Math Test 2 (Ch 8, 9, 10, 11)
how to effectively use pre-grouped base ten manipulatives for teaching place value.
- extra effort to confirm that students understand that a ten piece really is the same as 10 ones - be sure they have adequate experience with groupable models
how to effectively use groupable base ten manipulatives for teaching place value.
- start introducing language of "tens" by matching the objects (cups of tens and ones) - then to general phrase (groups of tens and ones" - eventually abbreviate to "tens" (four tens and seven)
single numeral
- student writes 36, views it as single numeral. - individual digits 3 and 6 have no meaning by themselves
benefits of invented strategies
- students make fewer errors - less reteaching is required - students develop number sense - invented strategies are the basis for mental computation and estimation - flexible methods are often faster than standard algorithm - algorithm invention is itself a significantly important process of "doing mathematics" - invented strategies serve students well on standardized tests
less reteaching
- teachers concerned when students invented strategy process is slow - but, the productive struggle builds network of ideas that is long-lasting and decreases time needed for reteaching
students make fewer errors
- this is because they understand their own methods - when students do not understand, they make mistakes and these systematic mistakes are hard to reverse
Think addition to solve a subtraction story would be effective for three of these problems. Which of the following would not be efficient?
. Lynn had a collection of 52 pencil and she gave 6 of them to her best friend. How many pencils does she have now?
Know the 5 levels of place value understanding*
1. Single numeral 2. position names 3. face value 4. transition to place value 5. full understanding
Assessing place value with the digit correspondence task helps the teacher recognize the student's level of understanding. According to Ross, which of the following statements represents a full understanding of place value when using the task with 36 blocks?
3 is correlated with 3 groups of ten blocks and 6 with 6 single blocks.
When presenting addition problems, which of the following would you use last?
356 + 127 =
Which of the following open number sentences represents partition division?
3 × = 18
Identify the equation below that represents stepwise strategy.
46 + 38 = 46 + 30 + 8 = 84
The integration of whole-number place-value involves using precise language. What statement below would confuse students about the groupings of tens and ones?
53 is the digits five and three
Three of the strategies described below reflect the use and knowledge of the base-ten pieces. Which of the following would not help the student solve using base-ten?
73 - 46, you give 3 to 73 = 76, subtract 46 = 36, so 36 - 3 = 27
Use 10 is a different strategy than Making 10. It does not require decomposition or recomposing a number. Identify the equation below that shows Use 10.
9 + 6 = student thinks 10 + 6 is 16 and 9 is one less so the answer is 15
Identify the reasoning strategy that is used in high performing countries that takes advantage of students' knowledge of combinations that make ten.
Take from 10
Which of the following statements about standard algorithms is true?
Teachers should spend a significant amount of time with invented strategies before introducing a standard algorithm.
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
full understanding
The 3 is correlated with 3 groups of ten blocks and the 6 with 6 single blocks.
transition to place value
The 6 is matched with 6 blocks and the 3 with the remaining 30 blocks but not as 3 groups of 10.
Why are teaching students about the structure of word problems important?
The structures help students focus on sense making and the development of the meaning of the operations.
When students see a story problem they generally focus on getting the answer. The contrast is to use context problems. Identify the problem below that would not be necessarily connected to children's lives.
The weather reporter said that the city had recorded 2.6 inches of precipitation for the month of May. What would the average rainfall be for the 4 days that it rained?
The zero and identity properties can often be challenging for students. Which of the following would help students understand the reason behind the products?
Use a number line and have students make 5 jumps of 0
Strategies for building a good lesson around a context problem include all of the following for student with the exception of which one?
Use only paper and pencil to solve
What is the best way to help students see the equal sign as a relational symbol?
Use the language "is the same as" when you read an equal sign.
counting strategies
Using object counting or verbal counting to determine the answer.
Which of the following instructional activities would be an important component of a lesson on addition with regrouping?
Using base-ten materials to model the problem
Which of the following statements about names for numbers is true?
When a student writes "three hundred fifty-eight" as "300508," the student may be at an early stage in moving accurately between oral three-digit numbers and written three-digit numbers.
When teaching computational estimation, it is important to:
accept a range of answers
examples of pregrouped models
base ten blocks, ten frames,
examples of groupable models
beans, coffee stirrers, cubes
Multiples of 10, 100, 1000, and occasionally other numbers, such as multiples of 25, are referred to as _____________ numbers.
benchmark numbers
pregrouped models
cannot be taken apart or put together
Each of the models below is an effective tool to support invented strategies for addition and subtraction except
chunking off
Which of the following strategies is a foundational strategy that must precede the learning of the others?
combinations of ten
A pre-place-value understanding of number relies on children:
counting by ones
Be able to describe the benefits of invented strategies and effective pedagogical methods that you as a teacher can employ to support the development of invented strategies among your students.*
describe
Know what is meant by an integrated place value model and how to effectively use groupable and pre-grouped base ten manipulatives for teaching place value.
describe
Which of the following is not a strategy for supporting students' learning of basic facts?
drill
serve students well on standardized tests
evidence suggests that students using invented strategies do as well or better on standardized tests than students who use standardized algorithms
Here are some general principles for guiding student's development of computational estimation except:
focus on answers not on methods.
Delia was asked to estimate 489 + 37 + 651 + 208. She said, "400 + 600 + 200 = 1200, so it's about 1200, but I need to add about 150 more for 80 + 30 + 50 + 0. So, the sum is about 1350." Which computational estimation strategy did Delia use?
front-end
Which of the following is not a common type of invented strategy for addition and subtraction situations?
high - low strategy
groupable models
models that most clearly reflect the relationship of ones, tens and hundred for which students can BUILD the ten from single pieces or units (10 beans in a cup: the cup is the same as the ten beans)
Models are important to guide students' conceptual understanding and the relationships of ones, tens, and hundreds. Identify the model below this is considered nonproportional.
money
The following statements are true about the benefits of invented strategies except:
more teaching is required.
When adding 10 on a hundreds chart, the most efficient strategy that demonstrates place value understanding is to:
move down one row directly below the number.
When subtracting 10 on a hundreds chart, the most efficient strategy that demonstrates place value understanding is to:
move up one row directly above the number.
flexible methods are faster
much less time than the time it takes to go through the steps using standard algorithm
Identify the strategy that relies on the student knowing specific facts to use this to "plus one or minus one."
near doubles
Teachers and students should orally refer to the manipulatives for ones, tens, and hundreds as:
ones, tens, and hundreds.
Which of the following is a common model to support invented strategies?
open number line
The three components of relational understanding of place value integrate:
oral names for numbers, written names for numbers, and base-ten concepts.
to support knowledge about the commutative property teachers should do what to help the students' focus on the relationship?
pair problems with same addends but in different orders
Be familiar with the 3 phases of fact fluency*
phase 1: counting strategies phase 2: reasoning strategies phase 3: mastery
What method below would students be able to infuse reasoning strategies, select appropriate strategies and become more efficient in finding the answer?
playing games
mastery
producing answers efficiently (quickly and accurately).
For problems that involve joining (adding) or separating (subtracting) quantities, which of the following terms would not describe one of the quantities in the problem?
product
When introducing place value concepts, it is most important that base-ten models for ones, tens, and hundreds be:
proportional (model for a ten is 10 times larger than the model for a 1).
integrated place value model
requires integration of new and sometimes difficult-to-construct concepts of grouping by tens with the procedural knowledge of how groups are recorded in our place-value system and how numbers are written and spoken
Problems that involve take away or take from involve a part of a quantity that is being removed from the start. Identify the name of the change problem structure that the start amount can be is the whole or the largest amount.
separate
position names
student correctly identifies the tens and ones positions but still makes no connections between the individual digits and the blocks
algorithm invention is an important process of "doing math"
students are involved in the process of sense making and building confidence
develop number sense
students who move to standard algorithms too early are unable to explain why they work
Invented strategies are:
the basis for mental computation and estimation.
face value
the student matches 6 blocks with the 6 and 3 block with the 3
What is the correct way to say 32 using base-ten language?
three tens and two ones.
Effective basic fact remediation requires three phases of intervention. Identify the statement below that would not be a part of an intervention.
Providing more fact drill and worksheets
What is the main reason for teaching addition and subtraction at the same time?
Reinforce their inverse relationship
Algorithms should have the following characteristics. Which of the follow does not belong?
Series of steps (memorized)
If a student was asked to count a container with 45 counters and you asked how many cups would you need if you placed 10 counters in each cup? What action below would provide the best evidence of the students' knowledge of place value?
Student goes back and counts to 10 and then starts again at 1
Computational estimation refers to which of the following?
Substituting close compatible numbers for difficult-to-handle numbers so that computations can be done mentally
Which of the following statements would not be evidence of about teaching the basic facts effectively?
Memorizing facts is important to mastering the facts.
Which of the following statements about reading and writing larger number is false?
After learning three-digit number names, students are easily able to generalize to larger numbers.
What type of problem structure does this phrase describe "the first factor represents the number of rows and the second factor represents the equal number found in each row"?
Array
Which of the following properties would this phrase describe "allows that when you multiply three numbers in an expression you multiply the first pair and then multiply that answer by the third"?
Associative
Number sense means that students have a grasp on the size of numbers. What does the term relative magnitude mean?
Number relationships - is it larger, smaller, close, or about the same
What are compatible pairs in addition?
Numbers that easily combine to equal benchmark numbers
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 and see if they are grouping the objects into groups of ten.
Three statements below support students in their development of fluency with basic facts. Identify the statement that does not support basic fact fluency.
Calculators can interfere with learning the basic facts and they should not be used until after the facts have been mastered.
Which problem structure is related to the subtraction situation "how many more?"
Comparison
Identify the problem structure that one group is a particular multiple of the other.
Comparison problems
Delia was asked to estimate 489 + 37 + 651 + 208. She said, "489 and 651 will be over 1000 since 489 is close to 500 and 651 is between 600 and 700. The 37 and 208 would be close to 240, so the solution will be around 1300." Which computational estimation strategy did Delia use?
Compatible numbers
Identify the statement below that would represent the child that has the level of understanding to work with the units of 10.
Counts tiles and makes 3 piles of tens and 1 pile of fives and says 10, 20, 30, and 5 more is 35
equal group problems involve three quantities. Which of the following would not be a part of equal group problem?
Difference between groups
Three of these statements are examples of effective formative assessment of basic facts. Identify the one that is often given as the reason given to use timed tests of basic facts.
Easier to implement
Making ten, known facts, derive unknown facts and double and one more group are examples of what effective basic fact teaching strategy?
Explicit reasoning
All of questions below would be ways to connect real-world ideas to support students understanding of place-value concept except which one?
How many numbers on a thousands chart?
Base-ten riddles are a method for showing equivalent representations. Identify the base-ten riddle what would not equal 42.
I have 20 ones and 2 ten. Who am I?
Which of the following student explanations uses the Making 10 strategy to solve 8 + 9?
I took 9 + 1 and added on 7 to get 17
The authors recommend strategies to guide students' problem solving skills. Identify the one that is often used by teachers and students but not always an effective approach.
Look for key words
reasoning strategy
using known information to logically determine unknown combination
the basis for mental computation and estimation
when invented strategies are the norm, mental computation is an intertwined mental skill
Be able to identify the structure of different addition problems, such as join, separate, part-part whole, and compare
worksheet
Be able to illustrate the various types of addition, subtraction, and estimation that were presented on the in-class worksheet (ie. split approach, jump method, think addition, compensation, etc.)
worksheet
Be able to visually depict, using counters, array, or grid paper, the: - commutative properties of multiplication - associative properties of multiplication - distributive properties of multiplication
worksheet
Be familiar with various reasoning strategies for addition, subtraction, multiplication, and division
worksheet
Which reasoning strategy below would require students to know their addition facts to effectively use it for subtraction facts?
"Think-addition" and "missing addend."
Which of the following equations illustrates the associative property for addition?
(2 + 5) + 4 = 2 + (5 + 4)
Which of the following equations illustrates the distributive property of multiplication over addition?
2(5 + 3) = 2 x 5 + 2 x 3
