Exam 1: Reasoning About Quantities

Carry out a quantitative analysis of the following problem situation by answering each of the questions that follow. Jennie got on the freeway at 2:00 PM, using the entrance closest to her home, and traveled at 55 mph to the College Avenue exit, where she turned off at 2:12 PM. Her roommate Cassie had finished her morning classes and was headed home at about the same time. Cassie entered the freeway from the College Avenue entrance at 2:08 PM and traveled to the home exit at 60 mph. At what time did Cassie arrive at the exit ramp to go home? A) What quantities here are critical? B) What quantities here are related? C) For what quantities do I know the value? D) For what quantities do I need to know the value?

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Question 1

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A) Jennie's starting time, Jennie's exit time, time Jennie traveled, speed Jennie traveled, distance Jennie traveled, Cassie's starting time, distance Cassie traveled, speed Cassie traveled, time Cassie traveled
B) All in part A are related, but in different ways.
C) Jennie's starting time, Jennie's exit time, speed Jennie traveled, Cassie's starting time, speed Cassie traveled
D) Time Jennie traveled, distance Jennie traveled (= distance Cassie traveled), time Cassie traveled (to get Cassie's exit ramp time)

List at least five relevant quantities that are involved with this problem situation. For each quantity, if the value is given, write it next to the quantity. If the value is not given, write the unit you would use to measure it. Pat and Li left the starting line at the same time, rumning in opposite direct. 400 -meter, oval-shaped race track. Pat was running at a constant rate of 1 minute. They met each other for the first time after they had been rumning $f$ minutes. How for had Pat nun when Li completely finished one lap?

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Question 2

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Sample answers (quantity, value or unit if value unknown; other units possible-e.g., seconds instead of minutes):
Length of track, 400 meters
Pat's speed, 175 meters per minute
Time until they meet for first time, 1.5 minutes
Distance Pat has traveled when they meet for first time, meters
Distance Li has traveled when they meet for the first time, meters
Li's speed, meters per minute
Time for Li to run one lap, minutes
Time for Pat to run one lap, minutes
Distance Pat has run when Li finished one lap, meters
(The above are relevant to one solution, but the following are quantities in the situation as well.)
Difference in time for one lap for Pat and Li, seconds or minutes
Difference in speeds, Pat and Li, meters per minute

Many teachers teach "key words" for solving word problems. What are the limitations of this strategy?

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Key words can be misleading-they only work part of the time. Students will not know what the answer means. Students may look for key words without trying to understand the problem or what it is asking, which may have a negative impact on their understanding long term.

The label on a can of chicken broth claims that its weight is 1.4 kg. Use your metric knowledge to tell how many milligrams this would be.

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Question 4

What are some advantages to using the metric system of measurement?

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Question 5

Carry out a quantitative analysis of the following problem situation by answering each of the questions that follow, and then solve the problem: A butcher had two pieces of bologna, A and B, with A weighing 3 1/3 times as much as B. After the butcher cut 1.8 pounds off A, A was still 2 1/3 times as heavy as B. How many pounds does piece B weigh? A) What quantities here are critical? B) What quantities here are related? C) For what quantities do I know the value? D) For what quantities do I need to know the value? E) Use a diagram to find the weight of B, in pounds.

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Question 6

How many grams are there in centigrams?

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Question 7

My sister can walk from school to home in 40 minutes. I can walk from school to home in 30 minutes. But today I stayed for some extra help, and my sister was already 2/5 of the way home when I started. If I walk at my usual speed, can I catch my sister before she gets home? If YES, exactly what fraction of the trip have we covered when I catch her? If NO, exactly what fraction of the trip have I covered when my sister gets home? In either case, write enough (words, numbers, drawings) to make your thinking clear.

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Question 8

How can the term "big" be quantified?

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Question 9

My brother and I go to the same school. My brother takes 50 minutes to walk to school, and I take 40 minutes. If he gets a 49-minute head start one day, can I catch him before he gets to school? Explain without referring to any short-cut in your explanation. (Hint: Do not do a lot of calculation.)

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Question 10

Use a diagram to solve the following problem: Silvia and Jesus are buying a new table and new chairs for their dining area. Chairs with armrests are $45; those with no armrests are $8.50 cheaper. The table is four times as much as a chair without armrests. If they buy a table and six chairs, two with armrests and four without, what is the total price they pay?

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Question 11

Give two quantities that one could have in mind when he/she says, "This has been a good day."

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Question 12

Name a metric unit that is analogous to a quart. Which is LARGER?

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Question 13

Consider the following problem situation: Two boats simultaneously left a pier and traveled in opposite directions. One traveled at a speed of 18 nautical miles per hour and the other at 22 nautical miles per hour. How far apart were they after 2.5 hours? List five relevant quantities that are involved in this problem. For each quantity, if a value is given, write it next to the quantity. If value is not given, write the unit you would use to measure it and its value if possible.

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Question 14

Use a diagram to solve the following problem: A combinded third- and fourth-grade elementary school classroom has 29 students. There are seven more third-graders than there are fourth-graders. How many students are there in each grade?

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Question 15

Some children, when asked to solve a story problem, try different operations on the numbers and then decide which one seems to give the best answer. What is the danger of solving problems in this way?

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Question 16

Would student motivation be difficult or easy to quantify? Explain. Tell how you might go about quantifying student motivation in this class.

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Question 17

Your elementary student brings in a picture and story about a fat cat brought to an animal shelter weighing a hefty 18.1 kilograms. You want to help them understand how heavy that cat is. A) How much did the cat weigh in pounds and ounces? B) What personal benchmark might be useful in understanding this weight?

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Question 18

Draw a diagram to represent the quantities in the following problem. Solve the problem and explain your solution process. Jacqui, Karen, and Lynn all own several bottles of nail polish. When they line them up, Jacqui has 12 more bottles than Karen and Karen has three times as many as Lynn. Together, they have 124 bottles of nail polish. How many bottles does each person have?

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Question 19

If a Pascal is some unit of measure, use your knowledge of metric prefixes to complete the following: 4 kiloPascals = _____ Pascals

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Question 20

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Exam 1: Reasoning About Quantities

Many teachers teach "key words" for solving word problems. What are the limitations of this strategy?

The label on a can of chicken broth claims that its weight is 1.4 kg. Use your metric knowledge to tell how many milligrams this would be.

What are some advantages to using the metric system of measurement?

How many grams are there in centigrams?

How can the term "big" be quantified?

My brother and I go to the same school. My brother takes 50 minutes to walk to school, and I take 40 minutes. If he gets a 49-minute head start one day, can I catch him before he gets to school? Explain without referring to any short-cut in your explanation. (Hint: Do not do a lot of calculation.)

Give two quantities that one could have in mind when he/she says, "This has been a good day."

Name a metric unit that is analogous to a quart. Which is LARGER?

Use a diagram to solve the following problem: A combinded third- and fourth-grade elementary school classroom has 29 students. There are seven more third-graders than there are fourth-graders. How many students are there in each grade?

Some children, when asked to solve a story problem, try different operations on the numbers and then decide which one seems to give the best answer. What is the danger of solving problems in this way?

Would student motivation be difficult or easy to quantify? Explain. Tell how you might go about quantifying student motivation in this class.

If a Pascal is some unit of measure, use your knowledge of metric prefixes to complete the following: 4 kiloPascals = _____ Pascals

Numeration Systems

Understanding Whole Number Operations

Some Conventional Ways of Computing

Using Numbers in Sensible Ways

Meanings for Fractions

Computing With Fractions

Multiplicative Comparisons and Multiplicative Reasoning

Ratios, Rates, Proportions, and Percents

Integers and Other Number Systems

Number Theory

What Is Algebra

A Quantitative Approach to Algebra and Graphing

Understanding Change: Relationships Among Time, Distance, and Rate

Further Topics in Algebra and Change

Polygons

Polyhedra

Symmetry

Tessellations

Similarity

Curves, Constructions, and Curved Surfaces

Transformation Geometry

Measurement Basics

Area, Surface Area, and Volume

Counting Units Fast: Measurement Formulas

Special Topics in Measurement

Quantifying Uncertainty

Determining More Complicated Probabilities

Introduction to Statistics and Sampling

Representing and Interpreting Data With One Variable

Dealing With Multiple Data Sets or With Multiple Variables

Variability in Samples

Special Topics in Probability

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