her mistake and show how she should have solved the problem

She added the numerator and denominator but should have just added the numerator.4/8 + 1/8 = 5/8

If 36,000 people make up 15% of a population, then what is the total population? Solve using any method

240,000 people

Divide 987 ÷ 4 (Give answer as whole number and remainder)

246 R 3

Use the divisibility test for 3 to determine whether the following numbers are divisible by 3 444,444,444

4+4+4+4+4+4+4+4+4=3(12), yes divisible by

Explain how to use the distributive property to make 42 ∙ 25 easy to calculate mentally. Write equations that correspond to your strategy

42 ∙ 25 = (40 + 2) = 40 ∙ 25 + 2 ∙ 25 = 1000 + 50 = 1050

Use trial division to determine whether the following numbers are prime 51

47 ÷ 2 = 23.5 47 ÷ 3 = 15. 6̅ 47 ÷ 5 = 9.4 47 ÷ 7 = 6.714 ... prime

Use trial division to determine whether the following numbers are prime 47

51 ÷ 2 = 25.5 51 ÷ 3 = 17 not prime

Convert the numeral to a numeral in base ten: 2321𝑠𝑖𝑥

553

Here is Macy's method for calculating 57 ∙ 50:"Six times 5 is 30, so 60 times 50 is 3000. Then, 3 times 5 is 15, so it's 3000-150, which is 2850."Write equations that incorporate Macy's method and that also show why her method is valid. Write your equations in the following form, being sure that each equation is equal to the one before it 57 ∙ 50 = some expression = some expression = 2850

57 ∙ 50 = (60 − 3) ∙ 50 = 60 ∙ 50 − 3 ∙ 50 = 6 ∙ 5 ∙ 100 − 3 ∙ 5 ∙ 10 = 30 ∙ 100 − 15 ∙ 10 = 3000 − 150 = 2850

Write the following numbers in fraction notation. Do not simplify 0.057

57/1000

Rewrite each of the following numbers as decimals 5 ∙ 10^3 + 7 ∙ 10^2 + 3 ∙ 10^−2

5700.03

Rewrite the fractions 3/4 , 1/6 , and 4/9 using the least common denominator. Then, write the fractions in order from least to greatest

6/36 , 16/36 , 27/36

Complete the stated operation Add. 4/7 + 2/7

6/7

Use the divisibility test for 3 to determine whether the following numbers are divisible by 3 7,591,348

7+5+9+1+3+4+8=37, not divisible by 3

Pete has 207 hot dogs and 117 hot dog buns. He wants to put the same number of hot dogs and hot dog buns on each tray. What is the greatest number of trays Pete can use to accomplish this?

GCF of 207 and 117: 9 9 trays

Rewrite each of the following numbers as decimals 8 ∙ 10^2 + 5 ∙ 10

850

Write the following decimals as fractions. Explain your reasoning (you need not write the fractions in simplest form) 0. 3̅ , 0. 6̅ , 4. 6

0. 3̅ = 1/3, 0. 6̅ = 2/3, 4. 6̅ = 1/4

Write the following decimals as fractions. Explain your reasoning (you need not write the fractions in simplest form) 0. 56̅̅̅̅, 0.00056̅̅̅̅, 1.11156

0. 56̅̅̅̅ = 56/99, 0.00056̅̅̅̅ = 56/99000, 1.11156̅̅̅̅ = 110045/99000

Lance is working on the multiplication problem 1.32 ∙ 0.476. Ignoring the decimal points, Lance multiplies 132 ∙ 476 and gets the answer 62,832. Lance can't remember the rule about where to put the decimal point in this answer to get the correct answer 1.32 ∙ 0.476. Explain how Lance can use reasoning about the sizes of the numbers to determine where to put the decimal point

1.32 is between 1 and 2. 0.476 is between 0.4 and 1. The product must be in the range from 1 ∙ 0.4 = 0.4 and 2 ∙ 1 = 2. Therefore, the answer must be 0.62832.

Use the standard algorithm to solve the following problems 0.086 + 1.26

1.346

Complete the stated operation Subtract. 1/4 − 1/6

1/12

Use the standard division algorithm to determine the decimal representation of 1/37. Determine whether the decimal representation repeats or terminates; if it repeats, describe the string of repeating digits

1/37 = 0. 027

A person's order at a restaurant cost $35 and they decided to leave an 8% tip. Use a percent table (go through 1%) to determine the amount of the tip

100% → $351% → $0.358% → $2.80

Complete the stated operation 8 − (−4)

12

Convert the base ten numeral to a numeral in base four: 105

1221𝑓𝑜𝑢𝑟

Use the standard algorithm to solve the following problems 93 + 77

170

Divide 5781 ÷ 6 (Give answer as a decimal)

963.5

What is the Sieve of Eratosthenes, and why does it work? Use this method to find all of the prime numbers between 2 and 25

A method of finding prime numbers

Classify the following problems as Add To/Take Away, Compare, or Put Together/TakeApart. Then, draw a strip diagram and solve the problem Rachel has 23 CDs. She got 18 more CDs. How many does she have now?

Add To - She has 41 CDs

Classify the following problems as Area, Array, Multiplicative Comparison, or Ordered Pair. Then, draw a math picture and solve the problem A painting canvas has a side of 3 feet. It has a total area of 21 𝑓𝑡2. What is the length of the other side?

Area - The canvas has side length of 3 ∙? = 21 → ? = 7 ft

Find the GCF and LCM of 2^5 ∙ 4^2 ∙ 5 and 2^3 ∙ 4^3 ∙ 11 without calculating the products. Explain your reasoning

GCF = 2^3 ∙ 4^2 This is the highest combination of prime factors shared between the two numbers. LCM = 2^5 ∙ 4^3 ∙ 5 ∙ 11 This is the smallest combination of prime factors that ensures that both numbers are a factor of the LCM.

Complete the table Fraction Notation: 5/25 Decimal notation: 0.4 Percent Notation: 375%

Fraction Notation: 5/25, 4/10=2/5, 375/100=15/4 Decimal Notation:.0.2,0.4,3.75 Percent Notation: 20%, 40%, 375%

Together, Frank, John, and David earned $15. They want to divide it equally, except that David should only get a half share, since he did half as much work as either Frank or John did (and Frank and John worked equal amounts). Write and solve a division problem to find out how much each person should get

Frank and John get $6. David gets $3

Determine whether the following statements are true or false. Modify each false statement to make it a true statement If a number is divisible by 3, then every digit of the number is divisible by 3

False, the sum of the digits is divisible by 3

Johnny says that 3 is a multiple of 6 because you can arrange 3 cookies into 6 groups by putting 1/2 of a cookie in each group. Discuss Johnny's idea in detail. In what way does he have the right idea about what the term multiple means, and in what way does he need to modify his idea

Johnny has the right idea that you multiply to find multiples, but we need to multiply bycounting numbers. 3 is a factor of 6, not a multiple

You and your brother both have jobs. You have every eighth night off. Your brother has every sixth night off. Both of you were off on July 1. When will the two of you have the same night off again?

LCM of 6 and 8: 24J uly 1 + 24 days = July 25th

Peter says that 6/6 is greater than 5/5 because 6/6 has more parts. Tamara says 6/6 is greater than 55 because 6 > 5 for both the numerator and the denominator. Are either student correct? If not, what is wrong with their reasoning? Draw math pictures to assist your explanation

Neither are correct. Looking at the math pictures, both largerectangles represent 1. Despite how we cut the pieces, the sameamount is shaded on both pictures.

Jamie calculates that 8 ÷ 3 = 2 remainders 2. When Jamie is asked to write her answer as a decimal, she simply puts the remainder 2 behind the decimal point: 8 ÷ 3 = 2.2. IsJamie correct or not? If not, explain why not, and explain to Jamie in a concrete way why the correct answer makes sense

No, 8/3 = 2. 6̅ . The remainder is 2/3 = 0. 6

Determine which of the following applies to the child's counting strategy. Explain your answer one two three five nine ten seven twenty eight a. The list of counting numbers is an ordered list: YES NO. Explain:

No, the numbers are not in order

Determine which of the following applies to the child's counting strategy. Explain your answer one two three five nine ten seven twenty eight c. The last number indicates the cardinality of this set: YES NO. Explain:

No, there are nine shells, but the last number is eight

Suppose you start with a fraction, and you add 1 to both the numerator and the denominator. For example, if you started with 2/5, then you'd get a new fraction 2+1/5+1 = 3/6. Is this procedure of adding 1 to the numerator and the denominator the same as adding the number 1 to the original fraction? (For example, is 2+1/5+1 equal to 2/5 + 1?) Explain your answer.

No, you need common denominators to add fractions. 2/5 + 1 = 2/5 + 5/5 = 7/5

s −8/0 =? defined? Write an equivalent multiplication expression to help explain why or why not.

No, −8 =? ∙ 0 is the equivalent multiplication expression and nothing in the question mark's spot will make this statement true so it is not defined

Decide if the following numbers are rational or irrational. Explain your reasoning. 0.65656565 ...

Rational, repeats

Decide if the following numbers are rational or irrational. Explain your reasoning. √64

Rational, √64 = 8

Sam used his calculator to calculate 123,123,123,123,123 divided by 3. Sam's calculator displayed the answer as: 4.104104104𝐸13Sam says that because the calculator's answer is not a whole number, the number123,123,123,123,123 is not evenly divisible by 3. Is he right? Why or why not? How do you reconcile this with Sam's calculator's display? Discuss

Sam is incorrect. 123,123,123,123,123 is divisible by 3. The number is too large so the calculatorputs the answer in scientific notation.

Write an equivalent expression using the distributive property so that the expression has no parenthesis (𝑥 + 2)(𝑦 − 𝑧 + 3)

𝑥𝑦 − 𝑥𝑧 + 3𝑥 + 2𝑦 − 2𝑧 + 6

Determine whether the following statements are true or false. Modify each false statement to make it a true statement If a number is not divisible by 5, then it is not divisible by 10

True

What is the only digit someone needs to look at to tell whether a given number is odd or even? Why?

We only need to look at the ones place. Any place higher than ones (tens, hundreds, etc.) can be split evenly into groups of 2

a. Write the decimal for 1/3 b. Write the decimal for 2/3. c. Add the two decimal answers from parts (a) and (b). d. Add 1/3 + 2/3. e. Comparing your answers from parts (c) and (d), what can you conclude?

a. 0. 3̅ b. 0. 6̅ c. 0. 9̅ d. 13 + 23 = 33 = 1 e. 0. 9̅ = 1