Mathematics: Representations and Computation
An electronics store is offering a 20% discount on a pair of noise-cancelling headphones that normally cost $169.79. What is the price of the headphones, before tax, after the discount is applied? A $149.79 B $135.83 C $33.96 D $166.39
$135.83 Convert the percentage to a decimal and multiply it by the original cost of the headphones to find the amount of the discount: 20% = 0.2 and (0.2)(169.79) = 33.96. Subtract the discount from the original price to find the new price: 169.79 - 33.96 = 135.83.
A sweater is on sale for $43.75. It is 30% off the original price. What is the original price? A $62.50 B $73.75 C $30.63 D $69.99
$62.50 Set up a ratio to solve this problem: 43.75/x = 7/10. Cross multiply so you have 43.75 * 10 = 7x. Divide both sides by 7 to isolate x. So, 437.5/7 = $62.50.
Sabrina finds a coat on sale for 18% off the original price of $85. She computes her potential savings in the following way: $85 X 0.18 Which of the following methods could Sabrina also use to correctly determine the potential savings? A $85/ 100 - 15 B 100 x $85/18 C $85 x 18 x 100 D $85 x 18/100
$85 x 18/100
What is the prime factorization of 36? A 2 × 3 B (2²) × (3²) C (2³) × (3²) D (2²) × (3³)
(2²) × (3²) The prime factors of a number are the prime numbers that divide the integer exactly. The prime numbers then can be multiplied together to equal that number. The prime factors of 36 are (2²)*(3²). For 36, the factor tree would be: 36 = 4 × 9 = (2²) × (3²).
Solve : 2/3 + 4/5 A 2/15 B 2/5 C 1 and 1/2 D 1 and 7/15
1 and 7/15
Solve: 2/3 - 1/6 A 3/3 B 1/6 C 12/3 D 1/2
1/2
Solve 1/3 + 2/5 A 6/5 B 3/5 C 1/15 D 11/15
11/15
n Anytown ISD, 13 out of every 20 students ride the bus. Which ratio compares the number of students who ride the bus to those who do not? A 7:13 B 7:20 C 13:7 D 13:20
13:7 This is the ratio that compares riders to non-riders. 13 out of 20 ride the bus. This means that the complement of this relationship, those who do not ride the bus, is 20 - 13 = 7. So, the ratio of those who ride the bus to those who do not ride the bus is 13:7. When writing ratios, remember that order is important and matters. 13:7 is not the same as 7:13.
At Memorial High School, 60 of the 240 freshmen students are on an athletic team sponsored by the school. If the ratio is the same for the sophomore class, how many of the 220 sophomores are NOT athletes? A 180 B 60 C 165 D 55
165 The ratio 60/240 = ¼ so ¾ of students are not athletes. Since this can be applied to the sophomores, ¾ × 220 = 165 sophomores.
What is the prime factorization of 18? A (2²) × 3 B 2 × 9 C (2²) × (3²) D 2 × (3²)
2 × (3²)
Which of the following is not a decomposed version of 4/9? A 2/9 + 2/9 B 1/3 + 1/9 C 1/9 + 1/9 + 1/9 + 1/9 D 2/6 + 2/3
2/6 + 2/3 When adding fractions, they need the same denominator and then numerators only are added. This expression assumes you add across both, which is incorrect. This expression simplifies to 1 so it is not equal to
Which of the following is not a decomposed version of 2 and 3/8 A 2 + 1/8 + 1/8 + 1/8 B 2/8 + 3/8 C 8/8 + 8/8 +3/8 D 16/ 8 + 1/ 8 + 2/8
2/8 + 3/8
Which expression can be used to solve the following word problem? John and Jose want to buy a pizza for dinner and then head to a movie. They will each pay for their movie ticket, which costs $12 each, and they will split the pizza cost of $9. John has $17 and Jose has $20. How much will Jose have left at the end of the evening? A 20 - (9/2 + 12) B 20 + 17 - 9 - 12 C 12 - (9/2 + 12) D 9/2 + 12
20 - (9/2 + 12)
Solve 6/7 - 2/3 A 4/21 B 14/ 18 C 32/21 D 4/3
214
What is 5/18 as a percent? A 5.18% B 0.278% C 27.78% D 51.8%
27.78% To convert a fraction to a percent, divide the numerator by the denominator and multiply by 100. 5 ÷ 18 = 0.2778 × 100 = 27.78%
What is the exponential notation of 1656 and its prime factors? A 2^3 × 3^2 × 23 B 8 × 9 × 23 C 2 × 2 × 2 × 23 × 3 × 3 D 1672^1
2^3 × 3^2 × 23
A pancake recipe requires 1 tablespoon of baking powder per 2 cups of flour. If 2 cups of flour make 4 pancakes, how many tablespoons of baking powder are needed to make 12 pancakes? A 1 B 6 C 3 D 9
3 1 tablespoon of baking powder per 2 cups of flour makes 4 pancakes -> 1 tablespoon of baking powder is used to make 4 pancakes. To make 12 pancakes, 3 tablespoons of baking powder are needed. To find this, divide 12 by 4 to the proportional increase in pancakes. 12 / 4 = 3 -> because the number of pancakes is 3 times as many as the recipe size of 4; multiply 1 tablespoon by 3 to get 3 tablespoons of baking powder.
A third-grade student draws the following array: dots in a 3 X 5 formation Based on this image, the student is most likely solving what problem? A 3 × 5 B 35 C 3 + 3 + 3 + 3 D 3 + 5
3 × 5 The array shown has 3 rows and 5 columns, which correctly models the problem 3 × 5.
Order these least to greatest: (¼)^3 , 3^-4 , 4^3 A 3-4 , (¼)3 , 43 B (¼)3 , 3-4 , 43 C 43 , (¼)3 , 3-4 D Not possible because (¼)3 and 3-4 are equal
3-4 , (¼)3 , 43 3-4 = ⅓ × ⅓ × ⅓ × ⅓ = 1/81 (¼)3 = ¼ × ¼ × ¼ = 1/64 43 = 4 × 4 × 4 = 64
The state sales tax is 7.5%. Which number could also represent 7.5%? A 0.0075 B 0.75 C 3/40 D 3/4
3/40 There is always the option of arriving at the answer by eliminating incorrect answer choices, but it is always a good idea to double-check the final choice. In the case of this question, 3/40 = 3 ÷ 40 = 0.075 = 75/1000 = 7.5/100 = 7.5%.
The varsity basketball team has 3 freshmen, 5 sophomores, 3 juniors, and 4 seniors. Approximately what percentage of the basketball team is comprised of sophomores? A 20% B 33% C 25% D 30%
33% A total of 15 students are on the basketball team: 3 + 5 + 3 + 4 = 15. There are 5 sophomores on the team. The ratio of sophomores to the whole team can be represented by 5:15 = 1:3. 1/3 = .33 or 33%.
Jenny is baking cookies. Each dozen cookies requires 2 cups of flour and 0.75 lbs of butter. How many cookies can she make with 6 cups of flour and 3 lbs of butter? A 36 B 48 C 12 D 3
36 She can make 36 cookies. 6 cups of flour allows for 6/2 = 3 dozen cookies to be made.
The ratio of Antone's height to arm span is the same as the ratio of his brother's height to arm span. About how tall is Antone? ANTONE: ARM SPAN: 36 IN HEIGHT: X IN ANOTEN'S BROTHER ARM SPAN: 56 IN HEIGHT: 60 IN A 42 inches B 34 inches C 40 inches D 39 inches
39 inches
Tom wants to mentally calculate a 20% tip on his bill of $40. Which of the following is best for Tom to use in the mental calculation of the tip? A 40 × (20/100) B 40 × .02 C 40 × (200/1000) D 40 × .1 × 2
40 × .1 × 2 Tom can quickly find 10% of 40 and then double it. In this case the answer is $8 because 10% of 40 is 4 and 4 × 2 is 8.
If the number 888 is written as a product of its prime factors in the form a3bc, what is the numerical value of a + b + c? A 222 B 8 C 42 D 37
42 The prime factorization of 888 is 23 \times× 3 \times× 37. 42 is the correct answer because 2 + 3 + 37 is 42.
Bill went to the store to purchase new clothes for the upcoming school year. Bill purchased 8 shirts, 4 pairs of shorts, and 2 pairs of pants. If a single outfit consists of one shirt and either one pair of shorts or one pair of pants, how many outfits can Bill create with the clothes he purchased? A 42 B 32 C 16 D 48
48 The answer can be found by multiplying the number of shirts by the number of pairs of shorts and pairs of pants. This would create the equation (8 shirts) × (4 pairs of shorts + 2 pair of pants) = 8 × (4+2) = 8 × 6 = 48.
Which math expression is not represented by the array below? A 48 ÷ 6 B 48 × 1/8 C 6 × 8 D 48 ÷ 8
48 × 1/8
A student uses a number line to solve a problem. Their work is shown below: Based on their work, the student was most likely solving which of the following problems? A 2 + 2 + 2 + 2 B 25 C 5 × 2 D 10 ÷ 2
5 × 2
Which of the following is not a decomposed version of 5 and 4/9? A 5/9 + 4/9 B 45/9+ 2/9 + 2/9 C 5 + 1/9 + 1/9 + 1/9 + 1/9 D 9/9 + 9/9 + 9/9 + 9/9 + 9/9 + 4/9
5/9 + 4/9 A decomposed fraction expression must equal the original fraction. When these fractions are added together, they equal \frac{9}{9}99, which is the same as 1. Since each whole unit would be \frac{9}{9}99, the whole number 5 becomes \frac{45}{9}945 not \frac{5}{9}95.
What is the scientific notation of 674.9723? A 6.749723 × 10-2 B 6.749723 × 102 C 6.749723 × 104 D 6.749723 × 103
6.749723 × 102
Below is an example of a student's work: 4/16 - 1/8 = 3/8 10/13 - 3/8 = 7/5 3/5 - 2/3 = 1/2 If the student continues making the same error, the student's most likely answer to the problem 9/16 - 3/4 would be: A 0 B 3/5 C 12/20 D 6/12
6/12
A runner is running a 10k race. The runner completes 30% of the race in 20 minutes. If the runner continues at the same pace, what will her final time be? A 67 minutes B 80 minutes C 60 minutes D 62 minutes
67 minutes To find the answer to this question, set up a ratio; remember that 30% = .3 and 100% = 1. Therefore, (20 / .3) = (x / 1) When you cross multiply to solve for x, you get the equation .3x = 20. Divide each side by .3 to isolate x and the answer is 66.66666. The best answer choice is 67 minutes.
Marsha asked all 72 children at recess and only ⅙ said that their favorite ice cream flavor was strawberry. Which of the following expression can be used to determine the number of children who told her strawberry was their favorite? A 72 × 0.33 B 72 × 33 C 72 × 0.166 D 72 × 16.66
72 × 0.166
There are 5 children in the Drake family. The oldest children are 12-year-old twins, and the youngest child is 4 years old. If the sum of all the children's ages is 47, what could be the ages of the other 2 Drake children? A 8 and 11 years old B 9 years old C 3 and 6 years old D 7 and 11 years old
8 and 11 years old The sum of the ages of the five children is 47. The twins are 12 (2 x 12 = 24) and the youngest child is 4 years old. So, for these three children we have used 24 + 4 = 28 of the sum of 47, leaving 19 as a sum for the ages of the two remaining children when each of the two children must be between the ages of 4 and 12. The only choice that sums to 19 is 8 and 11.
What value does 8 represent in the number 2.86 x 10^4? A 80 B 8 / 10 C 800000 D 8000
8000
Which expression can be used to solve the following word problem? Mr. Henry wants to purchase 24 hamburgers and 24 hotdogs for a Bar-B-Q he is having at his house. If hotdogs come in a package of 8 and hamburgers come in a package of 6, how many packages total of hamburgers and hotdogs will Mr. Henry have to buy? A 24/8 + 24/6 B 24(1/8 + 1/6) C 24(7/24) D all of the above
All of the above
The following word problem was given to Mr. Trout's fourth-grade class: The Hotel Vacay is hosting a Wintertime Brunch for families. Each child that attends gets to decorate a gingerbread house and use the ice slide 3 times. Every family gets 2 snowballs per person. If 108 people can be seated and there are an equal number of adults and children, how many gingerbread houses and snowballs do they need? Morgan solved the problem using this equation: 108/2 + 108 × 2 Julia solved the problem using this equation: 108 × 2 ½ Which child is correct? A Morgan B Julia C Both children D Neither child
Both children Both are correct. Morgan found the number of gingerbread houses and then the number of snowballs and added them. Julia determined that each parent-child pair gets a gingerbread house which is the same as ½ a house per person so she multiplied 2 ½ by the total number of seats.
Jessica is working on adding 8 to 25. She starts counting at 25, using her fingers to count 8 more numbers out loud. Which counting technique is she using? A Counting on B Skip counting C Counting backwards D Counting collections
Counting on
Mr. Sexton has been trying a variety of teaching methods to engage his class, but it seems to make things more out of control. How can he increase engagement while maintaining an orderly classroom? A Incorporate more activities that allow students to move around and burn energy. B Rearrange the seating chart to separate disruptive students. C Establish a daily procedure for class and vary the activities used for instruction. D Use a louder voice that commands attention.
Establish a daily procedure for class and vary the activities used for instruction.
A kindergarten class is beginning a unit on data collection. Which of the following would be the best first activity? A Give each student a collection of colored tiles to sort by color. B Have each student bring or draw a picture of their favorite pet and arrange them into a class graph. C Show the class a bar graph representing favorite fruits and have them tell you which fruit is the most favorite, least favorite, etc. D Any of the above would be an equally good first activity for this unit.
Give each student a collection of colored tiles to sort by color.
A teacher has presented adding fractions to her class and they have used manipulatives to gain a basic understanding. What is the next step in the learning process? A Give the students word problems with at least one fraction. B Give the students problems with one variable to solve. C Give the students fraction addition problems using numbers instead of manipulatives. D Give homework with 10-15 problems to reinforce the concept.
Give the students fraction addition problems using numbers instead of manipulatives.
When asked to answer the question what is ½ ÷ 2; Jon answered 1. Which is the best activity for Jon so he would better understand the problem and how to answer it? A Tell Jon that he has answered the question, "What is ½ of 2?" Then have him rework the correct problem. B Have Jon draw a picture to illustrate his thought process to you. C Tell Jon to think about having ½ a pizza. Ask him, "if he were to share the pizza equally with a friend, how much of the pizza would each of you get?" D Have Jon draw a picture to represent ½ a pizza, then divide it into two equal parts. Ask him how much of the whole pizza each part represents.
Have Jon draw a picture to represent ½ a pizza, then divide it into two equal parts. Ask him how much of the whole pizza each part represents.
In a first-grade class, the students have been working with manipulative materials and pictures as they investigate the concept of addition. Through both formative and summative assessments, the teacher has determined that the students are ready to move to more abstract (pencil and paper) ways to represent addition. How should she begin this process? A Have the children model pictorial representations of problems like 7 + 2 = 9 that include the numbers that represent each step. B Relate the symbolic representation of addition facts to models the children have created, modeled, or drawn in their math lessons. C Model one of the problems, 7 + 2 for example, for the children by writing: 7 + 2 = 9. Then have the students repeat the process with a different problem. D Give the students a page of one digit addition problems with sums of 10 or less and having them draw a picture to match the sum.
Have the children model pictorial representations of problems like 7 + 2 = 9 that include the numbers that represent each step.
When Mr. Troutner introduces scientific notation to his math class, Jerome asks why they are learning about science during math class. What is the best answer to his question? A Math is the most important subject and it is used in all other subjects; this is just one example. B This is not a science standard, it just shares a name. C Math and other academic subjects work together in the real world. D The school is encouraging cross-curricular activities and he is trying to implement them.
Math and other academic subjects work together in the real world.
A number pie is squared and then multiplied by 2, then 7 is subtracted and that result is divided by 4. Which of the following sequential steps would reverse that procedure? A Multiply the final number by 4, add 7, then divide by 2. Then take the square root of the result. B Multiply the final number by \frac{1}{2}21 and add 7. C Multiply the final number by 4, add 7, then divide the result by 4. D Take the square root of the final number, and then divide it by 2, then add 7 and multiply the result by 4
Multiply the final number by 4, add 7, then divide by 2. Then take the square root of the result.
What is the difference between -5^2 and (-5)^2? A One equals -25 and the other equals -10. B The answers are the same: -25. C One answer is positive and one answer is negative. D The answers are the same: 25.
One answer is positive and one answer is negative. -5^2 = -(5 × 5) = -25 because the negative sign is not inside the parentheses. (-5^)2 = (-5 × -5) = 25 because the negative sign is inside the parentheses.
Mrs. Luna tried flipping her classroom to teach common denominators, having students watch a lecture at home and then doing the homework practice during class. Many students did not watch the entire video because they thought they had the concept down after the first example. If she tries this again, how should she change her approach? A Have students create the video. B Keep students inside during recess if they did not watch the entire video so they can catch up. C Give prizes for those who watch the entire video. D Provide a notes outline that needs to be filled in as they watch the video.
Provide a notes outline that needs to be filled in as they watch the video.
Mrs. Marshall grades tests from the last unit and realizes most students are missing word problems because they do not identify the correct operation to use. They did demonstrate mastery on questions that provided the equation. How should she address this issue? A Provide a warm-up question each day and students must underline key terms that help decide what operation to use. B Address it through homework packets. C Reteach the unit using word problems. D Add a mini-unit covering math vocabulary and key terms.
Provide a warm-up question each day and students must underline key terms that help decide what operation to use.
Miss Kelly has been teaching fractions and believes her students understand composing and decomposing fractions through the activities they have done. What activity would be best to informally assess their knowledge before moving on to the next lesson? A Give a short pop quiz with fraction composition and decomposition B Provide a warm-up question that asks them to write one way to decompose 3/4 C Lead a discussion on fractions in the real world D Include fraction composition and decomposition in the homework tonight
Provide a warm-up question that asks them to write one way to decompose 3/4
Jessica starts to count by 3's: 3, 6, 9, 12, 15, 18, ... Which counting technique is she using? A Counting on B Counting backwards C Skip counting D Counting collections
Skip counting Skip counting is counting by something other than one, such as counting by 5's or 10's.
Mrs. Herschend decided not to give a test about ratios and instead had her students do a project to display their knowledge. She has decided that she will do this for every unit going forward. What is the main disadvantage to this approach? A Projects take more class time than giving a test. B Students need to practice test taking skills periodically. C Some students are not creative and projects are more stressful than tests. D Parents can help with projects and the students knowledge may not be displayed.
Students need to practice test taking skills periodically.
As Mr. Costa reviews probability for the test, he reminds students of the various ways to write probability: ratios, fractions, decimals, and percentages. To allow for practice, he divides the class into 4 groups and assigns each group one method. Every student answers the questions, but must write the response in the assigned way. Why is this not an ideal method? A Ratios are easier than decimals so it is an unequal assignment. B Working in groups is distracting for some students and they will not receive enough review. C Students only practice one method and do not review other methods. D Working in groups allows students to copy answers.
Students only practice one method and do not review other methods.
Which of the following situations might require the use of a common denominator? Select all answers that apply. A Multiplication of fractions B Division of fractions C Subtraction of fractions D Addition of fractions
Subtraction of fractions AND Addition of fractions
Mr. Harris assigns his students to create a factor tree of the products of a number. One student turns in the product factor tree of 32 shown here. Which of the following best describes the error in this factor tree? A The factor tree does not identify the factors of 2. B The factor tree does not correctly identify the factors of 32. C The factor tree identifies the sums and not the product factors of many of numbers. D There is no error in the tree.
The factor tree identifies the sums and not the product factors of many of numbers.
Which of the following is not equivalent to 1/2? A 1 / square root of 4 B 1/3 ÷ 2/3 C 50% D 2 ÷ square root of 4
The square root of 4 is 2, so this becomes 2 \div 2 = 12÷2=1
Mrs. DeLong knows that building relationships with her students is one of the best ways to increase academic performance. Today's warm-up question asks students to write a word problem with at least 2 fractions. What is the greatest benefit from this exercise? A Their word problem will likely include something important to them. B This will show where their mathematical weaknesses are. C It provides potential test questions. D It is cross-curricular with language arts.
Their word problem will likely include something important to them.
Before teaching multiplication, a teacher reviews skip counting on a number line. Students use different colored markers to show counting by 2s, 5s, and 10s. After introducing multiplication, they review their number lines and connect the concept to the jumps. Why did the teacher return to the number line as she taught? A This allowed students to connect prior knowledge to new concepts with a visual example. B This tied the warm-up to the exit ticket. C It provided remediation for struggling students. D Students could add decorations if they already understood multiplication.
This allowed students to connect prior knowledge to new concepts with a visual example.
A student is using an array to solve 12 ÷ 3. Which of the following shows the array that would be used to correctly solve this problem?
This array shows that when 12 items are divided into 3 equal columns, there are 4 items in each column. This correctly represents the problem 12 ÷ 3 = 4.
Mr. Wheeler has taught prime factorization to his class this week. In an effort to differentiate instruction, he decides to block off time to provide remediation for students with learning disabilities. What is the main problem with this approach? A He has not planned a fun activity for the other students in class. B This targets only one group of students, which may or may not have struggled with this specific concept. C It breaches confidentiality because they all have learning disabilities and the other students will know this. D This not an issue; Mr. Wheeler should use this approach.
This targets only one group of students, which may or may not have struggled with this specific concept.
Which of the following is the best way for elementary students to learn inverse operations? A Use word problems to illustrate how inverse operations work. B Give students a worksheet packet with progressively harder questions. C Use a number line to illustrate adding and subtracting the numbers in a fact family. D Read the definition and take notes about operation inverses.
Use a number line to illustrate adding and subtracting the numbers in a fact family.
Students in Mrs. Wilson's class have mastered multiplication and have been introduced to division. Mrs. Wilson gave a test over introductory concepts in division and found that a number of students struggled. Which of the following strategies is best to help improve the students' understanding of division? A Provide a new study guide with division problems and give a new test again in two days. B Use manipulatives to model division and connect it to multiplication. C Separate the class into groups and have at least one student that understands division in each group. The higher-level students can reteach the struggling students the concept through peer tutoring. D Teach that multiplication and division are opposites, and have students memorize times tables to make division easier.
Use manipulatives to model division and connect it to multiplication.
Which of the following is the best way for elementary students to be introduced to rectangular arrays? A Using manipulatives such as 10-blocks to create their own arrays B During math games, use an array as part of a question C Giving them sample problems with arrays D Watch an online video over arrays
Using manipulatives such as 10-blocks to create their own arrays
Students are asked to identify the answer to a problem without actually solving it. She asks them: What answer is best when given the expression: 1/10 × 7/60 ? 7/600 7/10 1 7/60 70/60 What concept does she want them to recognize? A When 2 fractions are multiplied, the product is smaller than either original fraction B Cross-simplifying fractions before multiplying C Base-10 multiplication D Multiplication tables
When 2 fractions are multiplied, the product is smaller than either original fraction
Students in a math class are working on the following problem: Julio can answer 3 math problems in 10 minutes. He completed his math homework after school, which consisted of 14 questions. Assuming he worked at the same rate the entire time, how long did it take Julio to complete his math homework? One student called the teacher over to see if she set up the problem correctly. The following was on her paper: 3/ 10 = x/14 Which of the following suggestions should the teacher give the student to guide them to realize their mistake? A Graph the equation. B Draw a picture representing the problem. C Write the units of each value in each ratio. D Re-write the ratios in decimal form.
Write the units of each value in each ratio.
Mr. Benjamin has been teaching his students to solve geometry word problems. If there is no figure drawn he encourages them to draw their own. Is this a good teaching practice? A Yes, because it helps students gain understanding of the problem. B No, because not all students are visual learners. C Yes, because some problems will require students to draw pictures so it is good for them to practice. D No, because this takes wastes students time. If the figures are not required they should not draw them.
Yes, because it helps students gain understanding of the problem.
A third-grade teacher is working with a small group of students on representing fractions. She asks students to use the method of their choice to represent ¾. All of the students choose to draw a circle divided into 4 equal parts and color 3 parts. She then asks students to represent the same fraction in another way. Which of the following is NOT a method that the students could use to represent ¾? A an array B a number line C a drawing of four items with three items shaded in D a fraction strip
an array Arrays are used to represent multiplication or division problems, not as a way to represent fractions.
A third-grade teacher is planning a lesson on representing multiplication facts. She wants students to be able to model multiplication problems using a variety of different representations. Which of the following includes ways that the students could model a basic multiplication fact? A part-part-whole models, geoboards, fraction strips B area models, number lines, fraction strips C arrays, equal-sized groups, number lines D area models, equations, line graphs
arrays, equal-sized groups, number lines
What learning progression should be used when teaching math concepts to sixth-grade students? A concrete to symbolic to real world B concrete to symbolic to abstract C symbolic to abstract to real world D symbolic to concrete to abstract
concrete to symbolic to abstract
A pre-K teacher has her students pretend to be a rocket ship as they squat down and then "blast off" as they say, "10, 9, 8, 7, 6, 5, 4, 3, 2, 1." What skill is the teacher promoting by doing this? A counting backwards B rote counting C one-to-one correspondence D using a number line
counting backwards By counting down from 10, students are practicing the skill of counting backwards.
A kindergarten teacher ends each day by having students march, jump, or run in place as they count out loud from zero to 30. What skill is she trying to promote by doing this? A counting on B counting by ones C number recognition D subitizing
counting by ones
A first-grade teacher gives each table 68 unifix cubes to count. She observes one table organizing the unifix cubes into groups of 10 and counting the sets of 10 that were made. What counting technique was this group utilizing?
counting collections The students have organized the groups into sets of 10 and counted the sets, which is an example of counting collections.
A second-grade teacher encourages her students to add two-digit to one-digit numbers by "grabbing the larger number in their mind" and using their fingers to count up to find the total. What counting technique are students using when they do this? A skip counting B addition algorithms C counting on D mental math
counting on By starting with the larger number in their head and counting up with their fingers, students are using a strategy of counting on.
Which of the manipulative materials below would be most suitable for teaching decimal notation to the hundredths place? Select all answers that apply. A tangrams B decimal squares C geoboards D base ten blocks E pattern blocks
decimal squares Decimal squares are tag-board pictures of 10x10 grids that have portions of the 100 smaller squares shaded. Students are asked to name the decimal represented by the shaded or unshaded area. They see that the sum of the shaded and unshaded areas always equals 100 hundredths or 1. Base ten blocks are hands-on manipulatives consisting of a large cube (made up of 1000 smaller cubes), a flat (10 x 10 grid or a 100 square), a long (1 x 10), and a unit cube (1 x 1). Base ten blocks allow the representation of decimals from 0.001 (the smallest cube) to 1 whole (the largest cube). AND base ten blocks Base ten blocks are hands-on manipulatives consisting of a large cube (made up of 1000 smaller cubes), a flat (10 x 10 grid or a 100 square), a long (1 x 10), and a unit cube (1 x 1). Base ten blocks allow the representation of decimals from 0.001 (the smallest cube) to 1 whole (the largest cube).
A third-grade teacher is introducing the idea of adding areas of smaller rectangles to make one larger rectangle. Which would be the most effective beginning activity? A demonstrating how to put two rectangles next to each other B leading a discussion about squares as special rectangles C identifying which shapes are rectangles D having the students explore rectangles that all have the same width
having the students explore rectangles that all have the same width
A second-grade teacher is introducing the idea of measuring using inches and centimeters. Which would be the most effective beginning activity? A giving the students rulers to measure one irregular object in both units B demonstrating conversion using linking cubes. C having the students find what objects are roughly an inch long in the classroom D asking each student to measure their hand in inches for homework
having the students find what objects are roughly an inch long in the classroom
A second-grade teacher is introducing the idea of adding different kinds of coins. Which would be the most effective beginning activity? A providing the problem using numbers and words B demonstrating how to add a nickel and a dime C identifying a coin as a penny or not D having the students use coins to represent a problem
having the students use coins to represent a problem
A math teacher plans her instructional delivery method on resolving the difficulty students have distinguishing between mode and median. She plans to have students first work alone calculating the mode and median of sets of performance results from the school track team. Next, her students will work in groups of 2 or 3 to discuss and interpret their results, and record a summary of the significance of the results on whiteboards. Finally, the groups will present their summaries to the class, along with a teacher-led discussion of the findings. By planning such an activity, the teacher demonstrates that she understands: A how technological tools and manipulatives can be used to assist students in developing mathematical thinking. B how to use a variety of questioning strategies that encourage mathematical discourse and help students analyze and evaluate their mathematical thinking. C how students' prior mathematical knowledge can be used to build conceptual links to new knowledge. D how to apply a variety of instructional delivery methods that can help students develop their mathematical thinking.
how to apply a variety of instructional delivery methods that can help students develop their mathematical thinking.
The mathematics teacher and art teacher work together to create an interdisciplinary lesson using tessellations, which are basic geometric shapes set to a repeating pattern. The students cover a large piece of poster board with the patterns they create. Which of the following mathematical concepts is most closely reflected in this activity? A infinity B number sense C perimeter D conservation
infinity
Which of the following activities is most effective in helping kindergarten students understand measurement of the lengths of small items, such as juice boxes or lunch boxes? A watching the teacher estimate the length of the item using a student's arm or leg B listening to a teacher explain how to use a ruler to measure the objects C tracing the items on construction paper and cutting the construction paper to have a two-dimensional replica of the item D placing same-size objects, such as Legos or cubes, next to the object and counting the number of objects
placing same-size objects, such as Legos or cubes, next to the object and counting the number of objects
Mrs. Johnson is teaching spatial reasoning to her eighth-grade resource math class. She wants to incorporate technology, so she finds a program that allows users to design a room. Her students create their dream bedrooms using 3D software, and then they print the bedroom designs and the furniture on separate papers. The students then move the furniture around the room until they like the designs. What shapes can the students use to recreate the furniture below? A set of rectangles B set of hexagons C set of trapezoids D set of squares
set of squares
A first-grade student is finding the value of a set of six nickels. What counting skill will this student most likely need in order to complete this task? A using a number line B skip counting C multiplication D counting by ones
skip counting In order to efficiently count the value of six nickels, the student will need to already be familiar with skip counting by fives.
Mr. Harris is planning to teach a unit on division to students for the first time. Which of the following would be the best first instructional lesson for Mr. Harris to present to his students? A solving a division problem using a concrete manipulative B presenting and solving a division problem using an abstract form or concept C having students memorize their multiplication tables D asking students their understanding of division prior to presenting the concept
solving a division problem using a concrete manipulative
A first-grade student is working on the problem 11 + 9. His teacher observes him as he draws 11 circles, then 9 more circles, and then counts the total. As he is counting, he counts one of the circles twice and gets an answer of 21. His teacher encourages him to try another strategy to check his work. Which of the following is a reasonable method that the student should choose? A using subtraction to work backwards to solve the problem B solving the problem using a number line C drawing another picture, but using squares this time D counting on his fingers
solving the problem using a number line
Joshua is learning about volumes of three-dimensional figures. First, his teacher explains what volume is. Then, she writes the formula for area of a cube on the board v = s3. Next, she has the students recite "the volume of a cube is the side length cubed". Finally, she has students take six-sided dice of various sizes and measure them to determine their volume. Which best describes the teaching method is the teacher attempting to use? A visual learning B task variety C small group instruction D auditory learning
task variety
A third-grade mathematics teacher noticed the students' performance on the recent assessment of the multiplication introductory unit was much lower than expected. Which of the following strategies should the teacher utilize to help the students understand the concept of multiplication? A use manipulatives to demonstrate that the product of 3 and 3 is the same as the sum of adding 3 together three times B have students complete the multiplication table in groups C have students complete a multiplication table at home with their parents D require students to memorize the multiplication table for numbers 1 through 9
use manipulatives to demonstrate that the product of 3 and 3 is the same as the sum of adding 3 together three times This is a great activity that uses concrete manipulatives to demonstrate the concept of multiplication. Students are able to scaffold their knowledge of addition into the concept of multiplication.
A fifth-grade teacher is beginning a unit on equivalent fractions with her students. If this is an introductory lesson, which of the following activities would be the most effective in helping the students understand the concept of equivalent fractions? A begin with the concept that 50¢ is ½ of $1; 25¢ is ½ of 50¢; 5¢ is ½ of 10¢ B compare pictures showing ½ of a variety of different objects C use pattern blocks to model different fractions equivalent to ½ D find as many fractions as possible equivalent to ½ in one minute
use pattern blocks to model different fractions equivalent to ½
Which of the following is the best activity for reviewing percentages with fifth-grade students? A writing percentages from decimal or fraction conversions B comparing percentages from their test scores throughout the year C coloring in 100-blocks to represent percent D using a variety of methods and scenarios to determine percentage
using a variety of methods and scenarios to determine percentage
Which of these is not a decomposed version of ⅚? A ⅙ + ⅙ + ⅙ + ⅙ + ⅙ B ⅔ + ⅙ + ⅙ C ⅓ + ⅙ + ⅙ + ⅙ D ⅓ + ⅓ + ⅙
⅔ + ⅙ + ⅙ ⅔ is equal to 4/6 and when added to ⅙ and ⅙ it equals 6/6 or 1.
