MATH
Get Something on the Table
“A blank stare solves nothing”
The Plan?
What is the smallest possible value for the
product of 2 integers that differ by 7?
To understand a question like this, I find it’s easiest to just try
something. This helps me see what the problem is asking:
“2 integers that differ by 7?” What 2 do I use? Use any! Just
get started!
Differ by 7
Product
2
9
2  9  18
1
8
1 8  8
-3
4
3  4  12
-7
0
7  0  0
-6
1
6 1  6
Let’s pick two more
8 is less than 18 …
Now I see the problem!
Let’s keep going!
Three distinct lines, all contained within a
plane, separate that plane into distinct regions.
What are all of the possible numbers of distinct
regions of the plane that could be separated by
any such three lines?
Let’s draw a box and
draw three lines …
Three lines drawn like
this create 4 regions …
The Plan?
What’s the problem asking? It’s hard to say. Draw a box and
draw three lines.
I start to see the
problem, and draw the
lines differently …
This way gives me 6
regions!
How else can I draw
the lines?
7 regions!
When the choir is arranged in rows of 5 people
each, the last row is one person short. When the
choir is arranged in rows of 6 people each, the
last row is still one person short. What is the
least possible number of people in the choir?
(29; 30; 56; 60; 99)
The Plan?
What’s the problem asking? There’s 5 chairs in each row (all
filled by people), except for the last row (where one chair is
empty). Draw this …
The choir is rearranged. Now there’s 6 chairs in each row (all
filled by people), except for the last row (where one chair is
empty). Draw this …
I start to see the problem …
5 Chairs / Row
6 Chairs / Row
X XXXX
X XXXXX
X XXXX
X XXXXX
X X X XO
X XXXXO
There can be
14 or 19 or 24 or 29
or 34 ... people in
the band.
There can be
17 or 23 or 29 or 35
... people in the
band.
There could be 29
people in the band.
Melissa had 3 fewer apples than Marcia.
Then, she gave 2 apples to Marcia. Now
how many fewer apples does Melissa have
than Marcia?
MELISSA
The Plan?
Forget formulas – use real things.
MARCIA
Melissa has 3 few
apples than Marcia
Melissa gave 2 to
Marcia
Melissa has 7 fewer
apples than Marcia.
The Plan?
Gary has turtles, cats, and birds for pets.
The number of birds he has is 4 more
than the number of turtles, and the
number of cats is 2 times the number of
birds. Of the following, which could be
the total number of Gary’s pets?
(14; 18; 20; 22; 26)
Try a single turtle. What does this mean?
Turtles
Cats
Birds
Total
1
5
10
16
nope
2
6
12
20
Yes!
but now I've got the feel of the problem
The Plan?
When x is divided by 7, the remainder is 4.
What is the remainder when 2x is divided
by 7?
Look for an easy number to get started.
Suppose I let x = 11
11  7  1 remainder 4
This is what the problem asked for –
In fact, that’s why I chose 11!
Now answer the problem!
22  7  3 remainder 1
Bus X travels 40 miles per hour for 2
hours; Bus Y travels 60 miles per hour for
1.5 hours. What is the difference, in miles,
between the number of miles traveled by
Bus X and the number of miles traveled by
Bus Y ?
Bus X:
40 mph for 2 hours
Bus Y:
60 mph for 1.5 hours
The Plan?
If the problem says “suppose you have a triangle”, draw
a triangle. If the problem says “suppose you have two
buses”, label two buses!
Forget formulas. Just draw a timeline.
40 m
1 hr
60 m
1 hr
40 m
1 hr
30 m
.5 hr
80 Miles
90 Miles
Sales for a business were 3 million dollars
more the second year than the first, and
sales for the third year were double the
sales for the second year. If sales for the
third year were 38 million dollars, what
were sales, in millions of dollars, for the
first year?
The Plan?
If the problem says “suppose you have a triangle”, draw
a triangle. If the problem says “suppose you have three
years”, label three years!
Forget formulas. Fill in what you know.
What do we know?
If sales for the third year were 38 million
dollars …
What else?
sales for the third year were double the sales
for the second year.
What else?
million dollars more the second year than
the first,year.
year 1
year 2
year 3
16
19
38
An industrial cleaner is manufactured using
only the 3 secret ingredients A, B, and C,
which are mixed in the ratio of 2:3:5,
respectively, by weight. How many pounds
of secret ingredient B are in a 42-pound (net
weight) bucket of this cleaner?
The Plan?
If the problem says “suppose you have a triangle”, draw
a triangle. If the problem says “suppose you have three
ingredients”, label three ingredients!
Where do I start? Since the ratio is 2:3:5, why not start
with that?
A
2
B
3
C
5
Total
10
4
6
10
20
Still not enough – let’s double everything!
8
12
20
40
Close. I need 42. I won’t be able to get it with whole
numbers, clearly. But I also see the nature of the
problem now.
2x
3x
5x
10x
Not enough – let’s double everything.
Total is 42
10 x  42
8.4
12.6
21
42
x  4.2
The 2 diagrams below show a
circle of radius 1 inch with shaded
sectors of angle x°, for 2 different
values of x. One of the following is
the graph in the standard (x,y)
coordinate plane of the area, y, of a
shaded sector with angle x°, for all
values of x between 0 and 360.
Which is that graph?
0
part of circle
shaded?
0
90
1/4
180
1/2
360
whole thing
angle
Now I have an idea
what to graph …
The Plan?
This is a hard one – wrestle with
what they’re asking for.
Read the question carefully. Look
at the two circles. What are they
asking?
A hint: if x is small, how much of
the circle does it “carve out”? How
about as x gets large?sking?
The lead of a screw is the distance the screw
advances in a straight line when the screw is
turned 1 complete turn. If a screw is 2½
inches long and has a lead of ⅛ inch, how
many complete turns would get it all the
way into a piece of wood?
1
2 "
2
The Plan?
If the problem says “suppose you have a triangle”, draw a
triangle. If the problem says “suppose you have a screw
going into wood”, draw a screw going into wood!” Make
sure to label everything.
What does the problem say about this screw going into the
wood?
1
2 "
2
1
"
8
1
1 turn  "
8
2
2 turns  "
8
4
1
4 turns  "  "
8
2
8
8 turns  "  1"
8
16 turns  2"
1
20 turns  2 "
2
1
21
8
2
1
x
x
21
2
8
x 5

8 2
2 x  40
x  20
If xy = 144, x + y = 30, and x > y,
what is the value of x – y?
The Plan?
Sometimes it’s easiest to just try some numbers and see
what works …
x
y
x+y
xy
20
10
30
200
Nope – too much. I need two different
numbers.
25
5
30
150
Still too much, but closer!
24
6
30
144
Bingo!
Now make sure you answer the right question!
x  y  18