Module 1: Posttest

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 the length of X?

18 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

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

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

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

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.

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

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

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

Commutative property

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.

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

I, II, and III

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.

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

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

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

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

No regrouping is required. OR Twice

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

Once

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.

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.

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

Twice

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.