Supporting Your Student

Homework Help:

If your student is having trouble with a homework problem. Recommend that they utilize the Homework

Help feature on the online textbook.

1-81. Rewrite each expression below without negative or zero exponents. Homework Help N

a. 4-1

7°

x-2

Each homework help may either provide a hint, define a term, or re-direct you to review a similar

problem that they have already completed in the course.

Preparing for Tests:

Tests are composed of both new materials from the unit, as well as material from previous units and

courses of study. Generally, students can expect that the questions based on previous units may be

more difficult than the questions related to the newer material since the student has had more time to

practice the material.

While preparing for an individual test, students should always review any study guides the teacher has

provided and the chapter closure questions carefully. In addition, it may be helpful for students to look

over the Parent Guide for that unit. This is available without a log in on the Parent Support page.

When your Student has an Extended Absence:

It is often difficult for students in any class to catch up after an illness that has kept them out of school

for more than a day or two. Although the in class work is exempt when a student is absent, I would

recommend that absent students still review the missed class work and ask their team or the teacher if

they have any questions. Absent homework should be made up. Students can coordinate with the

teacher to determine a deadline for absent work when the absence is more than a day or two. The

parent guide is also helpful for catching up absent students because it outlines the key ideas, and

provides additional practice.

Check Points:

The CPM program understands that students do not always develop a skill to a level of mastery

immediately. Students are introduced to concepts that they will not be expected to have completely

mastered until later units. In order to make sure that students are developing skills at the expected rate,

the book provides checkpoints. If a student reaches a checkpoint, and they do not feel confident with

their ability to complete that skill, they should find time to work with the teacher on that skill.

How to Find Additional Parent Resources:

Go to cpm.org. Then, use the Support tab to find Parent Support.

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Parent Support

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This parent support page gives you more information about how to help your student progress through

the course. One particularly helpful feature is the parent guide which is organized by chapter. To view

the parent guide, scroll down the page and select INT1:Parent Guide. I have included the parent guide

for unit 1 so that you can review it and determine if these may be something that is useful to you and

your student.

The Parent Guide provides an alternative explanation of key Ideas along with aantonal practice problems. The Parent Guide

resources are an r , 1/ chapter and strand. The Parent Guide Is edso avallabte as a printed copy for purchase attire CPM Web

Store or accessible free beicAv.

Core Connections (English) Core Connections (Espanol) Connection Series

C:Cl: Parent Guide Ca: Gufa pare pall Js on practice adicional MCt Pa nt Guide

CC2:Parent Guide CC.Z Gula pans padres con practice adicionat MC2: Parent Guide

CC3:Parent Guide CC3: Gufa pare padres con practice acticional AC Parent Guide

CCA: Parent Guide CCA: Gila pare padres con practice adicional CAC Parent Guide

CCG: Pc rent Guide CCG: Gate pare padres con practice adicionat GC Parent Guide

CCA2Lect Guide CCA2: Gufa pate padres con practice adtclonal A2C Parent Guide

Parent Gulc.) Cat Gina pare padres con practice adicional

INT2: Parent Guide INT2: Gin pare padres con practice acliclonal

INT1 Potent Guide ma Gula pare padres con practice adiclonat

Integrated Math 1 Ms. Valerie Weage

Birmingham Covington School Room 403

Email: VWeaae@birminaham.k12.mi.us

>. Core Connections Integrated 1 textbook (each student will be issued

one textbook for the school year)

> Online textbook through the Student eBook (https://sso.cpm.ore)

o Students will be given an enroll code

o Students will have a login and password

> Course Overview and Resources (http://cpm.oreint1)

o Introduction, Course Contents, Lesson Structure

o Homework Help Tool

o Parent Guide with Extra Practice

STUDY TEAM ROLES

Recorder/Reporter:

✓ Team spokesperson, shares findings

✓ Ensures each member records data

,( Reminds team to write down HW

Resource Manager

,7 Collects/returns team's supplies

✓ Calls teacher over for team questions

✓ Debriefs any absent teammates

Facilitator:

✓ Starts conversation for each task

✓ Ensures understanding of directions

✓ Organizes/assigns the team for the task

Task Manager:

✓ Enforces use of classroom norms

✓ Ensures task is completed on time

✓ Stops off task conversation

COURSE CONTENTS

0 Functions

Appendix A: Solving Equations

0 Linear Functions

0 Transformations & Solving

0 Modeling Two-Variable Data

0 Sequences

0 Systems of Equations

O Congruence & Coordinate Geom.

0 Exponential Functions

® Inequalities

O Functions & Data

0 Constructions & Closure

IN-CLASS EXERCISES are the bulk of the daily math

lessons. These exercises require students to work

with their "Study Teams" in a collaborative manner.

Each student is responsible to record their exercise

math data in their notebooks each day. Notebooks

will be evaluated throughout the course of the year.

"Notebook Checks" are unannounced to check on

the students' mathematical evidence and effort

from the in-class exercises. "Notebook Checks" may

be done individually, as a group, or one notebook

could be collected for the group. Groups are

responsible for making sure that all group members

are on track, completing the work, and recording

the groups work.

MATH NOTES are a designated area in the

fThOAND MENNS textbook at the end of each MATH N.Ttt? lesson that consolidate core

content ideas and provide definitions, explanations,

examples, instructions about notation,

formalizations of topics, and extensions or

applications of mathematical concepts. Students are

expected to copy the math notes in their notebooks.

HOMEWORK has been carefully designed to offer

students spaced practice with past material and to

help lay a foundation for future learning. Homework

for this course is identified with a "Review and

Preview" icon.

• All assignments are to be completed in a graph

composition, spiral notebook, or organized in a

binder. Work should be done in pencil.

• Title assignments: Section #, HW #, Exercise #'s

(Example: /-/-/ HW #1: 1-6 through 1-10)

• Each exercise must be completed with written

mathematical evidence and quality effort.

• Students must check their HW each night and

write correct solutions with a colored PEN next

to each problem (HW solutions are found on

classroom webpage).

• The online textbook offers homework help.

CHECKPOINTS are key homework problems

identified for determining if students are building

skills at the expected level. When students find that

they need help with these problems, worked

examples and practice problems are available in the

"Checkpoint Materials" at the back of their book.

CLASSROOM WEBPAGE contains the homework

calendar link, notes, and other pertinent resources.

http ://weage 17 . wiki s .birmingham k 1 2.mi .us/home

ENGAGE. INTERACT. PRACTICE. SMILE. SUCCEED.

LATE ASSIGNMENTS will be awarded partial credit

only and a late mark will appear in PowerSchool. An

assignment is considered late if it is not present in

the classroom when it is due. Students will not be

permitted to leave during class to collect an

assignment.

ABSENCE POLICY is that if a student is absent, they

are responsible for communicating with their Study

Team regarding classwork (it would be wise to

collect group contact info). Absent

assignments/assessments will show up as missing

and no points will appear inPowerSchool until they

are completed. Students are allowed two class

periods for each excused absence to make-up work

that was assigned on the day of an excused

absence. The return day is considered the first class

period. Students are required to submit any work

that was previously assigned and due on the day of

their excused absence on the day of their return.

Students who have an excused absence on the day

of a quiz or test will be required to make up that

quiz or test. Any assessment missed due to an

excused absence should be made up the following

day. Students who have an excused absence on a

Team Performance Task day are exempt from

completing the in class assignment. Optionally, the

student may arrange to make up the Team

Performance Task individually, if desired.

GRADING CATEGORIES are Homework (10%)

Teamwork (10%), Team Performance Tasks (10%),

93-100% A 83-87.996 B 73-773% C 63-673% and Individual Assessments (70%). 90-92.9%A- 80-823% B- 70:72.9%C- 60-62.9% D-

Extra Credit will not be provided. 88-89.9% B+ 78-79396 C+ 68-693% 13+ 59.996 & below

TYPES OF ASSESSMENTS

Team Performance Tasks: Students will complete these assessments in their study teams during a class

period. The teacher will randomly pick one group members work to score and return to the team for review.

Team Performance Tasks will be administered several days prior to the Individual Assessment. Study teams

should use the feedback from the Team Performance Task to prepare for the Individual Assessment.

Individual Assessments: Individual Assessments are announced approximately a week in advance and are

given the week following the end of a unit. In the meantime, the class will begin work on the next unit,

allowing students time to prepare for the individual test. Since we will have begun the next unit, students will

be expected to take the test on the previous unit on the day it is given, unless otherwise arranged with the

teacher in advance due to extenuating circumstances. While Individual Assessments will be spaced out

throughout the year following the closure of a unit, the Individual Assessments will essentially all be

cumulative. The Individual Assessments will be composed of both new material from the most recently

completed unit, as well as previous learned material in about equal proportions.

Cell Phones and Music

Cell phones are allowed in the classroom and are to be used only when given permission from the teacher.

Please talk to me if you have an extenuating circumstance and need to make a phone call. DO NOT feel free to

listen to music unless I have said that it is appropriate to do so. Any electronic abuse will result in teacher

taking electronic away for the rest of the day.

Additional Help

Even the best students get stuck or confused. Remember that sometimes 5 or 10 minutes of individual help

makes all the difference in understanding and learning. Please do not wait until the day of a quiz or test to get

extra help! Rather, talk to me as soon as you begin having difficulties.

• Feel free to either arrange a time in person, or contact me via email to set up a time to work together.

• There is an abundance of information online that covers most topics we'll be studying!

• The online textbook has homework help for just about every problem.

After you have reviewed the course syllabus with a parent/guardian, please sign below and return to Ms.

Weage by Friday, September 8th. Please let me know if you have any questions. This syllabus will be posted

online if you need to review it throughout the year.

Looking forward to a great year!

Student Signature:

Parent Signature:

Chapter 1

DESCRIBING FUNCTIONS 1.1.2 and 1.1.3

The main objective of these lessons is for students to be able to fully describe the key elements of the graph of a function. To fully describe the graph of a function, students should respond to these graph investigation questions:

Graph Investigation Question Sample Summary Statement What is the shape of the graph? The graph is a line/curve.

Is the function increasing or decreasing (reading left to right)?

As x gets bigger, y gets bigger, so the function is increasing.

What are the x- and y-intercepts? The graph crosses the x-axis at (2, 0) and the y-axis at (0,-3).

Are there any limitations on the inputs (domain) of the equation?

Only positive values of x are possible. Zero is also possible.

Are there any limitations on the outputs (range) of the equation? (Is there a maximum or minimum y-value?)

The smallest y-value is 0. There is no maximum y-value.

Should the points be connected? The given situation only makes sense for integer inputs, so the points should not be connected.

The more formal concepts of function and domain and range are addressed in Lessons 1.2.2 and 1.2.3.

For more infoimation, see the Math Notes boxes in Lesson 1.1.2. Student responses to the Learning Log in Lesson 1.1.3 (problem 1-27) can also be helpful.

Example 1

For the situation below, make an x y table, draw a graph, and describe the graph.

At the farmer's market, apples cost $0.50 each.

Note that the smallest possible number for the x --> y table is x = 0. You cannot buy a negative number of apples.

x (# of apples) 0 1 2 4 6 10 y (cost) 0 0.50 1.00 2.00 3.00 5.00

The graph is a discrete set of linear points because you can only buys whole numbers of apples. It starts at (0, 0) and increases from left to right. The inputs are limited to positive integers, and the outputs are 0 and multiples of $0.50.

Parent Guide with Extra Practice

0 2014 CPM Educational Program. All rights reserved.

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1. _ x+4 Y - 2

3. y=2x+1

Example 2

For the equation y = 2.x -2 , make an x y table, draw a graph, and fully describe the graph.

At this point there is no way to know how many points are sufficient for the x y table. Add more points as necessary until you are convinced of shape and location.

-2 2 3 —1.75 -1.5 0 2 6

Be careful with substitution using negative exponents when calculating values. The negative exponent moves across the

fraction bar to become positive so 2-2 = . 2/-

For example if x = -2 , y = 2-2 - 2 = - 2 = - = -1.75 .

The graph is a curve. From left to right, the function increases. The x-intercept is (1,0) . The y-intercept is (0,-1) . The points on the graph are connected. There are no limitations on inputs to the function. Outputs can be any value greater than -2.

Problems

For each equation or situation, make an x y table, draw a graph, and describe the graph.

2. Gasoline costs $4.00 per gallon. How much does it cost to buy x gallons of gas?

4. My science experiment starts with 5 bacteria and each hour the amount doubles. How many bacteria are there after x hours?

5. y = 5 - 2x 6. The product of two numbers is 12.

7. y = (0.5)x 8. My tomato plant was 5 cm tall when planted and grows 2 cm per week. How tall is my tomato plant after x weeks?

9. y=x2 - 4

2 Core Connections Integrated I

© 2014 CPM Educational Program. All rights reserved.

9. 8. 7. Y

10 5

5 lox # of weeks

—10

Chapter 1

Answers

1. 2.

3. ts' 5 8

5 X

of gallons

4.

Line; intercepts (-4, 0) and (0, 2); increasing function. Inputs can be any real number. Outputs are any real number. The points are connected.

cc!

1 1 1 1

—4-2 .— 2 4 # of hours

Curve; intercept intercept and starting point (0, 5); increasing function. Inputs can be any non-negative number. Outputs are greater than or equal to 5. The graph should be several disconnected points (but there are so many it will look connected).

A ray (proportional graph); intercept and starting point (0, 0); increasing function. Inputs can be any non-negative number. Outputs are greater than or equal to 0. The points are connected.

5.

Line; intercepts (2.5, 0) and (0, 5); decreasing function. Inputs can be any real number. Outputs are any real number. The points are connected.

Curve; intercept (0, 2); increasing function. Inputs can be any number. Outputs are greater than 1. The points are connected.

Inverse variation; no intercepts, decreasing function. Inputs can be any number except 0. Outputs are any number except 0. The points are connected except at x = 0.

Curve; intercept (0, 1); decreasing function. Inputs can be any real number. Outputs are greater than 0. The points are connected.

Ray; intercept and starting point (0, 5); increasing function; Inputs can be any non-negative number. Outputs are greater than or equal to 5 (but probably also less then or equal to 180).

U-shape; intercepts (2, 0), (-2, 0) and (0, —4); decreasing for x <0, increasing for x >0. Minimum value at (0, —4). Inputs can be any real number. Outputs are greater than or equal to —4. The points are connected.

Parent Guide with Extra Practice

© 2014 CPM Educational Program. All rights reserved.

FUNCTIONS 1.2.1 through 1.2.3

A relationship between the input values (usually x) and the output values (usually y) is called a function if for each input value, there is no more than one output value. Functions can be represented with an illustration of a "function (input—output) machine", as shown in Lesson 1.2.3 of the textbook and in the diagram in Example 1 below. Note: f(x) = 2x+ 1 is equivalent toy = 2x + 1.

The set of all possible inputs is called the domain, while the set of all possible outputs is called the range.

For additional information about functions, function notation, and domain and range, see the Math Notes box in Lesson 1.2.3.

Example 1

The inputs of a function are "x"s and the outputs are "f(x)"s Numbers are input into the function machine labeled f one at a time, and then the function performs the indicated operation on each input to determine its corresponding output. For example, when x = 3 is put into the function machine f at right, the function multiplies the 3 by 2 and then adds 1 to get the corresponding output, which is 7. The notation f (3) = 7 shows that the function named f connects the input 3 with the corresponding output 7. This also means the point (3, 7) lies on the graph of the function.

inputs x = 3

f (x) = 2x +1

f (3) =7 outputs

Example 2

a. If f (x) = x — 2 then f(11) =? f (1 1) = V11— 2

f (11) =

f (11) = 3

b. If g(x) = 3— x2 then g (5) = ? g (5) = 3— (5)2

g(5) = 3-25

g (5) = —22

c. If f (x) = 2x:35 then f(2) = ?

f (2) =

f (2) = —5

4 Core Connections Integrated I

2014 CPM Educational Program. All rights reserved.

Chapter 1

Example 3

A relationship in which each input has only one output is called a function.

g(x) is a function; each input (x) has only one output (y). g (-2) = 1, g(0) = 3, g(4) = —1, and so on.

f (x) is not a function: each input greater than —3 has two y-values associated with it. f(i) = 2 and f(i) = —2.

Example 4

The set of all possible inputs is called the domain, while the set of all possible outputs of is called the range.

In Example 3 above, the domain of g(x) is —2 x 4 , or "all numbers between —2 and 4". The range of g(x) is —1 5_ g(x) 3 or "all numbers between —1 and 3".

The domain of f(x) in Example 3 above is x —3 or "any real number greater than or equal to 3," since the graph starts at x = —3 and continues forever to the right. Since the graph of extends in both the positive and negative y (vertical) directions forever, the range is "all real numbers".

Example 5

For the graph at right, since the graph extends forever horizontally in both directions, the domain is "all real numbers". The y-values start at y = 1 and increase, so the range is y 1 or "all numbers greater or equal to V.

Parent Guide with Extra Practice 5

© 2014 CPM Educational Program. All rights reserved.

1. x = 2 --\

J * I f (x)= -2x+ 4

I I-

-,- f (x)= ?

4. f (x) = (5 — x)2

f(8) =?

13. 14. 15.

4X -

• - •

• •

.4 i i I 1 I I

Y 4—

_

-4 • - _

• - -4— •

Problems

Determine the outputs for the following function machines and the given inputs.

2. 3. x = -6 x = 9

1 1 j I I

F I( x ) • = I x - 2 IF

i f(x)= 47 +1

I LE 1 L f(x)=?

5. g(x) = x2 — 5 6. f (x) = 2x+7 x2-9

g (-3) = ? f(3) = ?

7. h(x) = 5 —IX 8. h(x) = NI 5 — x 9. f (x) = —x2 h(9) = ? h(9) = ? f (4) = ?

Determine if each graph below represents a function. Then state its domain and range.

10.

11.

12.

-4

-

-

6 Core Connections Integrated I

© 2014 CPM Educational Program. All rights reserved.

Chapter 1

Answers

1. f(2) = 0

4. f(8) = 9

7. f(9) = 2

10. Yes, each input has one output; domain is all numbers, range is —1 y 5 3

13. No; x = —1 has two outputs; domain is —4, —3,-1,0, 1, 2, 3, 4, range is —4,-3,-2, —1, 0, 1, 2

2. f(-6) = 8

5. g(-3) = 4

8. not possible

11. No, for example x = 0 has two outputs; domain is x —3, range is all numbers

14. Yes; domain is all numbers, range is y —2

3. f(9) = 4

6. not possible

9. f(4) = —16

12. Yes; domain all numbers, range is —3 5_ y 5 3

15. No, many inputs have two outputs; domain is —2 5_ x 5. 4 range is —2 5 y 5 4

Parent Guide with Extra Practice

C) 2014 CPM Educational Program. All rights reserved.

Example 1 (2xy3)(5x2y4

Reorder:

Using law (1):

Example 2 14x2y12

7x5y7

( 14

7

Using laws (2) and (5):

2.5.x-x2 .y3 -y4 Separate:

10x3y7

X2 r 12

,5 ,7 I

2y5 2x-3y- =

x3

LAWS OF EXPONENTS AND SCIENTIFIC NOTATION 1.3.1 and 1.3.2

Laws of Exponents

In general, to simplify an expression that contains exponents means to eliminate parentheses and negative exponents if possible. The basic laws of exponents are listed below.

(1) xa xb = xa+b Examples: x3 • x4 = x3+4 = x7

a x140 = x10-4 = x6 (2)

X = Xa—b Examples:

Xb

27 24 = 27+4 = 211

4 = 24-7 = 2-3 or 2 2-

(3) (xa b xab Examples: (x4 )3 = x4-3 = x12 2X3 )5 = 215 • X3.5 = 25 X15 = 32X15

(4) x° =1 Examples: 20 =1 (-3)o = 1 ( -)°

= 1 4

(5) x' = - Examples: x-3 = )r4 .

xn l x 3

1 = 1 1 1

16 4 Y Zr2 1

42 1

16

In all expressions with fractions we assume the denominator does not equal zero.

For additional information, see the Math Notes box in Lesson 1.3.2. For additional examples and practice, see the Checkpoint 4 materials.

8 Core Connections Integrated I

© 2014 CPM Educational Program. All rights reserved.

Using law (5): (2x3 )2

Using law (3): 1

22 (X3 )2

1 Using law (3) again:

4x6

8. (x3y2)3 9. (y 4 ) 2

(y 3 ) 2

7. 12m8

6m-3

14.

17. (3a2x3)2(2

20. (2x5y3 )3 oxy4 8 x7y12

23. (2x)-3

26. (2;)-2

(2a7)(3a2 )

6a3

18. x3y 4

y4 )

21. x-3

24. (2x3)°

)3

15.

( 4 3 x2) 13. (4xy2)(2y)3

5m3n )3 5

6x8y2 \ 2 12x3y7 )

2x-3

5-2.3

16.

19.

22.

25.

Chapter 1

Example 3 Example 4 3x2y4 )3 (2x3 ) 2

Using law (3): 33 .(x2)3 .(4)3

Using law (3) again: 27x6y12

Example 5

Simplify:

Separate:

Using law (2):

Using law (4):

10X7y3

15X-2y3

( 10 ) r 7

X \ 1 y3 \

\ y3

2 9 0 --x y 3

2x9 —2

x9

.1 = —2

x9 = 3 3 3

Problems

Simplify each expression. Final answers should contain no parentheses or negative exponents.

1.

4.

5 7

(y5 )2

2. b4 .b3 .b2 3.

5. (3a)4 6.

86.8-2

m8 m3

10. 15 x2y5

3x4y5 11. (4 c4 )(ac3)(3a5 c) 12. (7x 3y5 )2

Parent Guide with Extra Practice 9

© 2014 CPM Educational Program. All rights reserved.

Answers

1. y12 2. b9 3. 84

4. y10 5. 81a4 6. m5

7. 2mit 8. x9y6 9. y2

10. 5 x2 11. 12a6c8 12. 49x6y1°

13.

16.

32xy5

125713

14.

17.

64 x6

72a7x18

15.

18.

a6 x12

Y12

1116

19. xio

20. 16x1°y5 21. 1 x3 4)p

22. 2 x3

23. 1 24. 1 8x3

25. 3 25

26. 9

4x2

Scientific Notation

Scientific notation is a way of writing very large and very small numbers compactly. A number is said to be in scientific notation when it is written as the product of two factors as described below.

• The first factor is less than 10 and greater than or equal to 1.

• The second factor has a base of 10 and an integer exponent.

• The factors are separated by a multiplication sign.

• A positive exponent indicates a number whose absolute value is greater than 1.

• A negative exponent indicates a number whose absolute value is less than 1

Scientific Notation Standard Form

5.32 x 1011 532,000,000,000

2.61 x 10-15 0.00000000000000261

10 Core Connections Integrated I

© 2014 CPM Educational Program. All rights reserved.

Chapter 1

It is important to note that the exponent does not necessarily mean to use that number of zeros.

The number 5.32 x 1011 means 5.32 x 100,000,000,000. Thus, two of the eleven decimal places in the standard form of the number are the 3 and the 2 in 5.32. Standard form in this case is 532,000,000,000. In this example you are moving the decimal point eleven places to the right to write the standard form of the number.

The number 2.61 x 10-15 means 2.61 x 0.000000000000001. You are moving the decimal point to the left 15 places to write the standard form. Here the standard form is 0.00000000000000261.

For additional information, see the Math Notes box in Lesson 1.3.1.

Example 1

Write each number in standard form.

7.84x108 = 784,000,000 and 3.72x10 3 = 0.00372

When taking a number in standard form and writing it in scientific notation, remember there is only one digit to the left of the decimal point allowed.

Example 2

Write each number in scientific notation.

52,050,000 5.205x107 and 0.000372 = 3.72x10 4

The exponent denotes the number of places you moved the decimal point in the standard faint. In the first example above, the decimal point is at the end of the number and it was moved 7 places. In the second example above, the exponent is negative because the original number is very small, that is, less than 1.

Parent Guide with Extra Practice 11

C 2014 CPM Educational Program. All rights reserved.

Problems

Write each number in standard fond.

1. 7.85x10'1 2. 1.235x109 3. 1.2305x103 4. 3.89 x10-7 5. 5.28 x10-4

Write each number in scientific notation.

6. 391,000,000,000 7. 0.0000842 8. 123056.7 9. 0.000000502

10. 25.7 11. 0.035 12. 5,600,000 13. 1346.8

14. 0.000000000006

15. 634,700,000,000,000

Note: On your scientific calculator, displays like 4.35712 (or 4.357E12) and 3.65-3 (or 3.65E-3) are numbers expressed in scientific notation. The first number means 4.357 -1012 and the second means 3.65 -10-3 . The calculator does this because there is not enough room on its display window to show the entire number.

Answers

1. 785,000,000,000 2. 1,235,000,000 3. 1230.5

4. 0.000000389 5. 0.000528 6. 3.91x10"

7. 8.42x10 5 8. 1.230567x105 9. 5.02x10 7

10. 2.57x10' 11. 3.5x10 2 12. 5.6x106

13. 1.3468x103 14. 6.0x10'2 15. 6.347 x1014

12 Core Connections Integrated I

© 2014 CPM Educational Program. All rights reserved.

About CPM

CPM began as a grant-funded mathematics project in 1989 to write textbooks to help students

understand mathematics and support teachers who use these materials. CPM Educational Program is

now a nonprofit educational consortium of middle and high school teachers and university professors

that offers a complete mathematics program for grades 6 through 12 (Calculus) designed to engage all

students in learning mathematics through problem solving, reasoning, and communication.

CPM's Mission:

CPM's mission is to empower mathematics students and teachers through exemplary curriculum,

professional development, and leadership. We recognize and foster teacher expertise and leadership in

mathematics education. We engage all students in learning mathematics through problem solving,

reasoning, and communication.

CPM's Vision:

CPM envisions a world where mathematics is viewed as intriguing and useful, and is appreciated by all;

where powerful mathematical thinking is an essential, universal, and desirable trait; and where people

are empowered by mathematical problem-solving and reasoning to solve the world's problems.

The 3 Pillars of CPM: Cooperative Learning, Problem-Based Learning, and Mixed,

Spaced, Practice

Synthesis of Research on Cooperative Learning

2013 Introduction

As mentioned in the Executive Summary, there is almost no new research on the effectiveness of

cooperative learning because its effectiveness is now so widely acknowledged. There have been a few

papers summarizing previous research, the most complete of which is probably Ruthven (2011), which

looked extensively at results from TIMSS (Trends in International Mathematics and Science

Study). Smaller summative studies—for example, Slavin et. al. (2009) and Wittwer (2008)—are also

worth reading, but doing new definitive experimental studies is not seen to be worth the

expense. Recent studies extend to college-level physics, chemistry, and economics as well as to newly-

hired adult learners in industry, but there are few new broad educational studies as compared to five

years ago.

Most research in the past focused almost exclusively on individual intellectual attainment, where the

utility of group work was evaluated based on whether or not it improved the knowledge and skills of an

individual student in mathematics, physics, or medicine. Now researchers are moving beyond these

basic issues in different ways. Some researchers continue to look at the impact of cooperative learning

on individual attainment, but there is much more interest in understanding how certain group processes

enhance that learning. There is special interest in helping teachers encourage productive

processes. See, for example, Gillies (2004) comparing structured vs. unstructured groups, and Webb &

Mastergeorge (2003) who look at helping behaviors.

Part of this research that is applicable in schools is also being driven by research on educating

employees in the workplace, because employers, too, have recognized the effectiveness of learning in

teams. This realization also dovetails with the increasing number of demands by businesses for "people

who can work on a team". A cursory Internet search can uncover evidence from a variety of businesses

that are quite explicit about needing employees with these skills.

So the daily work in teams in the CPM classroom allows not only the deeper learning of mathematics,

but also the practice of important social skills as a prelude to becoming an effective member of the

workplace.

Research Findings

Does cooperative learning help students learn better?

It is unusual in educational research to see such unanimity of findings—in both individualistic settings

and randomized experiments. The consistency of these results over a wide span of age groups and a

wide set of topics indicates that a fundamental learning principle must be involved: social interaction

increases the ability to learn ideas as well as the ability to integrate these new ideas into existing

cognitive structures. The techniques for using collaborative learning groups can undoubtedly be

improved, and more is being learned all the time, but their overall efficacy is not in doubt.

History

In the 1970's and 1980's studies began on the effects of peer tutoring—that is, having older or more

able students tutor within classrooms. As was to be expected, students receiving the tutoring gained

significantly. Less expected was the discovery that students doing the tutoring gained even more. See

Dineen et. al. (1977), and Cohen et. al. (1982) for summaries of this research and Semb et. al. (1993) for

evidence that tutoring fosters longer-term retention.

Because both tutors and tutees were found to learn better by means of these conversations, many

people began to use them as an integral part of the learning experience. Teachers conceived of the

classroom as consisting of smaller cooperative learning groups in which every student would have a

chance both to tutor and to be tutored. The impacts of having such regular arrangements have been

larger than expected.

The effects of various forms of classroom cooperative learning groups (also known as small-group

learning or learning teams) have now been studied extensively for over 30 years. For thorough older

overviews of the research, the reader is directed to Sharan (1980), Davidson (1985), Qin et. al (1995),

Slavin (1996), and Springer et. al (1999). The most recent general articles are those cited above by

Slavin et. al. (2009) and Wittwer (2008). In two smaller studies, Tan (2007) looked at junior-high

students in Singapore and found no difference in average individual achievement for students in

cooperative learning groups when compared to those having whole-class instruction. In Hong Kong,

Cheng (2008) also found no differences for the same age group.

Other articles of general interest are Webb (1991), Yager et. al. (1986), Dees (1991), and Davidson &

Kroll (1991). The main result of all of these tens of thousands of hours of research is that cooperative

learning is a more effective model than direct instruction for students of all ages to learn most

concepts—and it is especially effective for students learning non-linguistic concepts (Qin, op. cit.).

Who does cooperative learning work for?

The short answer is almost every group that has been studied. As a brief cross-section of the results,

we mention the following: first graders learning math [Fuchs et al.(2002)]; eighth graders learning

science [Chi et al. (1994)1; junior high history and geography classes [Shachar & Sharan (1994)]; high

school geometry classes [Nichols (1996)1; pre-calculus students [Whicker et al. (1997)]; college level

classes in physics [Enghag et al. (2007)], chemistry [Overton & Potter (2011)], and economics [Yarnarik

(2007)]; and adults who are engineers [Cavalier et al. (1995)] or in management training [Nembhard et

al. (2009)1. These studies have been carried out not only in the U.S. but also in Great Britain, Australia,

Singapore, China, the Netherlands, and Turkey. All in all, the evidence is quite overwhelming.

Does cooperative learning work for high-ability students?

A commonly voiced concern by parents of high-ability students is that being part of a cooperative

learning group will interfere with their own child's learning. Stevens & Slavin (1995) addressed this

concern directly and concluded, after a two-year study in elementary school, that "gifted students in

heterogeneous cooperative learning classes had significantly higher achievement than their peers in

enrichment programs without cooperative learning". More recently, Carter et al. (2003) investigated

achievement gains of high-ability fifth-grade students in a science unit and found no significant

differences in the benefits to high-achieving students regardless of who they partnered with.

At the high-school level, Saleh et al. (2005) looked at students randomly assigned to homogeneous or

heterogeneous ability groups in a plant biology course, and the researchers concluded that "low-ability

students achieve more ... in heterogeneous groups ... whereas high-ability students show equally strong

learning outcomes in homogeneous and heterogeneous groups". In Hong Kong, Cheng et. al. (2008)

reported from a study of 367 groups that both "low and high achievers reported higher collective

efficacy than self-efficacy when group processes were of high quality" and concluded that "[the quality]

of group processes played a pivotal role."

Thus it appears that the achievement and learning of high-ability students is not hindered by their

participation in cooperative learning groups, and may, in fact, be increased because they have the

chance to act as tutors within the group.

Is teamwork valued in business?

[Note: The internet websites that are summarized in this section follow at the end of the academic

references.]

The most complete recent study was done by Google as Project Oxygen, which examined what traits are

important in a good manager. In a data-driven analysis—doubtless far more thorough than any an

academic researcher could ever afford to carry out—they found that four of the eight most important

skills for a manager involved the ability to work with and lead teams. A commentator, Paul Sohn,

observed that "many of the habits Google identified are the very same principles that make up good

management on invariably any organization."

In a memo on hiring, Cisco's first point about current college graduates noted their "strong interest in

working collaboratively in teams to reach a goal or solve a problem." In the Keller Graduate School of

Management list of five Traits of Effective Employees, #3 is "Effective employees work well with others."

Joe Hadzima, Chair of the MIT Enterprise Forum, asserts that one of the characteristic of a highly

effective entrepreneurial employee is that "The Right Stuff Employee is a true team player." There are

many other such quotes available—and no one talks about how they want to hire employees who prefer

working by themselves.

Returning to formal research results, Gillies has done a series of studies investigating the long-range

impact on students who work in cooperative groups. In Gillies (2000) she showed that first-grade

"children who have been trained to cooperate ... are able to demonstrate these behaviors in

reconstituted groups without additional training a year later." She followed up these results in Gillies

(2002) by showing that fifth-graders who had been trained in cooperative groups two years earlier were

"more cooperative and helpful than their untrained peers." So the impact of the ability to cooperate in a

group lasts well beyond the end of the year or the situation in which that learning occurred.

In a study of three high schools, Boaler (2008) found that students at the one school that extensively

employed teams for learning mathematics gained a key skill of effective employees: "[Students learned]

to treat each other in more respectful ways than is typically seen in schools" and were far more

helpful to classmates. In an equally positive benefit, "seniors at the end of high school they told us that

ethnic cliques ...did not form at their school because of the mathematics approach used." It should also

be noted that the students at this school learned more mathematics beginning at a lower level and

placed about twice the percentage of students in calculus than either of the other schools.

References

Boaler, J. (2008), Promoting 'relational equity' and high mathematics achievement through an

innovative mixed-ability approach. British Education Research Journal 34 (2): 167-194.

Carter, G., Jones, M. G., Rua, M. (2003). Effects of partner's ability on the achievement and conceptual

organization of high-achieving fifth-grade students. Science Education 87 (1): 94-111.

Cavalier, J. C., Klein, J. D., Cavalier, F. J. (1995). Effects of cooperative learning on performance, attitude,

and group behaviors in a technical team environment. ETR&D - Educational technology research and

development. 43 (3): 61-71.

Cheng, R.W., Sam, S., Chan, J. C., (2008). When high achievers and low achievers work in the same

group: The roles of group heterogeneity and processes in project-based learning. British Journal of

Educational Psychology, 78(2), 205-221.

Chi, M. T. H., DeLeeuw, N., Chiu, M. H., Lanancher, C. (1994). Eliciting self- explanations improves

understanding. Cognitive Science 18 (3): 439-477.

Cohen, P. A., Kulik, J. A., Kulik, C. L. C. (1982). Educational outcomes of tutoring — a meta-analysis of

findings. American Educational Research Journal 19 (2): 237-248.

Crouch, C. H., Mazur, E. (2001). Peer Instruction: Ten years of experience and results. American Journal

of Physics 69 (9): 970-977.

Davidson, N. (1985). Small group learning and teaching in mathematics: A selective review of the

research. In Learning to Cooperate, Cooperating to Learn, edited by R. Slavin et al. Plenum Press, New

York.

Davidson, N., Kroll, D. L. (1991). An overview of research on cooperative learning related to

mathematics. Journal for Research in Mathematics Education 22 (5): 362-365.

Dees, R. L. (1991). The role of cooperative learning in increasing problem-solving ability in a college

remedial course. Journal for Research in Mathematics Education 22 (5): 409-421.

Dineen, J. P., Clark, H. B., Risley, T. R. (1977). Peer tutoring among elementary students-educational

benefits to tutor. Journal of Applied Behavior Analysis 10 (2): 231-238.

Enghag, M. Gustafsson, P., Jonsson, G. (2007). From everyday life experiences to physics understanding

occurring in small group work with context rich problems during introductory physics work at

university. Research in Science Education 37 (4): 449-467.

Fuchs, L. S., Fuchs, D., Yazdian, L., Powell, S. R. (2002). Enhancing first-grade children's mathematical

development with Peer-Assisted Learning Strategies. School Psychology Review 31 (4): 569-583.

Gillies, R. M. (2000). The maintenance of cooperative and helping behaviours in cooperative

groups. British Journal of Educational Psychology 70 (1): 97-111.

Gillies, R. M. (2002). The residual effects of cooperative-learning experiences: A two- year follow-

up. Journal of Educational Research 96 (1): 15-20.

Gillies, R. M. (2004). The effects of cooperative learning on junior high school students during small

group learning. Learning and Instruction 14 (2): 197-213.

Nennbhard, D., Yip, K., Shtub, A. (2009). Comparing competitive and cooperative strategies for learning

project management. Journal of Engineering Education 98 (2): 181-192.

Nichols, J. D. (1996). The effects of cooperative learning on student achievement and motivation in a

high school geometry class. Contemporary Educational Psychology 21 (4): 467-476.

Overton, T.L., Potter, N.M. (2011). Investigating students' success in solving and attitudes towards

context-rich open-ended problems in chemistry. Chemistry Education Research and Practice 12 (3): 294-

302.

Qin, Z. N., Johnson, D. W., Johnson, R. T. (1995). Cooperative versus competitive efforts and problem-

solving. Review of Educational Research 65 (2): 129-143.

Ruthven, K. (2011). Using international study series and meta-analytic research syntheses to scope

pedagogical development aimed at improving student attitude and achievement in school mathematics

and science. International Journal of Science and Mathematics Education 9 (2): 419-458.

Saleh, M., Lazonder, A. W., De Jong, T. (2005). Effects of within-class ability grouping on social

interaction, achievement, and motivation. Instructional Science 33 (2): 105-119.

Semb, G. B., Ellis, J. A., Araujo, J. (1993). Long-term memory for knowledge learned in school. Journal of

Educational Psychology 85 (2): 305-316.

Shachar, H., Sharan, S. (1994). Talking, relating, and achieving - effects of cooperative learning and

whole-class instruction. Cognition and Instruction 12 (4): 313-353.

Sharan, S. (1980). Cooperative learning in small groups: Recent methods and effects on achievement,

attitudes, and ethnic relations. Review of Educational Research 50<: 241-271.

Slavin, R. E. (1996). Research on cooperative learning and achievement: What we know, what we need

to know. Contemporary Educational Psychology 21 (1): 43-69.

Springer, L., Stanne, M. E., Donovan, S. S. (1999). Effects of small-group learning on undergraduates in

science, mathematics, engineering, and technology: A meta-analysis. Review of Educational

Research 69 (1): 21-51.

Treisman, U. (1985) A study of the mathematical performance of Black students at the University of

California, Berkeley. Thesis, University of California, Berkeley.

Vigotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge,

MA: Harvard University Press.

Webb, N. M., Mastergeorge, A. M. (2003). The development of students' helping behavior and learning

in peer-directed small groups. Cognition and Instruction 21 (4): 361-428.

Whicker, K. M., Bol, L., Nunnery, J. A. (1997). Cooperative learning in the secondary mathematics

classroom. Journal of Educational Research 91 (1): 42-48.

Yager, S., Johnson, R., Johnson, D., and Snider, B. (1986). The impact of group processing on

achievement in cooperative learning groups. The Journal of Social Psychology 126 (3): 389-397.

Yamarik, S. (2007). Does cooperative learning improve student learning outcomes? Journal of Economic

Education 38 (3): 259-277.

Internet references to working as a part of teams

Cisco (2012) Attracting and Engaging the Gen Y Workforce

http://www.cisco.com/en/US/solutions/collateral/ns340/ns1176/business-of-

it/Trends_in_IT_Gen_Y_Flexible_Collaborative_Workspace.html

Engineering Schools.com: Top Ten Qualities of a Great Engineer http://engineeringschools.com/resources/top-10-qualities-of-a-great-engineer

Google (2011) Project Oxygen: Eight Habits of Effective Google Managers

http://www.nytimes.com/2011/03/13/business/13hire.html?pagewa nted=all

http://paulsohn.org/the-eight-habits-of-effective-google-managers/

Joe Hadzima, Seven Characteristics of Highly Effective Entrepreneurial Employees.

http://web.mit.ed u/e-club/hadzima/seven-characteristics-of-highly-effective-entrepreneurial-

employees.html

Keller Graduate School of Management of DeVry University, Traits of Effective Employees

http://smallbusiness.chron.com/traits-effective-employees-18302.html

Synthesis of research on problem-based learning

2013 Introduction

The research of the last seven years on Problem Based Learning (PBL) has also accepted the view that

PBL is a good strategy to help students from first grade through college to retain more knowledge from

their instruction than they do when they are simply lectured to and told a rule or procedure. It seems to

be especially true for students learning mathematics or science.

Additional research has simply confirmed this fact. Relatively few of the results are useful to CPM,

however, as a large fraction of current research is devoted to understanding what happens in a

computer-based PBL setting. Such studies have the significant advantage of being easier and much,

much cheaper to carry out than classroom-based research. Science and medical education are also the

beneficiaries of many of these studies.

New results relevant to CPM that are worth noting, however, are that assisted PBL is the most useful

variant of the technique, that it is important to do PBL in a cooperative learning environment, and that

students of virtually all learning abilities can profit from a PBL curriculum.

What is Problem Based Learning?

In the simplest terms, PBL means working on problems in order to develop an understanding together

with a procedure for solving them rather than practicing a procedure after being told. Trying to learn

something in the first place is a very different goal than trying to imitate a practice, and it needs

different methods. No single method is superior for all children and all topics and all cases. Every

successful program needs a mix of the methods, and every student needs both. No one can be asked to

discover the definition of a trapezoid, and no one can be told the concept of an unknown. The former is

a matter of social convention while the latter is such a deep concept that words are inadequate. At

different times, students need different opportunities, and learning different topics requires different

methods and different time frames.

But two very different classroom activities have been done under the name PBL for some time. In a

meta-analysis, Alfieri et al. (2011) looked at the results of 164 different studies that compared different

types of PBL with a traditional direct instruction format. One group of studies

involved unassisted discovery (here is a problem of a kind you have never seen before; figure out how to

solve it in any way you can) while the other group of studies used assisted discovery, where problems

are organized in a sequence to promote discovery of the answer. The conclusion was that assisted

discovery was better than direct instruction, which, in turn, was better than unassisted discovery. CPM

has always used assisted instruction. See also Hmelo-Silver (2004) or Prince & Felder (2006) for more

extensive discussions of types of PBL.

Why use Problem Based Learning?

Unfortunately, there is a strong belief on the part of many educators that students need only to be told

what to do, and if they are told properly and practice, they will learn the fact, skill, or concept. This

method certainly seems efficient at conveying knowledge. The trouble with this belief is that it is not

true, except when one is dealing with very young children or with topics that have no unifying structure,

such as names of state capitals. It is certainly not true in mathematics.

The problems with teaching by telling have been amply documented by many researchers in

mathematics and science at all levels and for most types of students, for almost as long as cooperative

learning has been studied. Carpenter et al. (1998) followed students in grades 1-3 for three years and

found that "students who used invented strategies before they learned standard algorithms

demonstrated better knowledge of base-ten number concepts and were more successful in extending

their knowledge to new situations than were students who initially learned standard algorithms." Similar

results were reported for students this age by Hiebert & Wearne (1996) and Cauley (1998). Some

research even indicates that being told rules before attempting to forge a personal understanding can

interfere with deeper learning.

For sixth graders, Hmelo et al. (2000) found that science design activities, which allow deeper

explorations of how systems work, helped students "learn more than students receiving direct

instruction." For eighth graders, Woodward (1994) reported that students who learned the reasons for

earth science phenomena "had significantly better retention of facts and concepts and were superior in

applying this knowledge in problem-solving exercises." Azer (2009) reported that "Saudi students from

the fifth, sixth and seventh grades perceived PBL in a positive way" following up on other work that he

had done with medical students.

In a survey article, McDermott & Redish (1999) have demonstrated that college level physics students

do not learn some very basic content through lecture, and this result has been duplicated with

thousands of students at many institutions ranging from very selective private institutions and large

state universities down through high schools. The work of Crouch & Mazur (2001), reporting on ten

years using Peer Instruction (an interactive method of teaching) for the introductory physics courses at

Harvard, shows "increased student mastery of both conceptual reasoning and quantitative problem

solving upon implementing Peer Instruction."

The work of Capon and Kuhn (2004) with adult students contrasting the outcomes of problem-based

learning with lecture and discussion showed that six weeks after instruction, the lecture group was

superior in the understanding of one concept and the two groups were equivalent in understanding of

the other. After 12 weeks, concept retention was equal, but the problem-based learning group was

superior in being able to explain what they had learned. Masek & Yannin (2012) showed that electrical

engineering students learned more about the principles and procedures in their first electrical

technology module.

These results are not only true in mathematics and the sciences. Cobb (1999) reports on a study of

students learning English in the Sultanate of Oman who learned vocabulary in two ways: by memorizing

dictionary definitions or by constructing their own definitions using the tools of lexicographers. "After 12

weeks, both groups were equal in definitional knowledge of target words, but lexicography group

students were more able to transfer their work knowledge to novel contexts."

Who should be taught using PBL?

At the same time that studies have demonstrated the failure of direct instruction for average students,

other studies have shown the advantages of PBL. Most of these studies have been done with gifted

students in K-12 or with older students studying engineering or medicine. See, for example, Albanese &

Mitchell (1993) for an extensive review of the medical literature on PBL, Prince (2004) for a briefer

summary on its uses with engineering students, and Dods (1997) or Gallagher & Stepien (1996) for

studies about gifted children learning with PBL.

These results confirm what has long been believed, that something akin to problem-based learning is

superior for learning, when it is appropriate for the students involved. These earlier studies focused on

students of ability—gifted elementary students or students in rigorous college programs. The implicit

assumption was that only a small minority of students could benefit from such an approach.

In the past 15 years, however, studies have found that well-designed PBL courses can benefit most, if

not all, students. Songer et al. (2002) reported on a study of 19 urban sixth-grade classes showing that

students in all classrooms made significant content and inquiry gains. Kahle et al. (2000) studied eight

middle schools in Ohio and showed that teachers who used PBL or a modified form of it for teaching

science "positively influenced urban, African-American science achievement."

The study of Marx et al. (2004) on approximately 8000 middle school students in the Detroit public

schools showed (1) statistically significant increases on test scores and (2) an increased effect for each of

the three years that students were in the program. At the college level, Hake (2002) reported on the

pre- and post-test gains for more than 6500 students in introductory physics classes, demonstrating the

large positive effect of interactive engagement.

In smaller studies, Sendag & Odabasi (2009), in a study of online learning for future mathematics

primary teachers, found that knowledge acquisition was not different for students in PBL experience,

but that critical thinking skillswere significantly improved. Similarly, Schneider et al. (2002) reported the

performance of 10th and 11th grade students enrolled in Problem-Based Science was significantly better

than matched groups on the National Assessment of Educational Progress science items, while Gallagher

& Stepien (1996) reported that gifted students in a PBL class acquired as much content as students in a

traditionally-taught class and acquired additional skills as well.

More recently, a small Singapore study of a seventh-grade mathematics class by Kapur (2010) concluded

that the students who engaged the "productive failure" of working on a complex problem followed by a

summary lecture by the teacher "significantly outperformed their counterparts" who had been taught

by a traditional "lecture and practice" method. But the students did not like the class as much. A

somewhat different result was found in another small study of sixth-grade students in Turkey by Demirel

& Turan (2010) which showed that PBL students both learned more and liked the experience better.

Dods (1997) reported that lecture tended to widen the coverage as compared to a PBL class for gifted

students in biochemistry, but "understanding and retention [were] promoted by PBL [emphasis added]."

A similar result was reported in the meta-analysis of studies by Dochy et al. (2003), which concluded

that "students in PBL gained slightly less knowledge, but remember[ed] more of the acquired

knowledge."

What these research pieces show is that the goals of long-term learning are better achieved by PBL and

that virtually all students can profit from this form of education. In particular, there is no need to

restrict this superior form of learning to the academically elite. This is why CPM structures its lessons so

that students are told as much as necessary for learning a topic, but based on the research cited above,

assumes that most of the learning—the quality learning—will take place while students are working on

problems.

How is Problem Based Learning Best Implemented?

The research in the past seven years has done nothing to contradict the earlier support of the central

concept of PBL, but in the intervening years the emphasis has changed to looking more carefully at what

other components must be present for a successful problem-based class. Furtak et al. (2012) has done a

comprehensive meta-analysis of various studies in science education, concluding that social

interaction (some form of cooperative learning or with a tutor) is an important component of problem-

based learning, a finding echoed by DeCaro & Rittle-Johnson (2012), which emphasized the role of

teacher control of activities. (Note: "teacher control" in this context means that the teacher is

responsible for ensuring that students are working well and on the mathematical topic—not that the

teacher is telling the students what to do.) The same results were found in an extensive German study

of 100 mathematics classrooms at the eighth grade level. Gruehn (2000) studied results from the 1997

TIMMS international comparison. In a very small, intense study, Yew & Schmidt (2012) found that for

college freshmen, "collaborative learning is significant in the PBL process, and may be more important

than individual study in determining students' achievement." For medical students, Schmidt et al.

(2011) concluded that learning in a PBL classroom requires both the social interaction of teams and also

individual learning.

So while PBL has been done with students working alone, it is clear that most students benefit from

collaboration.

References

Albanese, M.A., Mitchell, S. (1993). Problem-based learning — a review of literature on its outcomes and

implementation issues. Academic Medicine 68 (1): 52-81.

Alfieri, L., Brooks, P.J., Aldrich, N.J., Tenenbaum, H.R. (2011). Does Discovery-Based Instruction Enhance

Learning? Journal of Educational Psychology103 (1): 1-18.

Azer, S.A. (2009). Problem-based learning in the fifth, sixth, and seventh grades: Assessment of students'

perceptions. Teaching and Teacher Education 25 (8): 1033-1042.

Carpenter, T.P., Franke, M.L., Jacobs, V.R., Fennema, E., Empson, S.B. (1998). A longitudinal study of

invention and understanding in children's multidigit addition and subtraction. Journal for Research in Mathematics Education 29 (1): 3-20.

Cauley, K.M. (1988). Construction of logical knowledge — study of borrowing in subtraction. Journal of

Educational Psychology 80 (2): 202-205.

Cobb, T. (1999). Applying constructivism: A test for the learner-as-scientist. ETR&D - Educational technology research and development47 (3): 15-31.

Crouch, C.H., Mazur, E. (2001). Peer Instruction: Ten years of experience and results. American Journal of Physics 69 (9): 970-977.

DeCaro, M.S., Rittle-Johnson, B. (2012). Exploring mathematics problems prepares children to learn

from instruction. Journal of Experimental Child Psychology113 (4), 552-568.

Demirel, M., Turan, B.A. (2010). The Effects of Problem Based Learning on Achievement, Attitude,

Metacognitive Awareness and Motivation. Hacettepe University Journal of Education 38: 55-66.

Dochy, F., Segers, M., Van den Bossche, P., Gijbels, D. (2003). Effects of problem- based learning: a meta-

analysis. Learning and Instruction 13 (5): 533-568.

Dods, R.F. (1997). An action research study of the effectiveness of problem-based learning in promoting

the acquisition and retention of knowledge. Journal for the Education of the Gifted 20 (4): 423-437.

Fleischner, J.E., Manheinner, M.A. (1997). Math interventions for students with learning disabilities:

myths and realities. School Psychology Review 26 (3): 397-413.

Furtak, E.M., Seidel, T., Iverson, H., Briggs, D.C. (2012). Experimental and Quasi-Experimental Studies of

Inquiry-Based Science Teaching: A Meta-Analysis. Review of Educational Research82 (3): 300-329.

Gallagher, S.A., Stepien, W.J. (1996). Content acquisition in problem-based learning: Depth versus

breadth in American studies. Journal for the Education of the Gifted 19 (3): 275-275.

Gruehn, S. (2000). Unterricht und schulisches Lernen. Schiller als Quellen der Unterrichtsbeschreibung.

Munster, New York: Waxmann.

Hake, R. (2002). Lessons from the physics education reform effort. Conservation Ecology 5 (2): Art. No.

28, Jan.

Hiebert, J., Wearne, D. (1996). Instruction, understanding, and skill in multidigit addition and

subtraction. Cognition and Instruction 14 (3): 251-283.

Hmelo, C.E., Holton, D.L., Kolodner, J.L. (2000). Designing to learn about complex systems. Journal of the

Learning Sciences 9 (3), 247-298.

Hmelo-Silver, C.E. (2004). Problem-based learning: What and how do students learn? Educational

Psychology Review 16 (3): 235-266.

Kahle, J.B., Meece, J., Scantlebury, K. (2000). Urban African-American middle school science students:

Does standards-based teaching make a difference? Journal of Research in Science Teaching 37 (9): 1019-

1041.

Kamp, R.J.A., Dolmans, D.H.J.M., van Berke!, H.J.M., Schmidt, H.G. (2012). The relationship between

students small group activities, time spent on self-study, and achievement. Higher Education 64 (3):

385-397.

Kapur, M. (2010). Productive failure in mathematical problem solving. Instructional Science 38 (6): 523-

550.

Marx, R.W., Blumenfeld, P.C., Krajcik, J.S., Fishman, B., Soloway, E., Geier, R., Tali, R.T. (2004). Inquiry-

based science in the middle grades: Assessment of learning in urban systemic reform. Journal of

Research in Science Teaching 41 (10): 1063-1080.

McDermott, L.C., Redish, E.F. (1999). Resource letter: PER-1: Physics education research. American

Journal of Physics 67 (9): 755-767.

Pease, M.A., Kuhn, D. (2011). Experimental Analysis of the Effective Components of Problem-Based

Learning. Science Education 95 (1): 57-86.

Prince, M.J. (2004). Does active learning work? A review of the research. Journal of Engineering

Education 93 (3): 223-231.

Prince, M.J., Felder, R.M. (2006). Inductive teaching and learning methods: Definitions, comparisons,

and research bases. Journal of Engineering Education.95 (2), 123-138.

Rittle-Johnson B., Siegler R.S., Alibali, M.W. (2001). Developing conceptual understanding and

procedural skill in mathematics: An iterative process. Journal of Educational Psychology 93 (2): 346-362.

Rittle-Johnson, B., Alibali, M.W. (1999). Conceptual and procedural knowledge of mathematics: Does

one lead to the other? Journal of Educational Psychology 91 (1): 175-189.

Schmidt, H.G., Rotgans, J.I., Yew, E.H.J. (2011). The process of problem-based learning: what works and

why. Medical Education 45 (8): 792-806.

Schneider, R.M., Krajcik, J., Marx, R.W., Soloway, E. (2002). Performance of students in project-based

science classrooms on a national measure of science achievement. Journal of Research in Science

Teaching 39 (5): 410-422.

Sendag, S., Odabasi, H.F. (2009). Effects of an online problem based course on content knowledge

acquisition and critical thinking skills. Computers & Education 53 (1), 132-141>

Skemp, R.R. (1986). The psychology of learning mathematics (2nd ed.). Middlesex, England: Penguin

Books.

Sockalingam, N., Rotgans, J.I., Schmidt, H.G. (2011). The relationships between problem characteristics,

achievement-related behaviors, and academic achievement in problem-based learning. Advances in

Health Sciences Education16 (4): 481-490.

Songer, N.B., Lee, H.S., Kam, R. (2002). Technology-rich inquiry science in urban classrooms: What are

the barriers to inquiry pedagogy?Joumal of Research in Science Teaching 39 (2): 128-150.

Star, J.R., Rittle-Johnson, B. (2008). Flexibility in problem solving: The case of equation solving. Learning

and Instruction 18 (6): 565-579.

Woodward, J. (1994). Effects of curriculum discourse style on 8th graders' recall and problem-solving in

earth-science. Elementary School Journal 94 (3): 299-314.

Yew, E.H.J., Schmidt, H.J. (2012). What students learn in problem-based learning: a process

analysis. Instructional Science 40 (2): 371-395.

Synthesis of Research on Mixed, Spaced Practice

2013 Introduction

The biggest change in the research pertaining to CPM in the past six years has been the huge rise in

interest in what is now known as mixed, spaced practice. At the time that the previous review summary

was written, there were only a few dozen papers dealing with this topic. Now there are almost

2000. Many researchers are taking the idea seriously.

It has been known for many years that spacing out review sessions over time increases the long-term

retention of the knowledge; this is known as spaced practice. The new feature of research on the role of

review in student learning is mixed practice, where students deal with different kinds of problems during

a single review or homework session. But, as Dempster (1988) noted a quarter of a century ago, the

knowledge about the effectiveness of spaced practice was almost never utilized in designing

curricula. At that time, mixed practice was nowhere on the research agenda, and this new knowledge is

rarely used in designing textbooks either.

The major reason that people persist in using massed practice—lots of similar problems of the same

kind all at once—rather than spacing it out is that this kind of practice feels good immediately. Students

believe they have learned what they were supposed to learn because they can follow a pattern, and

teachers believe that they have taught it because they see students getting the right answers. So

everyone is happy on that day. The problem is that the effect fades away quickly.

Small children often learn by following patterns, and learn to engage in a practice without

understanding why they are doing something in the way they are doing it and usually never even

thinking that they should put the action in a larger context. This is fine when you are young, and these

personal histories of success are hard to ignore, even as cognitive scientists are discovering much more

about consolidation of memories. As Kornell & Bjork (2008) pointed out after a study on learning,

"Participants rated massing as more effective than spacing, even after their own test performance had

demonstrated the opposite" [emphasis added].

What does the research show about spaced practice?

By now, the "spacing effect" is an overwhelmingly well-documented phenomenon that shows that

learning is improved when the learning practice is spaced over time, rather than being massed, or

happening all at once and then being ignored. In the past 70 years, dozens of researchers of psychology,

workplace training and education have validated this "spacing effect." Researchers who study workplace

training refer to "distributed practice" or "spaced practice" (as opposed to "massed practice") as the

cause of the spacing effect while they seek methods of improving the effectiveness of training programs

or workers. Roughly speaking, as long as there is some latent memory of earlier learning of a skill,

delaying the reinforcement by spacing improves both transfer and long-term learning. See Carpenter et

al. (2012) or Son & Simon (2012) for good summary review articles about spaced practice.

Psychologists have verified the phenomenon in babies as young as three months of age in one study

[(Roveecollier et al., (1995)] and in numerous studies for school-age children up to adults and in areas as

diverse as rolling kayaks [(Smith & Davies, (1995)], aircraft recognition [(Goettl, (1996)] and learning

languages [(Bahrick & Phelps, ( 1987) and Bahrick et al., (1993)]. Because the spacing effect appears in

so many contexts, it appears, as Raaijmakers (2003, p. 432) commented, "that basic principles of

learning and retention are involved" [emphasis added].

Rohrer & Pashler (2010) commented that "the temporal dynamics of learning show that learning is most

durable when study time is distributed over much greater periods of time than is customary in

educational settings." Rohrer (2009) went further from his study of overlearning (unneeded practice)

and flatly states that "overlearning is an inefficient use of study time," and Rohrer & Taylor (2006)

lamented that "most mathematics textbooks rely on a format that emphasizes overlearning and

minimizes distributed practice." For further references, see Seabrook et at. (2005) on learning reading,

Vlach & Sandhofer (2012) on elementary age children learning science concepts, Rohrer & Pashler

(2007) on learning mathematics, and Bude et al. (2011) for a study on college students learning

statistics.

Even with all of this research, there is a significant reluctance to use spaced practice in the classroom. A

major reason is that this practice slows down the initial learning at the same time that it improves

long-term retention and transfer. Rohrer et al. (2005) pointed out in a study of geography students,

"The overlearners recalled far more than the low learners at the one-week test, but this difference

decreased dramatically thereafter." Other studies making the same findings are Karpicke & Roediger

(2007) and Vlach & Sandhofer (2012).

What does the research show about mixed practice?

The research on mixed practice—interweaving different types of mathematics problems in a single

homework session—is much newer, and fewer people have published studies about it. Rohrer & Taylor

(2007) found that for college students, "performance was vastly superior after mixed practice." In 2010,

Rohrer & Pashler found that "interleaving of different types of practice problems (which is quite rare in

math and science texts) markedly improves learning." An earlier result from the research by Hatala et

at. (2003), which focused on how to teach medical students to read ECGs, also showed support for

mixed practice and implies that students studying subjects other than mathematics and science can

benefit from this strategy. While all of these studies were done on people of college age or older, there

seems to be no reason to believe that similar effects would not be found for school-age students. In

fact, Rohrer (2009) provides a strong rationale for incorporating both spaced and mixed practice

regularly:

o Spacing provides review that improves long-term retention, and mixing [problem types]

improves students' ability to pair a problem with the appropriate concept or

procedure. Hence, although mixed review is more demanding than blocked practice,

because students cannot assume that every problem is based on the immediately

preceding lesson, the apparent benefits of mixed review suggest that this easily adopted

strategy is underused.

CPM has been using mixed, spaced practice for 24 years, and virtually all of our teachers believe that

this practice is central to improving long-term student learning.

References

Bahrick, H.P., Phelps, E. (1987). Retention of Spanish vocabulary over 8 years. Journal of Experimental

Psychology— Learning, Memory and Cognition 13 (2): 344-349.

Bahrick, H.P., Bahrick, L.E., Bahrick, A.S., Bahrick, P.E. (1993). Maintenance of foreign-language

vocabulary and the spacing effect. Psychological Science 4 (5): 316-321.

Benjamin, A.S., Tullis, J. (2010). What makes distributed practice effective? Cognitive Psychology 61 (3):

228-247.

Bude, L., Innbos, T., van de Wiel, M.W., Berger, M.P. (2011). The effect of distributed practice on

students' conceptual understanding of statistics. Higher Education 62 (1): 69-79.

Cantor, J., Engle, R.W. (1993). Working-memory capacity as long-term memory activation: an individual-

differences approach. Journal of Experimental Psychology: Learning, Memory and Cognition 19 (5):

1101-1114.

Carpenter, S.K., Cepeda, N.J., Rohrer, D., Kang, S.H.K., Pashler, H. (2012). Using Spacing to Enhance

Diverse Forms of Learning: Review of Recent Research and Implications for Instruction. Educational

Psychology Review 24(3): 369-378.

Cepeda, N.J., Vul, E., Rohrer, D., Wixted, J.T., Pashler, H. (2008). Spaced Effects in Learning: A Temporal

Ridgeline of Optimal Retention. Psychological Science 19 (11): 1095-1102.

Chen, Z., Mo, L. (2004). Schema induction in problem solving: A multidimensional analysis. Journal of

Experimental Psychology— Learning, Memory and Cognition 30 (3): 583-600.

Dempster, F.N. (1988). The spacing effect — a case study in the failure to apply the results of

psychological research. American Psychologist 43 (8): 627-634.

Fisher, D.L., Wisher, R.A., Ranney, T.A. (1996). Optimal static and dynamic training schedules: State

models of skill acquisition. Journal of Mathematical Psychology 40 (1): 30-47.

Gick, M.L., Holyoak, K.J. (1987). The cognitive basis of knowledge transfer. In S.M. Cormier & J.D.

Hagman (Eds.), Transfer of learning: Contemporary research and applications (pp. 9-46). San Diego:

Academic Press.

Gleason, M., Carnine, D., Valia, N. (1991). Cumulative versus rapid introduction of new

information. Exceptional Children 57: 353-358.

Grote, M.G. (1995). The effect of massed versus spaced practice on retention and problem-solving in

high-school physics. Ohio Journal of Science 95 (3): 243-247.

Goettl, B.P. (1996). The spacing effect in aircraft recognition. Human Factors 38 (1): 34-49.

Hatala, R.M., Brooks, L.R., Norman, G.R. (2003). Practice makes perfect: The critical role of mixed

practice in the acquisition of ECG interpretation skills. Advances in Health Sciences Education 8 (1): 17-

26.

Hesketh, B. (1997). Dilemmas in training for transfer and retention. Applied Psychology —An

International Review 46 (4): 317-339.

Karpicke, J.D., Bauernschmidt, A. (2011). Spaced Retrieval: Absolute Spacing Enhances Learning

Regardless of Relative Spacing. Journal of Experimental Psychology-Learning, Memory, and

Cognition 37 (5): 1250-1257.

Karpicke, J.D., Roediger, H.L. (2007). Expanding retrieval practice promotes short-term retention, but

equally spaced retrieval enhances long-term retention. Journal of Experimental Psychology-Learning,

Memory, and Cognition 33(4): 704-719.

Kornell, N., Bjork, R.A. (2008). Learning concepts and categories: Is spacing the "Enemy of

Induction"? Psychological Science 19 (6): 585-592.

Mayfield, K.H., Chase, P.N. (2002). The effects of cumulative practice on mathematics problem

solving. Journal of Applied Behavior Analysis 35 (2): 105-123.

Mizuno, R. (2002). The cause of the spacing effect: A test on the influence of reactivation

amount. Japanese Journal of Educational Psychology 46 (1): 11-20.

Pashler, H., Zarow, G., Triplett, B. (2003). Is temporal spacing of tests helpful even when it inflates error

rates? Journal of Experimental Psychology: Learning, Memory, and Cognition 29:1051-1057.

Piaget, J. (1957). Construction of Reality in the Child. London: Routledge & Kegan Paul.

Raaijmakers, J.G.W. (2003). Spacing and repetition effects in human memory: application of the SAM

model. Cognitive Science 27 (3): 431-452.

Rohrer, D. (2009). The effects of spacing and mixing practice problems. Journal for Research in

Mathematics 40 (1): 4-17.

Rohrer, D. (2009). Avoidance of overlearning characterises the spacing effect. European Journal of

Cognitive Psychology 21 (7): 1001-1012.

Rohrer, D., Pashler, H. (2007). Increasing retention without increasing study time. Current Directions in

Psychological Science 16 (4): 183-186.

Rohrer, D., Pashler, H. (2010). Recent Research on Human Learning Challenges Conventional

Instructional Strategies. Educational Researcher 39 (5): 406-412.

Rohrer, D., Taylor, K. (2006). The effects of overlearning and distributed practice on the retention of

mathematics knowledge. Applied Cognitive Psychology 20 (9): 1209-1224.

Rohrer, D., Taylor, K. (2007). The shuffling of mathematics problems improves learning. Instructional

Science 35 (6): 481-498.

Rohrer, D., Taylor, K., Pashler, H., Wixted, J.T., Cepeda, N.J. (2005). The effect of overlearning on long-

term retention. Applied Cognitive Psychology 19 (3): 361-374.

Roveecollier, C., Evancio, S., Earley, L.A. (1995). The time window hypothesis - spacing effects. Infant

Behavior & Development 18 (1): 69-78.

Salas, E., Cannon-Bowers, J.A. (2001). The science of training: a decade of progress. Annual Review of

Psychology 52: 471-499.

Schilling, M.A., Vidal, P., Ployhart, R.E., Marangoni, A. (2003). Learning by doing something else:

variation, relatedness and the learning curve. Management cience 49 (1): 39-56.

Schmidt, R.A., Bjork, R.A. (1992). New conceptualizations of practice - common principles in 3

paradigms suggest new concepts for training. Psychological Science 3 (4): 207-217.

Schroth, M.L. (1997). The effects of different training conditions on transfer in concept

formation. Journal of General Psychology 124 (2): 157-165.

Seabrook, R., Brown, G.D.A., Solity, J.E. (2005). Distributed and massed practice: From laboratory to

classroom. Applied Cognitive Psychology 19 (1): 107-122.

Shea, J.B., Morgan, R.L. (1979). Contextual interference effects on the acquisition, retention, and

transfer of a motor skill. Journal of Experimental Psychology— Human Learning and Memory 5 (2): 179-

187.

Skemp, R.R. (1986). The Psychology of Learning Mathematics (2nd ed.). Middlesex, England: Penguin

Books.

Smith, PJ.K., Davies, M. (1995). Applying contextual interference to the Pawlata roll. Journal of Sport

Sciences 13 (6): 455-462.

Smith, S.M., Rothkopf, E.Z. (1984). Contextual enrichment and distribution of practice in the

classroom. Cognition and Instruction 1 (3): 341-358.

Son, L.K., Simon, D.A. (2012). Distributed Learning: Data, Metacognition, and Educational

Implications. Educational Psychology Review 24 (3): 379-399.

Storm, B.C., Bjork, R.A., Storm, J.C. (2010). Optimizing retrieval as a learning event: When and why

expanding retrieval practice enhances long-term retention. Memory and Cognition 38 (2): 244-253.

Vlach, N.A., Sand hofer, C.M. (2012). Distributing Learning Over Time: The Spacing Effect in Children's

Acquisition and Generalization of Science Concepts. Child Development 83 (4): 1137-1144.