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

The following isosceles trapezoid is composed of 7
matches. Modify the position of three matches in order to obtain two
equilateral triangles.

Problems

Grades: 9th to 12th, 6th to 8th, 3rd to 5th

Mathematical Practices

Geometry

Reason abstractly and quantitatively.

Make sense of problems and persevere in solving them.

Draw construct, and describe geometrical figures and describe the relationships between them.

7.G.A.2, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2

Which decade in the 21st
century will contain no year that is divisible by 11?

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.

Multiply and divide within 100.

3.OA.C.7, 4.OA.B.4, CCSS.Math.Practice.MP1

Let A = 1, B = 2, and so on, with each letter equal to its position in the alphabet. The *lexivalue* of a word is the sum of the values of its letters. For example, ROMANS has a lexivalue of 18 + 15 + 13 + 1 + 14 + 19 = 80.

Now, do the following:

- Pick a number.
- Convert it into its representation in Roman numerals.
- Find the lexivalue for that Roman numeral.

For example, if you choose 11, that becomes XI in Roman numerals, and XI has a lexivalue of 24 + 9 = 33.

Are there any numbers for which the lexivalue is equal to the original number?

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

How
do I love thee? Let me graph the ways!

Can
you come up with one or more equations to graph a heart on the coordinate
plane? The equations can be rectangular, polar, or parametric.

**Bonus:**
Can you shift your heart so the graph or its interior includes the point (2, 14)?

Problems

Grades: 9th to 12th, 6th to 8th, 3rd to 5th

Mathematical Practices

Functions

Algebra

Make sense of problems and persevere in solving them.

Building Functions

Interpreting Functions

Reasoning with Equations and Inequalities

Creating Equations

HSA-CED.A.2, HSA-REI.D.10, HSF-IF.B.4, HSF-IF.C.7b, HSF-BF.B.3, CCSS.Math.Practice.MP1