EDCI 349 CHS. 7, 10, 11, 12

9. What role does 10 play in developing number sense?

" The number 10 plays such a large role in our numeration system and because two fives equals 10, it is very useful to develop relationships for the number 1 to 10 connected to the benchmarks of 5 and 10. The most common models for exploring benchmark numbers 5 and 10 (and effective templates for structing counting of manipulatives) are Five Frames and Ten Frames.

The integration of whole-number place value involves using precise language. What statement would confuse students about the groupings of tens and ones.

"53 is five tens and three"

3 components of relational understanding of place value integrate:

base-ten concepts, written names for numbers, and oral names for numbers.

A pre-place-value understanding of numbers relies on children

counting by ones

teachers and students should orally refer to the manipulatives for ones, tens , and hundreds as

cubes, longs, and flats.

2. What does the book say about teaching and using the standard algorithm?

make sure students know why they work and the students should be able to explain the steps

Always undestimate how much can be shared and always round up to the nxt mulitple of 10

pretend there are more sets or people

Number sense also includes having a grasp on the size of numbers. Relative Magnitude means?

refers to the size of the relationship one has with one another. (number relationships)—is it much larger, much smaller, close or about the same?

Doubling is when

students add two numbers and realize that the next two will have the same sum and so on.

which idea below is used for three-digit number development and should be extended to larger numbers

ten in any position makes a single thing in the next position to the left.

" Kenneth's work shows how he is partitioning

the factor 12 into 3×2×2. He multiplied the factors by the answer to the pervious equations. Kenneth broke numbers to create number that he can easily understand and multiply.

Which of the following statements about names for numbers is true?

there are many more errors saying the names of three-digit numbers than four-digit numbers

which is the correct way to say 32 using base-ten language?

three tens and two ones

Partitioning means

to divide into parts

Number Sense

you can think about different sized quantities and use numbers and relationships in multiple ways to estimate and solve problems. Understanding the magnitude (or size) of numbers.

When introducing place value concepts, it is important that base-ten models for ones, tens, and hundreds, be:

Proportional (model for a ten is 10 time larger than the model for a 1).

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

" A possible explanation is that young children may have many opportunities to use the word more but may have limited exposure to the word less. " Think about a mother taking to her child. She is more likely to ask her child if he or she wants more food. You never ask children how many less do you have.

Developing Whole-Number Place-Value Concepts

" A significant part of place value development includes students putting numbers together (composing) and taking them apart (decomposing) in a wide variety of ways as they solve addition and subtraction problems with two- and three-digit numbers. Without a firm foundation and understanding of place value, students may face chronic low levels of mathematics performance

5. What does the text say about developing the concept of 0 (zero)?

" Children need to discover the number zero and its value. It's a required standard of kindergartners. " Plays an important part in our base ten system. It is important for students to know zero has a meaning. " Three- and four-year old children can begin to use the word zero and the numeral 0 to symbolize that there are no objects in the set. " Zero is one of the most important digits in the base-ten system, and purposeful conversation about it and its position on the number line are essential.

6. Name some strategies (at least 3) used for calculation that are based on the ten-structure of the number system.

" Crossing A Decade using decade numbers " Make 10 Strategy (add on to get ten) " Down Under 10 or up over ten

2. What are some recommended research-based high-quality learning activities to use in the first six years?

" Enhance children's natural interest in mathematics and assist them in using mathematics to make sense of their world. " Build on children's experience and knowledge using familiar contexts. " Base mathematics curriculum and teaching practices on a solid understanding of both mathematics and child development. " Use formal and informal experiences to strengthen children's problem solving and reasoning processes. " Provide opportunities for children to explain their thinking about mathematical ideas. " Assess children's mathematical knowledge, skills, and strategies through a variety of formative assessment approaches.

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

" Pre-counter- Here the child has no verbal counting ability. A young child looking at three balls will answer "ball" when asked how many. The child does not associate a number word with a quantity. " Reciter- The child verbally counts using number words, but not always in the right order. Sometimes they say more number than they have objects to count, skip objects, or repeat the same number. " Corresponder- a child at this level can make a one-to-one correspondence with number and objects, stating one number per object. If asked "How many?" at the end of the count, they may have recount to answer.

10. What should children know when counting up to 20?

" Recognizing a set of ten plays a major role in children's early understanding of numbers between 10 and 20. " Children should able to orally say the number and know the number words in order. " The relations core. " Be able to tell one more, two more, and one less of the number. " Must have benchmark of ten.

11. What are the benefits for using a hundred's chart?

" The Hundreds Chart is an essential tool for every K-2 classroom. " Highlights number patterns (can see evens and odds) " They will learn and recognize how to count 2 digit numbers. " Shows the location of the numbers from 0-100.

10. What is important to keep in mind when teaching computational estimation?

- Use Real Examples -Use Estimation Language -Use a Context -Accept a Range of Estimates, Offer a Range as an Option -Focus on Flexible Methods, Not Answers Important teacher note: Do not reward or overemphasize the estimate that is the closest. It is already very difficult for students to handle "approximate" answers; worrying about accuracy and pushing for the one closest answer exacerbates this problem. Instead, focus on whether the answers given are reasonable for the situation or problem at hand. Offer ranges for answers that are estimates. Ask whether the answer will be between 300 and 400, 450 and 550, or 600 and 700

6. What are the skills required for verbal counting?

-By the end of kindergarten, children should be able to count to 100. -Verbal counting has at least two separate skills: 1. A child must be able to produce the standard string of counting words in order: "One, two, three, four.....". 2 .A child must be able to connect this sequence in a one-to-one correspondence with the objects in the set being counted. Each object must get one and only one count. As part of these skills, children should recognize that each counting number identifies a quantity that is one more than the previous number and that the new quantity is embedded in the previous quantity.

what are the models used to develop invented strategies?

-The Break Apart Strategy or split strategy (also called decomposition) -The Jump Strategy (similar to counting on or counting back) -The Take Away Strategy -The Shortcut Strategy (sometimes known as compensation)

8. What do we want children to learn regarding the concepts of more, less, and same? (p. 134)

-The concepts of "more," "less," and "same" are basic relationships contributing to children's overall understanding of numbers. For all three relationships (more/greater than, less/less than, and same/equal to), children should construct sets using counters (manipulatives) as well as make comparisons or choices between two sets. CCSS Standards for Math Practice states MP2. Students should be able to reason abstractly and quantitively. If a child is unable to indicate which one is more when shown two sets of number, they are at an educational risk. This is based on research.

12. In the early stages of number development what is the use of graphs

-The use is primarily for developing number relationships and for connecting numbers to real quantities in the children's environment. The graphs focus attention on tallies and counts of realistic things. Once a simple bar graph is made, it is very important to take the time to ask questions (e.g., "What do you notice about our class and our ice cream choices?"). " Equally important, graphs clearly exhibit comparisons between numbers that are rarely made when only one number or quantity is considered at a time.

invented strategies

-number oriented -left handed rather than right-handed. -a range of flexible options rather than "one-right way"

What are three main methods taught for computation discussed in chapter 11?

1. Direct Modeling- with guidance, this method can develop into an assortment of flexible and useful invented strategies, many of which can be carried out mentally. This method involves the use of manipulatives or drawings along with counting to directly represent the meaning of and operation or story problem. 2. Invented Strategies- refers to any strategy other than the standard algorithm or that does not involve the use of physical materials or counting by ones as. [ "Based on place value, properties of operations, and/or the relationship between addition and subtraction." ] 3. Standard Algorithms- an important part of what a student needs to learn. 4. Computational Estimation

3. What are the three foundational areas in mathematics content for early learners identified by the National Research Council?

1. The number core 2. The relations core 3. The operations core

which represents a full understanding of place value when using the task with 36 blocks?

3 is correlated with the 3 groups of 10 blocks and 6 with 6 single blocks

Students should solve multiplication and division problems with numbers appropriate for their grade level:

3rd: Fluently multiply and divide within 100 4th: Multiply a whole number up to 4 digits and find whole number quotients for up to a four-digit dividend with one-digit divisors 5th: Fluently multiply multi-digit whole numbers and find whole number quotients with up to four-digit dividends and two-digit divisors

11. Explain what students do when using the Front-End Method when estimating.

A front-end approach is reasonable for addition or subtraction when all or most of the numbers have the same number of digits.Notice that when a number has fewer digits than the rest, that number may be ignored initially. Also note that only the front (leftmost) number is used and the computation is then done as if there were zeros in the other positions

Clusters

An approach to multi digit multiplication that encourages students to use a series (or string) of facts and useful combinations they already know in order to explicitly guide students in figuring out more complex computations.

e. Array

An array is created according to the number of digits in the problem, label the sides with the factors decomposed by place value, multiply, and then add the partial products. In other words, it is a chart that places the numbers within the equation and separates the number according to place value and multiplying. Finally adding the final values to solve.

12. As a classroom teacher, how will you handle when siblings and family members are pushing the use of the standard algorithm when you are teaching invented strategies

Apply the same rule to standard algorithms as to all strategies: If you use it, you must understand why it works and be able to explain it. Let them make sense and explain why a certain procedure works. Reinforce the idea that just like the other strategies, it may be more useful in some instances than in others.

Multiples of 10, 100, and occasionally other numbers, such as multiples of 25, are referred to as

Benchmark numbers

Partition or Fair Sharing

Eileen's piggy bank has 783 pennies. She wants to share them equally with her 4 friends and herself. How many pennies will Eileen and each of her friends get? Partition or fair sharing uses base-ten materials

" Nick's work shows he used compensation

He manipulated the numbers in the problem to make it easier for him.As students like Nick begin partitioning numbers by place value, their strategies are often like the standard algorithm but without the traditional recording schemes. However, it is not a standard algorithm

what are the reasons for using an open number line ?

Helps with modeling student thinking" It can be jotted down anywhere. -It works with any numbers. -It eliminates confusion with tick marks and the spaces between them. -Students are less prone to making computational errors when using an empty number line

Compensation Strategies

In compensation strategies, students look for ways to manipulate numbers so that the calculations are easy. 27 x 4 is changed to an easier one, and then an adjustment or compensation is made. In 250 x 5 the first factor is cut in half and the second factor is doubled. This can be used when a 5 or a 50 is involved. Compensation strategies are dependent on the numbers involved, so they are not used for all computations. Very good for mental math and estimation.

f. Half-then-double

In this strategy, one factor is cut in half and the other factor is doubled. NOTE: This approach is often used when a 5 or 50 is involved. **Because these strategies are dependent on the numbers involved, they can't be used for all compensations.

" Briannon used the complete-number strategy.

It is the act of repeated addition through doubling of numerals to solve a multiplication number sentence. Notice the student knew that 32 groups of twelve is equal to the equation 35x12. He or she used repeated addition. The student broke the equation into 3 repeated addition trees then use addition of the three numerals to get final answer. She may need to see and hear about strategies other classmates developed to move toward a more efficient approach.

9. When teaching the subtraction standard algorithm what does the book say about zero?

It says to anticipate difficulties with zeros. Problems in which zeros are involved tend to cause special difficulties. The common errors that emerge when students "regroup across zero" are best addressed at the modeling stage.

Models are important to guide students' conceptual understanding and the relationships of ones, tens, and hundreds. Identify the model below that is considered nonproportional:

Money

In an environment that is appropriate for inventing strategies a teacher should:

Move unsophisticated ideas to more sophisticated thinking through coaching and strategic questioning. -Students need a classroom environment where they can act like mathematicians and explore ideas without fear. When students in your classroom attempt to investigate new ideas such as invented strategy use, they should find your classroom a safe and nurturing place for expressing naive or rudimentary thoughts. --climate for taking risks --testing conjectures --trying new approaches

Which of the following assessments can be used to determine students' understanding of base-ten development?

Observe students counting out a large collection of objects to see if they are grouping the objects into groups of 10.

what are the benefits of invented strategies?

Students make fewer errors because they understand their own methods. Less reteaching is required because the productive struggle in these early stages builds a meaningful and well-integrated network of ideas that is robust and long-lasting. Students develop number sense through strategies they understand. Invented strategies are the basis for mental computation and estimation Flexible methods are often faster than standard algorithms. Algorithm invention is itself a significantly important process of "doing mathematics." Invented strategies serve students well on standardized tests.

Complete Number Strategy

Students who use this strategy do not decompose numbers into decades or tens and ones. This is considered an early strategy based on repeated addition. This strategy is not efficient or useful. Encourage doubling instead of listing a string of addition. Doubling capitalizes on the distributive property.

a. Which comes next in the development of number sense?

The Reasoning Core

7. Cardinality

The number of elements in a set. It is being able to count successfully and understand that the last word stated is the amount of the set. " Children will learn how to count before they understand that the last counted word stated in a count indicates the amount of the set (how many you have total) or the set's cardinality. " Students are expected to have the cardinality principle, which is a refinement of their early ideas about quantity in kindergartner. " The child has learned cardinality if, after counting five objects, he or she answers the question "How many do you have in all?" with "Five".

Partitioning Strategy

The partitioning strategy by decades is the same as the standard algorithm except that students always begin with the largest values. This mental math strategy is very powerful.

"Short-cut Stragegy"

The shortcut strategy involves the flexible adjustment of numbers. For example, just as students used 10 as an anchor in learning their facts, they can move from numbers such as 38 or 69 to the nearest 10 (in this case 40 or 70) and then take the 2 or 1 off to compensate later

c. Complete numbers

When students list long columns of numbers to add them up and they do not decompose numbers into decades or tens and ones.

Subitizing

a process by which adults and children can look at a few objects and almost immediately know how many objects are present Means knowing quantity without having to count. For example, when you roll a die and immediately know that it is five without counting the dots, that ability to "just see it" is called subitizing. " A fundamental skill in the development of children's understanding of number and can be developed and practiced through experiences with patterned sets. " Children are able to see without counting.

8. What should we remember when teaching the standard algorithm? (name at least 3)

a.Good choice in some situations b.Require concrete models first. c.Explicit connections should be made between concept and procedure.

which of the following about reading and writing larger number is false?

after learning three-digit number names, students are easily able to generalize to larger numbers