Math Methods Test 3
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) = (9a)(9b)
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
