Teaching mathematics final

what fractional part of the set of dots is red? (Check all that apply)

- 1/4 - 3/12

Suppose that it is 2:45 PM, how many minutes until it is 4:30 PM?

105 Minutes

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

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

George begins a trip at 9:30 A.M. If he travels for 5 hours and 45 minutes, when will the trip be over?

3:15 PM

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

A quick way to distinguish the use of counting strategies from direct modeling is by observing whether the child is using his or her fingers?

False

A third grade teacher poses an elapsed time problem to her students. She asks them to find the time 4 hours and 30 minutes before 2:45 AM. One student, Shelby, excitedly explains that she has figured out a new way of solving this problem. She explains that because she has to cross the midnight or noon time marks, she makes 2:45 into 14:45 by adding 12 to the hours, and then subtracts 4 from the 14 and 30 from forty-five to get 10:15 PM.Which of the following responses BEST matches how you would respond?

I'd praise her, but point out that while her method works in this case, it might not always work

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

I, II, and III

ABCD is a quadrilateral. All of the following is true except? a. ABCD is a parallelogram b. Angle A is 90 degrees c. AB ≃AD d.AC ≃BD

If a and d is true, ABCD is a square.

Using the statements below which conclusions can you make EXCEPT for: Statement 1: Figure A is a rectangleStatement 2: Figure A is a parallelogram

If statement 2 is true, then statement 1 is true.

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., It is mathematically incorrect but I cannot explain why.

One of your students comes to class excited. She tells you that she figured out a theory that you never taught in class. She explains that she discovered that as the perimeter of a closed figure increases, the area also increases. What do you think about her theory?

It is sometimes true

According to the EC-6 vertical alignment chart, by what grade should students be able to: Compare two objects with a common measurable attribute to see which has more of/less of the attribute and describe the difference?

Kinder

Which of the following is true for both an isosceles trapezoid and a rectangle?

The two diagonals are the same length.

According to the van Hiele model an individual's levels of geometric thought are dependent of the types of instruction and activities that the individual has been exposed to and has engaged with.

True

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

According to the Carpenter (1999) readings, which problem type below involves the comparison of two or more disjoint sets?

compare problem

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

no regrouping is required, twice

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

subtract 3 from 13, then subtract 5 from that result

What is 3/4 + 1/3?

thirteen twelfths

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

Sally is comparing the fractions 3/8 and 1/2. Which of the following statement(s) are TRUE?

- 3/8 < 1/2 because four eighths is more than three eighths - 3/8 < 1/2 because when you cross multiply 8 x 1 is larger than 3 x 2.

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?

- Mary's method - Mark's method - Fanni's method

All of the following is true of all rhombi EXCEPT?

- The two diagonals are the same length. - All angles have the same measure.

Which of the following are properties of all rectangles?

- They have four right angles. - Opposite sides are parallel., - The diagonals have the same length. - The opposite sides have the same length.

Which of the following methods would feel MOST comfortable with to solve the following question? Sally begins a trip at 9:30 A.M. If she travels for 5 hours and 45 minutes, when will the trip be over?

- a time line - an analog clock and keep track of the time on piece of paper - apply the make 60 approach and keep track of time - None of these methods

Suppose that you are given the following growing pattern: 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

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

- decompose 18 to (4 + 14) so that you could add 4 to 16 to make 20 - decompose 16 to (2 + 14) so that you could add 2 and 18 to make 20

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 the other students -have them use multiplication

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

- join - separate

Problem type(s) where no change in quantities is described include:

- part part whole - compare

What problem type(s) is represented below: 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?

- separate problem - compare problem

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?

- separate problem - join problem

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.

According to the Carpenter (1999) readings, which problem type(s) below contextualize addition and subtraction operations as an "action"?

-Join problems -separate problems

Please find the following: 3/1000 + 2/100 + a/10 Assume "a" is a number between 0 and 9.

0.a23

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

11

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

122

How many tens are in 1,255?

125

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

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

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 the length of X?

18 units

According to the EC-6 vertical alignment chart, by what grade should students be able to: Illustrate that the length of an object is the number of same-sized units of length that, when laid end-to-end with no gaps or overlaps, reach from one end of the object to the other end?

1st grade

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

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

Which of the following sequences below is NOT an arithmetic pattern?

2, 4, 8, 16, 22

How would you write "two and fifty-two hundredths" in decimal format?

2.52

How would you write 0.205 in expanded form?

2/10 + 5/1000

Suppose that we have a set of counters where 16 are red and 8 are yellow. Suppose that we group the counters into larger equal-sized groups of size 8 by color, can you find another fraction that is equivalent to 16/24?

2/3

Suppose that we divide each equal-sized part of the fraction 1/4 by 2, can you find another fraction that is equivalent to 1/4?

2/8

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

2000+400+50+7

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

How would you write 24.325 in expanded form?

24 + 3/10 +2/100 + 5/1000

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 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?

29-6x3

Suppose that it is now 10:30 P.M. What time will it be in 3 hours and 45 minutes?

2:15 AM

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

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

Suppose that it is now 9:30 A.M. If lunch begins at 12:45 P.M., how much time do you have until lunch?Model response

3 hours and 15 minutes

How would you write 0.032 in expanded form?

3/100 + 2/1000

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

According to the EC-6 vertical alignment chart, by what grade should students be able to: Determine the solutions to problems involving addition and subtraction of time intervals in minutes, using pictorial models or tools such as 15-minute event plus a 30 minute event equals 45 minutes?

3rd grade

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?

3rd grade

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

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

41

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

44

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

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 grade

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

5

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

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

Suppose that it is now 9:30 P.M. What time was it 3 hours and 45 minutes ago?

5:45 PM

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

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 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

Suppose that it is now 8:30 P.M. What time was it 1 hour and 45 minutes ago?

6:45 PM

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 it is now 8:30 P.M. What time was it 1 hour and 15 minutes ago?

7:15 PM

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

Suppose that 6 pieces of candy represent 2/3 of a whole bag of candy. How many pieces of candy were in the whole bag?

9 pieces of candy

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

Suppose that we divide each equal-sized part of 3/6 by 3, can you find another fraction that is equivalent to 3/6?

9/18

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

All of the following is true of all parallelogram except?

Both diagonals have the same length.

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

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

According to the Carpenter (1999) readings, which problem type(s) below do NOT contextualize addition and subtraction operations as an "action"?

Compare problems

A teacher poses the following word problem to her students: Clara has 15 peanuts. Ellen has 9 peanuts. How many more peanuts does Clara have than Ellen? To solve this problem, Tommy sets up 15 objects and 9 objects. He then matches them in a 1 -to-1 correspondence until one set is used up. The answer is the number of unmatched objects remaining in the larger set. Which of the following strategies is Tommy using to solve this word problem? (Pick the best answer)

Direct modeling and matching

According to the van Hiele model individuals can skip levels if they have advanced mathematical skills.

False

According to the van Hiele model, a person can skip from level 0 to level 1 if they are smart enough.

False

In a word problem, the keyword "together" always means that the addition operation will be used.

False

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.) The most common answers that your students come up with are: Equation 1: G=3-L Equation 2: G-3=L Equation 3: G=L-3 Which equation is correct?

G=L-3

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.

A class of third grade students are learning about elapsed time. For example, it is now 9: 30 am. What time will it be in 3 hours and 45 minutes? One student says that it will be 12:75 am. Which of the following responses BEST matches how you would respond?

I'd ask him to explain how he figured this out and why he thinks it works.

A boy in your first grade class identifies a square as a rectangle. How would you respond?

I'd ask him, "What's a rectangle? What's a square?" and have the child remember the difference.

Your class is learning the parts of a whole definition of fractions. Scott, a second grade student, is asked to explain what the 5 in the fraction 2/5 means. He says that the "5 is the whole."How would you respond to Scott?

I'd let him use this explanation, as it works for all fractions.

Using the statements below which conclusions can you make: Statement 1: Figure A is a parallelogram Statement 2: Figure A is a rhombus

If statement 2 is true, then statement 1 is true.

Using the statements below, which conclusions can you make? Statement 1: Figure A is a rhombus. Statement 2: Figure A is a square.

If statement 2 is true, then statement 1 is true.

All of the following are advantages of using non-standard units of measurements EXCEPT?

It is easy to convert from one non-standard unit of measurement to another.

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 What are your thoughts on his reasoning process?

It is mathematically correct and works when multiplying all whole numbers

Celia states that she has found an easier way to divide fractions. For example, suppose that she is given 2 and 1/2 divided by 1/4. She says "2 and 1/2 is like $2.50 and 1/4 is like a $0.25, therefore this question is really like asking how many quarters there are in $2.50." What do you think about Celia's reasoning?

It is mathematically correct for all cases.

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.

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. " What do you think about William's reasoning?

It works in some cases, but it would not work in general.

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(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

A teacher poses the following word problem for her students to solve: Cindy had 6 pencils. Her mom gave her 4 more pencils. How many pencils does Cindy have now? Three of her students solve in the following manner. Ann puts up 6 fingers on one hand and 4 fingers on the other. She then counts her fingers, bending one slightly with each count "1, 2, 3, 4, 5, 6, 7, 8, 9, 10." June extends her fingers one at a time and says "7, 8, 9, 10" Sandra says "the answer is 10." Which of these students appears to be using counting strategies?

June

A teacher poses the following word problem for her students to solve: Sandy had 6 pencils. Her friend gave her 4 more pencils. How many pencils does Sandy have now? Three of her students solve in the following manner. Ann puts up 6 fingers on one hand and 4 fingers on the other. She then counts her fingers, bending one slightly with each count "1, 2, 3, 4, 5, 6, 7, 8, 9, 10." June extends her fingers one at a time and says "7, 8, 9, 10" Sandra says "the answer is 10." Which of these students appears to be using counting strategies?

June

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?

Kinder

According to van Hiele's Level of Geometric thought, children at what level focus on what shapes look like?

Level 0

According to van Hiele's Levels of Geometric thought, at what level might a child say that a three sided closed shape is NOT a triangle because it is "too skinny"?

Level 0

Billy's fifth grade teacher asks him to describe triangles. Billy says "a triangle is a shape that looks like a mountain." Which van Hiele level do you think is Billy at in regard to triangles?

Level 0

Mrs. Lee showed Figure A to her 2nd grade student, Jacob, and he said, "It is a square". When she turned the square 45 degrees (Figure B), Jacob said, "Now it is a rhombus because it looks like a diamond". What van Hiele level do you think is Jacob at in relation toRhombi?

Level 0

Sorting shapes based on what they look like would be a good activity for children at what van Hiele level?

Level 0

Asking children to list all the properties of squares would be LEAST appropriate for children at which level of van Hiele's model for squares?

Level 0: Visualization

Sylvia can name and recognize a rectangle by its appearance but sometimes struggles when the shape's sides are not oriented parallel to her. What level of van Hiele is she in regard to rectangles?

Level 0: Visualization

According to van Hiele's Levels of Geometric thought, at what level might a child think about a shape within shapes for the purpose of describing properties?

Level 1

According to van Hiele's Levels of Geometric thought, children at this level are able to describe the attributes of shapes and compare and contrast by finding similarities and differences between them?

Level 1

Cindy's fifth grade teacher asks her to describe a triangle. Cindy says "a triangle is a closed shape that has three straight sides and three angles." What van Hiele level do you think is Cindy at in regard to triangles?

Level 1

Fifth grade teacher, Mr. Flores, showed a figure below to his class and asked, "What is this? and how would you describe it to your friends?" Erick answered, "It is a rectangle and it has four sides, closed, two long sides, two shorter sides, opposite sides parallel, and four right angles..." What van Hiele level do you think is Erick at in regard to rectangles?

Level 1

Having children list the properties of shapes and figures would be a good activity for children at what level of van Hiele?

Level 1

Joan recognizes the properties of a square and a rectangle but makes the observation that "a square is not a rectangle because a rectangle has two long sides and two short sides." What level of van Hiele is Joan in regard to rectangles?

Level 1: Analysis

Ian described a rhombus as "a parallelogram with two adjacent sides congruent". What van Hiele level do you think is Ian at in regard to rhombi?

Level 2

Luke makes the observation that "all squares are rectangles, but not all rectangles are squares."

Level 2: Informal Deduction

Suppose that Andrew is at a van Hiele level 1 with his understanding of rectangles, which of the following shape might Andrew say is NOT a rectangle?

M

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

Cindy is comparing the fractions 3/12 and 5/20. Which of the following statement(s) are TRUE?

None of these statements are true

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

Placing focus mainly on computations

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 ten squares

A teacher poses the following word problem for her students to solve: Cindy had 6 pencils. Her mom gave her 4 more pencils. How many pencils does Cindy have now? Three of her students solve in the following manner. Ann puts up 6 fingers on one hand and 4 fingers on the other. She then counts her fingers, bending one slightly with each count "1, 2, 3, 4, 5, 6, 7, 8, 9, 10." June extends her fingers one at a time and says "7, 8, 9, 10" Sandra says "the answer is 10." Which of these students appears to be using number facts?

Sandra

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

Please fill in the blank: Join problems will _________ involve the addition operation to solve.

Sometimes

Please fill in the blank: Separate problems will _________ involve the subtraction operation to solve.

Sometimes

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

Sometimes true

Four students are discussing fractions as parts of a whole in real world contexts specifically, with the gender makeup of the 25 students in class.Student A proposes, we can use the fraction 13/12 since "13 is the number of boys in class and 12 is the number girls in class."Student B says that we can use the fraction 12/13 since "fractions are parts of a whole and the bigger number is always used in the denominator." Student C insists that "The whole in this context could be 25 because that is the total number of students in the class. So, we can use the fraction 12/25 for girls and 13/25 for boys."Student D responds, "I think you all are right."Which student(s) do you think is/are correct when using fractions to represent the gender make up of the class?

Student C

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.

Suppose that you have a square ABDC. All of the following relationships are true in all squares EXCEPT?

The correct answer is: AD and BD are perpendicular

Raymond, a third grade student, took a piece of paper and divided it into 6 equal-sized regions. He then shaded two of the regions green. Jian, another third grade student, took a piece of paper the same size as Raymond's and divided it into 12 equal-sized pieces. He then shaded four of the regions blue. You tell the class that both Raymond and Jian have shaded fractional parts of their paper. You then ask the class whether Raymond or Jian shaded more of their piece of paper.Bobby says that "Jian shaded more because he shaded 4 regions." How do you think about Bobby's reasoning?

The correct answer is: This is not mathematically correct.

What is 1/4 + 2/3?

The correct answer is: eleven twelfths

A class of third grade students are learning about elapsed time. For example, it is now 9: 30 am. What time will it be in 3 hours and 45 minutes? One student says that it will be 12:75 am. What do you think about this?

This is not mathematically correct

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

A class of third grade students are learning about elapsed time. For example, solving problems like: "It is now 7:30 am. What time will it be in 2 hours and 45 minutes?" One student says that it will be 9:75 am. What do you think about this?

This is not mathematically correct.

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.

According to the van Hiele model instruction should be matched with the individual's van Hiele levels for learning to take place.

True

According to the van Hiele model, a person can progression (or lack of it) from level to level depends more on the content and methods of instruction received.

True

According to the van Hiele model, if a student is at one level and instruction by the teacher is at a different level the desired learning and progress will not occur.

True

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.

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.

Suppose that Carol is at a van Hiele level 0 with her understanding of triangles, which of the shape below might Carol say is NOT a triangle?

X

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)

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

You want to split 1/4 pie evenly between two families. How much pie should each family get?

Over time, children's solution strategies for solving word problems transition from using:

a. direct modeling to counting strategies to number facts

Which approach below do you feel would be MOST effective for teaching children how to solve the question below?A birthday celebration begins at 9:30 A.M. If it lasts 5 hours and 45 minutes, when will it be over?

apply the make 60 approach and keep track of time

A teacher is working with a group of elementary students and wants to teach the concept of fractions using fraction circles. Which representational model would be MOST appropriate when using this tool?

area model

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

Billy's fifth grade teacher asks him to describe triangles. Billy says "a triangle is a shape that looks like a mountain." All of the following instructional activities are aligned with Billy's level of geometric understanding and would be appropriate for his level of geometric thought Billy EXCEPT?

classifying different types of triangles based on their properties

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

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

communitative property

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

Ian described a rhombus as "a parallelogram with two adjacent sides congruent". Which instructional activity is appropriate to Ian according to van Hiele theory (in relation to rhombi) EXCEPT for?

comparing shapes according to their characterizing properties.

Mrs. Lee showed Figure A to her 2nd grade student, Jacob, and he said, "It is a square". When she turned the square 45 degrees (Figure B), Jacob said, "Now it is a rhombus because it looks like a diamond". All of the following instructional activities would be appropriate for Jacob's van Hiele level of geometry thinking (in relation to Rhombi) EXCEPTfor?

comparing shapes according to their characterizing properties.

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. " How would you respond to William?

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

A teacher poses the following word problem for students to solve: 7 puppies are swimming in a pond. 8 more puppies join them. How many puppies were there then? Sandy says that the answer is 15. When the teacher asks her how she figured it out, Sandy says "Because 7 and 3 is 10, and 5 more is 15." Which of the following strategies does Sandy appear to be using?

derived facts

Cindy's fifth grade teacher asks her to describe a triangle. Cindy says "a triangle is a closed shape that has three straight sides and three angles." All of the following activities are aligned with Cindy's level of geometric thought and would be helpful for Cindy EXCEPT?

developing definitions for shapes

A teacher poses the following word problem for her students to solve: Julie had some pieces of candy. Her friend gave her 7 more pieces of candy. Then she had 12 pieces of candy. How many pieces of candy did Julie have before her friend gave her some more? Jimmy puts seven color counters on the table in front of himself. He then adds seven more counters to the original set and counts the total. He finds that the total is 14 rather than 12. He then puts the counters back with the unused counters and starts over with seven counters in front of himself. Seeing that he went over last time, he then adds five more counters this time. He counts and finds that the total is 12. He then recounts the counters that were added to the original 7 counters and responds, "Julie had five pieces of candy before her friend gave her more."

direct modeling with trial and error

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

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.

false

To multiply fractions you must first find the least common denominator?

false

Suppose that it is 3:30 PM and you want to find how many minutes until it is 4:45 PM. If you are using the make 60 approach, what is the first step that you would take?

find the number of minutes that it would take to get to 4:00

What is 3/8 + 1/4?

five eighths

Suppose that your friend tells you that he is 18 hooplas tall. What attribute is your friend measuring?

height

Suppose that your friend tells you that he is 18 hooplas tall. What unit of measurement did your friend use to measure the attribute?

hoopla

A third grade teacher poses an elapsed time problem to her students. She asks them to find the time 4 hours and 30 minutes before 2:45 AM. One student, Shelby, excitedly explains that she has figured out a new way of solving this problem. She explains that because she has to cross the midnight or noon time marks, she makes 2:45 into 14:45 by adding 12 to the hours, and then subtracts 4 from the 14 and 30 from forty-five to get 10:15 PM.What do you think about Shelby's reasoning?

it is mathematically correct.

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 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

A teacher is working with a group of elementary students and wants to teach the concept of fractions using fraction tiles. Which representational model would be MOST appropriate when using this tool?

linear model

Suppose that you have a closed number line having the length of one unit. The number line is divided into 8 segments that are equal in length. A red dot appears in the middle of the fifth segment. What location on the would the red dot represent?

none of these

Using the times given in the analog clocks below, how much time has passed between the start and end times? (Assume that times are during the same day)

none of these

A teacher is working with a group of elementary students and wants to teach the parts of whole definition of fractions by using color counter. Which attribute of color counters should the teacher emphasize?

number of counters in sets and subsets

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

once

What is 3/4 divided by 3?

one fourth

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?

part part whole

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

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

According to the Carpenter (1999) readings, which problem type below involves the relationship between a set and its two or more subsets?

part part whole problems

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.

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?

separate

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

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

Fifth grade teacher, Mr. Flores, showed a figure below to his class and asked, "What is this? and how would you describe it to your friends?" Erick answered, "It is a rectangle and it has four sides, closed, two long sides, two shorter sides, opposite sides parallel, and four right angles..." Which instructional activity is appropriate to Erick's van Hiele geometry thinking level in relation to rectangles?

sorting and resorting shapes by single attributes

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

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 one fourth

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 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 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.

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

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

What is 3/4 x 1/2?

three eighths

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

twice