Mathematics: Reasoning and Problem Solving
Sheila has a large collection of stickers. She gives ½ of her collection to Sue, ½ of what is remaining to Sandra, and then gave ⅓ of what was left over to Sarah. If she has 30 stickers remaining, how many stickers did she begin with? A 90 stickers B 270 stickers C 180 stickers D 120 stickers
180 stickers This is a problem that can be worked backwards. Sheila is left with 30 stickers after she gave ⅓ of what she had to Sarah. That means that 30 stickers represent ⅔ of what Sheila had before she gave any stickers to Sarah. 30 is ⅔ of 45; so, Sheila had 45 stickers before she gave any to Sarah. 45 is half of what was left when half of the collection was given to Sandra. This means that Sandra received 45 stickers and that Sheila had 90 stickers before she gave any to Sandra. 90 stickers is how many Sheila had after she gave ½ of what she had to Sue; this means that Sue received 90 stickers and that Sheila had 180 stickers before she gave any away to Sue.
Sixteen teachers placed a book order for books to be used in their classrooms. The bill, totaling $350, is to be shared equally. Which is the most appropriate representation for how much will each teacher pay? A 350/16 = 21 AND 7/8 B 350/16 = 175/8 C 350/16 = 21r 14 D 350/16 = 21.876 ---> 21.88
350/16 = 21.876 ---> 21.88 This situation involves money. We are looking for an answer that can be "translated" into money.
Which equation below models xª•xᵇ = xª⁺ᵇ? A 5³ • 5⁴ = 25⁷ B 5 • 5 • 5 + 5 • 5 • 5 • 5 = 5⁷ C 5³ • 5⁴ = 5⁷ D 5 + 5 + 5 • 5 + 5 + 5 + 5 = 5⁷
5³ • 5⁴ = 5⁷ xª is a multiplication problem where "x" identifies the factor to be multiplied "a" times. These are some rules to follow when you are modeling an algebraic expression: 1) choose numbers other than 0 or 1 to replace the variables; 2) choose a different number for each variable; 3) replace the variables with the sample numbers you have chosen; and 4) evaluate to see if the relationship/equation is true. So, for this problem, the following sample numbers have been chosen: x = 5; a = 3; b = 4. Now, let's see if 5³ • 5⁴ = 5⁷. 5³ = 5 • 5 • 5 and 5⁴ = 5 • 5 • 5 • 5. So, 5³ • 5⁴ = (5 • 5 • 5) • (5 • 5 • 5 • 5) = 5 • 5 • 5 • 5 • 5 • 5 • 5 = 5⁷. This tells us that 5³ • 5⁴ = 5⁷ is a good model of the equation.
If the sum of 2, 6, and n is 15; then the product of 6 and (n+3) is: A 10 B 7 C 60 D 42
60 60 is the correct answer. First n must be determined, then n must be plugged into the expression 6 x (n + 3) 6 x (7 + 3) = 60
A diagram of Layla's backyard is provided. The blue square represents a pool recently installed. Her backyard has a total area of 1,800 square feet. Which equation could be used to determine A, the area of Layla's backyard remaining for landscaping? A A=1800-2 (20 + 15) B A=1800 + (20 X 15) C A =1800 - (20 X 15) D A= 1800 + 2(20+15)
A=1800−(20 ×15) To find the area of the yard after the pool was installed, you would need to subtract the area taken up by the pool from the total area of the yard. The area of the yard taken up by the pool is 15 x 20 = 300 ft², so the area remaining left for landscaping is the difference.
Mr. Johns gave a test last week and Ginny missed one question. She answered that 14.5 people would ride on each bus rather than 15. Her parents would like a conference because she did the math problem correctly and should receive credit even though her answer was not reasonable. How should Mr. Johns handle this situation? A Agree to meet, but let them know in advance that the grade will not be changed. B Offer to send the test home for them to review. C Agree to meet, listen to their concerns, and then explain that one component of math is understanding reasonable answers. D Offer to call them because a conference is not necessary.
Agree to meet, listen to their concerns, and then explain that one component of math is understanding reasonable answers.
Mrs. Dobbs is teaching students to skip-count by 2s, 5s, and 10s in her second-grade class. Earlier in the year, she evaluated her students learning style and assigns them one task based on this evaluation. Visual learners have been given a number line and they are to draw the hops across the top. Auditory learners have been given a list of the even number to 20, numbers divisible by 5 to 50 and numbers ending in 0 up to 100. They are told to say them over and over aloud to memorize the skips. Kinesthetic learners have been given a large number line on the floor. They are jumping to the next number as they skip-count. What can Mrs. Dobbs do to improve her teaching? A Allow all students to participate in all three activities by rotating through them. B Allow for more creativity by letting students make an art project out of drawing or write a song as they recite numbers. C Require them to participate in the jumping activity first to burn off some energy and then choose one of the other activities. D Allow students to choose which activity they participate in.
Allow all students to participate in all three activities by rotating through them.
Mrs. Smith wants her students to understand the reasoning behind their steps as they solve algebraic problems such as 2x + 7 = 12. How can she best encourage understanding? A Ask students write an essay about how they solved their math problem. B Ask students to write the reasoning for each manipulation as they solve an equation. C Ask students draw pictures to accompany their solved problems. D All of the above options encourage understanding equally well.
Ask students to write the reasoning for each manipulation as they solve an equation.
Mr. Kim shares with his geometry class the triangle sum property - The sum of all angles in a triangle always add to 180\degree180°. Then, he asks the students to find the missing angle in the triangle below: What type of thinking are the students using to solve this problem? A Inductive reasoning B Critical thinking C Abductive reasoning D Deductive reasoning
Deductive reasoning With deductive reasoning, students start with a proven fact, rule, or definition to arrive at a conclusion.
A first-grade teacher has been working with students on counting by twos, fives, and tens. The students are doing well with the concept, but the teacher is concerned that they are just memorizing the order of the numbers rather than applying the skill. What is one way that the teacher can encourage students to apply skip counting to their daily lives? A Ask students to practice skip counting at home with a parent or sibling. B Practice skip counting daily with a quick video. C Give students a set of nickels and have them count by fives to find the total value. D Count students by twos when they are lined up in the hallway.
Give students a set of nickels and have them count by fives to find the total value.
Mrs. Blue wants her students to be able to write two column geometric proofs. Which is the most appropriate way to determine their mastery? A Have students write an essay about writing proofs. B Give students an open ended exam where they write multiple two column proofs. C Give students a multiple choice exam related to proofs. D Give students a proof with steps missing and ask them to write in the missing reasons.
Give students an open ended exam where they write multiple two column proofs.
A second-grade teacher is planning a lesson on measuring length using standard units. Which of the following would be an effective way to engage students in the lesson while allowing them to practice measurement strategies? A Asking students to predict the length of common household items. B A worksheet that includes measurements of toys the students like. C Using an online program that allows students to use an on-screen ruler to measure objects. D Going outside to measure various parts of the playground in inches, feet, or yards.
Going outside to measure various parts of the playground in inches, feet, or yards.
Mr. Romeo wants his students to understand that math is very important and used in the real world all the time. He wants them to understand that math is used outside of school on a regular basis. Which of the following is the best way to have students learn about this? A Make students do a hands on project that uses math. B Have students interview two adults about how they use math in their everyday life. C Have students teach their younger siblings to use math to solve problems. D Make students read a book about famous mathematicians.
Have students interview two adults about how they use math in their everyday life.
A third-grade class has been working on adding increments of time smaller than 60 minutes. The majority of students are able to correctly add 15- and 30-minute increments in both isolated problems and word problems. What activity could the teacher add to the next lesson to increase student engagement? A Give students an opportunity for extra credit if they complete a homework page on adding increments of time. B Move on to the next unit, as students have mastered this one already. C Challenge students to add lengthier times, such as 1 ½ hours. D Have students work in pairs to create a new daily schedule with 30 more minutes of recess, 15 more minutes of lunch, and 15 more minutes of PE.
Have students work in pairs to create a new daily schedule with 30 more minutes of recess, 15 more minutes of lunch, and 15 more minutes of PE. This activity will encourage students to apply their knowledge to a familiar real-world scenario and will likely be an appropriate challenge for students to complete.
A math teacher wants to introduce a lesson on the use of decimals and fractions. Which of the following strategies is most likely to increase the students' understanding of the concepts? A Have students complete a pre-instructional worksheet on the topic. B Repeat the lesson until students have committed the lesson to memory. C Highlight examples of decimal and fraction use from the students' lives. D Have students write down what they think is the purpose of decimals and fractions.
Highlight examples of decimal and fraction use from the students' lives.
Mrs. Wheelan is teaching geometric shapes and wants to use informal reasoning questions for discussion. What question is best to start with? A What geometric shape does not have any edges? B How do geometric shapes play a role in daily life? C What is your favorite shape and why? D What is a quadrilateral?
How do geometric shapes play a role in daily life? This is an open-ended question and it allows for real-world connections.
Maria solved a word problem and correctly gave 72 as the answer. Which of the following could not have been the question asked? A How many guests were in attendance at the party? B How many months did Alexa achieve perfect attendance last year? C What was the average temperature, in Fahrenheit, in the city in March? D What is the least common denominator of two numbers?
How many months did Alexa achieve perfect attendance last year? The number of months in a year can not exceed 12. Therefore, 72 can not be the answer.
Mrs. Scott's class has 18 girls and 10 boys. All students are required to play kickball. She makes two kickball teams with even numbers of girls and boys. The teams play off in a very close game. Which of the following questions could NOT be answered with the information provided? A How many girls are on each kickball team? B How many girls are in the class? C What percentage of the class is girls? D How many points will the winning team score?
How many points will the winning team score?
In Ms. Suzi's math class, she asks the students to look at the following pattern and find the next three terms. 1, 2, 4, 5, 7, 8, 10, 11, 13, ...1,2,4,5,7,8,10,11,13,... What type of thinking are the students using to solve this problem? A Abductive reasoning B Critical thinking C Inductive reasoning D Deductive reasoning
Inductive reasoning With inductive reasoning, students look for patterns and then make generalizations.
Which of the following statements is false? A Direct, formal proof is based on deductive reasoning. B Inductive reasoning is the basis of many hypotheses and conjectures. C Inductive reasoning never leads to a correct conclusion. D Indirect proof is based on deductive reasoning.
Inductive reasoning never leads to a correct conclusion. Inductive reasoning, or reasoning from specific examples to end with a more general conclusion, may or may not lead to a correct conclusion. It is incorrect to say that inductive reasoning never leads to a correct conclusion.
A sixth-grade student asked his teacher the value for 0 ÷ 0. What is the answer to this student's question? A It is infinity. B It is equal to 0. C It is indeterminate. D It is equal to 1.
It is indeterminate. In division, 6 ÷ 3 = 2 because 3 • 2 = 6; and any number divided by itself is equal to 1. In general, a ÷ b is asking the question, "how many groups of size b are there in a?" So, it is pretty well understood that 0 ÷ n = 0, where n is any number. There are 0 groups of size n in 0, and n • 0 = 0. It is also pretty well understood that division by 0 is not allowed; the mathematical term used is "undefined". But the problem of 0 ÷ 0 is rather different. It is a number divided by itself, so it seems that we could say that 0 ÷ 0 = 1. It also follows that 0 • 1 = 0. So, what is the problem? The problem is that one could say that 0 ÷ 0 = 99 because 99 • 0 = 0! In fact, we could say that 0 ÷ 0 = z, where z is ANY NUMBER you want it to be. So, because there is not a specific "answer" for 0 ÷ 0, it is said that 0/0 is indeterminate - unable to be determined.
The 4th-grade students are going on a field trip to the science museum. There are 4 classes with 23 students and 3 adults in each class. They will take 5 buses with an equal number of people on each bus. How many people will be on each bus? Which of the following statements about the solution must be true? A Because of the real-world context, the solution must belong to all the set of all rational numbers. Therefore the solution is acceptable. B Because of the real-world context, the solution must belong to all the set of all natural numbers. Therefore the solution is acceptable. C Because of the real-world context, the solution must belong to all the set of all rational numbers. Therefore the solution is unacceptable. D Because of the real-world context, the solution must belong to all the set of all natural numbers. Therefore the solution is unacceptable.
Because of the real-world context, the solution must belong to all the set of all natural numbers. Therefore the solution is unacceptable. Natural numbers are whole counting numbers beginning with 1 and used in real word problems. In this problem, 26 people from 4 classes → 104 people on 5 buses. Because this problem requires 20 ⅘ people to ride the bus, the solution is unacceptable.
Mr. Michaels is teaching his class about areas of complex figures. Students seem to be getting overwhelmed by figures made of multiple shapes. What should Mr. Michaels encourage his students to do? A Break down the figures into their simpler shapes before proceeding. B Double check their work for errors before moving on. C Tell them that this material is unlikely to be tested on state exams and not to worry about mastery as long as they get the main ideas. D Ask a friend for help when they get stuck.
Break down the figures into their simpler shapes before proceeding.
Ms. Monroe is teaching her students about counting money and change. In her morning class, she gives several word problems as practice. In her afternoon class, she has students run a school store and practice giving change. She finds that students in her afternoon class perform much better on the unit test. What could explain the difference? A The word problems were easier than giving change in a school store. B Students did not like the word problems. C Students found the school store engaging and learned the material better than students given word problems. D Students in her afternoon class are more intelligent than students in her first period class.
C Students found the school store engaging and learned the material better than students given word problems. Running the school store likely engaged the students and they found it interesting. The practice likely deepened their knowledge.
Mr. Marshall is a math teacher and a student council sponsor. He has encouraged student council to do a service project, but they are struggling with ideas. He decides to assign his math class a project where they research local non-profits. How can he align this project with the curriculum? A Ask for a numbers sheet that has 10 numerical facts about the agency B Have students review the budget for the agency C Teach a lesson on how math can be used to inform people about social issues and have students find numbers that can help tell the story of the agency D Require students to write 3 words problems about the agency they research
Teach a lesson on how math can be used to inform people about social issues and have students find numbers that can help tell the story of the agency
Mrs. Gupta is teaching her students about evaluating truth of statements. Jenny gives the example: All boys are smelly. John is a boy. Therefore, John is smelly. How should Mrs. Gupta respond? A Send Jenny to the principal's office for bullying. B Tell Jenny that while her reasoning is correct, her premises are wrong. Ask Jenny for another example with factually correct premises. C Reprimand the student in front of the class to prevent this situation in the future. D Commend Jenny on her good reasoning.
Tell Jenny that while her reasoning is correct, her premises are wrong. Ask Jenny for another example with factually correct premises.
Some friends had a contest to guess the number of M&M's in a jar without going over. Tina guessed 270, Terry guessed 50% more than Tina, Twila's guess was twice Tina's, and Toby guessed 500. There were actually 450 M&Ms in the jar. Which friend guessed closest to the correct answer? A Twila B Tina C Toby D Terry
Terry Tina guessed 270, Terry guessed 50% more than Tina - 270 + 135 = 405, Twila guessed 2 x 270 = 540, and Toby guessed 500. Terry's guess of 405 is 45 M&Ms under the actual number while Toby is 50 M&Ms over. So, Terry is the closest.
A third-grade student is asked to find the best estimated answer for the problem below by rounding to the nearest ten. 162 + 287 + 395 The student gets an answer of 840. Which of the following best describes the student's error? A The student rounded all three numbers up. B The student added the numbers first, then rounded the answer. C The student rounded to the nearest hundred instead of the nearest ten. D The student rounded all three numbers down.
The student added the numbers first, then rounded the answer. Adding the numbers first and then rounding the answer would lead to the student's answer of 840 rather than the correct answer of 850. When estimating totals, students should be taught to round first, then add.
A second-grade student is given the problem 38 + 94. The student's work is shown below: 38 + 94 __________ 1212 Which of the following best describes the student's error? A The student did not regroup or "carry" the tens value when adding 8 + 4. B The student made a simple calculation error when adding the digits in the tens place. C The student did not align the numbers by place value. D The student multiplied the numbers instead of adding.
The student did not regroup or "carry" the tens value when adding 8 + 4.
Sarah wants to buy a sandwich, chips, and a drink for lunch. The sandwich cost $3, the chips cost $1.50 and the drink costs $1.25. What information is needed to determine if Sarah has enough money to cover the cost of her lunch? A The total amount of money she has to spend on lunch. B The number of coins she has. C The number of ounces in her drink. D The total cost of her lunch.
The total amount of money she has to spend on lunch.
Many students reach an incorrect answer when multiplying two-digit numbers, as shown in the work here. Which of the following is mostly likely the error made by students? 17 x 13 = 321 + 170 = 491 A The product of 1 ten and 1 ten was recorded as 1 one hundred, but should have been recorded as 1 ten. B The 20 composed from the multiplication of 3 ones and 7 ones was not carried and therefore caused the error in the final answer. C The product of 1 ten and 7 ones was written in the wrong location requiring the student to add a zero to the second row. D The 4 hundred should have been 5 hundred because the student forgot to carry a ten from the previous column.
The 20 composed from the multiplication of 3 ones and 7 ones was not carried and therefore caused the error in the final answer. When 3 and 7 are multiplied, you get 21. This is decomposed to 20 and 1. The 20 needs to be carried, but the student did not write the 2 above the 1 in 17, causing an additional error.
Two standard pieces of 8.5 x 11 paper are rolled into cylinders - open at both ends. One is rolled so that the height is 8.5 inches and the circumference is 11 inches and the other is rolled so that the height is 11 inches and the circumference is 8.5 inches. Which of the following statements is true about these two cylinders and their volume? A The volume of B is greater than A. B The volume of A is greater than B. C Their lateral areas are equal. D Their volumes are equal.
The volume of A is greater than B. Since V = Bh, both volumes can be calculated with some minor conversions first. In Figure A, the base, a circle, has a circumference of 11 which means the radius is 5.5/π and the area of the base, B, is equal to: B = π(5.5/π)² . So, the volume of figure A becomes: V = π(5.5/π)2(8.5) ≈ 257.125/π u³. For figure B, h = 11, the circumference of the base is 8.5 which means the radius is 4.25/π, and B = π(4.25/π)². So, the volume for figure B becomes: V = πr²h = π(4.25/π)²(11) ≈ 198.7/π u³. Since 257.1/π > 198.7/π, this means that the volume of cylinder A > volume of cylinder B.
John made a circular garden in his backyard. The garden has a diameter of 20 feet. He used ⅓ of the garden for tomatoes, his favorite vegetable. He enclosed the entire garden with a picket fence that was 12 inches high. Which of the following questions could NOT be answered with the information provided? A What is the area of the tomato patch? B What is the volume of the dirt in the garden? C What is the area of John's garden? D How many feet of fence does John need?
What is the volume of the dirt in the garden?
A fifth-grade student was asked to multiply 15 and 35. His work is provided below. 13 x 15 --------- 1525 35 ------ 1560 As his teacher, what remediation would you plan on providing? A a remedial lesson on estimation and reasonableness B a remedial lesson on place value C a remedial lesson on two-digit multiplication D remedial flashcards to practice multiplication facts
a remedial lesson on place value
A kindergarten class is finishing a lesson on two-dimensional shapes. Which of the following would be the most beneficial activity that creates real-world connections for students to complete? A having students draw and cut the different 2D shapes using construction paper B a worksheet on which students match the name of the shape to its picture C showing students different pattern blocks and asking them to name the shape D a scavenger hunt in which students work in pairs to find examples of different shapes in the classroom
a scavenger hunt in which students work in pairs to find examples of different shapes in the classroom This activity would be the most beneficial for students because it allows them to apply what they have learned about 2D shapes to their daily lives and the objects that they see in the real world.
A tennis ball has a diameter of about 3 inches. What is the approximate volume of a cylindrical container if it holds three tennis balls? A about 64 in³ B 82 in³ C about 27 in³ D 108 in³
about 64 in³ To find the volume of a cylinder, the B (area of the base) is multiplied by the height. The tennis ball can is three tennis balls high or about 9 inches. B, the area of the base, would be the area of the circle with the diameter of the tennis ball, or 3 inches. If the diameter is 3 inches, the radius would be 1.5 inches and the area would be: B = A of circular base = πr² = π(1.5)² = π(2.25) ≈ 7.07 in². So, the volume of the cylinder would be: V = Bh ≈ 7.07(9) = 63.63 in³. 64 in³ is the best approximate answer to this question.
Mrs. Doloff's third-grade class has learned about ordering people according to age when given a word problem such as "John is older than Mei and Mei is older than JD. Who is oldest?" What is the next concept for Mrs. Doloff to teach about ordering? A adding algebraic terms so that each person is represented by a letter such as person A, B, C B teaching them to order based on height without numbers C teaching them to order based on height with numbers D adding numbers to the problem to solve for exact age
adding numbers to the problem to solve for exact age This is the next step in scaffolded learning.
Caitlin knows that all birds have a beak. Adam is a bird. Therefore, Caitlin concludes that Adam has a beak. What type of reasoning is Caitlin using? A deductive reasoning B informal reasoning C inductive reasoning D formal reasoning
deductive reasoning Deductive reasoning involves statements such as: every dog is happy; Sally is a dog; therefore Sally is happy. Caitlin is using exactly this type of reasoning.
When teaching geometric shapes, Mr. Gaines challenges his students to prove a statement right or wrong. He writes on the board, "All rectangles are parallelograms and all squares are rectangles; therefore, all squares are parallelograms". What type of thinking is trying to promote? A inductive reasoning B conjectured reasoning C empirical reasoning D deductive reasoning
deductive reasoning Deductive reasoning requires the students to think through two or more known statements to determine if the conclusion is true.
During a lesson on using models in mathematics, a teacher asks the students to figure out how many hours they spend on homework for all their classes each year. In asking this question, the teacher has asked the class to: A demonstrate their ability to use statistics with data. B demonstrate the use of symbols to represent mathematical quantities. C demonstrate an understanding of the estimation process. D demonstrate their proficiency with the use of proofs.
demonstrate an understanding of the estimation process. Students will need to apply several estimates to determine the amount of time they spend on homework each year.
A survey is taken of students in a math class to determine what pets the students have. 7 students have birds; 15 students have cats; 18 students have dogs. Some students have more than 1 animal. For example, 3 students have cats and dogs and 4 students have cats, dogs, and birds. All students have at least one of these three types of pets. Which of the following would be the best strategy to use to answer a question about how many total students are in the class? A work a simpler problem B work backwards C simply add all of the given numbers D draw a Venn diagram
draw a Venn diagram
Adam wants to determine how much to charge for an event. He looks through his records from old events to determine a reasonable price for the venue, the average price of catering, and thinks about other incidentals. He then solicits quotes from several people and places before setting a price for the event. What process is he using to create this budget? A informal reasoning B formal reasoning C inductive reasoning D deductive reasoning
formal reasoning Formal reasoning is used to answer questions and solve problems that have a single solution (a right answer) by using rules of logic and algorithms (systematic methods that always produce a correct solution to a problem) to reach a conclusion. This is what Adam did when planning the budget.
A resource math teacher spends time with his students each day reviewing the calendar, reading a clock, and recognizing coins. These activities help them to prepare for the real world by offering them necessary skills for independence. What type of instruction is he offering? A concrete manipulatives B abstract symbolic math C STEM (Science, Technology, Engineering, and Mathematics) D functional math
functional math Functional math helps to prepare students for vocations and independent living.
Colin is a child learning about animals. He notices that dogs have four legs and a tail. When he sees a cat he incorrectly calls it a dog. What type of reasoning is Colin using? A inductive reasoning B formal reasoning C informal reasoning D deductive reasoning
inductive reasoning Inductive reasoning or generalizing knowledge from one area to another is used to make predictions. This is what Colin is doing when he predicts that a cat is a dog.
Janine is trying to determine who to vote for in the class president race. She thinks that candidate A is friendlier to her, but candidate B is better at convincing adults to do things. What type of reasoning is she using when she decides who to vote for? A formal reasoning B inductive reasoning C deductive reasoning D informal reasoning
informal reasoning Informal reasoning is used to answer questions and solve problems that are complex and open-ended without a definitive solution by using everyday knowledge to synthesize information and reach a conclusion. Janine is using feelings rather than true logic in her decision making here.
A first-grade student is asked to find the total value of the following coins: 3 dimes, 1 nickel, and 4 pennies. The student's response is that the coins are worth $0.12. Based on this response, what concept does this student likely need help with? A how to correctly write the value of a set of coins B recognizing different coins and their respective values C understanding that different coins have different values D remembering to check their work
recognizing different coins and their respective values
Mallory has $2.16 in her pocket to buy an apple and a bag of chips. What information is needed to determine how much money Mallory will have left after she makes her purchase? A how many apples Mallory has in all B the cost of the apple and the chips C how many coins she used to make her purchase D how many coins she has remaining
the cost of the apple and the chips
Mr. Isaka wants to buy a new car for his wife and needs to borrow $16,500. The bank will loan him $16,500 that he must pay back in 48 equal monthly payments. The amount to be paid back will include the amount he borrowed plus interest. What other information is necessary to determine the amount of Mr. Isaka's monthly payment? A the interest rate that the bank charges B the kind of car Mr. Isaka is purchasing C the amount of Mr. Isaka's monthly salary D the amount of Mr. Isaka's down payment
the interest rate that the bank charges The interest rate will definitely affect the amount of money Mr. Isaka will have to repay the bank and, therefore, will impact the monthly payment.
A student is working through a double-digit multiplication problem and turns in the work pictured. Which of the following best describes the student's error? 15 x 19 ----- 135 25 ------ 385 A the student added the products incorrectly B the student carried over the hundreds value from the 9 and 15 multiplication C the student erred when multiplying 9 and 15 D the student's understanding of the base ten numerical system needs remediation
the student carried over the hundreds value from the 9 and 15 multiplication
A first-grade teacher wants to encourage her students to use addition and subtraction skills in their daily lives. Which of the following would be the most effective way to do this? A Use addition and subtraction flash cards as time fillers throughout the day. B Look for opportunities during the day to ask students an addition or subtraction problem. For example, "We have 18 students in our class, but I only see 15 in line. How many students must still be getting water?" C Ask students to take note of the times they use addition and subtraction throughout the day and at the end of the day have a few students share what they took note of. D Give students word problems that use the names of students from the class.
Look for opportunities during the day to ask students an addition or subtraction problem. For example, "We have 18 students in our class, but I only see 15 in line. How many students must still be getting water?" This is a good way for the teacher to point out real-world applications of addition and subtraction while having students practice the skill in context.
Does the following given statement: "a rectangle is a square" apply to all problems? A No, because this rectangle is not a square. B Yes, because all squares are rectangles. C No, because not all rectangles are squares. D Yes, because all rectangles are squares.
No, because not all rectangles are squares.
Emily purchases 2 canvases and 6 paintbrushes for her art project. Sammy purchases 3 canvases for her art project. What information is needed to find the number of paintbrushes Sammy must purchase in order for the two girls to spend the same amount of money on art supplies? A The cost of a canvas and a paintbrush. B The cost of a canvas. C The cost of a paintbrush. D The cost of a container of paint.
The cost of a canvas and a paintbrush.
Mr. Habib bought 8 gifts. If he spent between $2 and $5 on each gift, which is a reasonable total amount that Mr. Habib spent on all of the gifts? A under $10 B $32 C $45 D more than $50
$32 Mr. Habib spent at least $16 and at most $40: $16 if every gift cost exactly $2 and $40 if every gift cost exactly $5. So only amounts within this range are reasonable.
Mrs. Johnson lets her students choose between two word problems: Problem A: If you are digging for dinosaurs and need to fence off your dig site, what's the biggest site you can fence off with 40 ft. of fence? Problem B: What is the largest area you can create with 20 inches of rope? Mrs. Johnson finds a significant majority of her students chose to work Problem A. Which of the following is the most likely reason more students chose Problem A instead of Problem B? A Problem B is harder than Problem A. B Problem A requires a lower mathematical knowledge. C Problem B is less interesting than Problem A. D Students who work Problem A get a greater reward from Mrs. Johnson than students who work Problem B.
Problem B is less interesting than Problem A. This is the best answer. Students are more likely engaged when presented with a problem about digging for dinosaurs than a simple mathematical word problem. The way problems are presented can impact students' engagement in the learning process.
Mr. Miller has taught addition with two-digit numbers and rounding. His students are beginning to use this concept in word problems. He teaches them 3 methods to simplify the process: guess and check, make a list, and draw a picture. Is teaching 3 different strategies a good practice? A No, because it is overwhelming to students to have 3 choices. B No, because these are all visual methods of learning and does not help auditory and kinesthetic learners. C Yes, because this allows students to develop a strategy that works for them. D Yes, because students like to have choices as this gives them a sense of control.
Yes, because this allows students to develop a strategy that works for them.
A student asks a teacher when would knowing the likelihood of a six being rolled on a dice be useful in real life. Which of the following examples would be the most appropriate response for the student? A a teacher averaging a student's grade for the semester B a casino estimating the expected number of jackpot payouts over the next fiscal year C a builder cutting materials for a house D a farmer measuring the length of the fields to determine area
a casino estimating the expected number of jackpot payouts over the next fiscal year
A student asks a teacher when calculating percentages of numbers will be useful in real life. Which of the following examples would be the most appropriate response for the student? A a builder cutting materials for a house B a architect designing a building C a pharmacist measuring the correct amount of medication D a mother going shopping at a store sale
a mother going shopping at a store sale
