Reasoning (Quiz 1-5)

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?"

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?"

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

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

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

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

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.

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?

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?

Which of the following statements is false?

Inductive reasoning never leads to a correct conclusion.

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.

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.

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.

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.

Which of the following is not considered a higher-order thinking question?

What is the product of 6 and 3?

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?

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.

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 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

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

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: x=2 Which property should the student use to justify step 3?

addition property of equality

Using the student's work below, which property does the student still need to master? Step 1: 3x-7=7x+13x−7=7x+1 Step 2: 3x=7x-63x=7x−6 Step 3:10x=-610x=−6 Step 4: x=-\frac{6}{10}x=−106​

addition property of equality

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=45x+3x−2x=4 Allie's approach: (5x+3x)-2x=4(5x+3x)−2x=4 Diane's approach: 5x+(3x-2x)=45x+(3x−2x)=4

associative property of addition

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

When working on solving an equation, Josef rearranged the terms on each side of the equation so: Original equation: -2+3x=4-5x−2+3x=4−5x New equation: 3x-2 = -5x+43x−2=−5x+4 Which property did Josef use to allow him to make this change?

commutative property of addition

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

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

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.

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

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

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

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

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

Student work is shown below. Step 1: 4x+3=7x+24x+3=7x+2 Step 2: 3=3x+23=3x+2 Step 3: 1=3x1=3x Step 4: \frac{1}{3}=x31​=x Which property would justify the student work from Step 3 to Step 4?

multiplication property of equality

12π ÷ 9 is approximately equivalent to:

4

Mr. Kim shares with his geometry class the triangle sum property - The sum of all angles in a triangle always add to 180\degree180°. Then, he asks the students to find the missing angle in the triangle below: Deductive Reasoning What type of thinking are the students using to solve this problem?

Deductive reasoning

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.

What details should a teacher consider when choosing appropriate higher-order thinking questions for math?

Grade level and subject matter standards

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

Which equation below models xª•xᵇ = xª⁺ᵇ?

5³ • 5⁴ = 5⁷

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?

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) The area of the yard taken up by the pool is 15 x 20 = 300 ft², so the area remaining left for landscaping is the difference.

Mr. Johns gave a test last week and Ginny missed one question. She answered that 14.5 people would ride on each bus rather than 15. Her parents would like a conference because she did the math problem correctly and should receive credit even though her answer was not reasonable. How should Mr. Johns handle this situation?

Agree to meet, listen to their concerns, and then explain that one component of math is understanding reasonable answers.

Mrs. Dobbs is teaching students to skip-count by 2s, 5s, and 10s in her second-grade class. Earlier in the year, she evaluated her students learning style and assigns them one task based on this evaluation. Visual learners have been given a number line and they are to draw the hops across the top. Auditory learners have been given a list of the even number to 20, numbers divisible by 5 to 50 and numbers ending in 0 up to 100. They are told to say them over and over aloud to memorize the skips. Kinesthetic learners have been given a large number line on the floor. They are jumping to the next number as they skip-count. What can Mrs. Dobbs do to improve her teaching?

Allow all students to participate in all three activities by rotating through them.

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.

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.

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.