Teaching Math Exam 2
He decomposed the nine from 39 into five and four, and then added the five to the five from 25
39+25=? 30+20 = 50 5+5 = 60 4+60 = 64 In the second step, how did the student come up with adding 5+5 to 50?
Providing more fact drill and worksheets.
Effective basic fact remediation requires three phases of intervention. Identify the statement below that would not be a part of an intervention. Explicitly teaching reasoning strategies. Identification of student fact knowledge. Providing more fact drill and worksheets. Determining student's level of number sense and reasoning.
Product
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? Start Result Product Change
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.
How are addition and subtraction related? Explain.
Comparison problems
Identify the problem structure that one group is a particular multiple of the other.
Take from 10
Identify the reasoning strategy that is used in high performing countries that takes advantage of students' knowledge of combinations that make ten.
3+0=3=0+3
Illustrate the identity property of addition for whole numbers.
Explicit reasoning
Making ten, known facts, derive unknown facts and double and one more group are examples of what effective basic fact teaching strategy?
None needed
Place parentheses if needed to make the following equation true: 18 / 6 + 1 = 4
4 + (3*6) = 42
Place parentheses if needed to make the following equation true: 4 + 3 * 6 = 42
Commutative property of addition
State the name of the property illustrated. 2+[4+(6)]=2+[(6)+4]
Associative property of addition
State the name of the property illustrated. 2+[8+(9)]=(2+8)+(9)
associative property of addition
State the name of the property illustrated. 4+[9+(2)] = (4+9)+ (2)
Use only paper and pencil to solve
Strategies for building a good lesson around a context problem include all of the following for student with the exception of which one? Use physical materials and drawings to solve Discussion about multiple methods for solving Use only paper and pencil to solve Focus on few problems to solve
Her claim is false; consider the example when a=1 and b=2.
Sue claims the following is true by the distributive property, where a and b are whole numbers. 13(ab)=(13a)(13b). Is her claim true or false?
Look for key words
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.
more teaching is required.
The following statements are true about the benefits of invented strategies except: more teaching is required. basis for mental computation and estimation. students develop number sense. are faster than the standard algorithm.
Lynn had a collection of 52 pencil and she gave 6 of them to her best friend. How many pencils does she have now?
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 some pencils and she decided to sharpen 32 of them, she has 63 of sharpened and not sharpened altogether. How many are not sharpened? Lynn had a collection of 52 pencil and she gave 6 of them to her best friend. How many pencils does she have now? Lynn gave some of her pencil collection to the teacher for use as extra pencils. She counted 52 pencils before giving them away. Now she has 43, how many did she give to the teacher? Lynn had a collection of 52 pencils. She traded for more pencil with a friend and now she has 63. How many did she obtain in her trade?
Easier to implement
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. Know which facts students do and don't know. Integrates assessment into instruction. Easier to implement. More insights into which reasoning strategies are used.
First she separated 6 into 1+5. 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 5.
To find 9+6, a student says she thinks of 9+6 as 9+(1+5)=(9+1)+5=10+5=15. What property or properties is she using?
First she separated 12 into 1+11. 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 11.
To find 9+12, a student says she thinks of 9+12 as 9+(1+11)=(9+1)+11=10+11=21. What property or properties is she using?
Pair problems with same addends but in different orders
To support knowledge about the commutative property teachers should do what to help the students' focus on the relationship?
Numbers that easily combine to equal benchmark numbers
What are compatible pairs in addition?
Use the language "is the same as" when you read an equal sign.
What is the best way to help students see the equal sign as a relational symbol?
Reinforce their inverse relationship
What is the main reason for teaching addition and subtraction at the same time?
Playing games.
What method below would students be able to infuse reasoning strategies, select appropriate strategies and become more efficient in finding the answer? Fact drills. Time fact tests. Fact worksheets. Playing games.
Array
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"?
move down one row directly below the number.
When adding 10 on a hundreds chart, the most efficient strategy that demonstrates place value understanding is to:
356 + 127 =
When presenting addition problems, which of the following would you use last? 356 + 127 = 39 + 23 = 645 + 354 = 43 + 32 =
move up one row directly above the number.
When subtracting 10 on a hundreds chart, the most efficient strategy that demonstrates place value understanding is to:
(2+5)+4=2+(5+4)
Which of the following equations illustrates the associative property for addition? 2+5=5+2 0+7=5+2 2+5=7, and 7-5=2 (2+5)+4=2+(5+4)
2(5+3)=2x5+2x3
Which of the following equations illustrates the distributive property of multiplication over addition? 2(5+3)=2x5+2x3 2(5+3)=(2+5)x(2+3) 2(5+3)=5+2x3 2(5+3)=2x5+3
Drill
Which of the following is not a strategy for supporting students' learning of basic facts? Memorization Explicit strategy instruction Drill Guided invention
3x =18
Which of the following open number sentences represents partition division? 3x =18 3+6=9 x6=18 3x6=18
Memorizing facts is important to mastering the facts.
Which of the following statements would not be evidence of about teaching the basic facts effectively? Memorizing facts is important to mastering the facts. Fluency includes being able to select appropriate strategies and answer problems quickly and correctly. Story problems can help students develop fluency with the basic facts. It is important to explicitly teach students strategies for solving basic fact problems.
the distributive property of multiplication over addition
Which property is illustrated by the following? (6+3)(7+9)=(6+3)7+(6+3)9
the distributive property of multiplication over addition
Which property is illustrated by the following? (7+4)(9+8)=(7+4)9+(7+4)8
the associative property of multiplication
Which property is illustrated by the following? 6•(3•9)=(6•3)•9
the commutative property of multiplication
Which property is illustrated by the following? 6•(3•9)=6•(9•3)
the distributive property of multiplication over addition
Which property is illustrated by the following? 7(4+8)=7*4+7*8
the commutative property of multiplication
Which property is illustrated by the following? 7(4•8)=(4•8)7
"Think-addition" and "missing addend."
Which reasoning strategy below would require students to know their addition facts to effectively use it for subtraction facts? "Think-addition" and "missing addend.". Five as an anchor. Down under 10. Take from ten.
The structures help students focus on sense making and the development of the meaning of the operations.
Why are teaching students about the structure of word problems important?
Repeated addition
a student solved 4*15 by taking (5+5+5)+(5+5+5)+(5+5+5)+(5+5+5). What method is she using?
14/7=2 14/2=7 7*2=14 2*7=14
complete the fact family for 14 / 2 = 7
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.
how can the multiplication table be used to solve division problems?
Adding and subtracting the same number to the problem has the effect of adding 0 to the problem, which produces an equivalent problem.
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? 48 +86 \|/ 48+2 + 86-2 \|/ 50 + 84 ______ 134
Even though the answer is correct, the use of the equal sign is incorrect. Therefore, this should not be allowed.
A child is asked to compute 1+7+3+4+8 and writes 1+7=2+9=11+4=15+8=23. Noticing that the answer is correct, if you were the teacher how would you react?
He decomposed the nine from 39 into five and four, and then added the five to the five from 25.
A student found 39+25 by doing the following: 30+20 = 50 5+5+50 = 60 4+60 = 64 In the second step, how did they come up with adding 5+5 to 50?
partition
A students solves 18/3 by drawing 18 objects and putting them into sections of 3. What method are they using?
Difference between groups
Equal group problems involve three quantities. Which of the following would not be a part of equal group problem? Total of all groups Number of groups Difference between groups Size of each group
After factoring out 93, the other factor sums to 100. It is easier to multiply by 100.
Explain how the distributive property of multiplication over addition would be helpful to mentally perform the following computation. 93•28+93•72
A tens digit was not regrouped as 1 ten when the sum of the units digits was more than 9.
Explain what is wrong in the following: 157 + 45 ________ 192
One tens value should have been traded from the 4 in the minuend's ten position.
Explain what is wrong in the following: 46 - 19 ______ 37
The units minuend is subtracted from the units subtrahend.
Explain what is wrong in the following: 46 - 28 _______ 22
Partial sums are not in the correct place value position.
Explain what is wrong in the following: 75 +67 ______ 1312
5; associative property of addition
Fill in the blank to make a true statement and identify the property of whole-number addition that is illustrated. 9+(5+7)= (9+—)+7
3; commutative
Fill in the blank to make a true statement, and identify the property. 2+3=___+2
0; identity
Fill in the blank to make a true statement, and identify the property. 2+__= 2
3; commutative
Fill in the blank to make a true statement, and identify the property. 3+ (7+4)= (7+4) + ___
8; associative
Fill in the blank to make a true statement, and identify the property. 7+ (8+3) = (7+__) +3
the basis for mental computation and estimation.
Invented strategies are:
It does work all of the time. This is because 9,6, 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 2+6) or by adding down (begin with 9+6). He wants to know why and if this works all the time. How do you respond? 9 6 +2 _____
It does work all of the time. This is because 7,5, and 3 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 3+5) or by adding down (begin with 7+5). He wants to know why and if this works all the time. How do you respond?
(5 + 4 + 9) / 2 = 9
Place parentheses if needed to make the following equation true: 5 + 4 + 9 / 2 = 9
none needed
Place parentheses if needed to make the following equation true: 5 + 6 / 2 + 6 = 14
(20*10)+(20*7)+(3*10)+(3*7)
Rachel finds 45*36 the following way: 45*36 =(40+5)*(30+6) =(40*30)+(40*6)+(5*30)+(5*6) =1200+240+150+30 =1620. What would be the third line for the problem 23*17?
Calculators can interfere with learning the basic facts and they should not be used until after the facts have been mastered.
Three statements below support students in their development of fluency with basic facts. Identify the statement that does not support basic fact fluency. Timed tests are not effective and there are better ways to assess students' progress in learning basic facts. Calculators can interfere with learning the basic facts and they should not be used until after the facts have been mastered. Games and activities are effective ways to practice strategies and work toward mastery. The goal is not just quick recall, but also flexibility and use of strategies.
True
True or False? 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.
9 + 6 = student thinks 10 + 6 is 16 and 9 is one less so the answer is 15
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 = students takes 1 one six to make 9 at 10 and then adds 10 + 5 =15. 9 + 6 = student thinks 10 + 6 is 16 and 9 is one less so the answer is 15. 9 + 6= students doubles 6 to get twelve and then adds the 3 left over from 9 to make 15. 9 + 6= student knows that 8 + 6 is fourteen so they add one more to make 15.
Using base-ten materials to model the problem
Which of the following instructional activities would be an important component of a lesson on addition with regrouping? Adding basic facts with sums to ten. Demonstrating the commutative property of addition. Using base-ten materials to model the problem. Reviewing the concept of greater than and less than.
Open number line
Which of the following is a common model to support invented strategies? Geoboard. Hundreds chart. Sentence strip. Open number line.
Teachers should spend a significant amount of time with invented strategies before introducing a standard algorithm.
Which of the following statements about standard algorithms is true? Standard algorithms are the only method for adding and subtracting multidigit numbers. Most countries use the same standard algorithms in mathematics. Teachers should spend a significant amount of time with invented strategies before introducing a standard algorithm. Standard algorithms should be taught without the use of models (such as completely on a symbolic level).
Combinations of 10
Which of the following strategies is a foundational strategy that must precede the learning of the others? Making 10 Add zero Near doubles Combinations of 10
I took 9 + 1 and added on 7 to get 17.
Which of the following student explanations uses the Making 10 strategy to solve 8 + 9? I added 8 + 8 + 1 to get 17. I knew that 8 + 10 was 18, and then I took one off to get 17. I took 9 + 1 and added on 7 to get 17. I see that the number 8 is two away from 10 and 9 is one away from 10, so the answer is three away from 20: 17.
Comparison
Which problem structure is related to the subtraction situation "how many more?" Take away Start unknown Comparison Part-part-whole
the identity property of multiplication
Which property is illustrated by the following? (4+8)1=4+8
the zero multiplication property
Which property is illustrated by the following? (7+6)*0=0
the zero multiplication property
Which property is illustrated by the following? (9*2)0=0
the identity property of multiplication
Which property is illustrated by the following? 1*(3*9)=3*9
the commutative property of multiplication
Which property is illustrated by the following? 6*(3*9)=(3*9)*6
the commutative property of multiplication
Which property is illustrated by the following? 7(4•8)=7(8•4)
