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professional and secondary education at East Stroudsburg University. So
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Success Story
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School in Marianna, Florida, is a beloved Illuminations lesson plan
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Success Story
A 40‑inch straightedge
(without markings) is divided into four pieces. The length of each piece is an
integer number of inches. These four pieces, when used in tandem, can be used
to measure any integer length from 1 to 40 inches.
What are the lengths of the
pieces?
Problems
The rectangle shown consists of eight squares. The length of each side of each
square is 1 unit. The length of the shortest path from A to C using the lines
shown is 6 units.
How
many different six-unit paths are there from A to C?
Problems
A Friedman number is a number that can be
represented with an expression that uses only the digits in the number. In
addition, the expression can include +, –, ×, ÷, exponents and parentheses, but
nothing else. For instance, 25 is a Friedman number because it can be
represented as 5^{2}. A nice
Friedman number is a Friedman number for which the digits occur in the same
order in the expression as they do in the original number.
So, 343 is a nice Friedman number, because it can be represented by an
expression with the digits 3, 4, and 3 in the same order:
343 = (3 + 4)^{3}
The first
seven nice Friedman numbers are 127, 343, 736, 1285, 2187, 2502, 2592. Can you
find an expression for each of them?
Problems
Grades: 6th to 8th, 3rd to 5th
Expression/Equation
Algebraic Thinking
Apply and extend previous understandings of arithmetic to algebraic expressions.
Write and interpret numerical expressions.
Multiply and divide within 100.
3.OA.C.7, 5.OA.A.1, 6.EE.A.1
A 10 × 10 grid is painted
with three primary colors (red, yellow, and blue) and three secondary colors
(green, purple, and orange). The secondary colors are made by mixing equal
parts of the appropriate primary colors — that is, red and yellow are
mixed to make orange, red and blue to make purple, and yellow and blue to make
green.
The figure at left shows
squares that were painted red and blue. No other squares were painted either
red or blue.
Suppose that each small
square requires a quart of paint. Altogether, 31 quarts of red paint, 40 quarts
of blue paint, and 29 quarts of yellow paint were used to paint the entire
10 × 10 grid.
Given this information, can
you determine if there were more yellow or purple squares? And how many more?
Problems
Grades: 9th to 12th, 6th to 8th
Algebra
Expression/Equation
Reasoning with Equations and Inequalities
Analyze and solve linear equations and pairs of simultaneous linear equations.
8.EE.C.8b, HSA-REI.C.6
The factorial of n is the
product of all positive integers less than or equal to n. It is represented as n!.
An example with n = 8 is
shown below. With that in mind, can you find three sets of numbers (a, b,
c) such that a! × b! = c! and a < b < c < 25?
Problems
Grades: 3rd to 5th
Algebraic Thinking
Multiply and divide within 100.
3.OA.C.7
Tom was born on Thanksgiving Day.
On his seventh birthday, he noticed that Thanksgiving had never fallen on
his birthday. How old will he be when he finally has a Thanksgiving birthday?
Problems
A
regular octagon is inscribed inside a square. Another square is inscribed inside
the octagon. What is the ratio of the area of the smaller square to the area of
the larger square?
Problems
Grades: 9th to 12th, 3rd to 5th, 6th to 8th
Geometry
Measurement & Data
Ratio & Proportion
Circles
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.A.1, 3.MD.C.7b, 4.MD.A.3, HSG-C.A.3
Juliet bought 10 beads for
$18. The beads she bought are red, blue or silver. Red beads are $1 each, blue
beads are $2 each and silver beads are $5 each.
If she bought at least one of each, how many red beads did she buy?
Problems
Grades: 6th to 8th, 9th to 12th
Expression/Equation
Algebra
Analyze and solve linear equations and pairs of simultaneous linear equations.
Reasoning with Equations and Inequalities
Creating Equations
8.EE.C.8b, HSA-CED.A.3, HSA-REI.C.6, 8.EE.C.8c
Melanie has 4 different colored crayons and 2 different
boxes as shown below. How many different ways can Melanie put all 4 crayons
into the 2 boxes so that each box has at least 1 crayon?
Problems
Starting at 12:00 midnight, you wait a number of minutes that is a perfect square and
then look at a digital clock. The number you see (with the colon removed) is
also a perfect square.
What
is the first time after midnight that this happens?
Problems
Grades: 3rd to 5th
Measurement & Data
Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
3.MD.A.1
Every day in a non‑leap year, John took a different path from home to his favorite
store. He walked on the grid of streets shown at left, and he only walked north
or east along each street. His home is in the lower left corner of the diagram.
He started on January 1, and on December 31 he took the last possible path. At
what intersection is his favorite store located?
Problems
“Mom, look at that license plate,” Will said.
“What
about it?” his mother asked. It didn’t seem unusual to her. The plate consisted
of two sets of three digits, with the state logo between the sets.
Will said, “All six digits are different. And when you multiply the
first three digits, you get the same product as when you multiply the last
three digits.”
“So
you do,” his mother said. “How many plates like that do you suppose there are?”
“Well,
that’s the cool part,” Will replied. “The number of plates like that is equal
to the product of the first three digits.”
What
license plate might Will have seen?
Problems
Grades: 3rd to 5th
Num & Ops Base Ten
Algebraic Thinking
Use place value understanding and properties of operations to perform multi-digit arithmetic.
Multiply and divide within 100.
3.OA.C.7, 4.NBT.B.5
In this multiplication example, P, E, and T represent different digits. What is the value of the three-digit
number PET?
P E T
×
3
T T T
Problems
Grades: 3rd to 5th
Num & Ops Fractions
Algebraic Thinking
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
Multiply and divide within 100.
3.OA.C.7, 5.NF.B.5a
Ten
is a triangular number, because 10 objects can be neatly arranged in an
equilateral triangle.
In
general, a triangular number is a number that can be represented as a triangle
with one object in the first row and each subsequent row contains one more
element than the previous row. (From the picture above, you might notice that
1, 3, and 6 are also triangular numbers.)
Order
the digits 1 through 9 so that the sum of any two adjacent numbers is a triangular
number.
Problems
Sam the Squirrel found a basket of pine cones one
morning. He decided that every day he would double the number of pine cones in
the basket in the morning and eat 2 pine cones out of the basket in the afternoon.
At the end of the 3^{rd} day, there were 34 pine cones in the basket.
How many pine cones were in the basket when Sam the Squirrel found it?
Problems
Grades: 3rd to 5th
Algebraic Thinking
Generate and analyze patterns.
Multiply and divide within 100.
3.OA.C.7, 4.OA.C.5
If you rearrange the letters
S, T, O, and P, what is the probability that you’ll end up with a common English word?
Problems
Grades: 6th to 8th
Stats & Probability
Investigate chance processes and develop, use, and evaluate probability models.
7.SP.C.5, 7.SP.C.7a
How many different color
patterns can be created by placing the circles onto the 4 × 4 grid
such that each circle is placed on a square with the same number?
Problems
If 18 students occupy
of the seats in the classroom, how many students would occupy
of the seats in the room?
Problems
Grades: 3rd to 5th, 6th to 8th
Num & Ops Fractions
Ratio & Proportion
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
Understand ratio concepts and use ratio reasoning to solve problems.
5.NF.B.4a, 6.RP.A.1, 6.RP.A.3a, 4.NF.B.4b, 4.NF.B.4c, 5.NF.B.7a, 5.NF.B.7b, 5.NF.B.7c