High school teachers always have trouble finding good problems to match algebra and geometry content. As we look at problems now we must ask: 1.) Which of the eight Mathematical Practices does this problem employ?

2.) To what Common Core Standards does this problem apply?

Do you have problems that make the Common Core Standards come alive? Please share here!

When you have persevered to a solution go to the “Problem Solved” page and check your work. Remember all problems can be solved by different approaches!

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The following problem comes from Stella’s Stunners: http://ohiorc.org/for/math/stella

1. Cubic to Quadratic to Linear

One root of a certain third-degree equation is 1. When the cubic term of the equation is crossed off, the resulting quadratic equation has a root of 2. When the squared term is also crossed off, the resulting linear equation has a root of 3. Reconstruct the original third-degree equation, expressing it in the form ax^3 + bx^2 + cx = d, with all coefficients as integers.

The following problem comes from Stella’s Stunners: http://ohiorc.org/for/math/stella

2. Stack of Water Pipes

Three of the new water pipes for Main Street are stacked as shown. The external diameter of each pipe is 12 inches. How tall is the stack (h in the picture-to see follow link)?

The following problem comes from Stella’s Stunners: http://ohiorc.org/for/math/stella

3. Area Ratio, Circles and Square 1

Find the ratio of the area of the circle circumscribed about a given square to the area of the circle inscribed in that square.

The following problem comes from Stella’s Stunners: http://ohiorc.org/for/math/stella

4. Exponential Equation with 3’s

How many roots are there of the following equation? Find them.

3^(2x+2) − 3^(x+3) − 3^(x) + 3 = 0

The following problem comes from Stella’s Stunners: http://ohiorc.org/for/math/stella

5. Basketball Bounces

Levy Witherspoon is the star center of the boys’ varsity basketball team. If he drops the ball, it will bounce back to 8/10 of the height that he dropped it from. Levy reaches overhead and drops the ball from a height of eight feet. How many times will it bounce before it bounces up less than two feet (approximately the height of Levy’s knees)?

The following problem comes from Stella’s Stunners: http://ohiorc.org/for/math/stella

6. Sit-Ups

To make the team, you are going to have to do 89 sit-ups for the coach a week from today. You decide to work up to it. You will start by doing 3 sit-ups today (no sense rushing into things) and end on the 8th day with 89. You don’t know how many you will do tomorrow, but you decide that from the 3rd day on, the number of sit-ups you do will be the sum of what you did on the two preceding days. That is, the number you do on Wednesday will be the sum of the number you did on Monday and the number you did on Tuesday; the number you do on Thursday will be the sum of what you did on Tuesday and Wednesday, and so on. Find out how many sit-ups you should do tomorrow to make this work, so that you come out with 89 a week from today.

The following problem comes from Stella’s Stunners: http://ohiorc.org/for/math/stella

7. Rectangle Decomposed Into Squares

A rectangle is decomposed into nine squares whose bases measure 1, 4, 7, 8, 9, 10, 14, 15, and 18 units respectively. What are the dimensions of the rectangle?

The following problem comes from Stella’s Stunners: http://ohiorc.org/for/math/stella

8. Clairbuoyant

A fishing boat, the Clairbuoyant, sails 40 miles east, then 80 miles south, and finally 20 miles east again. How far is it from its starting point?

The following problem comes from Stella’s Stunners: http://ohiorc.org/for/math/stella

9. Sphere Packed in Box

A spherical basketball is packed in a cubical box into which it fits exactly. What percent of the volume of the box is used up by the basketball?

The following problem comes from Stella’s Stunners: http://ohiorc.org/for/math/stella

10. Driving to Cleveland

Zach and his family were driving into Cleveland for the ball game. Zach fell asleep when they were halfway along the way. When he woke up the distance they still had to go was half as far as they went while he was asleep. For what fraction of the way did Zach catch his Z’s?

The following problem comes from Stella’s Stunners: http://ohiorc.org/for/math/stella

Bonus Question:

The Lighthouse

You are sailing your splendid yacht, the Gemini, down the Hudson River into New York Harbor. It is a sunny afternoon, with a light breeze, and everything is going well. Your crew has been scrubbing down the bulkheads, polishing the brass hardware, and oiling up the teakwood decks. Your chef has begun preparing a splendid dinner on the afterdeck for you and your guests — life is good.

You are approaching the great gray George Washington Bridge, which spans the Hudson from Manhattan Island to New Jersey. Beneath the bridge, on the Manhattan side, is a famous little red lighthouse. You take a sighting of the lighthouse and observe that it is 15 degrees to port. You proceed for two minutes at your stately rate of 5 knots, and you observe that the lighthouse is now 29 degrees to port. How close will you come to the lighthouse as you pass under the bridge? (Data: 1 knot is approximately 6076 feet per hour.)

This “rich task” comes from http://nrich.maths.org/

Diminishing Returns

Take a look at the image below:http://nrich.maths.org/content/id/6700/sevensq.png

Work out what proportion of the image is coloured blue.

Try to provide a convincing explanation that your answer is right.

Imagine continuing the pattern towards the centre of the square:http://nrich.maths.org/content/id/6700/sevensq2.png

If this process could be continued forever, what proportion of the image would be coloured blue?

Try to provide a convincing explanation that your answer is right.

This “rich task” comes from http://nrich.maths.org/

http://nrich.maths.org/content/id/8001/curvyareas%206468.pdf

This “rich task” comes from http://nrich.maths.org

Charlie created a symmetrical pattern by shading in four squares on a 3 by 3 square grid:

Alison created a symmetrical pattern by shading in two triangles on a 3 by 3 isometric grid:

Choose whether you would like to work on square grids or isometric grids.

How many different symmetrical patterns can you make?

Here are some questions you might like to consider:

How many different patterns can you make if you are only allowed to shade in one… two… three… four cells?

How does the number of patterns with 6 cells shaded relate to the number with 3 cells shaded?

Can you make patterns with exactly one… two… three… four lines of symmetry?

Can you make patterns with rotational symmetry AND lines of symmetry?

Can you make patterns with rotational symmetry but NO lines of symmetry?

Can you make patterns using more than one colour?

Problems which relate to quadratics:

http://nrich.maths.org/6300

A problem from nrich that allows solutions from more than one perspective:

Choose four consecutive whole numbers.

Multiply the first and last numbers together.

Multiply the middle pair together.

Choose several different sets of four consecutive whole numbers and do the same.

What do you notice?

Can you explain what you have noticed? Will it always happen?

Algebra problem from nrich:

http://nrich.maths.org/2216