Mathematics Instruction and Processes Quiz Questions Flashcards
1
Q
A parent is complaining about the math homework. They feel that their ELL child is at a disadvantage because they cannot afford internet at home and the homework is best completed using online software. The teacher is providing students time in class to complete the work that requires online resources, however, this student has not been using it stating that he will do it at home. What is the best strategy the teacher should use to prepare for meeting with the parent?
A
Open communication with the parents that does not involve educational jargon.
Open communication that encourages parental involvement is best.
2
Q
A fifth-grade teacher is preparing to launch a unit focused on multiplying and dividing fractions. Which of the following concepts should he include on the pre-unit diagnostic test?
A
finding simplest form
Before learning to multiply or divide fractions, students must have a foundational understanding of how to manipulate fractions to find equivalent fractions and simplest form. If a student struggles with this skill, they will face additional challenges when attempting to multiply and divide fractions.
3
Q
A first-grade teacher is finishing a unit on place value and composing/decomposing numbers using hundreds, tens, and ones. Which of the following would help to ensure that students continue practicing this skill even after the unit is finished?
A
A
a “number of the day” that students model using hundreds, tens, and ones
B
an online game in which students identify the hundreds, tens, and ones place
C
counting each day of school by adding a popsicle stick to a jar and making groups of tens when applicable
D
all of the above
correct
All choices are appropriate ways to reinforce the concept of place value.
4
Q
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
Establish a daily procedure for class and vary the activities used for instruction.
Students will be more on task when provided a routine to follow .
5
Q
Ms. Nakaroti wants to teach her students about properties of points, lines, planes, and angles. Which of the following should she include in her planning for the unit?
A
Analyze the standards to determine learning objectives before she starts writing lesson plans.
The standards should always be consulted before beginning lesson planning.
6
Q
At the end of a lesson on factoring, Ms. Wilson gave her class an exit ticket. After she reviewed the responses on the exit ticket, Ms. Wilson realized that many of her students were still struggling with the concept of factoring. Which of the following strategies would be best for Ms. Wilson to use in her next lesson on factoring to help the students solidify their conceptual understanding of factoring?
A
using manipulatives to show factoring as the reverse, or un-doing, of distribution
This activity uses concrete manipulatives to demonstrate the concept of factoring. Students can use prior knowledge of distribution to make connections to factoring.
7
Q
Mr. Meadows is a third-grade teacher in a low performing school. There is a high rate of absenteeism and low rate of students doing homework. He makes a public star chart where students get a sticker for each assignment they complete. Which of the following learning theories best matches the use of a star chart?
A
behaviorism learning theory
Behaviorism has to do with students learning new behaviors based on the response they get to current behaviors. The students are receiving positive reinforcement through the use of the star chart. This is the correct answer.
8
Q
According to the TEKS, which of the following is an appropriate skill for a second-grade student to master during a unit on numbers and operations?
A
Students will be able to place a given whole number in the correct position on an open number line.
9
Q
Ms. Stevens’ third-grade class is completing an assignment in which they circle the larger of two fractions. While observing students as they work, she notices that one student, Ava, is consistently circling the smaller fraction on each pair. When Ms. Stevens asks Ava to explain her thought process on one of the problems, Ava states that “⅛ is greater than ¼, because 8 is bigger than 4.”
Based on this comment, which of the following would be the best way for Ms. Stevens to support Ava?
A
providing Ava with fraction bar manipulatives to model two fractions before identifying the larger fraction
10
Q
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.
11
Q
Which of the following would be the most beneficial activity to include in a seventh-grade lesson introducing circumference of a circle?
A
having students measure the distance around various circular objects using a string or measuring tape
12
Q
A third-grade teacher is introducing the concept of multiplication using manipulatives, pictures, and arrays. Last year several parents asked why their child wasn’t learning to multiply the same way that they did. What should the teacher do in order to address potential questions this year?
A
Send an email to parents that explains how multiplication will be introduced and include tips for practicing these skills with their child.
This keeps parents informed by explaining ahead of time how multiplication will be taught, and it allows parents the option to become involved in their child’s learning by giving tips on practicing multiplication strategies with their child.
13
Q
Which of the following would be the most beneficial teaching tool to use during a fifth-grade lesson on comparing numbers with decimals?
A
a number line with marks placed at 0.25 intervals
A number line divided into 0.25 intervals would help students compare numbers with decimals. Students would be able to plot the numbers with decimals at approximate points on the number line, then use this visual representation to determine which number is larger.
14
Q
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.
This is an excellent activity to begin a unit on data collection. After sorting, students can begin to answer questions like, “What color of tile do I have the most of?”, and “And the least of?” They can even begin comparing what they have with what another student has.
15
Q
Which of the following would be the most beneficial activity for kindergarten students who are practicing subitizing skills?
A
a math station where students repeatedly roll a die, recording the number that they roll each time
Subitizing is the ability to quickly recognize the amount of objects in a set without counting each object. By rolling a die and recording the value, students are reinforcing this skill as they gradually learn to recognize the value they rolled without having to count each dot.
16
Q
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
Students need to practice test taking skills periodically.
Students need to practice these skills for standardized testing.
17
Q
What learning progression should be used when teaching math concepts to third-grade students?
A
concrete to symbolic to abstract
Students need to be introduced to topics through manipulatives and concrete examples. Then, students can move on to symbolic representations such as drawings to represent equations. Finally, they can move to abstract that involves only numbers and variables in equations.
18
Q
A third-grade teacher is planning an introductory lesson on perimeter. Which of the following would be the most appropriate to include in the lesson?
A
an activity in which students are given the dimensions of the school garden and asked to work in groups to determine how much fencing they would need to enclose the garden
19
Q
Ms. Davis teaches a fifth-grade math class primarily composed of English language learners (ELL). Which of the following can support her ELL students?
Select all answers that apply.
A
Make a word wall.
Use gestures, pictures, and models to explain terms.
20
Q
Maria has recently moved from Mexico City to the U.S. She is a secondary student who speaks little English, but who came from her school in Mexico City with excellent grades. Which of the following would be the most appropriate accommodation for Maria’s math teacher to use with Maria?
A
Pair Maria with another student who speaks Spanish, to clarify instructions in Spanish as needed.
21
Q
Below is the outline of a mathematics activity a third-grade teacher recently completed with her students.
Learning Goal: The learner will be able to identify and create models of the following fractions: ⅓, ¼, ½
Lesson Plan:
Ask the students if they have ever eaten a pizza and show them a fake pizza in slices in a pizza box. Have students discuss the different ways a pizza could be divided.
Students sit with a partner and complete a worksheet titled, “Equal Parts or Unequal Parts?”
Using manipulatives, the teacher uses the document camera to show students how to take a whole pie and turn it into the following fractions: ⅓, ¼, ½.
Students sit with a partner and complete a worksheet titled, “Draw the Pizza Pie.”
Have each group come up to the board one at a time and draw one of the fraction pies with a dry erase marker.
Teacher Notes:
Around half of the groups draw the correct fractions on the board. The rest draw pies that have equal parts, but the incorrect fraction. When I graded both worksheets, 60% of partner pairs demonstrated mastery of the material.
Based on the teacher’s notes following the lesson, which of the following steps should the teacher take to improve the lesson plan and increase student understanding of fractions?
A
include an opportunity for students to work with the manipulatives by themselves and with a partner
22
Q
Below is the outline of a mathematics activity a third-grade teacher recently completed with her students.
Learning Goal: The learner will be able to identify and create models of the following fractions: ⅓, ¼, ½
Lesson Plan:
Ask the students if they have ever eaten a pizza and show them a fake pizza in slices in a pizza box. Have students discuss the different ways a pizza could be divided.
Students sit with a partner and complete a worksheet titled, “Equal Parts or Unequal Parts?”
Using manipulatives, the teacher uses the document camera to show students how to take a whole pie and turn it into the following fractions: ⅓, ¼, ½.
Students sit with a partner and complete a worksheet titled, “Draw the Pizza Pie.”
Have each group come up to the board one at a time and draw one of the fraction pies with a dry erase marker.
Teacher Notes:
Around half of the groups draw the correct fractions on the board. The rest draw pies that have equal parts, but the incorrect fraction. When I graded both worksheets, 60% of partner pairs demonstrated mastery of the material.
The lesson plan shows the teacher understands the importance of:
A
engaging students in the lesson’s content.
23
Q
A second-grade teacher is finishing up a lesson on using graphs to analyze data and make predictions. She wants to make sure that students continue to use this knowledge instead of learning it as an isolated skill. Which of the following would be the best way for the teacher to achieve this?
A
Plan a science lesson in the upcoming weeks in which students graph the daily temperature using bar graphs.
24
Q
A first-grade teacher is planning a group project in which students will work in groups of three to create a survey question to ask their peers, collect data from the class, and create a picture graph using the data collected. Which of the following should the teacher do to help ensure successful group collaboration?
A
Assign roles to each group member, such as recorder, speaker, and materials manager.
25
A pre-K teacher is having students work in groups to count a set of small toys. She notices that one group of students has started playing with the toys instead of counting them. What would be an appropriate first step to take in this situation?
Remind students in the group of the expectations of the activity.
26
Mrs. Hoyt is excited to be teaching fifth-grade math this year. She would like to encourage her students' independent skills. How can she facilitate this in her classroom?
allow for a variety of choices when it comes to assignments
27
A sixth-grade teacher is beginning a unit on probability. She utilizes the following steps in planning her unit:
1. Determine the necessary prerequisite skills.
2. Begin planning probability activities that involve the collection of data.
3. Determine what the students already know by using a KWL chart.
4. Plan the final assessment for the unit.
What is the best order for the teacher to organize these steps?
IV, I, III, II
28
Which of the following corresponds to the progression of stages of learning about a new mathematical concept from most basic to most advanced?
concrete, representational, abstract
29
A third-grade teacher is teaching a whole group lesson on counting money. After the lesson is complete, she asks students to form groups of four and find the total of different sets of bills and coins that she will display on the board. During the group activity, there is a lot of commotion and off-task behavior, and several students begin working independently instead of with their group. What could the teacher have done differently to improve the group activity?
assigned groups ahead of time and established clearer expectations for group members
30
A first-grade class has been working on place value for several days. The teacher notices that some students are still struggling with the basic concept, some students are improving but still need additional practice, and some students have caught on quickly and are becoming bored. She plans to work with students in small groups while the rest of the class works in stations or independent work. What would be the most appropriate way to group students in this scenario?
homogeneously
Homogeneous grouping is appropriate for this scenario because it allows the teacher to target specific skills with each group or provide enrichment opportunities for students who have mastered the skill.
31
A kindergarten teacher is working with a small group of students on adding single digit numbers. She shows students the problem 3 + 5, then counts out and attaches a set of three unifix cubes and a set of five unifix cubes. She then joins the sets of unifix cubes together, counts the total of eight unifix cubes, and writes 3 + 5 = 8. Next, she gives each student in the group a set of unifix cubes and writes the problem 4 + 2. The teacher models how to count and attach a set of four cubes and a set of two cubes, join the two sets together, and count to find the total of six. While she models this, students do the same with their unifix cubes. Finally, the teacher writes 3 + 4 and has students count and attach their cubes independently while she observes them.
Based on this description, what strategy is the teacher using?
gradual release
This teacher is following the "I do, we do, you do" approach from the gradual release model.
32
Students in Ms. Cook's fourth-grade class are learning about different types of angles. Students are given a worksheet with several different angles and asked to identify each angle as acute, obtuse, or right. Ms. Cook notices that one student, Amari, is correctly labeling all right angles, but is labeling all acute angles as obtuse and vice versa.
Based on this observation, Ms. Cook can best help Amari by:
reviewing the definitions of acute and obtuse angles and providing a mnemonic device to remember each definition.
33
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?
use pattern blocks to model different fractions equivalent to ½
Since this is an introductory activity, concrete, proportional manipulative materials like this should be used for concept development. It is important not to rush past this step and to use a variety of different materials to develop and reinforce understanding of this concept.
34
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:
how to apply a variety of instructional delivery methods that can help students develop their mathematical thinking.
35
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?
having the students explore rectangles that all have the same width
36
Mr. Fischer, a bilingual teacher, teaches a mathematics class composed of native English speakers and English language learners (ELLs). He has introduced a new topic with new vocabulary words in which he presented the vocabulary words with several examples. Which of the following strategies should Mr. Fischer use next to check each student's understanding of the vocabulary words?
having students write a definition for each term in their own words in their native language
37
Ms. Colon, a new fifth-grade teacher, is planning her math lessons for the grading cycle. She thinks of all of the topics she needs to teach and makes discrete daily lessons. Each unit has an opening pre-test. Each lesson has instruction, guided practice, and independent practice. Which of the following are methods she should incorporate into her lesson planning?
Select all answers that apply.
Instead of making single lesson plans, first create a thematic unit around which to frame her lessons.
Plan time each day for students to explain concepts they have learned to their peers.
Plan each lesson with a closure activity.
38
Ms. Trask wants to create an authentic assessment to test her students about angles in triangles. Which of the following should she do first?
Look at the standards to determine what a meaningful task that students could complete to demonstrate their knowledge.
39
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?
task variety
The teacher is using task variety because she is presenting the same material in multiple ways.
40
Mrs. Jones is teaching a lesson on slope-intercept form. She requires each student to find the slope and y-intercept of a set of graphs, then put them into a formula that describes the graph. The students work one problem at a time and Mrs. Jones circulates to check their work. If a student has the correct answer, Mrs. Jones gives them a checkmark and they move on to the next question. If the student has the wrong answer, Mrs. Jones directs them to the incorrect portion of their work and they revise their answer. Mrs. Jones continues to circulate the room until all students have finished the assignment. Which of the following learning theories best matches the activity Mrs. Jones uses with her students?
Behaviorism learning theory
The students are learning by receiving feedback with positive reinforcement.
41
A second-grade teacher is introducing the idea of measuring using inches and centimeters. Which would be the most effective beginning activity?
having the students find what objects are roughly an inch long in the classroom
42
Ms. Miller is a student teacher in a fourth-grade classroom. She has heard that group work is important so she wants to plan for group activities. On her first day student teaching, she briefly says "today we'll be doing group work about fractions" before she sends the students to stations. How could she best improve her teaching?
Give a whole group lesson on fractions before breaking into groups.
43
Which of the following does NOT foster a learning environment that provides all students with opportunities to develop and improve their mathematical skills?
implementing unpredictable routines to keep students' attention
Students often respond best when routines are predictable. Some students may become confused by routines that vary from day to day.
44
A second-grade teacher is planning a group activity in which students will sort 3D shape models based on their defining attributes. How should the teacher plan on grouping students for this activity, and why?
heterogeneously, so that struggling students can learn from their peers and other students can benefit from explaining their reasoning
45
Mr. Davis is teaching his first-grade students how to solve problems with unknown addends, such as 5 + __ = 9. Through observations and student assignments, he finds that four of his students are consistently adding the two numbers in the missing addend equations, resulting in incorrect answers.
Which of the following is the best way for Mr. Davis to provide support for these four students?
using small group instruction, teach students how to use counters and part-part-whole diagrams to solve problems with unknown addends
46
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?
infinity
The tessellations will continue in infinity. The teacher is introducing a mathematical concept that does not end, but repeats continually. This is the concept of infinity.
47
Base ten blocks are commonly used by teachers to illustrate the concept of:
place value.
Base ten blocks are a common manipulative used for this concept.
48
Mariela is a third-grade student in Mr. Miller's math class. Mr. Miller has noticed that Mariela is struggling with comparing fractions with the same denominator. What should Mr. Miller use when working with Mariela on this concept?
fraction tiles
49
A kindergarten teacher is planning a lesson on two-dimensional shapes. Which of the following manipulatives would be the most effective to use when teaching students about the different qualities of specific shapes?
attribute blocks
Attribute blocks are an effective manipulative to use when teaching students about two-dimensional shapes because they are models of different 2-D shapes such as squares, triangles, and rectangles.
50
A first-grade class is finishing a unit on counting sets of coins. Which of the following would be an effective use of technology at the end of the unit?
an online assessment in which students select the set of coins needed to purchase different items
51
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?
solving the problem using a number line
A number line is a good way for first-grade students to represent addition and subtraction problems. Using the number line would be an effective way for the student to check his work.
52
Which of the following manipulatives are most appropriate for kindergarten students learning about two-dimensional shapes?
attribute blocks
53
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?
arrays, equal-sized groups, number lines
54
A math teacher is incorporating the use of technology to enhance her unit on charts and graphs. The class worked together to write questions to guide their data collections. The students wrote questions like, "What is your favorite color?" and "What is your favorite dessert?" They have collected responses from their classmates and are ready to organize their data to make graphs. What technological tool could the students use to organize their data and create their graphs?
a spreadsheet
55
Which of the following is not an appropriate instructional strategy to promote student's use of mathematical language?
use everyday language
56
A third-grade teacher is planning a lesson on representing data using dot plots. She plans to introduce the concept of dot plots, show examples, and create a class dot plot that shows how many siblings the students have. Which of the following would be the best way to incorporate technology into this lesson?
an online program that allows students to plot their data point on a dot plot
57
A prekindergarten teacher is planning small group lessons on one-to-one correspondence. Which of the following manipulatives would be the most effective to use when introducing this concept?
counters
Counters are a helpful manipulative to use when teaching one-to-one correspondence, as students can touch each counter as they count out loud.
58
Mr. Stiles is introducing measurement to his third-grade class. He has rulers, stopwatches, scales, and graduated cylinders available for them to use. Based on previous lessons, he knows that most students do not know how to use these tools correctly. What is the best introductory lesson for this unit?
providing students time to explore the items and then creating a K-W-L chart
59
At the onset of a new unit, a teacher wants to ensure all students can recall previously-used mathematics terminology and apply it to the new unit. Which of the following strategies would best help students extend previously learned concepts into future discussions?
Constructing a word wall that students can refer to throughout the unit
60
Anytown School District provides a 50 multiple-choice question mathematics assessment to all students. The students complete the assessment, the tests are scored, and the scores are compared throughout the school district. Which of the following mathematics component is most likely the goal of this type of assessment?
accuracy
61
Which of the following activities would be most effective in helping first-graders understand partitioning 2-dimensional shapes into equal parts?
cutting out different shapes and having students fold them into 2 or 4 equal parts
62
Ms. Daniels is a second-grade teacher who notices that several of her students are struggling to determine the value of an unknown addend in an equation. She plans to reteach this concept in small groups to the students who are struggling. Which of the following manipulatives would enhance Ms. Daniels' small group lessons on this topic?
Cuisenaire rods
63
A teacher draws a 10x8 grid on the board without any "X"s in the grid. The teacher then writes "X"s in one and a half rows of the grid. The teacher asks the students to create an equation to represent blocks left on the grid, if the "X"s represent the blocks that have been removed.
Which of the following is the best equation in response to the teacher's question?
80 – 15
There are 80 blocks on the grid. If 15 are removed, then the equation 80 – 15 represents this.
64
A first-grade teacher is planning a lesson on solving problems with an unknown addend, such as 3 + __ = 8. She knows that students have struggled with this concept in previous years and is looking for a way to engage students with technology while still improving their understanding of the concept. Which of the following could she do in order to achieve this?
Use an interactive part-part-whole model that allows students to drag items from the "whole" to the "parts" to find the unknown addend.
65
Two color counters are often used to model addition and subtraction of integers. The red counters represent negative integers; the yellow represent positive integers.
If the counters above were used to model the addition, what would be the result?
-2
The problem pictured above is -7 + 5 = -2. Using the chips, a red and a yellow chip are paired together to form a "zero-pair"–a model of -1 + 1 = 0. Pairs are matched and removed from the group leaving two red chips unmatched. This results in a representation of an answer of -2.
66
A local school district requires all fourth graders to complete a 20-minute test containing 200 multiple-choice mathematics problems. Students are not expected to complete all of the questions, but are graded on the number they are able to get correct in the time limit. What component of mathematics fluency is most likely the goal of this type of assessment?
rate
The school district is assessing rate because they are seeing how many questions students can answer in a given time frame.
67
Jayce is a first-grade student struggling with comparing two-digit numbers. Which two of the following manipulatives could his teacher use to provide support for Jayce?
Select all answers that apply.
Unifix cubes
base ten blocks
68
A second-grade teacher is planning an introductory lesson on ordering numbers on open number lines and wants to incorporate technology into the lesson. Which of the following would be the most effective use of technology in this scenario?
an interactive number line on which students can drag and drop numbers to the correct place
69
A teacher engages her class in a discussion of the coordinate plane. The students are asked to identify the quadrants, the coordinate axes, and the mathematical notation for various points in the plane. Students are asked to develop a way to quickly identify the quadrant in which various points lie. Which of the following objectives is the teacher most likely trying to address with this lesson?
developing precise mathematical language when expressing mathematical ideas
70
In a fourth-grade class, students are making cookies to learn about fractions. They are using a set of measuring cups and spoons for their measurements. This learning is best described as:
formal standard measurement.
Standard measurements include formal tools, such as a measuring cup, which have a known and standardized quantity per measurement. In other words, every scoop using a 1 cup measuring cup will equal 1 cup.
71
A kindergarten teacher is working on adding one-digit numbers with a small group of students. Which of the following would NOT be a manipulative that the teacher might plan to use?
base ten blocks
72
Mrs. Adamson's student asks her how much space a cube takes up. Mrs. Adamson said to answer this question, the student would need to calculate the volume of the cube. Which of the following measurable attributes is the formula for a cube based upon?
length
73
A recent formative assessment shows that 80% of Ms. Parker's first-grade class has not mastered the skill of finding the value of unknown addends. In her lesson on this concept, Ms. Parker taught students to "count up" from one number to the other to find the unknown addend. Based on results of the formative assessment, Ms. Parker should:
reteach the concept to the whole class using manipulatives.
74
What activity would be most effective at developing student knowledge of fact families?
providing students with three numbers and prompting them to create numerical sentences that relate the numbers
75
A kindergarten teacher is planning a lesson in which students will measure the length of various items in the classroom using nonstandard units of measurement. Which of the following would be best suited for measuring crayons, books, and computers during this lesson?
paper clips
76
Marcus is a first-grade student working on place value task cards at a math center. Each task card shows an image of base ten manipulatives, and students are to write the corresponding number in standard form on their recording sheet. Marcus selects a card that shows 1 hundreds block and 3 ones cubes. On his recording sheet, he writes that the number shown on the card is 13.
Based on this scenario, which of the following tools would be the most effective to provide to Marcus?
a dry-erase place value chart
77
A third-grade teacher notices that some of her students are struggling with putting fractions in order from least to greatest. What would be the most appropriate manipulative to help students with this concept?
fraction strips
78
Which of the manipulative materials below would be most suitable for teaching decimal notation to the hundredths place?
Select all answers that apply.
decimal squares
base ten blocks
79
Mr. Zammit is teaching his class about shapes. One of his students incorrectly labels all rectangles as squares and all rectangular prisms as cubes. Which of the following should Mr. Zammit do in this situation?
Select all answers that apply.
Make an analogy to help the student understand his mistake, for example calling every rectangle a square is like calling every fruit an apple.
Require his student to use correct mathematical vocabulary.
80
A teacher is planning a unit on comparing and ordering numbers. Which of the following activities could the teacher use to support students' learning of this concept?
Students play a card game in which students arrange themselves in order according to the number on the card they are holding.
81
Mrs. Cooper wants to reinforce a concept she is teaching her sixth-graders by having students use their calculators to solidify their conceptual understanding. In which of the following activities would the use of a calculator be most beneficial to conceptual understanding?
having students graph lines with different y-intercepts then determine how b changes the graph of y=mx+b
82
Ms. Hobbs is planning to introduce long division to her students for the first time. Which of the following would be the best first instructional step for this topic?
quiz students on their understanding of basic division before proceeding
83
Mr. James plans to assess his students' knowledge during a unit on linear functions and would like feedback from the students on how well they feel they are learning the concepts. Which of the following assessment would be the most appropriate for Mr. James to use during the unit?
Daily open-ended formative assessment in which the students complete a problem, with justification and any questions they still have about the material from that day.
84
A mathematics teacher gives her class a two-question clicker quiz at the end of each class period and tabulates their answers according to their mathematical understanding, misconceptions, and error patterns. If her goal is improvement in her students' mathematical proficiency, her best use of the data would be to use it to:
inform upcoming instructional strategies.
85
Ms. Rose's class is learning about order of operations. While practicing this new concept, a student simplifies the following expression.
8−(4+3)^2×6
8−16+9×6
8−16+54
−8+54
46
Which of the following best describes the student's error?
The student evaluated the power of exponents before adding within the parentheses.
86
Mrs. Smith is teaching 2 digit by 2 digit multiplication to her class. By assessing her class with exit slips, she notices that 10 of her 22 students have achieved the same solution, seen here. What should Mrs. Smith teach tomorrow?
She should reinforce placeholder zeros.
87
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?
the student erred when multiplying 10 and 15
88
Mr. Barrios is teaching a unit on multiplication to his fifth-grade class. On the very first day he gives an exit slip with the following problem on it:
123.456×789=_______
Every single student gets the question correct. How should he adjust his teaching?
Teach more advanced multiplication content to challenge his students.
89
A second-grade teacher gives students a short assessment on subtracting two- and three-digit numbers and collects the following data:
Based on this data, the teacher should plan to reteach concepts related to:
regrouping or "borrowing"
90
Which of the following best describes a high-stakes assessment, such as state mandated exams?
formal summative assessment
91
A fifth-grade student was asked to multiply 15 and 35. His work is provided below.
As his teacher, what remediation would you plan on providing?
a remedial lesson on place value
92
Formative assessments provide information that can be used for which of the following?
to change and improve instruction
93
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?
The student did not regroup or "carry" the tens value when adding 8 + 4.
94
Which of the following is the greatest benefit of using a variety of assessment methods?
It allows the students several different ways to demonstrate what they have learned and can do.
95
Which of the following statements best describes a formative assessment?
Formative assessments measure what students know along the way.
96
Which of the following would be an appropriate time for a classroom teacher to use a formative assessment?
as a closing activity at the end of a class period
when students are involved in a cooperative group project
as the teacher is introducing a new concept
all of the above are good times for a formative assessment
97
Mrs. Rogers is teaching her students about volumes of regular rectangular prisms. She has students work with pieces of wood and measure their length, width, and height to determine volume. She asks students to analyze nets of prisms to determine their volume. While students have been successful using pieces of wood, they seem to be having issues using nets. How should Mrs. Rogers intervene?
She should demonstrate how to cut out and assemble the nets and then ask students to determine area of the nets in 3D form.
98
Ms. Ma is teaching her students about subtraction. On her exit ticket she asks "What is the difference between 21 and 14?" Several students answer 35. Another question she asks is "25-6 =___" to which every student answers 19. What should Ms. Ma teach tomorrow?
a lesson on math vocabulary focusing on keywords in word problems
99
Rather than give a unit test, Mrs. Kirby decides to assign a major project to her students. They are provided a rubric that sets the expectations and guidelines. Students will be given 2 class periods to work on it and the rest must be completed at home. Students will then present their projects in class. What is the main advantage to giving a project rather than a test?
Projects require higher level thinking and can demonstrate greater concept mastery than tests.
100
A teacher is monitoring her class while the students are involved in a group activity exploring the size of angles in a set of triangles. She moves from group to group, pausing and watching the group dynamics. What the teacher is doing can best be described as:
informal formative assessment
101
A first-grade teacher administers the following pre-assessment on place value to the 20 students in the class.
Based on these results, which of the following strategies should the teacher implement in future lessons?
modeling with base ten blocks
102
What is the primary goal of summative assessments?
measuring student achievement
103
A class of sixth-grade students is given the following problem:
1/2 + 3/4 =
Many of the students arrive at the answer:
1/2 + 3/4 = 4/6 = 2/3
What should the teacher NOT consider with respect to remediation?
Students need more work reducing fractions.
104
Nora is a student in Ms. Perez's sixth-grade math class. She is working on an assignment in which students are asked to put sets of integers and rational numbers in order from least to greatest. The following are two sets of numbers from the assignment, along with Nora's answers:
Which of the following skills should Ms. Perez plan to review with Nora?
locating negative numbers on numbers lines
105
A teacher is preparing a lesson on addition. Her sample problems are formatted as follows:
12 + 3 = ________
5 + 6 = ________
102 + 8 = ________
What is the best way she improve the structure of her problems to help all types of learners in her classroom?
Give problems in multiple formats, including writing in vertical format and giving word problems.
106
Students are asked to solve the word problem below.
The school carnival is coming up and Jenny and Sarah plan to sell cupcakes. Since the school carnival is a fundraiser, Jenny and Sarah's parents make a donation to their cupcake booth to get them started. Jenny starts with a $5 donation and sells her cupcakes for $3 each. Sarah starts with a $10 donation and sells her cupcakes for $2 each. How many cupcakes do Jenny and Sarah have to sell for their profits to be equal?
One student's response is "Zero, because if Jenny sells her cupcakes for more money, then she will always have more profit." Which of the following activities could help the student realize his misconception?
graphing the scenarios
107
The semester exam administered to students at the end of the term is considered to be a:
summative assessment.
108
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?
recognizing different coins and their respective values
109
Mr. Marks gives his students a pop quiz on graphing on the coordinate plane. Sixty percent of his students fail the quiz. What should he do next?
Reteach the (x,y) coordinate structure and axes to the whole class.
110
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 parent going shopping at a store sale
111
Who is credited with creating much of what we consider geometry?
the Greeks
112
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?
Going outside to measure various parts of the playground in inches, feet, or yards.
113
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?
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?"
114
Mr. Feeny has been teaching fifth-grade math for thirty years. He will only accept answers from his students that follow his algorithmic procedures. If a student determines a correct solution by any method other than the way they were taught in class they will not receive credit. How could Mr. Feeny improve his teaching practice?
Teach students multiple varied ways to achieve the right answer and accept any correct answer as long as there is mathematically reasonable supporting work.
115
Which of the following is NOT considered a benefit of cash?
security
116
Mr. Erikson has his friend Ted, who is an architect, come present to the class about how he uses math in his job. What is Ted likely to talk about?
How geometric figures are a part of most buildings.
117
A student asks the teacher who invented the number system we use today. Which of the following answers would be most appropriate?
The base-ten number system was developed by the Hindu-Arabic civilizations.
118
A sixth-grade teacher discovers that each student in his class receives an allowance from their parents. Which of the following examples would best demonstrate to the students the power of saving their allowance instead of spending all of their allowance?
Show students the expected return of 5% allowance savings over a 10-year period.
119
Which of the following civilizations is most closely associated with the development of algebra?
Arabian
120
Using money in mathematical examples is a good strategy to promote student engagement in activities. A first-grade teacher decides to begin teaching about place value by using money, specifically with the example of 10 pennies = 1 dime and 10 dimes = $1. Why is this strategy probably not a good beginning strategy?
Select all answers that apply.
The relationships above are too abstract for young learners.
The coins are not proportional with respect to shape and size.
Most young learners would rather have 8 pennies rather than 1 dime.
121
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 scavenger hunt in which students work in pairs to find examples of different shapes in the classroom
122
Mr. Jones is hosting a career day for his sixth-grade class. Jeremy tells him that his father cannot come present because his job has nothing to do with math because he did not attend college. Mr. Jones asks Jeremy what his dad does for a living. Jeremy says his dad is a carpenter. What should Mr. Jones tell Jeremy?
Carpentry involves a lot of math including accurate measurements, determination of angles, and geometry.
Carpentry involves algebra to determine how much raw material to buy.
Mathematics is used by people throughout their lives, whether or not they attend college.
All of the above.
123
Interest is best defined as:
the cost associated with borrowing from the bank which issues a credit card.
124
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 casino estimating the expected number of jackpot payouts over the next fiscal year
125
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?
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
126
In a unit on personal finance, a sixth-grade teacher wants students to be able to identify the difference between fixed and variable costs. Which of the following examples would best highlight this difference?
categorizing the expenses of a local restaurant into expenses that depend on the number of customers and expenses that do no not depend on the number of customers
127
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?
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.
128
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?
Students found the school store engaging and learned the material better than students given word problems.
129
One pound of apples costs $2.02, $3.80, and $1.95 at three different stores. Which of the following expressions is a reasonable way to estimate the average cost for 4 pounds of apples?
4(2+4+2)/3
To find an estimate for the average cost of the apples, first round the prices of one pound of apples:
$2.02→$2, $3.80→$4, $1.95→$2. Then, find the average by adding the prices and dividing them by the number of prices:
2+4+2/3
. Lastly, multiply by 4 to calculate the average for 4 pounds of apples:
4(2+4+2)/3
130
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?
Agree to meet, listen to their concerns, and then explain that one component of math is understanding reasonable answers.
131
Student work is shown below.
Step 1: 4x+3=7x+2
Step 2: 3=3x+2
Step 3: 1=3x
1/3=x
Which property would justify the student work from Step 3 to Step 4?
multiplication property of equality
The multiplication property of equality states that the resulting statement is still equal after the quantities on both sides of an equal sign are multiplied by the same amount. In working from step 3 to step 4, the student multiplied both sides by 1/3.
132
A second-grade teacher is planning a lesson to review three-dimensional shapes. Students have already learned the attributes of three-dimensional shapes and the necessary vocabulary, such as faces, edges, and vertices. Which of the following questions could the teacher include in her lesson that would be most likely to encourage higher-order thinking?
A 3D shape has at least one face that is a rectangle. What are some of the 3D shapes that it could be?
133
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?
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.
134
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?
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.
135
Anytown School District wants all elementary students to be able to use computational strategies fluently and estimate appropriately. Which of the following learning objects best reflects this goal?
Students evaluate the reasonableness of their answers.
136
A third-grade teacher is working with a small group on comparing fractions. The teacher asks students to write ½ and ¾ and put the correct symbol between the two fractions. The students correctly write ½ < ¾ . The teacher asks the students how they know that ½ is less than ¾ , but none of the students offer an answer. Which TWO steps would be appropriate for the teacher to take next to encourage higher-order thinking?
Select all answers that apply.
Allow wait time for the students to process their answer.
Ask students to use manipulatives or drawings to model the two fractions.
137
Ms. Jones, a fourth-grade teacher, asks her students to find all the factor pairs of the number 48. After the students work independently for 2 minutes, which of the following would be the next best instructional step for Ms. Jones to use to assess her students' conceptual understanding?
circulate and listen while students discuss their answers with their seat partner before calling on a few pairs of students to explain and justify their answer
138
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?
Yes, because this allows students to develop a strategy that works for them.
139
Jaylene is trying to solve the equation 5x+7=72. She raised her hand to get help from her teacher. When the teacher comes over, the following is written on her paper:
Jaylene wants to know if her first step is correct. Of the following, which is the best teacher response to Jaylene?
The first step is incorrect; you must apply the subtraction on both sides of the equal sign.
140
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?
deductive reasoning
141
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?
$32
142
Mr. Swan wants to ensure that his students truly understand the material he is teaching. When students get questions incorrect on a test, he presents them the opportunity to correct their answers for half credit. He asks students questions such as "what if I changed this number?" and "why did you do this?" What process is Mr. Swan trying to get his students to engage in?
metacognition
Metacognition is reflecting on one's thought process to deepen understanding. This is what Mr. Swan is attempting to do.
143
There are 512 students at Rockmore Elementary. You estimate that between 32 and 36 percent of them will drink chocolate milk at lunch. What is a reasonable way to estimate the number of students that will drink chocolate milk at lunch?
(32+36/2⋅100) x (512)
144
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:
demonstrate an understanding of the estimation process.
145
What details should a teacher consider when choosing appropriate higher-order thinking questions for math?
Grade level and subject matter standards
146
Which of the following is not considered a higher-order thinking question?
What is the product of 6 and 3?
147
12π ÷ 9 is approximately equivalent to:
4
148
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=1800−(20 ×15)
149
After a lesson on rounding and estimation, a teacher tells students that the football concession stand has purchased 590 candy bars to sell for the 6 football home games this year. The teacher asks the students to estimate the average number of candy bars that will be sold at each home game. Which of the following would be the correct estimation?
100
An estimate is finding an approximation of a value. Estimates are used to quickly find an answer that is close, but probably not precise. 590 can easily be rounded to 600 which is divisible by 6. This means that about 100 candy bars will be sold per game.
150
A third-grade teacher notices that a student got nine out of ten multiplication problems correct, but on the missed problem they wrote 24 × 3 = 27. What would be the best step for the teacher to take next?
Ask the student if 27 seems like a reasonable answer to 24 × 3.
151
A kindergarten teacher is planning a lesson on comparing two numbers using "greater than" and "less than." After introducing the phrases "greater than" and "less than," she writes a 4 and 8 on the board and asks students to think about which number is greater. Which of the following activities should the teacher use next to promote and assess students' mathematical reasoning skills?
Ask students to explain why they think one number is greater than the other.
152
Using the student's work below, which property does the student still need to master?
Step 1: 3x−7=7x+1
Step 2: 3x=7x−6
Step 3: 10x=−6
Step 4: 4x=− 6/10
addition property of equality
153
While traveling to its destination, a plane flies at 570 miles per hour for 830 miles and 145 miles per hour for 204 miles. What is a reasonable way to calculate the total flight time?
830/570 + 240/145
To find the total flight time, first find the time it took to fly 830 miles at 570 miles per hour. Then, find the time it took to fly 204 miles at 145 miles per hour. To find the total time, add the two expressions together.
154
After instruction, a teacher poses a question. After asking the question, the teacher immediately chooses someone who raises their hand to answer the question. Which of the following would have been a better method for the teacher to take?
Tell students to turn to their seat partner and discuss their thoughts.
155
Use the student work shown below to answer the question:
Step 1: 3x+2=16−4x
Step 2: 7x+2=16
Step 3: 7x=14
Step 4: 4x=2
Which property should the student use to justify step 3?
addition property of equality
156
Which equation below models xª*xᵇ = xª⁺ᵇ?
5³ * 5⁴ = 5⁷
157
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?
Allow all students to participate in all three activities by rotating through them.
158
Use mental math to solve the problem below.
James has saved $35.25. He wants to save his money to buy a bicycle that is listed as $85.00. If sales tax is 8%, about how much more must he save to purchase his bike, including tax?
$60
The math used: 8% is close to 10% so sales tax on $85.00 is around $8.50. So $85.00 + $8.50 = $93.50 total. (Notice this is an overestimate so James' target will be a bit more than he actually needs). James needs to save about $93. If he has saved about $35, he will need an additional $58. ($93 - $35 = $58).
Therefore, if rounded up this would be the best choice: $60. When dealing with money, generally an overestimate is more reasonable.
159
Mr. Kim shares with his geometry class the triangle sum property - The sum of all angles in a triangle always add to
180
°
180°. Then, he asks the students to find the missing angle in the triangle below:
Deductive Reasoning
What type of thinking are the students using to solve this problem?
Deductive reasoning
With deductive reasoning, students start with a proven fact, rule, or definition to arrive at a conclusion.
160
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?
inductive reasoning
161
When solving an equation, Allie and Diane chose to take different approaches in their first steps. Which property ensures that they are both correct?
Original equation: 5x+3x−2x=4
Allie's approach: (5x+3x)−2x=4
Diane's approach: 5x+(3x−2x)=4
associative property of addition
162
Mrs. Wheelan is teaching geometric shapes and wants to use informal reasoning questions for discussion. What question is best to start with?
How do geometric shapes play a role in daily life?
163
A first-grade teacher is working with a small group of students on skip counting by tens. The students are able to recite numbers from 10 to 100 while skip counting, but they struggle when asked what ten more than 40 is. Which of the following strategies would help students improve their mathematical reasoning skills related to this concept?
Guide students in highlighting multiples of ten on a number line or hundreds chart as they skip count out loud.
164
Which of the following statements is false?
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.
165
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?
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.
166
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?
adding numbers to the problem to solve for exact age
This is the next step in scaffolded learning.
167
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?
deductive reasoning
168
Of the following higher-order thinking questions, which would be the most appropriate in a fourth-grade classroom?
Using the information you have collected, create a bar graph to display the data.
169
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?
What is the volume of the dirt in the garden?
170
A first-grade teacher has created three different ramps by stacking large blocks and laying pieces of cardboard from the tops of the block towers to the floor. One ramp is built with two large blocks, one with three, and one with four. After students have had a chance to roll a ball down each of the ramps, the teacher decides to ask students a question. Which of the following questions is most appropriate to ask at this point in the experiment?
"What have you noticed about how the ball rolls differently with the different ramps?"
171
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?
informal reasoning
172
Maria solved a word problem and correctly gave 72 as the answer. Which of the following could not have been the question asked?
How many months did Alexa achieve perfect attendance last year?
173
One pound of apples costs $2.02, $3.80, and $1.95 at three different stores. Which of the following expressions is a reasonable way to estimate the average cost for 4 pounds of apples?
4(2+4+2)/3
To find an estimate for the average cost of the apples, first round the prices of one pound of apples: $2.02→$2, $3.80→$4, $1.95 →$2. Then, find the average by adding the prices and dividing them by the number of prices: 2+4+2/3
Lastly, multiply by 4 to calculate the average for 4 pounds of apples: 4(2+4+2)/3
.
174
When working on solving an equation, Josef rearranged the terms on each side of the equation so:
Original equation: −2+3x=4−5x
New equation: 3x−2=−5x+4
Which property did Josef use to allow him to make this change?
commutative property of addition
Since −2+3x=3x−2 and 4−5x=−5x+4, Josef used the commutative property of addition that says a+b=b+a.
175
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?
draw a Venn diagram
176
Out of the following higher-order thinking questions, which one would not be appropriate in a 1st-grade classroom?
Given an example problem, have students evaluate the method that was used when solving the problem.
177
While traveling to its destination, a plane flies at 570 miles per hour for 830 miles and 145 miles per hour for 204 miles. What is a reasonable way to calculate the total flight time?
830/570 + 204/145
178
A kindergarten teacher reads the following problem to a student: "Anna has 4 crayons, and her friend gives her 2 more. How many crayons does she have now?" The student puts out 4 counters, then adds 2 more counters and counts to get a total of 6. What would be the best question for the teacher to ask next to encourage higher-order thinking?
"How did you know that you needed to add 4 and 2?"
179
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?
formal reasoning
180
