Math Methods Test 3

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Students should learn the relationship between multiplication and division. Explain this relationship

- If a / b =​ c, then b*c = a - If a*b = ​c, then c / b = a - Division with remainder 0 is the inverse of multiplication and vice versa.

Describe all pairs of whole numbers whose sum and product are the same.

0 and 0 2 and 2

Amy says that dividing a number by 1/2 is the same as taking half of a number. How do you​ respond?

1/2 of a number x is equivalent to x/2 but dividing a number x by 1/2 is equivalent to 2x

Fifteen light bulbs were in a chandelier. Four​-fifths of the bulbs were shining. What fraction of the light bulbs were not​ shining?

1/5 of the light bulbs are not shining.

The cost of having a plumber spend h hours at your house if the plumber charges ​$30 for coming to the house and ​$50 per hour for labor

20+40h

Matt has nine times as many stickers as David. How many stickers must Matt give David so that they will each have 250 ​stickers?

200

A 6 ft board is to be cut into three​ pieces, two​ equal-length ones and the third 6 in. shorter than each of the other two. If the cutting does not result in any length being​ lost, how long are the​ pieces?

26

A ball is dropped from a height of 10 ft and bounces 83 % of its previous height on each bounce. How high off the ground is the ball at the top of the 6th ​bounce?

3.3

The cost of having a plumber spend h hr at your house if the plumber charges ​$30 for coming to the house and​ $x per hour for labor

30+hx

The amount of money in cents in a jar containing some nickels and d dimes and some quarters if there are 6 times as many nickels as dimes and twice as many quarters as nickels.

340d

A car trip took 7 hours at an average speed of 57 mph. Mentally compute the total number of miles traveled

399 miles

The sum of three conscecutive even whole numbers if the greatest is x

3x-6

Eight years from now a​ girl's age will be 3 times her present age. Find the​ girl's age now

4

Fifteen light bulbs were in a chandelier. One​-fifth of the bulbs were shining. What fraction of the light bulbs were not​ shining?

4/5 of the bulbs were not shining

A car trip took 6 hours at an average speed of 69 mph. Mentally compute the total number of miles traveled

414 miles

For a particular​ event, 806 tickets were sold for a total of ​$5943. If students paid ​$6 per ticket and nonstudents paid ​$9 per​ ticket, how many student tickets were​ sold?

437

A student uses​ front-end estimation to estimate the product of two numbers as 2800. List a pair of possible factors.

47 and 74

For a particular​ event, 923 tickets were sold for a total of ​$2287. If students paid ​$2 per ticket and nonstudents paid ​$3 per​ ticket, how many student tickets were​ sold?

482 tickets

Pawel's total earnings after 3 yr if the first year his salary was s​ dollars, the second year it was ​$7000 higher and the third year it was twice as much as the first year.

4s+7000

The sum of four consecutive integers if the greatest integer is x

4x-6

The temperature t hr ago if the present temperature is 50 degrees F and each hour it drops by 6 degrees F.

50+6t

At the beginning of a​ trip, the odometer registered 52,500. At the end of the​ trip, the odometer registered 59,321. How many miles were traveled on this​ trip?

6821 mi were traveled on the trip.

In a college there are 16 times as many students as professors. If together the students and professors number 8 comma 500​, how many students are there in the​ college?

8,000

Matt has nine times as many stickers as David. If David has d stickers and Matt has m​ stickers, and Matt gives David 13 ​stickers, how many stickers does each have in terms of​ d?

9d-13

About 4020 calories must be burned to lose 1 lb of body weight. Estimate how many calories must be burned to lose 6 ​lb, to the nearest thousand.

About 24,000 calories must be burned to lose 6 lb

Which of the following is important to do before students learn the formal​ algorithms?

Address misconceptions

Which of the following is a good explanation for how to add​ fractions?

Add​ equal-sized parts - finding a common denominator can help to solve the problem

Explain how the distributive property of multiplication over addition would be helpful to mentally perform the following computation 45*25+45*75

After factoring out 45​, the other factor sums to 100. It is easier to multiply by 100.

Establishing a culture where students are making their own conjectures develops their skills at justification. Which of the following would foster this​ culture?

Always, sometimes or never mathematical statements

A student​ says, "My answer must be wrong - my answer got​ bigger." Which of the following responses will best help the student understand why the answer got​ bigger?

Ask them to explain the meaning of 8​ ÷ 2, using cutting ribbon as a​ context, and then ask them to​ re-explain to you using 8​ ÷ 1/2 still using cutting ribbon as a context.

Which of the following best describes how to teach multiplication involving a whole number and a​ fraction?

A​ "fraction times a whole​ number" and a​ "whole number times a​ fraction" are conceptually​ different, so they should be taught separately

Two classes were given a same test. In class​ A, 10 out of 15 students​ passed, and in class​ B, 12 of 18 passed. One of students in class B claimed that the classes did equally well. How could you explain the​ student's reasoning?

Because the fraction of students passing in each class was 2/3 the argument could be made that they did equally well on the test

Two classes were given a same test. In class​ A, 16 out of 20 students​ passed, and in class​ B, 24 of 30 passed. One of students in class B claimed that the classes did equally well. How could you explain the​ student's reasoning?

Because the fraction of students passing in each class was 4/5 the argument could be made that they did equally well on the test

The way we write fractions with a top and bottom number is a convention. What method focuses on making sense of the parts rather than the​ symbols?

Begin by using words​ (i.e., one-fourth)

Suppose a student argued that 0/ 0 = 1 because every number divided by itself is 1. How would you help that​ student

By the definition of​ division, 0/ 0 ​= x​ if, and only​ if, 0=0 times x has a unique solution. But the last equation is true for all whole numbers x. Because the equation has no unique​ solution, 0/ 0 is not meaningful

All the following are recommendations for effective fraction computation instruction except​

Carefully introduce procedures

Which of the following statements about multiplication strategies is true​?

Cluster problems use multiplication facts and combinations that students already know in order to figure out more complex computations.

Which of the following strategies would you like students to use when determining which of these fractions is greater than 7/8 or 5/6?

Compare how far from 1

All the following are representative of how algebraic thinking is integrated across the curriculum except​

Composing and decomposing shapes

All of the methods below would work to support​ students' knowledge about what is happening when multiplying a fraction by a whole number​ except

Compute with a calculator

Locating a fraction on a number line can be challenging but is very important. Which is a common error that students make in working with the number​ line?

Count the tick marks that appear without noticing any missing ones

The teachers have identified three manipulatives to use when teaching fractional concepts. Each teacher intended to select one manipulative to show each fraction model. Which teacher succeeded in selecting manipulatives for each​ type?

Denise selected​ tangrams, color​ tiles, and number lines

What is the primary reason to not focus on specific algorithms for comparing two​ fractions?

Developing number sense about relative size of fractions is less likely

Ally described her problem solving process as taking 6*10 equals 60 and then 4*6 equals​ 24, and 60 plus 24 is 84. ​Ally's process is known as what property of​ whole-number multiplication?

Distributive Property of Multiplication over Addition

Dave purchased a ​$39,000 life insurance policy at the price of ​$32 per​ $1000 of coverage. If he pays the premium​ quarterly, how much is each​ installment?

Each installment is ​$312.

Sue purchased a ​$26,000 life insurance policy at the price of ​$36 per​ $1000 of coverage. If she pays the premium in 12 monthly​ installments, how much is each​ installment?

Each installment is ​$78

The goal is to rename a fractional amount. What is the concept that requires the use of many contexts and​ models?

Equivalent fractions

Felisha was asked to share one cookie among four people

Felisha used a fair share model by splitting the cookie into four parts.

What is a problem with learning only designated​ (standard) algorithms for fraction​ operations?

Follow a procedure in a short​ term, but not retain

Language plays and important role in thinking conceptually about division. Identify the statement below that would not support students thinking about the problem 4​ ÷ 583

Four goes into 5 how many​ times?

Delia was asked to estimate 489​ + 37​ + 651​ + 208. She​ said, "400​ + 600​ + 200​ = 1200, so​ it's about​ 1200, but I need to add about 150 more for 80​ + 30​ + 50​ + 0.​ So, the sum is about​ 1350." Which computational estimation strategy did Delia​ use?

Front-end

Which of the following analyzes how the pattern is changing with each new element in the​ pattern?

Geometric growing patterns

All of the following statements are​ research-based recommendations for teaching and learning about fractions except one. Which​ one?

Give greater emphasis to specific algorithms for finding common denominators

Research findings support all of the following fraction teaching ideas but one. Which of the following is the unsupported​ method?

Give students area models that are already partitioned and ask them to record the fractional amount shaded

Sue claims the following is true by the distributive​ property, where a and b are whole numbers. 9​(ab) = (9​a)(9​b)

Her claim is​ false; consider the example when a=1 and b=2

A student claims that to divide a number with the units digit 0 by​ 10, she just crosses out the 0 to get the answer. She wants to know if this is always true and why and if the 0 has to be the units digit. How do you​ respond?

If the number has three digits with the units digit​ 0, we have ab0 divided by 10equalsab since ab times 10equalsab0. This will not work if the 0 digit is not the units digit

A student asks why she should learn the standard long division algorithm if she can get a correct answer using repeated subtraction. How do you​ respond?

If the repeated subtraction algorithm is done with large multiples of the​ divisor, the repeated subtraction can be quite efficient.​ However, if a student uses repeated subtraction by subtracting small multiples of the​ divisor, the process can be very time consuming.

Identify which statement below would not be considered a common or limited conception related to fractional​ parts?

Knowing that answers can be left as fractions rather than writing them as mixed numbers

When we add two fractions with unlike denominators and convert them to fractions with the same​ denominator, must we use the least common​ denominator? What are the advantages of using the least common​ denominator?

No, but the advantage is that it is easier to write in simplest terms if the least common denominator is used

When the least common denominator is used in adding or subtracting​ fractions, is the result always a fraction in simplest​ form?

No, the resulting rational number could have a numerator and denominator with a common factor greater than 1.

What form of algebraic reasoning is the heart of what it means to do​ mathematics?

Noticing generalizations and attempting to prove them true

On a 16​-day ​vacation, Glenn increased his calorie intake by 1700 calories per day. He also worked out more than usual by swimming 3 hours per day. Swimming burns 504 calories per​ hour, and a net gain of 3500 calories adds 1 lb of weight. Did Glenn gain at least 1 lb during his​ vacation?

No​, Glenn did not gain at least 1 lb while on vacation.

Research recommends that teachers use one of the following to support​ students' understanding that fractions are numbers and they expand the number system beyond whole numbers

Number Lines

There are multiple contexts that can guide students understanding of fractions. Which of the following would involve shading a region or a portion of a group of​ people?

Part-whole

Which of the following best describes the relationship between iterating and​ partitioning?

Partitioning is finding the parts of a​ whole, whereas iterating is counting the fractional parts.

Arithmetic and algebra are closely connected. Identify the reason below that best describes​ why?

Place value and operations are generalized​ rules; a focus on algebraic thinking can help students make connections across problems and strengthen understanding

Which instructional method does not support purposeful teaching of mathematical​ properties?

Providing opportunities for students to name and match properties to examples

All the following are reasons that data and algebra are good topics to integrate except​

Real data can be gathered and used to see if the data​ covary, for example in a linear​ manner, which builds knowledge of both algebraic and statistics.

Guiding students to develop a recording scheme for multiplication can be enhanced by the use of what​ tool?

Recording sheet with​ base-ten columns

When students use the break apart of decomposition strategy with​ division, what must they​ remember?

Remember that you cannot break apart the divisor

Connor was asked to solve the problem 4*25 and came up with 100 as an answer. When asked to describe his problem solving process he said he counted by​ 20's four times to get​ 80, and then added five to itself four times to get 20.​ So, 80+20 is 100.

Repeated Addition Model Skip Count Model

What division approach is good for students with learning disabilities that allows them to select facts the already​ know?

Repeated subtraction

If the number of professors in a college is P and the number of students​ S, and there are 16 times as many students as​ professors, write an algebraic equation that shows the relationship.

S=16P

If the number of professors in a college is P and the number of students​ S, and there are 20 times as many students as​ professors, write an algebraic equation that shows the relationship.

S=20P

Which of the following can be presented to students that will open opportunities for them to​ generalize?

Set of related problems

Which model below would not provide a clear illustration of equivalent​ fractions?

Show an algorithm of multiplying the numerator and denominator by the same number

Identify the example below that represents a​ relational-structural approach for the problem 8​ + 4​ = n​ + 5

Since 4 is one more than 5 on the other​ side, that means n is one less than 8

Providing students with many contexts and visuals is essential to their building understanding of equivalence. More examples of linear situations are needed to make comparisons more visible. Which of the following would not be best to model on a number​ line?

Slices of pizza eaten

Computational estimation refers to which of the​ following?

Substituting close compatible numbers for​ difficult-to-handle numbers so that computations can be done mentally

Which of the following options would be misleading for student understanding of​ fractions?

Tell students that fractions are different from whole​ numbers, so the procedures are also different.

Using a​ calculator, Sarah multiplied by 5 when she should have divided by 5. The display read 350. What should the correct answer​ be?

The correct answer is 14

Using a​ calculator, Sarah multiplied by 10 when she should have divided by 10. The display read 800. What should the correct display​ be?

The correct display should be 8

Andrew finds the product of 7*12. Which properties does he use to determine the correct​ answer?

The distributive property of multiplication over addition for whole numbers

Rachel is asked to calculate 45*36. What property does she use to determine the correct​ answer?

The distributive property of multiplication over addition for whole numbers

An estate of ​$554,000 is left to three siblings. The eldest receives 6 times as much as the youngest. The middle sibling receives ​$14,000 more than the youngest. How much did each​ receive?

The eldest sibling received ​$405,000​, the middle sibling received ​$81,500​, and the youngest sibling received ​$67,500

A theater has 50 rows with 28 seats in each row. Estimate the number of seats in the theater

The estimate for the number of seats in the theater is 1500

Explain why 10 cents is one - tenth of a​ dollar, yet 6 minutes is one - tenth of an hour. Why should these one - tenths not be​ equal?

The fractions are each 1/10 and are​ equivalent, but in context with the different​ bases, they represent different quantities.

Explain why 25 cents is one dash fourth of a​ dollar, yet 15 minutes is one dash fourth of an hour. Why should these one dash fourths not be​ equal?

The fractions are each one fourth and are​ equivalent, but in context with the different​ bases, they represent different quantities

Pick a number. Double it. Multiply the result by 3. Add 36. Divide by 6. Subtract your original number. Is the result always the​ same? Write a convincing argument for what happens

The result is always 6. Let the original number be x. The operation appears as​ follows:[(2x)3 + 36]/6x=6

A student writes ​a(bc)=​(ab)(ac). How do you​ respond?

The student is applying the distributive property of multipication over addition to multiplication. The student is incorrect in​ his/her extension of the distributive property.

A student argued that a pizza cut into 12 pieces was more than a pizza cut into 6 pieces. How would you​ respond?

The student was probably thinking that more pieces meant more pizza. The amount of pizza did not change and only the number of pieces changed

A student argued that a pizza cut into 8 pieces was more than a pizza cut into 4 pieces. How would you​ respond?

The student was probably thinking that more pieces meant more pizza. The amount of pizza did not change and only the number of pieces changed.

A popular brand of pen is available in 8 colors and 4 writing tips. How many different choices of pens do you have with this​ brand?

There are 32 different choices of pens with this brand

A new model of car is available in 4 exterior colors and 2 interior colors. Use a tree diagram and specific colors to show how many color schemes are possible for the car.

There are 8 different color schemes for this new model of car.

​Five-sevenths of the students at a nearby college live in dormitories. If 6000 students at the college live in​ dormitories, how many students are there in the​ college?

There are 8,400 students in the college

A designer has designed different​ tops, pants, and jackets to create outfits. How many different outfits can the models wear if she has designed the following​ pieces? three ​tops, eight ​pants, six jackets

There are a total of 144 different outfits

Make a statement about a person or an environment and use fractions in each. Explain why your statements are true. Choose the correct answer below

There are three birds on a​ wire, two of them are​ blackbirds; hence 2/3 of the birds are blackbirds. This is true because the fraction 2/3 represents the ratio of blackbirds to the total number of birds

Sixteen light bulbs were in a chandelier. Three​-fourths of the bulbs were not shining. How many light bulbs were not​ shining?

There were 12 ​bulb(s) that were not shining

Students were divided into 10 teams with 12 on each team.​ Later, the same day students were divided into teams with 2 on each team. How many teams were there​ then?

There were 60 teams of 2 students each.

Students were divided into 8 teams with 9 on each team. Later the same day students were divided into teams with 8 on each team. How many teams were​ there?

There were 9 teams

All the following are reasons that data and algebra are good topics to integrate except​:

There​ isn't enough time in the year to address​ everything, so it is more efficient to teach these two together.

The benefits of using a rectangular area to represent multiplication of fractions include all the following except​ which?

They are easy for students to draw

What statement below best describes​ functions?

They describe a relationship between two variables and may be linear or not.

When adding fractions with like denominators it is important for students to focus which key​ idea?

Units are the same

Teaching fractions involves using strategies that may not have been part of a​ teacher's learning experience. What is a key recommendation to teachers from this​ chapter?

Use multiple​ representations, approaches,​ explanations, and justifications

Algebraic thinking includes several characteristics. Which of the following statements is not a part of algebraic​ thinking?

Using manipulatives to reason about situations

How would you explain to children how to multiply 356 times 685​, assuming they know and understand multiplication by a single digit and multiplication by a power of​ 10?

Using the distributive property of multiplication over​ addition, first multiply 356 by 6 and the result by 10 squared​; then multiply 356 by 8 and the result by​ 10; then multiply 356 by 5 and add all the numbers obtained together

Which of the following is not representative of the current thinking about arithmetic and algebra in the elementary​ classroom?

Variables are not appropriate for​ elementary-age students; a box is a more concrete representation

Mathematical modeling is appropriate for investigating real challenges. Which of these examples requires some mathematical​ modeling?

What would be the better​ deal, buy-one-get-one half​ off, 25%​ off, buy-two-get-one-free?

Do you think it is valuable for students to see more than one method of doing computation​ problems? Why or why​ not?

Yes. Some students may find one algorithm easier to understand or to use than others and therefore it will be easier for him or her to remember or reproduce.

When teaching computational​ estimation, it is important​ to:

accept a range of reasonable answers

Mixed​ numbers

can be changed into fractions or​ "improper" fractions and added.

An important concept in working with repeating patterns is for the student to identify​ the:

core of the pattern

One way to effectively model multiplication with large numbers is​ to

create an area model using​ base-ten materials

A box contains 89 ​coins, only dimes and nickels. The amount of money in the box is ​$5.75. How many dimes and how many nickels are in the​ box?

dimes -26 nickels - 63

Conceptualizing the symbol for equal as a balance can support​ students' understanding​ of:

equality or inequality

A common misconception with set models​ is:

focusing on the size of the subset rather than the number of equal sets

Writing fractions in the simplest terms means to write it​ so

fraction numerator and denominator have no common whole number factors.

If g is the number of girls in a class and b the number of boys and if there are sixteen more girls​ (g) than boys​ (b) in a​ class, write an algebraic equation that shows this relationship.

g=b+16

Children as early as first grade can explore functional thinking by​ using

input-output activities

Mathematical models are useful in both real life and mathematics​ because:

models such as​ equations, graphs, and tables can be used to analyze empirical​ situations, to understand them​ better, and to make predictions.

Proficiency with division requires​ understanding

place​ value, multiplication, and the properties of the operations.

The amount of bacteria after n min if the initial amount of bacteria is q and the amount of bacteria triples every 20 sec.​ (Hint: The answer should contain q as well as​ n)

q(3^3n)

Using contextual problems with fraction division works in providing students with an image of what is​ being:

shared or partitioned

Suppose there are c chairs and t tables in a classroom and there are 23 more chairs than tables. Write an algebraic equation relating c and t

t+23

A critical aspect of understanding divisions of fractions​ is

the divisor is the unit

A parent died and left an estate to four children. One inherited 1/4 of the​ estate, the second inherited 1/4 ​and the third inherited 3/8. How much did the fourth​ inherit?

the fourth child inherited 1/8

A fraction by itself does not describe the size of the whole. A fraction tells us​ only

the relationship between part and whole

Sixteen light bulbs were in a chandelier. one - half of the bulbs were not shining. How many light bulbs were not​ shining?

there were 8 bulbs that were not shining

Students need experiences with variables that​ vary, and pairs of variables that​ covary, early in the elementary curriculum. It is important to emphasize​ the

variable stands for the number of

If a school has w women and m men and you know that there are 100 more men than​ women, write an algebraic equation relating w and m

w+100


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