Mathematics 4020 Core Activity Standard Grade leveljesseshupesadventures.weebly.com/uploads/1/3/1/2/... · 2019. 8. 20. · Mathematics 4020 Core Activity Below are some objectives

Mathematics 4020 Core Activity

Below are some objectives from the Common Core State Standards for Elementary Mathematics. Can you identify the grade levels?

Standard Grade level

1. Understand units of liquid volume and mass.

2. Distinguish between defining and non-defining attributes of shapes.

3. Make histograms.

4. Know what a hexagon is.

5. Begin finding areas of polygons.

6. Learn metric prefixes.

7. Use the mean absolute deviation.

8. Convert between various measurement units.

9. Draw pictographs.

10. Understand the difference between two- and three-dimensional shapes.

11. Know what a cylinder is.

12. Compute volumes of objects.

Name Physical Object(van Hiele level 0)

Careful definition(van Hiele level 1)

Picture

line segment

angle

right angle

scalene triangle

isosceles triangle

equilateral triangle

right triangle

square

rectangle (that is not a square)

parallelogram

Name Physical Object(van Hiele level 0)

Careful definition(van Hiele level 1)

Picture

rhombus

kite

trapezoid

isosceles trapezoid

Mathematics 4020, Homework 1

1. Which of the following figures are not quadrilaterals? Explain why.

a. !b. !!!!!! c. !!!!!d.

!!!!!!!!!!!!!!!!!!

2. Give as many (mathematically precise) names as possible for the figures:

a. ! ! b. ! c.

3. How many rectangles are in this figure?

4. Divide a rectangle (not square) into four congruent right triangles.

5. On a sheet of isometric graph paper (i.e., paper tiled by equilateral triangles instead of squares; available from Canvas), draw the following shapes, if possible. Some will not be possible!

a. isosceles (but not equilateral!) triangle.b. squarec. rectangle that is not a squared. rhombus that is not a squaree. parallelogram that is not a rhombusf. right triangleg. isosceles right triangleh. trapezoid

Geometry Labs Section 1 Angles 3

LAB 1.1 Name(s) Angles Around a Point

Equipment: Pattern blocks

Place pattern blocks around a point so that a vertex (corner) of each block touches the point and no space is left between the blocks.The angles around the point should add up to exactly 360°.For example, with two colors and three blocks you can make thefigure at right.Use the chart below to keep track of your findings.

Every time you find a new combination, circle the appropriatenumber on the list below.Cross out any number you know is impossible.If you find a possible number that is not on the list, add it.

Since the two-colors, three-blocks solution is shown above, it is circled for you.

Colors: How many blocks you used:all blue 3 4 5 6all green 3 4 5 6all orange 3 4 5 6all red 3 4 5 6all tan 3 4 5 6all yellow 3 4 5 6two colors 3 4 5 6 7 8 9 10 11 12three colors 3 4 5 6 7 8 9 10 11 12four colors 3 4 5 6 7 8 9 10 11 12five colors 3 4 5 6 7 8 9 10 11 12six colors 3 4 5 6 7 8 9 10 11 12

How many solutions are there altogether? __________

Discussion

A. Which blocks offer only a unique solution? Why?B. Why are the tan block solutions only multiples of 4?C. Explain why the blue and red blocks are interchangeable for the purposes of this activity.D. Describe any systematic ways you came up with to fill in the bottom half of the chart.E. How do you know that you have found every possible solution?F. Which two- and three-color puzzles are impossible, and why?G. Which four-color puzzles are impossible, and why?H. Why is the five-color, eight-block puzzle impossible?I. Which six-color puzzles are impossible, and why?

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LAB 1.2 Name(s) Angle Measurement

Equipment: Pattern blocks, template

1. What are the measures of the angles that share a vertex at the center ofa. A Chrysler symbol? ___________b. A Mercedes symbol? ___________c. A peace sign? ___________d. A clock, between consecutive hours? ___________e. A cross? ___________

2. Find the measures of all the angles for each of the pattern blocks shown below.Write the angle measures in these shapes.

3. One way to measure angles is to place smaller angles inside larger ones. For example,six copies of the small angle on the tan pattern block fit inside the figure below.Thisfigure, called a protractor, can be used to measure all pattern block angles.a. Mark the rest of the lines in the

figure with numbers.b. Use it to check the measurements

of the pattern block angles.c. Using the tan pattern block,

add the 15° lines between the 30° lines shown on the protractor.

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30

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4 Section 1 Angles Geometry Labs

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4. Use the protractor on your template to measure the angles in the trianglesbelow. For each one, add up the angles.Write the angle sum inside each triangle.

5. Use the protractor on your template to draw the following angles in the spacebelow.a. 20°b. 50°c. 100°

Discussion

Which of the questions in Problem 1 did you find most difficult to answer? What made the others easier?

LAB 1.2 Name(s) Angle Measurement (continued)

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LAB 1.3 Name(s) Clock Angles

What angles do the hour and minute hands of a clock make with each other atdifferent times? Write an illustrated report.

Start by figuring it out on the hour (for example, at 5:00).Then see if you can answer the question for the half-hour (for example, at 5:30).Continue exploring increasingly difficult cases.

Remember that the hour hand moves, so, for example, the angle at 3:30 is not 90°but a little less, since the hour hand has moved halfway toward the 4.Use the clocks below for practice. Cut and paste clocks into your report.

Discussion

A. How many degrees does the hour hand travel in an hour? In a minute?

B. How many degrees does the minute hand travel in an hour? In a minute?


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LAB 1.4 Name(s) Angles of Pattern Block Polygons

Equipment: Pattern blocks

1. Find the sum of the angles for each pattern block, and record them below.a. Green __________b. Orange __________c. Blue __________d. Tan __________e. Red __________f. Yellow __________

2. Which blocks have the same sum?

3. Use pattern blocks to make a polygonsuch that the sum of its angles is thesame as the sum for the hexagon.Sketch your solution in the space atright. How many sides does it have?

4. Use pattern blocks to make a polygonsuch that the sum of its angles is lessthan the sum for the hexagon but more than the sum for the square.Sketch your solution in the space atright. How many sides does it have?

5. Use pattern blocks to make a polygonsuch that the sum of its angles is greaterthan the sum for the hexagon. Sketchyour solution in the space at right.How many sides does it have?

6. For each number of sides from 3 to 12, make a pattern block polygon and findthe sum of its angles. Sketch your solutions on a separate sheet of paper and fillout the table below.

Sides Sum of the angles

3

4

5

6

7

Sides Sum of the angles

8

9

10

11

12

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7. If a pattern block polygon had 20 sides, what would be the sum of its angles? Explain.

8. If the sum of the angles of a pattern block polygon were 4140˚, how manysides would it have? Explain.

9. If the sum of the angles of a pattern block polygon were 450˚, how many sideswould it have? Explain.

10. What is the relationship between the number of sides of a pattern blockpolygon and the sum of its angles? Write a sentence or two to describe therelationship, or give a formula.

LAB 1.4 Name(s) Angles of Pattern Block Polygons (continued)

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LAB 1.5 Name(s) Angles in a Triangle

Equipment: Template

Types of triangles

Obtuse: contains one obtuse angleRight: contains one right angleAcute: all angles are acute

1. Could you have a triangle with two right angles? With two obtuse angles?Explain.

2. For each type of triangle listed below, give two possible sets of three angles. (Insome cases, there is only one possibility.)a. Equilateral: ______________, _______________b. Acute isosceles: ______________, _______________c. Right isosceles: ______________, _______________d. Obtuse isosceles: ______________, _______________e. Acute scalene: ______________, _______________f. Right scalene: ______________, _______________g. Obtuse scalene: ______________, _______________

3. If you cut an equilateral triangle exactly in half, into two triangles, what arethe angles of the “half-equilateral” triangles? _______________

4. Which triangle could be called “half-square”? _______________

5. Explain why the following triangles are impossible.a. Right equilateral

b. Obtuse equilateral

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6. Among the triangles listed in Problem 2, which have a pair of angles that add up to 90°?

7. Make up two more examples of triangles in which two of the angles add up to 90°. For each example, give the measures of all three angles.

______________, _______________

8. Complete the sentence:“In a right triangle, the two acute angles . . . ”

9. Trace all the triangles on the template in the space below and label them by type: equilateral (EQ), acute isosceles (AI),right isosceles (RI), obtuse isosceles (OI), acute scalene (AS),right scalene (RS), half-equilateral (HE), obtuse scalene (OS).

LAB 1.5 Name(s) Angles in a Triangle (continued)

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LAB 3.3 Name(s) Walking Convex Polygons

Equipment: Pattern blocks, template, unlined paper

Definitions:

A polygon is a closed figure with straight sides.An n-gon is a polygon with n sides.A convex polygon is one where no angle is greater than 180°.

1. Is there such a thing as a nonconvex triangle? If so, sketch one in the space at right; if not,explain why not.

2. Is there such a thing as a nonconvex quadrilateral? Sketch one, or explain why there’s no such thing.

3. For each number of sides from 3 to 12, make a convex pattern block polygon.Use your template to record your solutions on a separate sheet.

In this activity, you will give each other instructions for walking polygons.As anexample, here are instructions for walking the pattern block trapezoid.We will usea scale of one step for one inch:

Walk forward one step from the starting point;Turn right 60°;Walk forward one step;Turn right 60°;Walk forward one step;Turn right 120°;Walk forward two steps;Turn right 120°.

The final turn is not necessary, since the whole perimeter has been walked bythen. However, it is traditional to include it, because it brings the walker backexactly to the starting position.

4. Mark the remaining steps and angles on the figure.

Note: The turn angle is also called the exterior angle. It is not usually the same asthe interior angle, which is usually called simply the angle.

One step

60

38 Section 3 Polygons Geometry Labs

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Geometry Labs Section 3 Polygons 39

5. Write another set of instructions for walking the trapezoid, starting at another vertex.

6. On a separate paper, write instructions for walking each of the other pattern blocks.

7. Take turns with a partner, following each other’s instructions for walkingdifferent blocks.

8. Make a two-block convex pattern block polygon, such as theone in the figure at right, and write instructions below forwalking it.

Discussion

A. Write alternate instructions for walking the trapezoid with one or more ofthese additional constraints:

Walking backward instead of forwardMaking left turns instead of right turnsStarting in the middle of the long side

B. What is the relationship between the turn angle and the interior angle? Whenare they equal?

C. How can you tell whether a polygon walk is clockwise or counterclockwise byjust reading the walk’s instructions?

LAB 3.3 Name(s) Walking Convex Polygons (continued)

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Mathematics 4020

Terminology again:

1. Vertex angle: the angle between sides (measured on the inside!) measured at a vertex.2. Central angle: the angle formed between line segments drawn from consecutive vertices to the

center point of a regular polygon. This definition doesn’t even make sense for a non-regular polygon. Why not?

3. Exterior angle: the angle formed on the outside by extending one of the sides of a polygon.

For a regular n-gon:For a regular n-gon:

Vertex angle

Central angle

Exterior angle

Find the vertex angle at one of the points of this star: Two different ways?

Reflection 4: Paper folding

No rulers allowed! Pencils only... Explain your steps for each of these on a separate sheet of paper, clearly, and in complete sentences.

1. Draw a perfectly straight line segment on your patty paper.2. Somewhere off that line, draw a point. Now (no other tools!) draw a line through that point exactly

perpendicular to the first line.3. Now draw a line through that point exactly parallel to the first line.4. Make an exact 45 degree angle.5. Divide your original line segment into two perfectly equal parts (still no rulers!).6. Divide your original line segment into four perfectly equal parts.7. Make an exact 22.5 degree angle.8. (Bonus) Extra challenging: Make an exact 60 degree angle.9. (Bonus) Extra challenging: Divide your original line segment into three perfectly equal parts.

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Mathematics 4020: Tessellation and Angle Measures

1. What is the most precise name you can give for this shape?

2. Divide the left copy of the shape into the smallest number of triangles possible. Show how to use this to find the angle sum.

3. Divide the right copy of the shape into the same number of triangles as the shape has. Show how to use this to find the angle sum.

4. Susan has a bag of regular dodecagons. No matter how hard she tries, she cannot form a tessellation with them alone. Show why. If she were allowed to use one other type of regular polygon, could she make a tessellation? Which one?

Measurement with non-standard units:

1. Name as many non-standard items as you can that could be used to measure. What sort of things would you measure with them?

2. What are the advantages of using non-standard units?

3. What are the disadvantages? Refer back to some of your answers to question 1.

Mathematics 4020: Angle sums

Use this table to record some figures:

Number of sides Angle sum Vertex angle, if regular

3

4

5

6

8

12

n

Mathematics 4020: Tiling Activity

Most of the tiled floors that we see are tiled with squares. But some other shapes work as well. Fill in the table below with your guesses about which ones will work, then check by tracing copies of the shapes on the other side onto a clean sheet of paper.

Shape Does it tile (your guess)? Does it tile (after checking)?

equilateral triangle

scalene triangle

rectangle (not square)

parallelogram (not rectangle)

trapezoid (not isosceles)

dart

regular pentagon

regular hexagon

regular octagon

Reflection 5: Convex regular polyhedra

A regular polyhedron is a polyhedron whose faces are all identical regular polygons. A convex regular polyhedron is one that doesn’t dent out. Formally: a polyhedron is convex when any line segment that starts and ends on the inside of the polyhedron stays on the inside.

Any convex regular polyhedron has a net: it can be cut along some of its edges, then unfolded to lie flat on a sheet of paper. Then the unfolded shape can be traced, cut out, and folded into a copy of the original.

Answer the following questions clearly and in complete sentences on another sheet of paper.

1. What is the smallest number of faces that meet at a single vertex of a polyhedron?2. What is the greatest number of equilateral triangles that may be arranged around a single point?3. What is the greatest number of squares that may be arranged around a single point?4. What is the greatest number of regular pentagons that may be arranged around a single point?5. What is the greatest number of regular hexagons that may be arranged around a single point?6. What is the greatest number of regular octagons that may be arranged around a single point?7. Which regular polygons can be the faces of a convex regular polyhedron? Justify carefully.8. For each of the regular polygons listed in part seven, draw the possible corners of regular

polyhedra with those faces. Just draw what the arrangement looks like around a single vertex.9. You should have a total of _____ pictures for #8. Did you get it right?10.There actually are only exactly ______ convex regular polyhedra (of a given size). So the corners

you drew in 8 can actually be realized.

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Mathematics 4020 homework on polygons and tesselations

1. Explain why a regular octagon will not tessellate by itself. A tessellation can be made using regular octagons and another type of regular polygon. Which one? Explain.

2. Draw a pentagon that will tesselate the plane by itself. Draw a bit of your tessellation.3. A central angle of a regular n-gon has measure 12 degrees. What is n?4. A vertex angle of a regular n-gon has measure 170 degrees. What is n?5. The sum of vertex angles in an n-gon is 3420. What is n?6. Repeat problem 3 for a central angle of 5 degrees.7. Repeat problem 4 for a vertex angle of 176 degrees.8. With full justification, find the measure on angle A below.

A

Math 4020 Triangle Activity

As you came in today, you should have picked up a card with a triangle on it. Donʼt show your triangle to anyone else!

1. Use as many adjectives (preferably at least some precise mathematical) to describe your triangle. Just write down your adjectives here.

2. Pair up with one other student. Each member of the pair should read his/her adjectives to the other. Based on your partnerʼs description, draw your partnerʼs triangle here:

3. Check your partnerʼs picture. Is it exactly right? How did you check? Still: donʼt show your triangle to anyone else, especially your partner!

4. Now try harder. Using whatever methods you can, try to describe your triangle to your partner in such a way that s/he can draw an exact copy. Write out your description below.

5. Draw your partnerʼs triangle, using the new description. Is it better?

6.Which of the triangles below is a right triangle?

7. The word “congruent” is used to mean that in a strict geometric sense, two triangles are the “same”. Try to write a definition of what you think congruent means.

8.What does the previous question have to do with the activity on the front?

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Mathematics 4020 Homework on Measurement

1. Give your best estimates for the following measurements:

English Metric

Temperature on a warm day

Length of a pencil

Weight (mass) of a textbook

Area of our classroom floor

Volume of our classroom

Capacity of a swimming pool

2. An inch is 2.54 centimeters. A yard is 36 inches. Which is longer, a yard or a meter?3. One square mile is 640 acres. One mile is 5280 feet. If a square lot is one acre, how long are its

sides?4. 1 mile is about 1.6 kilometers. How fast in miles per hour is 100 km/h?5. Convert 50 miles per hour into feet per second.

LAB 5.1 Name(s) Introduction to Symmetry

1. These letters are line symmetric:

These letters are not:

a. Explain the difference.

b. Draw the line or lines of symmetry on each letter in the first group. (In somecases, there are two lines.)

2. Show how each of the five capital letters not shown in Problem 1 can bewritten two ways: line symmetric, or not.

3. Organize the lowercase letters in two lists below: line symmetric, and not.

Line symmetric:

Not line symmetric:

4. Below are some figures that have rotation symmetry and some figures that donot. Explain the difference.

Have rotation symmetry

Do not have rotationsymmetry

70 Section 5 Symmetry Geometry Labs

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Geometry Labs Section 5 Symmetry 71

5. Organize the capital letters in two lists below: those with rotation symmetryand those without.

Rotation symmetry:

No rotation symmetry:

6. Organize lowercase letters in two lists below: those with rotation symmetryand those without.

Rotation symmetry:

No rotation symmetry:

7. Artist Scott Kim finds ingenious ways to adapt letters so that he can designsymmetric words. Describe the symmetry of the design below.

From Inversions, by Scott Kim.

8. On a separate sheet of paper, write symmetric words. In this case, symmetrycan be interpreted in several ways, using capitals or lowercase letters.Thefollowing list gives possible interpretations.a. Words that have a horizontal line of symmetry, such as COB.b. Words that have a vertical line of symmetry, such as TOT. (To make this easier,

you may write words vertically, with each letter beneath the previous one.)c. Pairs of words that are mirror images of each other, such as box and pox (using a

horizontal mirror).d. Words that can be read upside down, such as SOS.e. Pairs of words where one word is the other word upside down, such as MOM

and WOW.f. Palindromes, such as RADAR or RACE CAR (ignoring the space).

LAB 5.1 Name(s) Introduction to Symmetry (continued)

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Discussion

A. Line symmetry is also called mirror symmetry, reflection symmetry, bilateralsymmetry, and flip symmetry. Explain why each of these words is appropriate.

B. What happens if a line-symmetric figure is folded along the line of symmetry?

C. For the figures in Problem 4, draw the lines of symmetry across the figures thatare line symmetric, and draw the center of symmetry in those that haverotation symmetry. Note that some of the figures have both kinds of symmetry.

D. The figures in Problem 4 with rotation symmetry include examples of 2-fold,3-fold, 4-fold, 5-fold, and 6-fold rotation symmetry. Classify each figureaccording to the specific type of rotation symmetry it has. Explain what itmeans for a figure to have “n-fold” rotation symmetry.

E. If a figure looks unchanged when you rotate it around a point by 180°, it has aspecial kind of rotation symmetry that is also called half-turn symmetry, centralsymmetry, and point symmetry. Explain why each of these words is appropriate.

F. What is your definition of the word symmetry? Compare it with other students’definitions and the dictionary definition.

G. Which letters have both line and rotation symmetry? What else do they have in common?

H. Some letters are ambiguous and can be written either with or without linesymmetry. Others can be written either with or without rotation symmetry.• Draw and explain examples of these ambiguities.• Explore how different fonts accentuate or detract from the symmetry of

individual letters.

LAB 5.1 Name(s) Introduction to Symmetry (continued)

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LAB 5.2 Name(s) Triangle and Quadrilateral Symmetry

Equipment: Dot or graph paper, isometric dot or graph paper

Eight types of triangles

Equilateral (EQ) Half-equilateral (HE) Right scalene (RS)Right isosceles (RI) Acute isosceles (AI) Obtuse scalene (OS)Acute scalene (AS) Obtuse isosceles (OI)

Eight types of quadrilaterals

Square (SQ): a regular quadrilateralRhombus (RH): all sides equalRectangle (RE): all angles 90˚Parallelogram (PA): two pairs of parallel sidesKite (KI): two distinct pairs of consecutive equal sides, but not all sides equalIsosceles trapezoid (IT): exactly one pair of parallel sides, one pair of opposite equal sidesGeneral trapezoid (GT): exactly one pair of parallel sides General quadrilateral (GQ): not one of the above

1. Find two examples of each kind of figure listed above, triangles andquadrilaterals, on dot or graph paper. Label them.

2. On each figure, draw the lines of symmetry if there are any.

3. On each figure, mark the center of symmetry if there is any rotation symmetry.

4. Fill out the table below by entering the types of triangles and quadrilaterals inthe appropriate spaces (some spaces may be empty; others may have more than one entry).

Rotation symmetry

Line symmetry None 2-fold 3-fold 4-fold

No lines

One line

Two lines

Three lines

Four lines

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Discussion

A. Is there a relationship between the numbers of lines of symmetry and the nature of the rotation symmetry? Explain.

B. For each empty cell in the chart in Problem 4, explain why it is empty.Comment on whether any figures exist with that kind of symmetry (not triangles or quadrilaterals).

C. In some cells, there are two or more figures. Do those figures have anything in common besides symmetry? Discuss.

D. You may have heard that “a square is a rectangle, but a rectangle is notnecessarily a square.” Does the classification of the quadrilaterals by symmetry throw any light on this statement? Can you use the classification by symmetry to find more statements of this type?

LAB 5.2 Name(s) Triangle and Quadrilateral Symmetry (continued)

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LAB 5.3 Name(s) One Mirror

Equipment: Mirror, template

Using your template on a piece of unlined paper, draw a triangle. Then place themirror on it in such a way as to make a triangle or quadrilateral. For example,starting with an equilateral triangle ABC it’s possible to make a different equilateraltriangle, a kite, or a rhombus.Check off the figures you make in this manner in the table below, and draw the mirror line on the triangle as a record of how you did it.Also indicate which side you are looking from with an arrow. On the table, mark impossiblefigures with an X.

Figures made By using the mirror onTriangles EQ AI RI OI AS RS HE OS

Equilateral (EQ)

Acute isosceles (AI)

Right isosceles (RI)

Obtuse isosceles (OI)

Acute scalene (AS)

Right scalene (RS)

Half-equilateral (HE)

Obtuse scalene (OS)

Quadrilaterals EQ AI RI OI AS RS HE OS

General

Kite

General trapezoid

Isosceles trapezoid

Parallelogram

Rhombus

Rectangle

Square

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Discussion

A. What polygons besides triangles and quadrilaterals can be made with a triangleand a mirror?

B. Which figures on the table cannot be made with a triangle and a mirror? Why?

C. What do all the figures that can be made have in common?

D. Which line-symmetric figures cannot be made? Why?

E. Which figures can be made from any triangle?

F. What is a good strategy for finding a way to create a given shape with atriangle and mirror?

G. How must the mirror be placed on the original triangle so the resulting figureis a triangle? Why?

H. It is never possible to make an acute isosceles triangle from an obtuse isoscelestriangle. However, it is sometimes possible to make an obtuse isosceles trianglefrom an acute one.Why?

LAB 5.3 Name(s) One Mirror (continued)

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LAB 5.4 Name(s) Two Mirrors

Equipment: A pair of mirrors hinged with adhesive tape, pattern blocks, template

In this lab, you will investigate what happens to reflections of pattern blocks as youchange the angle between the mirrors.

1. Using your mirrors and one blue pattern block, make the figures below.To record your work, draw the mirrors’ position on the figures.

2. For each example in Problem 1, what is the angle between the mirrors?Explain.

3. What is the next figure in the sequence? Make itwith a block and mirrors, and draw it in the space at right. Is there another figure after this one?

4. Using a tan pattern block, set up the mirrors so you can see exactly 2, 3, 4, 5,and so on copies of it, including the original block (no matter where you lookfrom). For each setup, what is the angle between the mirrors?

5. Calculate the following angles: 360°/3, 360°/4, and so on, up to 360°/12.

6. How can you get the mirrors to form each of the angles you listed in Problem 5without using a protractor or pattern blocks? (This may not really be possible forthe smallest angles. Hint: Look at the reflections of the mirrors themselves.)

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7. Which of the angles from Problem 5 yield only mirror-symmetric designswhen a pattern block is placed between the mirrors? Which angles can yieldeither symmetric or asymmetric designs?

Symmetric only:

Symmetric or asymmetric:

8. On a separate sheet of paper, create your own designs using one or morepattern blocks and the hinged mirrors. Record your designs with the help of the template. For each design, make a note about the angle you used andwhether your design is symmetric.

Discussion

A. Set up the mirrors at an angle of 120°, and place a blue pattern block near onemirror and far from the other, as shown on the figure below. Note that, dependingon where you are, you will see two, three, or four blocks (including the originaland its reflections).Write 2, 3, and 4 on the figure to indicate where you wouldlook from to see that many blocks.

B. Compare the results of using even and odd numbers for n when using theangle 360°/n between the mirrors. Discuss both the reflections of the mirrorsthemselves and the reflections of other objects.

C. Note that, when you use an odd value for n, the middle reflections of themirrors create the illusion of a two-sided mirror, in the sense that if you look atthem from the left or the right you will see a mirror. However, if you lookfrom a point that is equidistant from the two real mirrors, you may see differentobjects on the left and right of that middle virtual mirror! (Try the setupshown on the figure below.)

Mirror Mirror

Mirror Mirror

LAB 5.4 Name(s) Two Mirrors (continued)

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D. If you allow yourself to stand anywhere, what range of angles allows you to seethree copies of a block (including the original)? Four copies? Five?

E. Look at the reflection of your face in the mirrors when they make a 90° angle.Wink with your right eye.What happens in the reflection? Can you explainhow and why this is different from your reflection in one mirror?

F. Are some of the reflections you see in the hinged mirrors reflections ofreflections? Reflections of reflections of reflections? If so, which ones?

G. Write the letter F and look at its reflections in the hinged mirrors.Whichcopies of it are identical to the original and which are backward?

LAB 5.4 Name(s) Two Mirrors (continued)

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LAB 5.5 Name(s) Rotation Symmetry

Equipment: Circle geoboards, Circle Geoboard Paper

1. Make these designs on your circle geoboards.

Each of the designs has rotation symmetry: If you rotate the geoboard around itscenter, the design returns to its initial position before you have done a full turn.For example, figure b above would look the same after one-third of a full turn. Itreturns to its initial position three times as you turn it (counting the final time afterthe full 360° turn).The design is said to have 3-fold rotation symmetry.

2. Each design in Problem 1 has n-fold symmetry for some n.What is n in each case?a. _______ b. _______ c. _______ d. _______ e. _______ f. _______

3. Which of the designs in Problem 1 also have line symmetry? How many linesof symmetry does each one have?

4. On a separate sheet of paper, make designs with n-fold rotation symmetry forvarious values of n. Some should have line symmetry and some not. Recordand label your designs.

a b c

d e f


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Discussion

A. It is possible to make the mirror image design for each of the designs inProblem 1. In some cases, the mirror image design will be different from theoriginal one. It is said to have different handedness.Which of the designs havedifferent handedness from their mirror images?

B. Is it possible to have n-fold rotation symmetry for any number n? What would1-fold rotation symmetry mean? What about 0-fold? 2.5-fold? –2-fold?

LAB 5.5 Name(s) Rotation Symmetry (continued)

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Today we’re talking measurement of three-dimensional

shapes.

List some sample units of measurement that may be

used to describe how much paper is necessary to wrap

a gift.

Your units should have been units that you have used

before. After all, the paper that you’re using to wrap

will be flat until you wrap it.

A box of cereal has dimensions 5 cm by 15 cm by 25

cm. How much paper would be needed to wrap it? Be

sure to include appropriate units.

The “Family Size” cereal box has all dimensions twice

as large as the cereal box in the last problem. How

much paper is needed to wrap it? How does the answer

compare to the answer for the regular size?

Summary:

The measurement of the area on the surface of a three-

dimensional object is called surface area (!). It is

entirely analogous to area (think about wrapping the

shapes in graph paper), and measured in the same

units. It scales the same way that area does: by the

square of the magnification factor.

There are books with formulas for surface area. We do

not need them. To find the surface area of an object,

we must think of a way of “unwrapping” the object

into a net. This is now a collection of two-dimensional

shapes, and the area is computed then.

Example: Find the surface area of a regular tetrahedron

of side length 4 inches. Include appropriate units.

It’s a little harder to find ways to flatten out shapes

with curves in them. We’ll deal with that next time.

Meanwhile, there is another way of measuring three-

dimensional shapes. In computing area, we are

answering the question of how many unit squares can

be cut up and reassembled to cover a figure exactly.

For a three-dimensional shape, we may ask the

question: “How many unit cubes are necessary to fill

this shape exactly?” (again, we allow the unit cubes to

be cut into as many oddly shaped pieces as necessary

to fill the shape).

This measurement is called volume. Write down a few

reasonable guesses at units for volume.

What is the volume of the regular size cereal box?

Recall that its dimensions were 5 cm by 15 cm by 25

cm.

What is the volume of the “Family Size” cereal box?

What is the relationship between the volumes of the

two cereal boxes? Answer the second question with a

clear, complete sentence.

Capacity is another measurement of the amount a

three-dimensional shape can hold. This amount can be

described using units of capacity, like gallons or

ounces (in the English system), or liters (in the metric

system). There is no hard-and-fast rule when to

measure the “inside” of a shape using volume or

capacity. Generally, if the inside is empty and going to

be filled with a liquid or a gas, we use capacity. If we

are talking about how much space a solid object takes

up, we use volume. But don’t take this too far: the

capacity of an aquarium may be given in gallons, or its

volume given in cubic feet.

By the way – which is larger, a gallon or a cubic foot?

Use your intuition.

Since capacity and volume are measuring the same

spaces, it would be useful to be able to convert

between them. In the metric system, this is easy:

1 cubic centimeter = 1 mL

Using metric prefixes, give the capacity unit

corresponding to a cubic meter.

Wouldn’t it be great if there were a similar fact in the

English system? Like 1 cubic inch equals one ounce?

But there isn’t. By the way, here is the mess of what

the English capacity system looks like:

3 teaspoons = 1 tablespoon

2 tablespoons = 1 fluid ounce

8 fluid ounces = 1 cup

2 cups = 1 pint

2 pints = 1 quart

4 quarts = 1 gallon

And it goes on and on with things like barrels and

hogsheads.

If you want to find out how many gallons are in a

cubic foot, you need to convert everything to metric

first, then go back. Why don’t you try that for

homework (due next class)? The last piece of the

puzzle you need is that there are about 3.79 liters to a

gallon.

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Mathematics 4020 HW on Surface Area and Volume, part 1!!1. This silo is 40 feet tall from bottom to top, and 8 feet wide. Find its surface area and volume (ignore

the “knob” on top and the ladder”).!!

� !

!!2. A standard 8.5” x 11” sheet of paper can be folded into a cylinder (without top and bottom) two

ways. Which of the two ways (“tall” or “short”) has the greater surface area? Which has the greater volume?!!!!!!!!!!!!!!3. The Great Pyramid of Giza is a right square pyramid whose base is 230 m on each side. The height of the apex from the ground is 146 m. Find the surface area and volume of the Great Pyramid.

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Geometry Labs Section 10 Similarity and Scaling 133

LAB 10.1 Name(s) Scaling on the Geoboard

Equipment: 11 ! 11 geoboard, dot paper

The figure shows four houses. House a is the original, and the others have beencopied from it following very exact rules.

1. What is the rule that was used for each copy?

2. Among the three copies, two are distorted, and one is a scaled image of the original.a. Which one is scaled?b. How would you describe the distortions?

When two figures are scaled images of each other, they are said to be similar. Similarfigures have equal angles and proportional sides.The sides of one figure can be obtainedby multiplying the sides of the other by one number called the scaling factor.

3. What is the scaling factor that relates the two similar houses in the figure above?

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134 Section 10 Similarity and Scaling Geometry Labs

4. Make a geoboard triangle such that the midpoint of each side is on a peg.Make your triangle as different as possible from those of your neighbors.Join the midpoints with another rubber band, making a smaller triangle.a. Describe the resulting figure, paying attention to equal segments and

angles, congruent triangles, parallel lines, and similar figures.

b. Find the slopes of the lines you believe to be parallel, the length of the segments you believe to be equal, and the scaling factor for the figures you believe to be similar.

5. Make a geoboard quadrilateral such that the midpoint of each side is on a peg.Make your quadrilateral as different as possible from those of your neighbors.Stretch another rubber band around consecutive midpoints, making a smallerquadrilateral.a. Describe the resulting figure, paying attention to equal segments and angles

and parallel lines.

b. Find the slopes of the lines you believe to be parallel and the length of thesegments you believe to be equal.

Discussion

A. The rules discussed in Problem 1 can be expressed in terms of coordinates(assuming the origin is at the bottom left peg) or in terms of measurements.Explain.

B. For any pair of similar figures, there are two scaling factors relating them.How are the factors related to each other?

C. Try to make a triangle so that two of its sides have midpoints on pegs, but thethird does not.What happens? Explain.

D. Problems 4 and 5 are special cases of general theorems that apply to themidpoints of any triangle or quadrilateral. State the theorems. Explain how the result in Problem 5 is a consequence of the result in Problem 4.

LAB 10.1 Name(s) Scaling on the Geoboard (continued)

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LAB 10.2 Name(s) Similar Rectangles

Equipment: 11 11 geoboard

The figure above shows geoboard rectangles nested inside each other.

1. Explain why the two rectangles on the left are similar but the two rectangleson the right are not.

2. Draw the diagonal from the bottom left to the top right of the larger rectanglein a. Note that it goes through the top right vertex of the smaller rectangle.Repeat on b. How is it different?

3. The rectangles in a are part of a family of similar geoboard rectangles. If we includeonly rectangles with a vertex at the origin (the bottom left peg of the geoboard),the family includes ten rectangles (five horizontal ones and five vertical ones—twoof the vertical ones are shown in the figure).a. List the rectangles in that family by listing the coordinates of their top right vertex.

b. What is the slope of the diagonal through the origin for the vertical rectangles?

c. What is the slope of the diagonal for the horizontal rectangles?

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4. Including the one in Problem 3, there are ten families of similar rectangles(including squares) on the geoboard.Working with your neighbors, find themall, and list all the rectangles in each family. (Again, assume a vertex at the origin,and use the coordinates of the top right vertex for the list. Only consider families with more than one rectangle.)

5. Working with your neighbors, find every geoboard slope between 1 and 2.(Express the slopes both as fractions and as decimals.) Counting 1 and 2, thereare seventeen different slopes.

Discussion

A. Problem 2 is an example of using the diagonal test for similar rectangles. Explain.

B. In Problems 3 and 4, how does symmetry facilitate the job of listing the rectangles?

C. What are the advantages of fractional versus decimal notation in Problem 5?

D. Explain how to use the answers to Problem 5 to create lists of geoboard slopesin the following ranges.a. Between 0.5 and 1b. Between –1 and –2c. Between –0.5 and –1

LAB 10.2 Name(s) Similar Rectangles (continued)

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LAB 10.3 Name(s) Polyomino Blowups

Equipment: Interlocking cubes, 1-Centimeter Grid Paper, Polyomino Names Reference Sheet

The figure above shows a bent tromino (a) and three copies of it.The b copy hasbeen doubled horizontally, the c copy has been doubled vertically, and the d copyhas been doubled in both dimensions.

1. Which of the doubled figures is similar to the original? Explain.

2. For each polyomino with area less than 5, draw the polyomino and threecopies on grid paper following the pattern in the figure above.As you work,record the perimeter and area of each stretched polyomino in the table.

Perimeter AreaDoubled Doubled

Original Horiz. Vertic. Both Original Horiz. Vertic. Both

Monomino

Domino

Bent

Straight

Square

l

int

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3. Study the table for patterns relating the numbers in the different columns.Write down what you observe.

4. The most important patterns are the ones relating the original figure with thesimilar figure. State the patterns for perimeter and area.

5. Write down a prediction about what may happen to perimeter and area if youtriple, and quadruple, a polyomino in both dimensions.

6. Test your prediction by drawing a few polyominoes and their tripled andquadrupled copies.Write down your conclusions.

The following problems are puzzles involving similar polyominoes.

7. a. With your interlocking cubes, make tiles in the shape of the l and t tetrominoes.

b. On grid paper, draw the five doubled tetrominoes (doubled in both dimensions!).c. Use your tiles to cover each doubled figure. Record your solutions.

8. Repeat Problem 7 with the following shapes, being sure to multiply thedimensions both horizontally and vertically.a. Tripled tetrominoes, using l and t tilesb. Doubled pentominoes, using P and N tilesc. Tripled pentominoes, using P and L tiles

LAB 10.3 Name(s) Polyomino Blowups (continued)

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Discussion

A. What is the relationship between the scaling factor and the ratio of perimeters?

B. What is the relationship between the scaling factor and the ratio of areas? Why is this answer different from the answer to Question A?

C. How many tetrominoes does it take to tile a tripled tetromino? How manypentominoes does it take to tile a tripled pentomino? Explain.

D. How many polyominoes does it take to tile a polyomino whose area has beenmultiplied by k? Explain.

LAB 10.3 Name(s) Polyomino Blowups (continued)

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LAB 10.4 Name(s) Rep-Tiles

Equipment: Interlocking cubes, 1-Centimeter Grid Paper, Polyomino Names Reference Sheet, template

A shape is a rep-tile if it can be used to tile a scaled copy of itself. For example,the bent tromino is a rep-tile, as you can see in the figure below.

1. The l tetromino and the P pentomino are rep-tiles. Use cubes and grid paper to tile the doubled and tripled versions of the shapes with the original. Keep arecord of your solutions.

2. How many original shapes did you need to tile the following?a. The doubled figuresb. The tripled figures

3. Which other polyominoes of area less than or equal to 5 are rep-tiles? Working with your neighbors, find the tilings to support your answer.

4. Classify the pattern blocks into two groups: those that are rep-tiles and those that are not. (Use the template to make your drawings.)

5. Using each triangle on the template, show that it is a rep-tile.

6. In most cases, it takes four copies of a triangle to show it is a rep-tile.Illustrate the following exceptions.a. Show that two right isosceles triangles tile a larger right isosceles triangle.b. Show that three half-equilateral triangles tile a larger half-equilateral triangle.

7. Classify the quadrilaterals on your template into two groups: those that are rep-tiles and those that are not.

Discussion

A. When a figure is scaled by a factor k, its area is multiplied by ____. Explain.

B. Prove that any triangle is a rep-tile.

C. What are the scaling factors in Problem 6?

D. Find a triangle that can tile a scaled copy of itself using five tiles.


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LAB 10.5 Name(s) 3-D Blowups

Equipment: Interlocking cubes

In three dimensions, similar solids are obtained by scaling in all three dimensions.The figure below shows an example, with the scaling happening in three steps:width, depth, and height.

1. The following questions refer to the figure above.a. Which two solids are similar? b. What is the scaling factor?c. What are their surface areas?d. What is the ratio of the surface areas?e. What are their volumes?f. What is the ratio of the volumes?

2. Working with your neighbors, make other pairs of similar solids with yourinterlocking cubes using a scaling factor of 2, and fill out the table below.Use only a few cubes in the smaller solid!

Surface area VolumeOriginal Blowup Ratio Original Blowup Ratio

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3. Repeat Problem 2 with a scaling factor of 3. Use even fewer cubes in thesmaller solid!

4. Choose one of the solids in the figure below, different from your neighbors’choices.Without using physical cubes, find the surface area and volume forblowups of your solid, and fill out the table.You may use sketches.

Discussion

A. When you use a scaling factor k, what happens to each individual unit cube? How does this determine the ratio of surface areas? The ratio of volumes? Explain.

B. Write a formula for the surface area A of a scaled solid as a function of theoriginal surface area A0 and the scaling factor k. Repeat for the volume.

C. Write a formula for the volume V of the blown-up versions of the solid youstudied in Problem 4 as a function of the surface area A and the scaling factor k.Compare your formula with those of your neighbors.

LAB 10.5 Name(s) 3-D Blowups (continued)

Surface area VolumeOriginal Blowup Ratio Original Blowup Ratio

Scaling Surface Area Volumefactor area ratio Volume ratio

1

2

3

4

5

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Mathematics 4020: Applications of Similarity

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2. In the right hand figure, if angle RSQ is congruent to angle RPT, how long are RQ and PT?

3. Luis and Maria are standing on the edge of a cliff. The canyon in front of them is too wide to get across, but they would like to know how far it is across. Design a method using congruent triangles to figure the distance across without crossing the canyon or throwing anything over it.

4. Good! Now Luis and Maria have hiked further and run into another canyon. The canyon is more than 20 feet across, but the trail that Luis and Maria are on is very narrow. The canyon is on one side, and a raging river full of piranhas is on the other. The trail is at most 10 feet wide. Can they find the width of the canyon without falling into it or the river?

5. I have measured the triangles ABC and DEF. Angles A and D, B and E, and C and F are congruent. Side AB is 4 cm long and side DE is 12 cm long. If triangle ABC has area 12 cm, what is the area of triangle DEF?

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More Probability Practice

1. How many three digit numbers are there?

2. How many three digit numbers have no digits in common?

3. How many three digit numbers have all three digits the same?

4. What is the probability that a three digit number has exactly two digits the same?

5. I have four pairs of pants, five shirts, and six pairs of socks. How many ways can I get dressed?

6. There are four Scrabble tiles in a bag.

a. Experiment: Draw one tile at a time and place the tiles in a row. How many outcomes are there?

b. Experiment: Draw three tiles, one at a time, and place the tiles in a row. How many outcomes?

c. How many sets of three letters (in no particular order) can be made from the tiles in the bag?

7. In Santaʼs bag are three toys for boys, four toys for girls, and two lumps of coal.

a. Draw a probability tree for the experiment: draw two toys.

b. What is the probability that both toys are for boys?

c. What is the probability that both toys are for girls?

d. What is the probability of getting a lump of coal?

e. What is the probability of not getting a lump of coal?

8. a. In a family there are two children. What is the probability that both are boys?

b. In a family there are three children. What is the probability that only one is a boy?

c. In a family there are four children. What is the probability that exactly two are boys?

Mathematics 4020: Complex Experiments

A bag contains 4 white marbles, 3 green marbles, and 3 red marbles. The following experiment is performed: draw out two marbles.

1. How large is the sample space of this experiment?

2. How many distinguishable (that is, different) outcomes does this experiment have?

3. Draw a probability tree for the experiment, labelling the branches with their probability.

Define the events: A: getting a match; B: getting at least one red marble; C: getting no white marbles.

4. Find:

a. P(A)

b. P(B)

c. P(C)

5. Describe, in ordinary English words, the event A ∩ B. Find its probability.

6. Describe, in ordinary English words, the event A ∪ B. Find its probability.

7. Can you find a relationship among P(A), P(B), P(A ∩ B), P(A ∪ B)?

8. Write in symbols the event “the marbles drawn do not match”. What is its probability?

9. Write in symbols the event “no red marble was drawn”. What is its probability?

Mathematics 4020 Reflection 4

Make up counting problems that lead to the computations below. Explain the main features of your problem (choices being used up, order does/doesnʼt matter, and so on). Use this sheet and class time to discuss the problems, but turn in your answers typed on a separate sheet of paper.

1. 9·9·9·9·9

2. 27

3. 12!

4. 12·11·10·9·8·7

5. (12·11·10·9)/(4·3·2·1)

6. 27·6·6

7. (5·4)/2 + (6·5)/2

8. Below is a picture of part of the Avenues. Supposing that Shipp Park is exactly at 4th Ave and I St and Memory Park is at 7th Ave and A St, how many route are there from Shipp Park to Memory Park that stay along the marked streets and always move towards the goal (that is, you only go North and West)? Then make up a problem involving milkshakes that has the same answer and explain how to link the two problems. Bonus problem: how many of the routes from Shipp Park to Memory Park go past Fifth Ave Park?

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Mathematics 4020 Homework on Numerical Summary

1. For each part, create a list of 5 numbers (not all the same) so that

a. the mean is less than the median

b. the median is less than the mean

c. the median equals the mode

d. the mode is greater than the mean

2. Create a list of 5 numbers with standard deviation zero. Explain (donʼt compute).

3. All twelve players on a basketball team played in a game. Their total score was 78. The star player scored 23 points. How many points did the other players average?

4. Draw side-by-side box-and-whisker plots to compare the following classes. Include the five-number summary for each.

Class A: 91, 63, 65, 73, 65, 86, 96, 75, 75, 79, 84, 72, 80

Class B: 87, 72, 95, 89, 69, 79, 56, 64, 66, 67, 89, 47

Mathematics 4020 Core Activity Standard Grade leveljesseshupesadventures.weebly.com/uploads/1/3/1/2/... · 2019. 8. 20. · Mathematics 4020 Core Activity Below are some objectives

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