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Success Story
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School in Marianna, Florida, is a beloved Illuminations lesson plan
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How do I love thee? Let me plot the ways!
A heart is drawn on a coordinate plane by plotting the following points and connecting them:
- The coordinates of the points are:
(
n
,
n
), (
n
- 3,
n
+ 3), (
n
- 6,
n
), (
n
- 9,
n
+ 3), (
n
- 12,
n
), (
n
- 12,
n
- 3), (
n
- 6,
n
- 9), and (
n
,
n
-3).
- The coordinates of one point are (2, 14).
- All coordinates are positive integers.
What is the value of
n
?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
The Number System
Geometry
Make sense of problems and persevere in solving them.
Apply and extend previous understandings of numbers to the system of rational numbers.
Graph points on the coordinate plane to solve real-world and mathematical problems.
5.G.A.1, 5.G.A.2, 6.NS.C.6b, 6.NS.C.6c, 6.NS.C.8, CCSS.Math.Practice.MP1
In the diagram at left, three different line segments each divide a quarter-circle
into two regions of equal area. Rank those three segments from shortest to
longest.
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Attend to precision.
Make sense of problems and persevere in solving them.
CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP6
The triangle at left lies on a flat surface and is pushed at the top vertex. The
length of the congruent sides does not change, but the angle between the two
congruent sides will increase, and the base will stretch. Initially, the area
of the triangle will increase, but eventually the area will decrease,
continuing until the triangle collapses.
What
is the maximum area achieved during this process? And, what is the length of
the base when this process is used to create a different triangle whose area is
the same as the triangle above?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Geometry
Measurement & Data
Attend to precision.
Make sense of problems and persevere in solving them.
Similarity, Right Triangles, and Trigonometry
Understand and apply the Pythagorean Theorem.
Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
3.MD.C.5a, 3.MD.C.5b, 3.MD.C.6, 3.MD.D.8, 8.G.B.7, HSG-SRT.C.8, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP6
A prime number is a natural number greater than 1 whose only factors
are 1 and itself. Can you place the digits 1–9 into the nine boxes (one
digit per box) so that the sum of every row and every column is a prime
number?
What if the prime sums for each row and column have to be different?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Algebraic Thinking
Make sense of problems and persevere in solving them.
Gain familiarity with factors and multiples.
4.OA.B.4, CCSS.Math.Practice.MP1
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: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Expression/Equation
Algebra
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
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, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2
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
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Attend to precision.
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, CCSS.Math.Practice.MP6
A
pocket watch is placed next to a digital clock. Several times a day, the
product of the hours and minutes on the digital clock is equal to the number of
degrees between the hands of the watch. (The watch does not have a second
hand.) As you can see, 10:27 is not one of those times — the angle between
the hands is not 270°. If fractional minutes aren’t allowed, find the
times at which the product of the hours and minutes is equal to the number of
degrees between the hands.
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Measurement & Data
Model with mathematics.
Make sense of problems and persevere in solving them.
Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
3.MD.A.1, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP4
The numbers 1 through 9 are
placed along the sides of the following triangle so that each side has the same
sum. However, three of the nine numbers are covered. What number is in the
circle with the question mark?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th, High School
Mathematical Practices
Make sense of problems and persevere in solving them.
CCSS.Math.Practice.MP1
Place a number in each of the following empty boxes so that the sum
of the numbers in any 3 consecutive boxes is 2013. What is the number
that should go in the box with the question mark?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th, High School
Mathematical Practices
Num & Ops Base Ten
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
Use place value understanding and properties of operations to perform multi-digit arithmetic.
3.NBT.A.2, 4.NBT.B.4, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2
Fill in each square with a digit 1-9, and fill in each circle with an operator +, –,
×, ÷. Use each digit and each operator exactly once. The resulting equation
should be true. Use the standard order of operations (PEMDAS).
What
is the largest possible three-digit result that can be obtained on the right
side of the equation?
Problems
Grades: 3rd to 5th, 9th to 12th, 6th to 8th
Num & Ops Fractions
Algebraic Thinking
Num & Ops Base Ten
Mathematical Practices
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
Write and interpret numerical expressions.
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
Generalize place value understanding for multi-digit whole numbers.
Develop understanding of fractions as numbers.
Multiply and divide within 100.
Make sense of problems and persevere in solving them.
CCSS.Math.Practice.MP1, 3.OA.C.7, 3.NF.A.1, 3.NF.A.3c, 4.NBT.A.2, 4.NF.B.4a, 5.OA.A.1, 5.NF.B.3
Algebra exercises often ask students to "Find
n
." But you won't find
n
in this brainteaser!
Create an equation of the form
c
=
ab
such that:
- When written out in English, none of the numbers
a,
b, or
c
contain the letter
n
.
- a,
b, and
c
are all integers.
- c
has the largest value possible.
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Algebraic Thinking
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
Multiply and divide within 100.
3.OA.C.7, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2
“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: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Num & Ops Base Ten
Algebraic Thinking
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
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, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2
Take two sheets of 8.5 by 11 inch paper. Roll one into a short cylinder and the other into a tall cylinder. Does one hold more than the other?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Measurement & Data
Geometry
Make sense of problems and persevere in solving them.
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
Geometric Measurement and Dimension
Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
Solve real-world and mathematical problems involving area, surface area, and volume.
6.G.A.2, 8.G.C.9, HSG-GMD.A.3, 5.MD.C.4, 5.MD.C.5a, 5.MD.C.5b, CCSS.Math.Practice.MP1
Imagine that you bought a Beanie Baby® for $6, sold it for $7, bought it back for $8, then sold it for $9. How much profit did you make?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Measurement & Data
Algebraic Thinking
Make sense of problems and persevere in solving them.
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
Use the four operations with whole numbers to solve problems.
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
3.OA.D.8, 4.OA.A.3, 4.MD.A.2, CCSS.Math.Practice.MP1
Two players each roll an ordinary six-sided die. Of the two numbers showing, the smaller is subtracted from the larger. If the difference is 0, 1, or 2, player A gets 1 points. If the difference is 3, 4, or 5, Player B gets 1 point. The game ends after 12 rounds. The player with the most points wins the game. Is the game fair?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Stats & Probability
Mathematical Practices
Using Probability to Make Decisions
Look for and make use of structure.
Attend to precision.
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
Investigate chance processes and develop, use, and evaluate probability models.
7.SP.C.7a, HSS-MD.B.6, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, CCSS.Math.Practice.MP6, CCSS.Math.Practice.MP7, HSS-MD.B.5a
Suppose you love chocolate. The top of each cookie is covered with the same thickness of chocolate. If you wanted to choose the cookie with more chocolate, which would you pick?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Make sense of problems and persevere in solving them.
CCSS.Math.Practice.MP1
Would you rather work seven days at $20 per day or be paid $2 the first day and have your salary double every day for a week?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Functions
Num & Ops Base Ten
Algebraic Thinking
Look for and make use of structure.
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
Interpreting Functions
Generalize place value understanding for multi-digit whole numbers.
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
3.OA.D.9, 4.NBT.A.2, HSF-IF.A.3, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, CCSS.Math.Practice.MP7
If all grape juice concentrates are the same strength, which recipe would you expect to have the strongest grape taste?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Ratio & Proportion
Num & Ops Fractions
Attend to precision.
Make sense of problems and persevere in solving them.
Analyze proportional relationships and use them to solve real-world and mathematical problems.
Understand ratio concepts and use ratio reasoning to solve problems.
Extend understanding of fraction equivalence and ordering.
Develop understanding of fractions as numbers.
3.NF.A.3d, 4.NF.A.2, 6.RP.A.1, 6.RP.A.3a, 7.RP.A.3, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP6