Skip counting is important in the development of fluency in which of the following skills:

I. calculation

II. number sense

III. multiplication and division

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?

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

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?

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?

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?

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

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?

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?

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)

The relations below demonstrate what property of addition and multiplication of real numbers:

a + b = b + a

OR

ab = ba

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?

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?

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?

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

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?

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?

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?

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

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?

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?

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?

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

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?

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?

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?

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

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?

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?

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?

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?

Suppose that you are given the following growing pattern

14, 26, 38, ...

Which of the following numbers is a term in the pattern?

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?

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?

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?

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?

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?

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

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?

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?”

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?

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

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

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?

Suppose that you are given the following growing pattern:

1, 3, 5, 7, 9, ...

What is the 15th term of this sequence?

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?

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?

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?”

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?

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?

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?”

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?

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?

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

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?”

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?

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?

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.”

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

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?

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?

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

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?

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?

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?

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?

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

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

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?

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?

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

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

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?

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?

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?

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?

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?

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

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?

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

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?

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)

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

Using the statements below, which conclusions can you make?

Statement 1: Figure A is a rhombus.

Statement 2: Figure A is a square.

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?

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?

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

Mrs. Lee showed Figure A to her 2^nd 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) EXCEPT for?

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

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?

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

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?

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?

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?

All of the following is true of all rhombi EXCEPT?

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?

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

All of the following is true of all parallelogram except?

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?

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.

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?

Using the statements below which conclusions can you make EXCEPT for:

Statement 1: Figure A is a rectangle

Statement 2: Figure A is a parallelogram

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?

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?

Which of the following are properties of all rectangles?

Using the statements below which conclusions can you make:

Statement 1: Figure A is a parallelogram

Statement 2: Figure A is a rhombus

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?

Mrs. Lee showed Figure A to her 2^nd 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”.

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?

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?

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?

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?

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

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?