TO INFINITY AND BEYOND . . .
GOING BEYOND ANSWER GETTING
CATHY SHIDE, CONSULTANT
1
OBJECTIVES
• Integrate the math practices
with word problems
• teachers and students going
beyond “answer getting”
• Use different modes of
representation to solve problems
with a focus on Fractions, Ratios,
and Percents
2
“TAPE DIAGRAM”
“A drawing that looks like a
segment of tape, used to illustrate
number relationships. Also known
as a strip diagram, bar model,
fraction strip, or length model.”
Also referenced in “Visual Fraction
Model” definition.
- CCSSM (Glossary) p. 87
3
WHY NOT KEY WORDS?
• Takes away reason for reading
• Does not develop process for problem solving
• Takes away creating equations and finding all the
information
• Takes away from thinking “What else do I know to
solve the problem? What else may I need?”
4
EXPLORE
• Caroline has 46 dolls in her collection. She has 12
more than her cousin, Tammy. How many dolls
does Tammy have?
• Alicia had $6 more than Bobby. If Bobby had $10,
how much did they have altogether?
• Pedro and Gina started out with an equal number
of coins. Pedro lost 12 coins, and Gina collected
another 36 coins. How many more coins did Gina
have than Pedro?
PROBLEM #1
Cathy and Joan started out
with the same number of coins.
Cathy lost 12 coins and Joan
gained 36. How many more
coins does Joan have than
Cathy?
6
1. After reading the
problem, mark the who
or what and unit bars.
2. Re-read each sentence
and adjust the unit bars
accordingly. Cathy’s
loses 12 and Joan gains
36
3. Record the ?, what
students need to find.
Cathy’s coins
Joan’s coins
Cathy’s
coins
Joan’s
coins
12
36
12
Cathy’s
coins
Joan’s
coins
36
?
4. Students solve and try to 12 + 36 = 48
record an equation if
CL + JG = ?
possible.
What Cathy lost plus what Joan gained
equals how many more coins Joan has.
7
Example from the ISAT ER problem from 5th grade.
6.RP.3
Students were creating spirit
necklaces to sell for a fundraiser.
A necklace takes twice as many
purple beads as white beads and
4 times as many purple beads as
black beads. One necklace takes
28 beads. What is the number of
each color of beads?
9
7.RP.3
A class had 32 students and
twenty-five percent were
boys. When some new boys
joined the class, the
percentage of boys
increased to 40%. How many
new boys joined the class?
10
7.RP.3
Two students were running for
school president. Student A
received 65% of the votes
and had 900 more votes than
Student B. How many
students voted?
11
5.NF.4
The fundraising committee
made 400 pizzas. The
students sold 5/8 of the pizzas
and took 1/5 of the
remainder for a party. How
many pizzas did the
committee have left to sell?
12
GROUP PROBLEM SOLVING
Work with your colleagues to create:
• A manipulative model with your color tiles
• A tape diagram (bar model) of your
problem
• An equation
• A verbal description of your thought process
• What other questions can be answered
about your situation/problem?
13
WHAT DO YOU KNOW?
WHAT CAN YOU ANSWER?
A cran-apple mixture is made
up of 3 parts apple juice and
1 part cranberry juice. The
company will use 5 gallon
containers for the cran-apple
mixture.
14
WHAT ARE THE MATH PRACTICES?
•Look at your Math Practices
•What practices have you
been engaged in?
15
ANSWER GETTING VS. LEARNING
MATH
• USA:
How can I teach my kids to get the answer to this
problem?
Use mathematics they already know. Easy, reliable, works with
bottom half, good for classroom management.
• Japanese:
How can I use this problem to teach the mathematics
of this unit?
Phil Daro, Writer of CCSS in Mathematics, Slide 16,
http://www.cmcmath.org/resources/downloads/Daro%20PS%20Conference.p
16
POSING THE PROBLEM
• Whole class: pose problem, make sure
students understand the language, no hints at
solution
• Focus students on the problem situation, not
the question/answer game. Hide question
and ask them to formulate questions that
make situation into a word problem
• Ask 3-6 questions about the same problem
situation; ramp questions up toward key
mathematics that transfers to other problems
Phil Daro, Writer of CCSS in Mathematics, Slide 80,
http://www.cmcmath.org/resources/downloads/Daro%20PS%20Conference.p
17
WHAT PROBLEM TO USE?
•
•
•
•
Problems that draw thinking toward the
mathematics you want to teach. NOT too
routine, right after learning how to solve.
Ask about a chapter: what is the most
important mathematics students should take
with them? Find a problem that draws
attention to this mathematics
Begin chapter with this problem (from lesson
5 thru 10, or chapter test). This has diagnostic
power. Also shows you where time has to
go.
Also near end of chapter, while still time to
Phil Daro, Writer of CCSS in Mathematics, Slide 81,
respond
http://www.cmcmath.org/resources/downloads/Daro%20PS%20Conference.p
18
WHICH IS BIGGER?
• 1/3 or 1/2
3/5 or 6/10
 3/4 or 7/8
 10/16 or 5/8
 2/3 or 6/12

• Katie collected 20 seashells.
Diego collected 3 times as
many seashells as Katie. What
was the total number of
seashells collected?
• Corey paid $84 for a kite and a
CD player. The CD player cost
6 times as much as the kite.
What is the price difference
between the two items?
• Of the 60 students in the fifth
grade, we know that 3/5 are girls.
We also know that 1/2 of the girls
have blue eyes and one-fourth of
the boys have blue eyes. How
many of the students in the fifth
grade have blue eyes?