Math Methods Test 2
Give examples from real-world situations where an estimate, rather than an exact answer, is sufficient.
Checking whether change received from a purchase is accurate Determining the amount of tip for a waiter in a restaurant
Which of the following strategies is a foundational strategy that must precede the learning of the others?
Combinations of 10
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
Which statement illustrates how Connor might use repeated addition to find 6 times 53 correctly?
Count by 50 to 300. Add 3 six times, which equals 18. Add 300 and 18.
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
In new math textbooks, there is an emphasis on mental mathematics and estimation. Explain why these topics are important for today's students.
Mental math and estimation help students to know whether answers that appear on their calculators are reasonable.
Each of the models below is an effective tool to support invented strategies for addition and subtraction except:
chunking off
which problem structure is related to the subtraction situation "how many more"?
comparison
Here are some general principles for guiding student's development of computational estimation except
focus on answers not on methods.
Tom's diet allows only 1800 calories per day. For breakfast, Tom had skim milk (90 calories), a waffle with no syrup (120 calories), and a banana (119 calories). For lunch, he had one half cup of salad (185 calories) with mayonnaise (110 calories) and tea (0 calories). Then he had pecan pie (570 calories). Can he have dinner consisting of steak (250 calories), one half cup of salad without mayonnaise, and tea?
yes
Students were divided into 8 teams with 9 on each team. Later the same day students were divided into teams with 4 on each team. How many teams were there
18
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.
Identify the equation below that represents stepwise strategy
46 + 38 = 46 + 30 + 8 = 84
write whole number in expanded form : 5440
5000+400+40
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
A car trip took 9 hours at an average speed of 63 mph. Mentally compute the total number of miles traveled.
567 miles
A theater has 16 rows with 28 seats in each row. Estimate the number of seats in the theater
600
Suppose you had a balance of $2194 in your checking account and wrote checks for $95, $257, $404 and $523. Estimate the four check amounts and the beginning balance to the nearest hundred to estimate the new balance.
$900
Shannon is asked to determine the number of stickers in 4 packs of stickers if each pack has 15 stickers. Which statement illustrates finding 4 times 15, using Shannon's strategy?
(5+5+5)+(5+5+5)+(5+5+5)+(5+5+5)
Describe all pairs of whole numbers whose sum and product are the same
0 and 0 2 and 2
Of the problems below, identify the missing-addend problem similar to the one Arriel solved
Robert saved $65 toward the $100 iPod he wants to buy. How much money does he have to add to his savings to purchase the iPod?
In order to finish her homework quickly, an elementary student does her estimation problems by using a calculator to find the exact answers and then rounds them to get her estimate. What do you tell her
She would not be learning estimation skills if she is using the calculator first. One important use of estimation is to determine if the answer is reasonable
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
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
Is the front-end estimate for addition before adjustment always less than the exact sum? Explain why or why not
The front-end estimation is almost always less than the exact sum because an adjustment is usually needed. The only case where the estimation is the exact sum is when all of the front-end numbers are followed by zeros
What is the correct way to say 32 using base-ten language?
Three tens and two ones
When teaching computational estimation, it is important to
accept a range of reasonable answers.
Students should learn the relationship between multiplication and division. Explain this relationship.
-Division with remainder 0 is the inverse of multiplication and vice versa -If a times b=c, then c divided by b=a -If a divided by b=c, then b times c=a.
A designer has designed different tops, pants, and jackets to create outfits. How many different outfits can the models wear if she has designed the following pieces? seven tops, nine pants, two jackets
126 different outfits
A popular brand of pen is available in 5 colors and 3 writing tips. How many different choices of pens do you have with this brand
15
Dave purchased a $28,000 life insurance policy at the price of $31 per $1000 of coverage. If he pays the premium quarterly, how much is each installment?
217
About 4040 calories must be burned to lose 1 lb of body weight. Estimate how many calories must be burned to lose 6 lb, to the nearest thousand
24,000
When presenting addition problems, which of the following would you use last?
356 + 127 =
A car trip took 6 hours at an average speed of 65 mph. Mentally compute the total number of miles traveled
390 miles
How are addition and subtraction related? Explain
Addition and subtraction are inverses of each other; that is, they "undo" each other. That is, if the number 8 had the number 3 added to it, followed by the subtraction of the number 3, the addition and subtraction operations would cancel each other out.
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
Shannon is asked to determine the number of stickers in 4 packs of stickers if each pack has 15 stickers. Which model does she illustrate to find 4 times 15?
A repeated-addition model.
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
Suppose a student argued that 0 divided by 0equals1 because every number divided by itself is 1. How would you help that student?
By the definition of division, 0 divided by 0 = x if, and only if, 0=0 times x has a unique solution. But the last equation is true for all whole numbers x. Because the equation has no unique solution, 0 divided by 0 is not meaningful.
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
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.
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
Do you think it is valuable for students to see more than one method of doing computation problems? Why or why not
Yes. Some students may find one algorithm easier to understand or to use than others and therefore it will be easier for him or her to remember or reproduce.
Equal group problems involve three quantities. Which of the following would not be a part of equal group problem?
Difference between groups
Ariel uses a Set (Partition) Model to solve 18/ 3, creating three groups of 6 out of 18 M&Ms she has drawn. If she uses the same method, how would she illustrate 12/ 4?
Draw 12 M&Ms and separate them into four groups
Which of the following is not a strategy for supporting students' learning of basic facts?
Drill
A student claims that to divide a number with the units digit 0 by 10, she just crosses out the 0 to get the answer. She wants to know if this is always true and why and if the 0 has to be the units digit. How do you respond?
If the number has three digits with the units digit 0, we have ab0 divided by 10equalsab since ab times 10equalsab0. This will not work if the 0 digit is not the units digit.
Multiples of 10, 100, 1000, and occasionally other numbers, such as multiples of 25, are referred to as _____________ numbers.
benchmark
A student asks why she should learn the standard long division algorithm if she can get a correct answer using repeated subtraction. How do you respond?
If the repeated subtraction algorithm is done with large multiples of the divisor, the repeated subtraction can be quite efficient. However, if a student uses repeated subtraction by subtracting small multiples of the divisor, the process can be very time consuming
A pre-place-value understanding of number relies on children
counting by ones.
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
To find 9+5, a student says she thinks of 9+5 as 9+(1+4)=(9+1)+4=10+4=14. What property or properties is she using?
First she separated 5 into 1+4. Then she used the associative property to get the 9 and 1 together. Next, she added the 9 and 1. Finally she added 10 and 4
In the video, Connor was asked to solve the problem 39+25 twice, the first time when the problem was in a context of baseball cards, and the second time when the problem was not in a context. When Connor solved the problem the second time he described the process in these steps: In the second step, how did Connor come up with adding 5 + 5 to 50? Choose the correct answer below.
He decomposed the nine from 39 into five and four, and then added the five to the five from 25
Which of the following is not a common type of invented strategy for addition and subtraction situations?
High-Low strategy
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
Which of the following equations illustrates the distributive property of multiplication over addition?
2(5+3) = 2 times 5+2 times 3
John claims that he can get the same answer to the problem below by adding up (begin with 4 plus 5) or by adding down (begin with 7 plus 5). He wants to know why and if this works all the time. How do you respond?
It does work all of the time. This is because 7,5, and 4 are whole numbers, so the commutative and associative properties hold and allow the three numbers to be added regardless of their order.
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
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?
Which of the following statements would not be evidence of about teaching the basic facts effectively?
Memorizing facts is important to mastering the facts
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
Identify the strategy that relies on the student knowing specific facts to use this to "plus one or minus one."
Near doubles
On a 13-day vacation, Glenn increased his calorie intake by 1350 calories per day. He also worked out more than usual by swimming 3 hours per day. Swimming burns 362 calories per hour, and a net gain of 3500 calories adds 1 lb of weight. Did Glenn gain at least 1 lb during his vacation
No, Glenn did not gain at least 1 lb while on vacation
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.
Which of the following is a common model to support invented strategies?
Open number line
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
Which conceptual model did she use to solve the problem. Choose the correct answer below.
Missing-Addend
What method below would students be able to infuse reasoning strategies, select appropriate strategies and become more efficient in finding the answer?
Playing games
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
Connor was asked to solve the problem 4 times 25 and came up with 100 as an answer. When asked to describe his problem solving process he said he counted by 20's four times to get 80, and then added five to itself four times to get 20. So, 80plus20 is 100. Which strategies did Connor apply to solve the problem? Select all that apply
Repeated Addition Model Skip Count Model
Algorithms should have the following characteristics. Which of the follow does not belong?
Series of steps (memorized)
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
Joey has 2 toy cars and receives an additional 5 cars for his birthday. Describe where the numbers given in the problem should be placed in the number bond to correctly represent this problem
The 2 and 5 both go in the part circles
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?
A new model of car is available in 4 exterior colors and 2 interior colors. Use a tree diagram and specific colors to show how many color schemes are possible for the car.
There are 8 different color schemes for this new model of car
Students were divided into 10 teams with 12 on each team. Later, the same day students were divided into teams with 2 on each team. How many teams were there then?
There were 60 teams of 2 students each
How would you explain to children how to multiply 387 times 645, assuming they know and understand multiplication by a single digit and multiplication by a power of 10
Using the distributive property of multiplication over addition, first multiply 387 by 6 and the result by 10 squared; then multiply 387 by 4 and the result by 10; then multiply 387 by 5 and add all the numbers obtained together.
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
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
Teachers and students should orally refer to the manipulatives for ones, tens, and hundreds as
ones, tens, and hundreds
The three components of relational understanding of place value integrate:
oral names for numbers, written names for numbers, and base-ten concepts.
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)
What is the main reason for teaching addition and subtraction at the same time?
reinforce that inverse relationship
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
At a volleyball game, the players stood in a row ordered by height. If Evan is shorter than Gina, Mona is taller than Gina, and Stefano is taller than Mona, who is the tallest and who is the shortest?
stefano - tallest evan - shortest
Invented strategies are
the basis for mental computation and estimation
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 open number sentences represents partition division?
3 × = 18
At the beginning of a trip, the odometer registered 52,278. At the end of the trip, the odometer registered 59,570. How many miles were traveled on this trip?
7292 miles
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
A student uses front-end estimation to estimate the product of two numbers as 5600. List a pair of possible factors
76 & 83
Using a calculator, Sally multiplied by 10 when she should have divided by 10. The display read 800. What should the correct display be
8
Sue purchased a $29,000 life insurance policy at the price of $36 per $1000 of coverage. If she pays the premium in 12 monthly installments, how much is each installment?
87
Using a calculator, George multiplied by 5 when he should have divided by 5. The display read 225. What should the correct answer be?
9
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
Mary has 5 roses in her garden. If she wants to give her mom a dozen roses for Mother's day, how many more roses does she need? Which type of model is being used in this problem?
The Missing Addend Model
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
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
