MATH 351 MIDTERM

Describe the three types estimation and give an example of how they could be used in teaching computational estimation.

- Measurement estimation: Determine an approximate measure without making a precise measurement. - Quantity estimation: approximating the number of items in a collection. - Computational estimation: Determining a number that approximates a computation that we cannot or do not need to determine precisely. They both can be used in teaching computational estimation because it is teaching them how to estimate with reason in different ways.

Identify two or more ways that an open number line can be used to support the use of invented strategies.

-Helps show students thinking -Less computational errors

A student who has place value understanding at the face value level, when asked to explain the digits of the number 45, would most likely

1. Be unable to identify the meaning behind the individual digits, and would see the number as one unit. 2. Be able to identify the digit in the ones place and in the tens place, but be unable to relate the meaning of the two digits to two separate amounts. 3. Match up four blocks to go with the 4 digit and five blocks to go with the 5 digit. 4. Verbalize that the 4 represents forty and the 5 represents five units. 3

When it comes to beginning grouping activities

1. Because students usually understand counting by ones, teachers should skip directly to grouping by ten. 2. Teachers should let students experiment with showing amounts in groups until they, perhaps, come to an agreement that ten is a useful-sized group to use. 3. Students should only work with very small items that can easily be bundled together. 4. Teachers should not worry about having students verbalize the amounts they are grouping. 2

Making the transition from base-ten to standard language

1. Can be made more confusing by using base-ten materials when verbalizing the number names. 2. Should not include the teacher using a mix of base-ten and standard language, 3. Should not include a discussion of the "backwards" names given to the teens, as they can be confusing. 4. Can be made less difficult by using a word wall to provide support for ELLs and students with disabilities. 1

Using base-ten language

1. Is demonstrated when the teacher says "We have fifty-three beans." 2. Can be helpful for students who are ELLs because many other countries routinely use base-ten language. 3. Is frequently confusing for students, and it is best avoided. 4. Looks only like this format: ____ tens and ___ ones. 2

Which of the following is NOT an example of a proportional model that can be used for place value?

1. Money 2. Beans in cups of ten and single beans 3. Base ten blocks 4. Stir straws bundled in groups of ten and with single straws 1

Modern technology has made computation easier. Identify the true statement below.

A) But mental computation strategies can be faster than using technology. B) And recent studies have found that a very low percentage of adults use mental math computation in everyday life. C) And mental computation contributes to diminished number sense. D) So the ability to compute fluently without technology is no longer needed for most people. A

Which one the statements below is not a part of the progression of a child's understanding of ten?

A) Child understands 30 as 3 tens. B) Child understands ten as ten ones. C) Child understands ten as a unit with materials or representations. D) Child solves tasks involving tens with using materials or representations. A

Problems with the join and separate structures, with the start or initial amount unknown, tend to be the hardest for students to understand and accurately solve. Identify the reason for they are more challenging for children to use.

A) Children can model the physical action. B) Children can act out the situation. C) Children cannot use counters for the initial amount. D) Children cannot grasp a quantity represents two things at once. C

What statement identifies the importance of using contextual problems as a primary teaching tool?

A) Children demonstrate procedural fluency. B) Children identify the structure of the situations. C) Children construct richer understanding of the operations D) Children connect problems to school mathematics. C

The concepts of more, less and same are basic relationships that contribute to children's understanding of number. Identify the one relationship that if a child is unable to use they may be at educational risk.

A) Choose the set that is equal B) Choose the set that is more. C) Choose the set that is less. D) Choose the set that can be subitized. B

The learning trajectory for counting has levels of thinking. What level below would represent a child that can count objects and know which have been counted and which have not, and respond to random arrangements.

A) Corresponder. B) Reciter. C) Counter and Producer. D) Skip Counter. C

Here is a possible list of the kinds of things children should learn about any number up to 20 while in pre-k or kindergarten. All of them are appropriate EXCEPT:

A) Count and know the number words in their order. B) Reading three digit numbers. C) State more and less by one and 2. D) Relates to the benchmark numbers. B

A blank number line and numeral cards can be used to support the development of numeral identification and what other concept?

A) Counting sequence. B) Place value. C) One-to-one correspondence. D) Cardinality. A

Which of the following is a description of Number Sense?

A) Counting tells how many things are in a set. B) Thinking about different sized quantities and use number relationships. C) Relating through comparing quantities. D) Operations with numbers based on the world around us. B

The general approach for teaching the subtraction standard algorithm is the same as addition. What statement below would not be a problem when using the standard algorithm for addition?

A) Develop the written record. B) Begin with models. C) Trades made after the column in the left has been done. D) Exercises with zeros. D

Remainders have an effect on all of the following EXCEPT:

A) Discarded leaving a smaller whole-number answer. B) Rounded to the nearest whole number for an approximate answer. C) Fractional part left over and not included in the whole-number answer. D) Force the answer to the next highest whole number. C

The key word strategy sends a wrong message about problem solving. Identify the statement below that would be offered in support of the key word strategy.

A) Encourages children to ignore meaning and structure of problems. B) Many problems do not have key words. C) Encourage children to use a list of key words with corresponding operations. D) Many problems have key words that may be misleading. C

Research-based recommendations for high quality learning activities in the first six years of life include all of the following EXCEPT:

A) Enhance children's natural interest in mathematics. B) Use formal and informal experiences. C) Provide experiences for children to become procedural fluent. D) Assess children's mathematical knowledge. C

While developing students' understanding of the relationships for numbers 10 through 20, all of the following should be kept in mind EXCEPT:

A) Even though students experience numbers up to 20 regularly in real life, it should not be assumed that they would automatically extend the relationships they learned for numbers 1 to 10 to bigger numbers. B) These relationships are just as important as the ones involving numbers 1 to 10. C) Children should learn that there is a set of ten involved in any number between 10 and 20. D) While learning about these relationships, students should develop a complete understanding of the concept of place value. D

Complete this statement, "Constructing models of arrays draws attention to..."

A) Factors connection with rows and columns. B) Factors and product. C) Number of rows and columns. D) Connection with measurement of area. A

What strategy for computational estimation after adding or subtracting do you adjust to correct for digits or numbers that were ignored?

A) Front-end. B) Rounding. C) Compatible numbers. D) Over and under. A

There are important things to remember when teaching the standard algorithm. Identify the statement that does not belong.

A) Good choice in some situations. B) Require written record first. C) Require concrete models first. D) Explicit connections are made between concept and procedure. B

A hundreds chart is an essential tool for early exposure to numbers to 100. All of the reasons below are benefits of using hundreds chart EXCEPT:

A) Highlight number patterns. B) Recognize and count with two digit numbers. C) Conceptualize numbers greater than 100. D) Locations of numbers. C

Often siblings and family members are pushing the use of the standard algorithm while students are learning invented strategies. What is the course of action for a teacher?

A) Insist on invented strategies. B) Require students demonstrate both standard and invented strategies. C) Expect them to be responsible for the explanation of why any strategy works. D) Memorize the steps. C

What statement below is the description of a Part-part-whole?

A) Involve comparison of two quantities. B) Two parts that are conceptually or mentally combined into one whole. C) The change is the amount being removed. D) Initial action of joining start amount, a change amount and resulting amount. B

An open number can be used effectively for thinking about addition and subtraction. All of the reasons below support the use of an open number line EXCEPT:

A) Is less flexible than a numbered line. B) Eliminates confusion with hash marks and spaces. C) Less prone to computational errors. D) Helps with modeling student thinking. A

The models listed below are used to support the development of invented strategies EXCEPT:

A) Jump strategy. B) Split strategy. C) Take-Away strategy. D) Shortcut strategy. D

Students who have learned this strategy for their "basic facts" can use it effectively with solving problems with multidigit numbers.

A) Jump strategy. B) Shortcut strategy. C) Think addition strategy. D) Split strategy. C

Complete this statement, "The early stages of number development, the use of graphs is primarily for..."

A) Learning the counting sequence. B) Connecting numbers to real quantities. C) Bench mark numbers. D) Part-Part-Whole. B

Complete the statement, "A mental computation strategy is a simple..."

A) Left-handed method. B) Invented strategy. C) Standard algorithm. D) One right way. B

Which problem represents the compare, difference unknown structure?

A) Maryann had 3 library books before she checked out 2 more. How many did she have all together? B) Maryann had 5 library books before she returned 2 of them. How many does she have now? C) Maryann had 4 nonfiction books and 2 fiction. How many books does she have? D) Maryann had 9 library books and Jim had 4. How many more did Maryann have than Jim? D

Identify the problem that represents the join, result unknown structure.

A) Maryann had 3 library books before she checked out 2 more. How many did she have all together? B) Maryann had 5 library books before she returned 2 of them. How many does she have now? C) Maryann had 4 nonfiction books and 2 fiction. How many more nonfiction books does she have? D) Maryann had 9 library books and Jim had 4. How many more did Maryann have than Jim? A

Which problem represents the separate, start unknown structure?

A) Maryann had 3 library books before she checked out 2 more. How many did she have all together? B) Maryann had 5 library books before she returned 2 of them. How many does she have now? C) Maryann had some nonfiction books and 2 fiction. She now has 8 books. How many did she begin with? D) Maryann had 9 library books and Jim had 4. How many more did Maryann have than Jim? C

Why is it significant for students to use think-addition for subtraction rather than take-away?

A) Mastering subtraction facts. B) Mastering symbolic representations. C) Mastering problem solving strategies. D) Mastering model-based solutions. A

The use of model-based problems that students can use counters, bar diagrams, or number lines helps with their problem solving skills. Identify the statement below that describes what the use of bar diagrams can demonstrate.

A) Modeling two parts in two separate piles. B) Measuring distances from zero. C) Generating mean-making space. D) Breaking the shape apart to show multiple parts C

One of the best ways for children to think of real quantities is to associate numbers with measures of things. What concept would not emerge from using estimation?

A) More or less than ________. B) Closer to ________ or to ________. C) Is ________ reasonable? D) Exactly how many ________. D

Cultural differences are evident in algorithms. What reason below supports teaching for mathematics?

A) Notational algorithms. B) Customary algorithms. C) Mental algorithms. D) Invented algorithms. D

Experience with number relationships will guide children beyond counting by ones to solve problems. Name the relationship 10 plays in developing number sense.

A) One and two more. B) One and two less. C) Part-part-whole. D) Benchmark numbers. D

Verbal counting requires separate skills. Identify the statement that represents one of them.

A) One-to-one correspondence. B) Last word names the quantity. C) Greater than, less than and equal to relationships. D) Anchor numbers. A

The National Research Council identified three foundational areas in mathematics content for early learners. Identify the core that is more applicable for experienced learners.

A) Operations core. B) Relations' core. C) Number core. D) Reasoning core. D

The Common Core State Standards states that student should learn a variety of strategies. These strategies should be based on all of the following EXCEPT:

A) Place value. B) Sophisticated thinking. C) Properties of operations. D) Prior to the standard algorithm. B

Children explore quantity before they can count. What is the word for children being able to "just" see how many there are without counting?

A) Quantifying.B) Relating. C) Subitzing. D) Enumerating C

Identify the reason why the equal sign can confuse children.

A) Relational symbol not operations. B) Means the answer is coming up. C) Operations symbol not relational. D) Means the answer is the same as. A

All of the following provide an example of a method used for computation EXCEPT:

A) Standard algorithms. B) Student-invented strategies. C) Discourse modeling. D) Computational estimation. C

Which of the following is a true statement about standard algorithms?

A) Students will frequently invent them on their own if they are given the time to experiment. B) They cannot be taught in a way that would help students understand the meaning behind the steps. C) In order to use them, students should be required to understand why they work and explain their steps. D) There are no differences between various cultures. C

Children who can count successfully orally, but do not understand that the last word stated is the amount of the set are not demonstrating what principle?

A) Subitize. B) Quantify. C) Enumerate. D) Cardinality. D

Complete the statement, "When creating a classroom environment appropriate for inventing strategies..."

A) The teacher should immediately confirm that a student's answer is correct, in order to build his/her confidence. B) The teacher should attempt to move unsophisticated ideas to more sophisticated thinking through coaching and questioning. C) The teacher should discourage student-to-student conversations in order to provide students with a quiet environment to think. D) The teacher should encourage the use of naked numbers as a starting point. B

All of the statements below are considered a benefit of invented strategies EXCEPT:

A) They require one specific set of steps to use them, which makes them easier to memorize. B) They help reduce the amount of needed re-teaching. C) Students develop stronger number sense. D) They are frequently more efficient than standard algorithms. A

Which problem is an example of the equal groups, number of groups unknown structure?

A) This month, Barry saved 8 times as much as last month. Last month, he saved $3. How much did Barry save this month? B) Barry's sandwich shop offers 3 kinds of meat and 2 kinds of bread. How many different sandwiches could he make if he uses one meat and one kind of bread for each? C) Barry saved $12 and Jill saved $6. Barry saved how many times as much money as Jill? D) Barry saved $24 total, and he saved $6 each month. For how many months had he been saving? D

Which of the following represents a true statement about the concept of 0?

A) Understanding of 0 comes easily understood by small children. B) Developing understanding of 0 is important with its role in the base-ten number system. C) Counting involves touching an object, 0 is rarely included in the count. D) Discovering 0 is not useful for small children. B

The following statements are true regarding computational estimation EXCEPT:

A) Use the language of estimation- about, close, just about. B) Focus on flexible methods. C) Focus on answers. D) Accept a range of estimates. C

The ten-structure of the number system is important to extend students thinking beyond counting. All of the activities below reference a strategy for calculation EXCEPT:

A) Using decade number. B) Odd or even. C) Up over 10. D) Add on to get to 10. B

All of the statements below are related to teaching multiplication and division EXCEPT:

A) What to do with remainders. B) Symbolism as a way to record thinking. C) Physical models, drawings and equations. D) Think addition as multiplication and take away as division. D

Graphs clearly exhibit comparisons between numbers. What question below would require a comparison response?

A) Which snack is the most liked? B) How many people like apples as a snack? C) How much is the difference between apples and oranges? D) Which snack is the least liked? C

A good lesson built on a context or related to a story would have all of the qualities listed below EXCEPT:

A) Words, pictures and numbers are used to explain solution. B) Students can easily find a solution with mental mathematics. C) Students can find mistakes in other's written solutions. D) Designed to anticipate and develop mathematical models of the real world. B

Reading and writing numerals is similar to teaching children the alphabet and reading. Identify the statement below that would be developmentally inappropriate for early learners.

A) Worksheets with numbers to trace. B) Writing numbers on interactive whiteboard. C) Tracing numbers with shaving cream. D) Pushing numbers on calculator. A

Identify the statement that is describes the importance for children to know the relationship between addition and subtraction.

A) Writing symbolic equations. B) Using the associative property. C) Mental mathematics. D) Using the same models or pictures. D

When using place-value mats, drawing _________ in the ones place will make it very clear to students how many ones there are so they can avoid recounting the ones.

x's or ten frame

How do the commutative and associative properties relate to children's understanding of the basic facts?

Commutative is the flip flop problems, doesn't matter what order they are in (addition and multiplication) - Helps them understand basic facts by reducing the number of facts that the child has to learn. Can help with mental math as you can add them in any way. They can add the easiest way in their head (making 10).

T-F: Base ten blocks are the only material that should be used to model place value concepts.

False

T-F: Non Proportional models should be used only after students understand that ten units makes a "ten."

False

Discuss why the concept of "less" is more difficult than "more."

The reason that the concept of "less" is more difficult than "more" is thought to be because children are less likely to be asked things such as "which Is less". Many teachers and parents will frequently ask the child, "which is more?", exposing them to the concept and practice of "more, which is beneficial but creates a lack of understanding of the concept "less". In order to expose children to the understanding of "less" parents and teachers should ask "which is less" as well as "which is more".

T-F: According to NCTM, it's not necessary for students to have fully developed place value understandings before giving them opportunities to solve problems with two and three-digit numbers.

True

T-F: In order to help students to understand the way the two digits of a number and a base ten model of it are related, models of tens should be grouped on the left, and units should be on the right, to reflect the structure of the numeric version.

True

Identify the first three levels of thinking from the trajectory for counting and give an example of what the child demonstrate at that level.

1. Precounter: No verbal counting ability from the child. If you show this child a set of something, they will name the object rather than the quantity when asked "How many are there?" For example, Teacher: "How many dogs are there?" Child: "Dogs". 2. Reciter: Can verbally count, but not always in the correct order. A child at this level may look like this: Teacher: "How many dogs are there?" Child: "One, two, four, three..." 3. Corresponder: Can verbally count and make one-to-one correspondence with objects. A child at this level may look like this: Teacher: "How many dogs are there?" Child: "One, two, three, four, five". Child may need to recount to verbalize the answer.

When helping students to conceptualize numbers with 4 or more digits, which of the following is NOT true?

1. Students should be able to generalize the idea that 10 in any one position of the number results in one single thing in the next bigger place. 2. Because these numbers are so large, teachers should just make due with the examples that are provided in textbooks. 3. Models, such unit cubes, can still be used. 4. Students should be given the opportunity to work with hands-on, real-life examples of them 2

When students are being introduced to three-digit numbers

1. The process should be quite different from introducing students to two-digit numbers. 2. They have normally not yet mastered the two-digit number names. 3. They frequently struggle with numbers that contain no tens, like 503. 4. Their mistakes when attempting to write numeric examples should not be discussed, in order to avoid embarrassment. 3

A _______ is an important tool that can hang on a wall, be displayed on a smart board, or can be given to students as paper copies, which students can use to discover numerous place-value-related patterns

100's chart

Statements/questions from a teacher can support children's understanding of why one can't divide by 0 EXCEPT:

A) "What happens when you take these 25 pennies and divide them into 0 groups?" B) "Just memorize that you can't divide by 0." C) "Can you show me how to share 8 apples between no people?" D) "Put 12 blocks in 0 equal groups." B

An important variation of part-part-whole activities is referred to as missing-part activities. Identify the statement below that describes a missing part activity.

A) 2-3-4.5. B) Making a bar of two color connecting. C) A dot plate showing 5 and say, "I wish I had seven." D) Saying a 10 fact, six and four equals 10. C

When a problem has a number that is a multiple of ________ or close to ________ it is an example of a problem that you leave one number intact and subtract from it.

A) 85 - 3568. B) 85 - 6469. C) 85 - 2970. D) 85 - 56 C

All of the statements below represent the differences between invented strategies and standard algorithms EXCEPT:

A) A range of flexible options. B) Left-handed rather than right-handed. C) Number oriented rather than digit oriented. D) Basis for mental computation and estimation. D

All of the following could be examples of invented strategies for obtaining the sum of two-digit numbers EXCEPT:

A) Adding on tens and then ones (For example, to solve 24 + 35, think 24 + 30 = 54 and 5 more makes 59.) B) Using nicer numbers to estimate (For example, to solve 24 + 47, think 24 is close to 25 and 47 is closer to 45 so 24 + 47 = 25 + 45 = 70.) C) Moving some to make 10 (For example, to solve 24 + 35, move 6 from 35 to make 24 + 6 and then add 30 to the remaining 29.) D) Adding tens and adding ones then combining (For example, to solve 24 + 35, think 20 + 30 = 50 and 4 + 5 = 9 so 50 + 9 = 59.) B

What is the importance of students knowing the commutative property?

A) Applies to addition and subtraction. B) Helps students master basic facts because, if they really understand it, it reduces the number of individual facts they have to memorize. C) Should be demonstrated with problems that have the same sums but different addends. D) Is a term that even very young students should memorize? B

What number property is illustrated by the problem 16 × 12 = 16(10 + 2) = 160 + 32 = 192?

A) Associative. B) Commutative. C) Identity. D) Distributive D

Which problem is an example of the comparison, product unknown (multiplication) structure?

A) Barry's sandwich shop offers 3 kinds of meat and 2 kinds of bread. How many different sandwiches could he make if he uses one meat and one kind of bread for each? B) Barry saved $12 and Jill saved $6. Barry saved how many times as much money as Jill? C) This month, Barry saved 8 times as much as last month. Last month, he saved $3. How much did Barry save this month? D) Barry saved $24 total, and he saved $6 each month. For how many months had he been saving? C

Multiples of 10, 100, and sometimes 25 are called ______, which work especially well with hundreds charts and number lines to help students find the distances between numbers.

Benchmark Numbers

Because students can often hide their lack of conceptual understanding, a more in-depth assessment tool, _______ , can be used to determine what they really know.

Diagnostic assessment

Present and discuss two reasons for using contextual problems to teach addition, subtraction, multiplication and division.

It is more relatable to the students as these problems are put in more "real life" scenarios and they are more interesting than just seeing naked numbers. Along with that, it makes the problems easier to understand as they are working on sense-making with the problem rather just rules.