MATH 392 TEST 2
Explain how the distributive property of multiplication over addition would be helpful to mentally perform the following computation. 98x19+98x81
After factoring out 98, the other factor sums to 100. It is easier to multiply by 100.
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
In the video, opens in a new tab, 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: 30+20=50 5+5+50=60 4+60=64 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.
In the videoopens in a new tab, 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: 30+20=50 5+5+50=60 4+60=64 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.
In the video, opens in a new tab, Freddie was asked to solve 400−150. During his second attempt, he is unable to answer correctly. What is Freddie's misconception?
He fails to make exchanges correctly in the standard algorithm for subtraction.
Sue claims the following is true by the distributive property, where a and b are whole numbers. 12(ab) = (12a)(12b) Is her claim true or false?
Her claim is false; consider the example when a=1 and b=2.
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
John claims that he can get the same answer to the problem below by adding up (begin with 2+5) or by adding down (begin with 6+5). He wants to know why and if this works all the time. How do you respond? 6 5 2 + ______
It does work all of the time. This is because 6,5, and 2 are whole numbers, so the commutative and associative properties hold and allow the three numbers to be added regardless of their order.
John claims that he can get the same answer to the problem below by adding up (begin with 1+8) or by adding down (begin with 9+8). He wants to know why and if this works all the time. How do you respond? 9 8 1+ _______
It does work all of the time. This is because 9,8, and 1 are whole numbers, so the commutative and associative properties hold and allow the three numbers to be added regardless of their order.
Which of the following statements would not be evidence of about teaching the basic facts effectively?
Memorizing facts is important to mastering the facts.
d.) 56-19=47
One tens value should have been traded from the 5 in the minuend's ten position.
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
b.) 85+36=1111
Partial sums are not in the correct place value position.
Watch the video, opens in a new tab of Arriel solving the fair share problem. Which conceptual model did she use to solve the problem?
Partition
What method below would students be able to infuse reasoning strategies, select appropriate strategies and become more efficient in finding the answer?
Playing games
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
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
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 property is illustrated by (6•5)0=0?
The zero multiplication property
Each of the following equations is an example of one of the properties of whole-number addition. Fill in the blank to make a true statement, and identify the property. a. 4+5=___+4 b. 6+(2+7)=(2+7)+____ c. 5+___=5 d. 9+(3+6)=(9+___)+6 a. Complete the equation below and identify the related property. b. Complete the equation below and identify the related property. c. Complete the equation below and identify the related property. d. Complete the equation below and identify the related property.
a. The completed equation is 4+5=5+4. The related property that permits the completion of the equation is the commutative property of addition. b. The completed equation is 6+(2+7)=(2+7)+6. The related property that permits the completion of the equation is the commutative property of addition. c. The completed equation is 5+0=5. The related property that permits the completion of the equation is the identity property of addition. d. The completed equation is 9+(3+6)=(9+3)+6. The related property that permits the completion of the equation is the associative property of addition.
Multiplication facts that most children have memorized can be stated in a table. Complete parts (a) through (c) below. a.) Find some patterns and explain why they occur in the table. Select all that apply. b.) How can the multiplication table be used to solve division problems? c.) Consider the odd number 35 shown in the multiplication table. Consider all the numbers that surround it. Note that they are all even. Does this happen for all odd numbers in the table? Explain why or why not.
a.) Moving diagonally down and to the right, the difference between entries increases by 2. This pattern occurs because, starting at row a and column b, the first difference is (a+1)(b+1)−ab, the second difference is (a+2)(b+2)−(a+1)(b+1), and the difference between differences is (a+b+3)−(a+b+1)=2. b.)Find the row for either factor, if possible. Within this row, find the entry corresponding to the other factor. The quotient is the number of the corresponding column. c.)Yes. The only way for a product of two numbers to be odd is if both factors are odd. Therefore, the surrounding row(s) and column(s) will contain products of even numbers, which are even.
Which of the following equations illustrates the associative property for addition?
(2+5)+4=2+(5+4)
Which reasoning strategy below would require students to know their addition facts to effectively use it for subtraction facts?
"Think-addition" and "missing addend"
In the video, Rachelo calculates 45x36 in the following way: 45x36 =(40+5)x(30+6) =(40x30)+(40x6)+(5x30)+(5x6) =1200+240+150+30 =1620 Using Rachel's strategy, what would the third line be for the problem 23×17?
(20x10)+(20x7)+(3x10)+(3x7)
In the video, opens in a new tab, Shannon is asked to determine the number of stickers in 4 packs of stickers if each pack has 15 stickers. Which statement illustrates finding 4×15, using Shannon's strategy?
(5+5+5)+(5+5+5)+(5+5+5)+(5+5+5)
Watch the video, opens in a new tab of Andrew solving the multiplication problem 7×12. Although Andrew did not show his work, what might it have looked like based on his explanation of his problem solving process?
(7x10)+(7x2) 70+14 84
Which of the following equations illustrates the distributive property of multiplication over addition?
2(5+3)=2x5+2x3
Which of the following open number sentences represents partition division?
3 x =18
Illustrate the identity property of addition for whole numbers. Which of the following is an example of the identity property of addition?
3+0=3=0+3
When presenting addition problems, which of the following would you use last?
365+127=
Write the complete fact family for 21/7=3.
7x3=21 3x7=21 21/7=3 21/3=7
Fill in the blank to make a true statement and identify the property of whole-number addition that is illustrated. 9+(7+2) = (7+2)+___ Then Identify the property.
9 Commutative Property of Addition
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
In the video, opens in a new tab, 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 4×15?
A Cartesian-product model
a.) 156+45=191
A tens digit was not regrouped as 1 ten when the sum of the units digits was more than 9.
Tira, a fourth grader, performs addition by adding and subtracting the same number. She added as follows. How would you respond if you were her teacher? 26+67 26+4 and 67-4 30+63=93
Adding and subtracting the same number to the problem has the effect of adding 0 to the problem, which produces an equivalent problem.
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.
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 of the following strategies is a foundational strategy that must precede the learning of the others?
Combinations of 10
State the name of the property illustrated. 2+[6+(7)] = 2+[(7)+6]
Commutative property of addition
State the name of the property illustrated. 3+[4+(5)]=3+[(5)+4]
Commutative property of addition
State the name of the property illustrated. 3+[8+(6)]=3+[(6)+8]
Commutative property of addition
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
Equal group problems involve three quantities. Which of the following would not be a part of equal group problem?
Difference between groups
Which of the following is not a strategy for supporting students' learning of basic facts?
Drill
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
A child is asked to compute 2+5+4+3+11 and writes 2+5=4+7=11+3=14+11=25. Noticing that the answer is correct, if you were the teacher how would you react?
Even though the answer is correct, the use of the equal sign is incorrect. Therefore, this should not be allowed.
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+15, a student says she thinks of 9+15 as 9+(1+14)=(9+1)+14=10+14=24.What property or properties is she using?
First she separated 15 into 1+14. 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 14.
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.
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?
What are compatible pairs in addition?
Numbers that easily combine to equal benchmark numbers
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 property is illustrated by 8•(7•9)=(8•7)•9?
The associative property of multiplication
Which property is illustrated by 3(4•5)=(4•5)3?
The commutative property of multiplication
Which property is illustrated by 3(4•5)=3(5•4)?
The commutative property of multiplication
Which property is illustrated by 8•(7•9)=8•(9•7)?
The commutative property of multiplication
Which property is illustrated by 8•(7•9)=(7•9)•8?
The commutative property of multiplication
Which property is illustrated by 3(4+5)=3•4+3•5?
The distributive property of multiplication over addition
Which property is illustrated by (3+4)(6+5)=(3+4)6+(3+4)5?
The distributive property of multiplication over addition
Which property is illustrated by (8+7)(1+9)=(8+7)1+(8+7)9?
The distributive property of multiplication over addition
Which property is illustrated by 1•(7•9)=7•9?
The identity property of multiplication
Which property is illustrated by (4+5)1=4+5?
The identity property of multiplication
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.
In the video, opens in a new tab, Freddie was asked to solve the problem 400−150 three times. Which time did he solve the problem correctly?
The third try, using the base-ten blocks
c.) 66-47=21
The units minuend is subtracted from the units subtrahend.
Which property is illustrated by (1+9)•0=0?
The zero multiplication property
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.
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
Explain how the model below can be used to illustrate each of the following addition and subtraction facts in parts a) through d). [16] [9][7] a.) 9+7=16 b.) 7+9=16 c.). 7=16-9 d.). 9=16-7
a. When 9 and 7 are put next to each other, it is equal to the same length as 16. b. If 9 and 7 are put together on top of 7 and 9 put together, they both equal 16. c. Take the length 9 away from 16 and the length left is equal to 7. d. Take the length 7 away from 16 and the length left is equal to 9.
Place parentheses, if needed, to make each of the following equations true. a.) 4+3x2=14 b.) 6/2+1=4 c.) 5+4+9/9=2 d.) 5+6/2+2=10
a.) Parentheses are needed. The corrected equation is (4+3)•2=14. b.) Parentheses are not needed to make the equation true. c.) Parentheses are needed. The corrected equation is (5+4+9)÷9=2. d.) Parentheses are not needed to make the equation true.
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
Invented strategies are:
the basis for mental computation and estimation
