Math 4370 Final

What is the 10th term is the following arithmetic sequence: 5, 9, 13, 17, ...

41

An effective representation for addition that will help students later with measuring length is to:

show addition by joining distances of on a number line

How many times must you regroup to find 303 divided by 3?

no regrouping is required

How would you write 2, 457 in expanded form using numbers?

b. 2000 + 400 + 50 + 7

At the school carnival, Carol sold 3 times as many raffle tickets as Sam. If the two of them sold 152 tickets all together, how many raffle tickets did Carol sell?

114 tickets

Problem types where no change in quantities is described include:

Part-part-whole and compare

On the hundreds chart, if you start at 12 and repeat the pattern "down two rows, then right two columns" three times, what number do you end at?

78

Suppose that you are told that 15 miles is 3/5 of the total distance from your house to your job. How far is your job from your house?

25 miles

Suppose that you gave 24 pieces of candy to your friend Billy. If these pieces of candy represented 6/8 of the total pieces of candy that you had, how many pieces of candy did you have before you gave some to Billy

32 pieces of candy

According to the EC-6 vertical alignment chart, by what grade should students be able to: represent decimals, including tenths and hundredths, using concrete and visual models and money?

4th

What problem type(s) does the word problem below illustrate? "There are 18 students in Miss Lily's class. Eight are boys and the rest are girls. If three girls transferred to another school this semester. How many girls are there in Miss Lily's class now?"

Part part whole and separate

Using the parts of a whole definition of fractions and the area model, how can seven-eighths be found? Choose the best answer.

Taking the area of a whole, dividing it into eight equal-size areas, and considering seven of those equal-sized areas in the context of the whole

Using the set model and base 10 blocks to find 124/4 would result in:

d. four groups consisting of three rods and one unit each

According to the EC-6 vertical alignment chart, by what grade should students be able to: represent and solve addition and subtraction of fractions with unequal denominators referring to the same whole using objects and pictorial models and properties of operations?

5th

According to the EC-6 vertical alignment chart, by what grade should students be able to: recognize instantly the quantity of structured arrangements?

1st grade

Suppose that you are given the following growing pattern 14, 26, 38, ... Which of the following numbers is a term in the pattern?

122

According to the EC-6 vertical alignment chart, by what grade should students be able to: Represent and solve addition and subtraction word problems where unknowns may be any one of the terms in the problem.

2nd Grade

According to the vertical alignment chart, by what grade should students be able to: generate and solve problem situations for a given mathematical number sentence involving addition and subtraction of whole numbers within 1,000?

2nd grade

Using the area model and emphasizing place value, which of the following products could be represented with the following: The area of a rectangle composed of six squares with side lengths of 10 units, three rectangles having the dimensions of 5 x 10 units, two rectangles having the dimensions of 2 x 10 units, and one rectangle having the dimensions of 5 x 2 units.

32 * 25

According to the EC-6 vertical alignment chart, by what grade should students be able to: represent fractions greater than zero and less than or equal to one with denominators of 2, 3, 4, 6, and 8 using concrete objects and pictorial models, including strip diagrams and number lines?

3rd grade

George had 25 pencils. Thirteen are red and the rest are yellow. George gives eight of his yellow pencils away. How many yellow pencils does George have now?

4 pencils

If you are given the size of groups shared equally amongst a certain number of groups and asked to find the total, this is called a "fair share" problem.

False

In a mathematics lesson on counting, Teacher A asked her students to write a sequence of numbers that represents skip counting by 5. Here are three sequences from Katie, John, and Wendy. Katie: 11, 15, 19, 23... John: 6, 11, 16, 21... Wendy: 1, 5, 10, 15... Whose answer best represent the ideas of skip counting by 5?

John

What problem type do we have below: Billy had some stickers. He gave 30 stickers to George. Now Billy has 35 stickers left. How many stickers did Billy have to begin with?

Join

What problem type(s) does the word problem below illustrate? "Jenny has some marbles. She gave Kate 13 marbles. Now Jenny has 19 marbles. How many marbles did Jenny give away?

separate

Alice gave ½ of her candy to Billy. Billy gave 1/4 of the candy that he received from Alice to Carter. How could you find what fraction of Alice's candy Carter received?

take one half and multiply by 4

On the hundreds chart, on the diagonal from left to right, numbers increase by?

11

All of the following are effective methods for helping children understand the meaning of the equal sign EXCEPT:

Placing focus mainly on computations

Sally is learning to skip count by four. If 1 is the first number in the sequence, what is the fifth number in the sequence?

17

using grid paper julie shades a rectangle that is 8 units wide and 12 units long. Randy shades a rectangle that is 4 units wide and 6 units long. Randy wants to create a rectangle that has the same area as Julie's. In order to do this he decides to extend the length of his original rectangle by X units long. What is X?

18 units.

Rachel had sixteen dollars. For her birthday she got twenty-eight more dollars but spent twenty-five on a new pair of shoes. How much money does she have now?

19

According to the EC-6 vertical alignment chart, by what grade should students be able to: Use relationships to determine the number that is 10 more or 10 less than a given number up to 120.

1st grade

On the hundreds chart, if you start at 5 and repeat the pattern "down one row" five times, what number do you end at?

55

According to the EC-6 vertical alignment chart, by what grade should students be able to: Recognize the difference between additive and multiplicative numerical patterns given in a table or graph.

5th grade

Using grid paper Sylvia shades a rectangle that is 12 units wide and 8 units long. Billy begins to create a rectangle that is 16 units wide that he will shade. How long must Billy's rectangle need to be in order for his rectangle to be equal in terms of area with Sylvia's rectangle?

6 units

By how much will the value of 4,372 increase if the 3 is replaced with a 9, the 7 replaced with an 8, and the 2 is replaced with an 8?

616

Sandy lives 19 miles directly North of Cindy. Sue lives 65 miles directly South of Cindy. How far apart do Sue and Sandy live from each other?

84 miles

Using the area model, how could you represent the product of 5 and 4?

On grid paper, drawing a rectangle 5 units wide and 4 units long. The product would be the area of the rectangle.

You give your students the following equation and ask them to solve: "Blank box minus eight equals 21" One of your students, Sammy, says "the blank box is 13 because 21 - 8 equals 13." What are your thoughts on the Sammy's reasoning?

This is not mathematically correct

What problem type(s) does the word problem below illustrate? "Stella has some dolls. Her parents gave her 8 more dolls on her birthday. Now Stella has 23 dolls. How many dolls did she have before her birthday?"

join

What problem type(s) are represented below: Julia has 25 kittens. Eight are male and the rest are females. She gives four of the female kittens to her friend. How many female kittens does she have left?

part part whole problem

A teacher is working with a group of first grade students on exploring the concept of ten. Which of the following activities would be the most effective in helping the students grasp the concept of grouping by ten?

putting one bean in each square of a grid containing 10 squares

An effective representation showing the relationship between addition and multiplication is using equal-sized groups of objects being combined to form a larger group. Which representational model is this?

set model

Suppose you have a balanced scale. On the left side we have two cube weights and one cylinder weight. On the right side you have four cube weights. The weights on the scale are balanced. If each cube weight weighs 3 pounds and each cylinder weight weighs N pounds. Which of the following processes could be used to represent the solving for the weight of the cylinder?

subtract two 3-pound cubes from both sides

Using the parts of a whole definition of fractions and using the set model, 3/8 can be represented by which of the following processes? Choose the best answer.

taking a whole set of objects, dividing it into eight sets having the same number of objects, then grouping three of those sets together within the context of the whole

Using the parts of a whole definition of fractions, what does the numerator in the fraction a/b represent? Choose the best answer below.

the number of equal-sized parts that we are considering from a whole that has been divided into _b_equal-sized parts

The expression 80n could be used to represent the following scenarios?

the total cost in cents of n candy bars at a cost of 80 cents for each candy bar

The commutative property of addition states that when adding three or more numbers the order of addition does not matter. For example, to find 3 + 6 + 8, I can add the 6 and the 8, then add the 3 or I can add the 3 and 6, and then add the 8.

true

Using the set model and base 10 blocks, 53 x 4 can be represented with?

two flats, one long and two units

One of your students said, "If 7 divided by 0, the result should be 7 because if you divide by nothing, you don't divide and so you would still have 7." Which of the following responses are mathematically correct?

- I would explain that there is no answer because there is no number times zero that will give you seven., - I would explain that the answer cannot be seven because seven divided by one is seven., - I would explain that any number divided by 0 is "undefined" and that is a rule that they will need to remember.

Skip counting is important in the development of fluency in which of the following skills: I. calculation II. number sense III. multiplication and division

1,2,3

A day prior to his birthday Robert had 7 toy cars left in his collection. This year his parents gave him some more toy cars for his birthday. He now has 19 toy cars. If his parents gave him 3 more toy cars this year than last year, how many toy cars did his parents give to him last year?

9 toy cars

William says that he has discovered a method that always works when comparing fractions. He states that he just looks at the denominator and "the fraction with the larger denominator is smaller because the whole is divided into more pieces, therefore each piece is smaller. "

I'd praise him, but point out that while this explanation works in some cases, it might not always work for all fractions

Using the linear model, how can you represent the product of 5 and 4?

On a number line, starting at 0, take 5 jumps having the size of 4 units each

You are teaching multiplication by 10. One of your students, Andrew, says that all you have to do is add a 0 to the right of the last digit of any number to get the answer. What do you think about Andrew's solution and process?

While it works with some numbers, it does not work with all numbers.

In the video "Reasoning About Division," Ms. Simpson helps children divide larger numbers by making connections to all of the following concepts EXCEPT?

a. commutative property of division

What problem type(s) are represented below: At the school carnival, Carol sold 3 times as many raffle tickets as Sam. If the two of them sold 152 tickets all together, how many raffle tickets did Carol sell?

compare problem, part-part-whole problem

When you divide a fraction by another fraction the answer is always smaller than one.

sometimes true

Using the parts of a whole definition of fractions and the set model, how would you find 2/3 of 18? Choose the best answer.

taking a set of 18 objects, dividing them into three equal-sized groups, then grouping two of those groups together within the context of the whole

Using the parts of a whole definition of fractions and the linear model, how can three-fifths be found? Choose the best answer.

taking the length of a whole, dividing it into five equal length segments, and joining three of those segments in the context of the whole.

Which of the following sequences below is NOT an arithmetic pattern

2,4,8,16,22

According to the vertical alignment chart, by what grade should students be able to:locate the position of a given whole number on an open number line?

2nd Grade

Using the area model and emphasizing place value, which of the following products could be represented with the following: The area of a rectangle composed of six squares with side lengths of 10 units, two rectangles having the dimensions of 9 x 10 units, three rectangles having the dimensions of 6 x 10 units, and one rectangle having the dimensions of 9 x 6 units.

39x26

What is the 15th term in the following arithmetic sequence? 2,5,8,11...

44

Ms. Roberts is working with her 5th grade students to develop expressions based on problem situations. She poses the following scenario: Billy has six times as many books as George. What expression can be used to represent this situation? Use the variable B to represent the number of books that Billy has and the variable G to represent the number of books that George has. Which of the following expressions could be used?

B=6G

The associative property of addition states that when adding three or more numbers, the order of the numbers does not matter. For example, 3 + 6 + 8 is the same as 6 + 8 + 3.

False

Using the parts of a whole definition of fractions and the linear model, how can you find 2/3 of 18? Choose the best answer.

Using a strip that is 18 units long, dividing it into three strips that have the same length, and then finding the combined length of two of those strips measured side by side

Using the area model and emphasizing place value, how could you represent 43 x 36?

With the area of a rectangle composed of twelve squares with side lengths of 10 units, four rectangles having the dimensions of 6 x 10 units, three rectangles having the dimensions of 3 x 10 units, and one rectangle having the dimensions of 6 x 3 units.

Which of the following word problems BEST illustrates what 1/4 divided 1/2 means?

You are making some homemade taffy and the recipe calls for 1/4 cups of butter. How many sticks of butter will you need? (Each stick = 1/2 cup)

Using the make 10 approach, which of the following could be possible steps for adding 16 and 18?

a. decompose 16 to (2 + 14) so that you could add 2 and 18 to make 20 Correct b. decompose 18 to (4 + 14) so that you could add 4 to 16 to make 20 Correct

The relations below demonstrate what property of addition and multiplication of real numbers a, b, and c? (a + b) + c = a + (b + c) OR (ab)c = a(bc)

associative property

A teacher is working with a group of first grade students on exploring the concept of counting by twos. Which of the following activities would be the most effective in helping the students grasp the concept of counting by twos?

coloring in the even numbers on a hundreds chart

What problem type(s) does the word problem below illustrate? "Luck, Mike, and Sally are friends that collect Pokemon card. Luck has 18 Pokemon cards. Mike has 9 more cards than Luck. If Mike gives 7 of his cards to Sally, how many cards does Mike have now?"

compare and separate

Using the parts of a whole definition of fractions and the area model, how can you find 3/4 of 16? Choose the best answer.

representing the whole with a region having the area of 16 square units, dividing it into four equal-sized parts having the same area, then grouping three of those equal-sized regions

Using the parts of a whole definition of fractions, what does the denominator in the fraction a/b represent? Choose the best answer below.

the number of equal sized parts that the whole is being divided into

Problem type(s) where a change in quantities is described include:

Join and Separate

What problem types do we have below? Sylvia has 20 more marbles than Rudy. Rudy has 30 marbles. If Sylvia gives 7 of her marbles away, how many marbles will Sylvia have left?

Compare and separate

Ms. Sierra is working with her 5th grade students to develop expressions based on problem situations. She poses the following scenario: Gary works three hours less than Landry. What equation can be used to represent this situation? (Let G represent the number of hours worked by Gary and L represent the number of hours worked by Landry.) Which equation is correct?

G=L-3

Your students are working on whole number subtraction with "regrouping." You have given them the problem: 36 -19 One of your students says she has come up with a much simpler method. She explains that 6 - 9 equals -3, and 30 - 10 equals 20, and -3 + 20 equals 17. What do you think about this student's solution method?

It is mathematically correct.

One of your students said, "If 7 divided by 0, the result should be 7 because if you divide by nothing, you don't divide and so you would still have 7." What are your thoughts on his reasoning process?

It is mathematically incorrect and I can explain why.

What problem type(s) do we have below? Hope had $36. She earned $15 more for babysitting. She then used $7 to buy a poster. How much money did Hope have left?

Join and Separate

What problem types are represented below? Rachel had sixteen dollars. For her birthday she got twenty-eight more dollars but spent twenty-five on a new pair of shoes. How much money does she have now?

Join problem, Separate problem

Using the make 10 approach, how would you find 13 - 8?

Subtract 3 from 10, then subtract 5 from that result

The relations below demonstrate what property of addition and multiplication of real numbers: a + b = b + a OR ab = ba

commutative property

What problem type(s) is represented in the word problem below? A day prior to his birthday Robert had 7 toy cars left in his collection. This year his parents gave him some more toy cars for his birthday. He now has 19 toy cars. If his parents gave him 3 more toy cars this year than last year, how many toy cars did his parents give to him last year?

Join problem, compare problem

According to the EC-6 vertical alignment chart, by what grade should students be able to: count a set of objects up to at least 20 and demonstrate that the last number said tells the number of objects in the set regardless of their arrangement or order?

K

You give your second grade students to following equation: "18 plus 17 equals blank box plus 29" George says "Twenty-nine is two more than 27, so the missing number must be two more than 18" What do you think about George's reasoning?

This is not mathematically correct

What problem type do we have below? William and Sally have 160 pennies in a jar. 83 of the pennies belong to William. How many pennies belong to Sally?

join

How many tens are in 1,255?

125

Using base-ten concepts from Module 1 and the concept of equivalence, find: ___ + 5 = 22

17

How many times must you regroup to find 1001 - 301?

once

How many times must you regroup to find 110 divided by 5?

twice

1, 5, 9, 13, 17, ... Which of the following processes could be used to find the 15th term of this sequence?

- add four to 17, then continue to add four to the previously found term nine more times, - using the expression 1 + 4(n-1) substitute 15 for n and solve, - multiply 4 times 10, then add that to 17

Just after Valentine's Day, Ms. Kay gives her class with 24 students the following mathematics question to solve: During Valentine's Day, each student in our class exchanged a Valentine's card with every student in the class. How many cards were exchanged in total in our class? Which of the following are mathematically correct ways of solving?

- have them count every card that was exchanged - have them add all the cards that each student gave to all of the other students - have them use multiplication

Suppose that you are given the following growing pattern: 14, 26, 38,... Which of the following processes could be used to find whether 60 is a term in the sequence?

- subtract 14 from 60, then divide that difference by 12. If the quotient is a whole number, then 60 is a term in the sequence., - using the equation 14 + 12(n-1) = 60, solve for n. If n is a whole number, then 60 is a term in the sequence., - subtract 12 from 60, then repeatedly continue to subtract 12 from the result. If 14, 26, and 38 are part of that sequence, then 60 is a term of the pattern.

Just after Valentine's Day, Ms. Kay gives her class with 24 students the following mathematics question to solve: During Valentine's Day, each student in our class exchanged a Valentine's card with every student in the class. How many cards were exchanged in total in our class? What is the solution for this problem?

. 552

Julia has 25 kittens. Eight are male and the rest are females. She gives four of the female kittens to her friend. How many female kittens does she have left?

13 kittens

Randy has 15 puppies. Josie has 4 more puppies than Randy. If Josie gives 6 of his puppies away, how many puppies does Josie have now?

13 puppies

Suppose that you are given 12 square units of area and are told that this represents 3/4 of the whole area. What is the area of the whole

16 square units

Lily uses 9 units, 4 flats, and 7 rods to represent a whole number. If she subtracts 8 units, 2 flats, and 4 rods, what does the result represent?

231

Suppose that you are given the following growing pattern: 1, 3, 5, 7, 9, ... What is the 15th term of this sequence?

29

William is practicing skip counting by 3. He skips six times and ends at 29, which process below could be used to find the number William started with?

29 - 6 x 3

According to the EC-6 vertical alignment chart, by what grade should students be able to: explain that the unit fraction 1/b represents the quantity formed by one part of a whole that has been partitioned into b equal parts where b is a non-zero whole number?

3r

William uses 3 units, 5 flats, and 5 rods to represent a whole number. If he subtracts 9 rods and 2 units, what does the result represent?

461

According to the EC-6 vertical alignment chart, by what grade should students be able to: Represent multi-step problems involving the four operations with whole numbers using strip diagrams and equations with a letter standing for the unknown quantity.

4th grade

What digit is in the tens place of 1,255?

5

One of your students says that he has found an easier way to do addition. For example, 32 + 27 + 46. Using the meaning of place value he explains that this sum is the same as saying that you have 9 tens and 15 ones. Then, since 15 ones is one ten and 5 ones, he explains that you can regroup to make a total of 10 tens and 5 ones. Then since 10 tens is one hundred, you have 105. What are your thoughts on his reasoning process?

It is mathematically correct and works when adding all whole numbers.

Philip says that he has found an easier way to multiply single digit and multidigit whole numbers. For example, to find 8 x 243 he multiplies 8 times 200, 8 times 40, and 8 times 3. He then adds those partial products together

It is mathematically correct and works when multiplying all whole numbers.

When calculating two-digit addition, such as 12 + 15, Mary, Jacob, Mark, and Fanni proposed the following solution methods: Mary: Subtract 8 from 15 and join it with 12 to make 20. Then add 7. Jacob: Subtract 5 from 12 and join it with 15 to make 20. Then subtract 7. Mark: Add 2 and 5 to make 7 in the ones place of the solution. Then add 1 and 1 to make 2 in the tens places of the solution. Fanni: Add 1 and 1 to make 2 in the tens place of the solution. Then add 2 and 5 to make 7 in ones place of the solution Which of the above methods are mathematically correct?

Marks method, Mary's method and Fanni's method

Three elementary teachers are having a discussion on effective methods for helping children conceptualize the meaning of the equal sign. Mr. A has his students solving equations such as 4 + 5 = ___ - 1. Mr. B has his students solve equations that look like 7 + 5 = ____. Mr. C has his students solve equations that look like 7 + ___ = 13. Mr. D has his students work with a balance where they add and subtract values from both sides and keep the scale balanced. Whose method would LEAST likely help students build a good understanding of the equal sign?

Mr. B

Bobbie is learning to count in his kindergarten classroom. He points to counting bears one at a time and says the sequence of number names. He says "one, two, three, four, five, ..." as he points to each bear. When his teacher asks him to show her the quantity 4. Bobbie points to the fourth bear. Which counting concept has Bobbie probably not yet have attained?

cardinality

Randy has 15 puppies. Josie has 4 more puppies than Randy. If Josie gives 6 of his puppies away, how many puppies does Josie have now?

compare, separate