math problem

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Question description

©

Noyce Foundation 2014.

This work is licensed under a Cr

eative Commons Attribution‐NonC

ommercial‐NoDerivatives 3.0

Unported License (

http://creativecommons.org/lic

enses/by‐nc‐nd/3.0/deed.en_US

).

Problem

of

the

Month

Diminishing

Return

The Problems of the Month (POM)

are used in a variety of ways t

o promote problem

solving and to foster the first

standard of mathematical practi

ce from the Common

Core State Standards: “Make sense

of problems and persevere in

solving them.” The

POM may be used by a teacher to

promote problem solving and to

address the

differentiated needs of her stude

nts. A department or grade le

vel may engage their

students in a POM to showcase problem solving as a key aspect o

f doing

mathematics. POMs can also be us

ed school wide to promote a pr

oblem‐solving

theme at a school. The goal is f

or all students to have the ex

perience of attacking

and solving non‐routine problems

and developing their mathemati

cal reasoning

skills. Although obtaining and justifying solutions to the pro

blems is the objective,

the process of learning to solve

problem is even more important

.

The Problem of the Month is stru

ctured to provide reasonable ta

sks for all students

in a school. The structure of

a POM is a shallow floor and a h

igh ceiling, so that all

students can productively engage, struggle, and persevere. The

Primary Version is

designed to be accessible to all

students and especially the ke

y challenge for grades

K – 1. Level A will be challeng

ing for most second and third

graders. Level B may

be the limit of where fourth and

fifth‐grade students have succ

ess and

understanding. Level C may stret

ch sixth and seventh‐grade stu

dents. Level D may

challenge most eighth and ninth‐

grade students, and Level E sho

uld be challenging

for most high school students. These grade‐level expectations

are just estimates and

should not be used as an absolut

e minimum expectation or maximu

m limitation for

students. Problem solving is a learned skill, and students may

need many

experiences to develop their reasoning skills, approaches, stra

tegies, and the

perseverance to be successful. T

he Problem of the Month builds

on sequential levels

of understanding. All students should experience Level A and th

en move through the

tasks in order to go as deeply a

s they can into the problem. T

here will be those

students who will not have access

into even Level A. Educators

should feel free to

modify the task to allow

access at some level.

Overview

In the Problem of the Month,

Diminishing

Return,

s

tudents use number operations,

organized lists, and probability

to solve problems. The mathem

atical topics that

underlie this POM are knowledge o

f number sense, comparison, su

btraction,

division, rates, rational number

s, and conditional probability.

The mathematics in

this POM includes converting repeating decimals to fractions an

d using algebraic

reasoning.

In the first level of the POM, s

tudents are presented with a si

tuation that asks the

students to determine the number of helpers needed to carry dri

nks. Each helper

carries a uniform amount. The stu

dents determine the number of

helpers that are

©

Noyce Foundation 2014.

This work is licensed under a Cr

eative Commons Attribution‐NonC

ommercial‐NoDerivatives 3.0

Unported License (

http://creativecommons.org/lic

enses/by‐nc‐nd/3.0/deed.en_US

).

needed to do the job and compare

how two classes complete the j

ob with different

numbers of helpers. Each class ne

eds helpers to carry drinks,

but the number of

drinks the two sets of helpers c

an carry differs. So students

determine which class

needs more helpers. In Level B,

students have a given amount o

f money and a list of

possible toys to purchase. Their

goal is to determine a list o

f toys whose prices sum

to the exact amount provided. I

n Level C, students are asked t

o consider a problem

that involves rate. Two people

can mow the lawn at differing r

ates. The challenge is

to determine a time when the first worker has twice as much law

n left to mow as

the second worker. In Level D, t

he student determines how much

weight a person

has lost if each month that pers

on loses weight at a diminishin

g level. In Level E,

students are asked to find the e

xact theoretical chance of a sp

ecified outcome,

knowing three conditional probable events expressed as repeatin

g decimals.

Problem of the Month Diminishing Return Page 1

© Noyce Foundation 2014.

This work is licensed under a Creative Commons Attr

ibution-NonCommercial-NoDerivatives 3.0 Unported

License (

http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US

).

Problem of the Month

Diminishing Return

Level A

Some classes are going out for a picnic lunch. The teachers bo

ught

drinks in packs for their classes.

Thirty‐three students are i

n Mrs. Browne’s class.

Mrs. Browne bought six‐packs for her class. She

needs helpers, so she pick

s students to carry one

six‐pack each.

Twenty‐two students are in Mrs. Robinson’s class.

Mrs. Robinson bought four‐packs for her class. She

needs helpers, so she pick

s students to carry one

four‐pack each.

Which teacher had to pick more helpers?

Show how you found your answer.

Problem of the Month Diminishing Return Page 2

© Noyce Foundation 2014.

This work is licensed under a Creative Commons Attr

ibution-NonCommercial-NoDerivatives 3.0 Unported

License (

http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US

).

Level B

Amusement Center Store

Yo Yo $1.22 Doll $2.75 Duckie $1.

85 Tractor $5.97 Airplane $6.47

Ball $2.16 Racecar $7.13 Dog $4.57 Jump Rope $1.46 Car $5.18

Elephant $3.16 Bear $4.89 Xylophone $7

.11 Tank $6.45 Checkers $4.77

Boat $8.04 Train $6.71 Jacks $2.31 Tr

uck $6.21 Whistle 98¢ Pinwheel 87¢

Mia has earned $43.94 worth of tokens playing games at the amus

ement

center. The store in the amusement center has the toys picture

d above

for sale. Mia plans to buy toys and donate them to a local cha

rity for

needy children. The tokens are

only good in this store, so she

plans to

spend all the tokens. What combinations of toys can she buy in

order to

spend all the tokens?

Show how you found your solution.

Is your solution the only p

ossible answer? Explain.

Problem of the Month Diminishing Return Page 3

© Noyce Foundation 2014.

This work is licensed under a Creative Commons Attr

ibution-NonCommercial-NoDerivatives 3.0 Unported

License (

http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US

).

Level C

Maxine and Sammie have lawns that are the same size. Maxine ca

n mow

her lawn in 24 minutes

and Sammie can mow his lawn in 36 minute

s.

After how many minutes will Sammie have twice as much lawn to m

ow

as Maxine?

Maxine and Sammie have to also m

ow their parking strips that ar

e the

same size. Maxine can mow her parking strip in 6 minutes and S

ammie

can mow his parking strip in 9 minutes. After how many minutes

will

Sammie have twice as much grass to mow as Maxine?

Problem of the Month Diminishing Return Page 4

© Noyce Foundation 2014.

This work is licensed under a Creative Commons Attr

ibution-NonCommercial-NoDerivatives 3.0 Unported

License (

http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US

).

Level D

Rollie was successful in losing w

eight. He had a target weight

in mind.

He went on a diet for three mont

hs. Each month, he would lose

one‐

third of the difference between his current weight and his targ

et weight

plus an additional 3 pounds. At

the end of 3 months, he was ju

st 3

pounds over his target weight. How many pounds did he lose in

those 3

months?

Explain how you arrived at your solution.


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