GEOMETRY

4th Grade Strand 3: Geometric Sense (GS)

GS01 (Shapes and Properties)Use attributes of geometric shapes and properties of parallel and perpendicular to identify, name, compare and sort geometric shapes and figures; recognize geometric shapes in the surrounding environment; draw simple shapes using tools (1.3.1, 1.3.2, 1.3.4)

Item Format(s):

Stimulus Attributes:

Mathematical Vocabulary and Terms:

Item Characteristics:

a) Items will assess knowledge beyond the level of simply labeling.

b) Items may ask students to identify properties (component shapes, parallel lines, perpendicular lines, number of sides, edges, faces, vertices) of 2-dimensional and 3-dimensional figures.

c) Items may ask students to describe properties (component shapes, parallel lines, perpendicular lines, number of sides, edges, faces, vertices) of 2-dimensional and 3-dimensional figures.

d) Items may ask students to compare 2-D and 3-D figures according to their properties (component shapes, parallel lines, perpendicular lines, number of sides, edges, faces, vertices).

e) Items may ask students to classify 2-D and 3-D figures according to their properties (component shapes, parallel lines, perpendicular lines, number of sides, edges, faces, vertices).

f) Items may ask students to sort different geometric figures according to given properties (component shapes, parallel lines, perpendicular lines, number of sides, edges, faces, vertices).

g) Short-Answer items may ask students to draw a simple 2-dimensional shape having certain properties (component shapes, parallel lines, perpendicular lines, number of sides, edges, faces, vertices).

Content Strand 3: Geometric Sense

GS02 (Location) Locate or describe the location of objects on a number line, map, or coordinate grid in the first quadrant (1.3.5)

Item Format(s):

Stimulus Attributes:

Mathematical Vocabulary and Terms:

Item Characteristics:

a) Items may ask students to identify an object on a location grid based on ordered pairs or coordinates.

b) Items may ask students to describe the relative location of objects on a number line or coordinate grid.

c) Items may ask students to draw objects on a location grid based on ordered pairs or map coordinates.

Content Strand 3: Geometric Sense

GS03 (Symmetry, Congruence & Similarity) Identify, describe, and compare symmetrical, congruent, and similar figures (1.3.3)

Item Format(s):

Stimulus Attributes:

Mathematical Vocabulary and Terms:

Item Characteristics:

a) Items may ask students to identify symmetrical, congruent, or similar figures in either 2-dimensional illustrations or illustrations of real-life objects.

b) Items may ask students to draw symmetrical, congruent, or similar figures in either 2-dimensional illustrations or illustrations of real-life objects.

c) Short-Answer items may ask students to complete a picture or design over a line of symmetry.

Content Strand 3: Geometric Sense

GSO4 (Transformations)Identify and draw simple geometric transformations using translations (slides), reflections (flips), or rotations (turns) (1.3.6)

Item Format(s):

Stimulus Attributes:

Mathematical Vocabulary and Terms:

Item Characteristics:

a) Items may ask students to identify a particular transformation.

b) Items may ask students to draw a particular transformation of a figure on a grid.

Transformations:

7th Grade Content Strand 3: Geometric Sense  (back)

GS01 (Properties & Relationships) Identify and use geometric properties and relationships to draw and describe shapes, figures, and simple scale drawings (1.3.1, 1.3.4, 1.3.7)

Item Format:

Stimulus Attributes:

Mathematical Vocabulary and Terms:

Item Characteristics:

a) Items may ask students to identify and/or draw geometric shapes and figures using geometric properties such as parallel, diagonal, intersecting, perpendicular, measures of angles, obtuse, acute, and right angles, isosceles and equilateral, quadrilateral, degrees in circle.

b) Items may ask students to describe shapes and figures using geometric properties such as parallel, diagonal, intersecting, perpendicular, measures of angles, obtuse, acute, and right angles, isosceles and equilateral, quadrilateral, degrees in circle.

c) Items may ask students to describe shapes using relationships among geometric properties.

d) Items may ask students to compare and classify shapes and figures according to their geometric properties.

e) Items may ask students to sort different geometric shapes and figures according to common geometric properties.

Content Strand 3: Geometric Sense

GS02 (Location) Identify and describe location of objects on a coordinate grid (1.3.3)

Item Format:

Stimulus Attributes:

Mathematical Vocabulary and Terms:

Item Characteristics:

a) Items may ask students to describe the location of objects on two-dimensional coordinate grids using integers.

b) Items may ask students to plot and label shapes on coordinate grids using integers.

Content Strand 3: Geometric Sense

GS03 (Symmetry, Congruence & Similarity) Demonstrate an understanding of symmetry, congruence, and similarity; draw symmetric (1.3.4, 1.3.5)

Item Format:

Multiple-Choice and Short-Answer items may be used to test this learning target.

Stimulus Attributes:

Mathematical Vocabulary and Terms:

Item Characteristics:

a) Items may ask students to identify, describe, or compare congruent, symmetrical, or similar figures in either illustrations of geometric figures or real-life objects using geometric properties.

b) Items may ask students to draw congruent, symmetrical, or similar figures.

c) Items may ask students to draw one or more lines of symmetry in given geometric figures.

d) Items may ask students to complete a picture or design given the line of symmetry.

Content Strand 3: Geometric Sense

GS04 (Transformations) Demonstrate an understanding of and apply simple geometric transformations using combinations including translations (slides) or reflections (flips) or rotations (turns) (1.3.6)

Item Format:

Stimulus Attributes:

Mathematical Vocabulary and Terms:

Item Characteristics:

a) Items may ask students to describe the location of transformed objects on 2-dimensional coordinate grids.

b) Items may ask students to identify or describe the relative location of objects to one another on a 2-dimensional coordinate or location grid (map), using ordered pairs or map coordinates (i.e., letter and number combinations) and in terms of mathematical transformations.

c) Items may ask students to identify, describe, or classify objects, with or without a grid, in terms of whether they are translated (slid), reflected (flipped), rotated (turned).

Questions 4th 5th and 6th grades

Math Practice Problems for Fourth Grade in Geometry

4G-20) Answer:  95 inches

A map of a National Forest in Alaska  has a scale of one inch for every 75 miles.  The perimeter of the forest is 7,125 miles.  How many inches would represent the perimeter on the map? Write to help explain your best thinking using words, numbers, or pictures.

4G-19)   Answer:  120
How many times does X occur in the diagram below?Write to help explain your best thinking using words, numbers, or pictures.

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4G18)   Answer:   71
How many times does the letter X occur in the diagram below?   Write to help explain your best thinking using words, numbers, or pictures.

             

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4G-17)   Answer:  36
How many times does the letter X appear in the diagram at the right?  Write to help explain your best thinking using words, numbers, or pictures.

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4G-16) Answer: 4x4 and 3X6

There are two rectangles whose dimensions are whole numbers and whose area and perimeter are the same number. Find them. Write to help explain your best thinking using words, numbers, or pictures.

4G-15) Answer: 15

For Gilda's party, the Hoagie House prepared a huge sub sandwich on a 7-foot long Italian roll. Gilda wants to feed 16 friends. How many cuts must she make?  Write to help explain your best thinking using words, numbers, or pictures.

4G-14) Answer: 23

Carla delivers packages in a secret underground building. Today she started on the first floor and went up 7 floors. Then she went down 3 floors, and then up 16 floors. She went up 2 more floors to the top floor of the building. How many floors were in this secret underground building?  Write to help explain your best thinking using words, numbers, or pictures.

4G-13) Answer: (diagram)

Lia and Carlos went on a bike ride with Anita. They left Anita's house and went 3 blocks to Maple Street and turned right. They went straight ahead 6 blocks to Pine and turned left. They went 4 blocks to Elm and turned left. Then they went straight ahead 4 blocks when Anita stopped, and said she wanted to go back to her house. What was the shortest way back to Anita's house? Write to help explain your best thinking using words, numbers, or pictures.

4G-12) Answer: 54

Sasha's family is having a big reunion on July 4. They borrowed tables and are setting them up outside. They can put three chairs on each side of the tables. If they set up 8 tables in a long row, with the tables together on one side, how many family members can sit down to eat? Write to help explain your best thinking using words, numbers, or pictures.

4G-11) Answer: 65

Penny works in the Package Palace. It is Penny's job to stamp the sides of packages that are not touching the floor and not touching another package. Today there are 25 packages on the floor. Penny put the packages in 5 stacks, and the sides of the stacks touch. How many sides of packages must Penny stamp? Write to help explain your best thinking using words, numbers, or pictures.

4G-10) Answer: 14 inches

Alice watched an ant hunting for food. The ant was traveling in a straight line toward a crumb on the floor. Suddenly it turned around and crawled 9 inches back to Alice's shoe. Another ant joined it and together they hurried 11 inches back toward the crumb. The two ants stopped and came back 2 inches toward Alice's shoe. Finally they turned around and raced 5 inches to the crumb. How far was the crumb from Alice's shoe? Write to help explain your best thinking using words, numbers, or pictures.

4G-9) Answer: 10

Sandy and Mario are going into Tuneville Center to hear a band play. They can get in through two different entrances. Each entrance goes into a lobby that has a snack bar in it. There are five doors going from the lobby into the room where the band will play. How many different paths can Sandy and Mario take to get from outside Tuneville Center to their seats in the band room?  Write to help explain your best thinking using words, numbers, or pictures.

4G-8) Answer: (diagram)

"Follow me" said Marcy to Lea. The two girls left Lea's home. They walked forward 4 blocks on Mystery Road and turned left on Wax Way. They went forward 8 blocks and turned right onto Sandy Street. Lea was puzzled by now, but she kept following Marcy down Sandy Street for 3 blocks. Marcy stopped there and went into a house. Lea followed her. "Surprise! Happy birthday, Lea!" called the girls' friends when Lea came into the house. Can you show the path from Lea's home to the house where the birthday party was?  Write to help explain your best thinking using words, numbers, or pictures.

4G-7) Answer: 180

The Wizard at Windows is washing all the windows in the building at 946 Third Street. There are 9 floors in the building and 12 offices on each floor, 6 offices across the back of the building and 6 offices across the front of the building. The outside wall of each office has 2 windows. How many windows does the Wizard have to wash?   Write to help explain your best thinking using words, numbers, or pictures.

4G-6) Answer: alike (4 sides, opposite sides equal and parallel ,all angles equal); different (rectangles may have 2 longer sides)

Describe a square and a rectangle. How are they alike? How are they different?   Write to help explain your best thinking using words, numbers, or pictures.

4G-5) Answer: South

Suppose that you are facing west. If you turn left three times, then twice to the right, what direction is behind you? Write to help explain your best thinking using words, numbers, or pictures.

4G-4) Answer: 50 inches

Ryan made 9 cuts in a board to divide it into equal 5-inch-long pieces. How long was the board?  Write to help explain your best thinking using words, numbers, or pictures.

4G-3) Answer: South 10 blocks

Ellen leaves school and walks 25 blocks south. Then she turns east and walks 16 blocks. Then she turns north and walks 15 blocks. Then she turns west and walks 16 blocks. Is she north, south, east, or west of the school and how far?  Write to help explain your best thinking using words, numbers, or pictures.

4G-2) Answer: 130 inches

Jake plans to make 12 cuts in a board. There will be a 10-inch interval between cuts. How long is the board?  Write to help explain your best thinking using words, numbers, or pictures.

 

4G-1) Answer: (clockwise) Kim, Mike, Mary, Dave

Mary, Dave, Kim and Mike are sitting around a table. Kim is sitting across from Mary. Mary is sitting at Dave's right.  Where is each person sitting?  Write to help explain your best thinking using words, numbers, or pictures.

Math Practice Problems for Fifth Grade in Geometry

5G-15) Answer: 35 cubes

The tower below is made up of five horizontal layers of cubes with no gaps.  How many individual cubes are in the tower?  Write to help explain your best thinking using words, numbers, or pictures.

5G-14) Answer: 22 Squares

Each of the boxes in the figure below is a square. Using the lines of the figure, how many different squares can be traced? 

5G-13) Answer: $6

Mrs. Smith is taking a log to the lumberyard. The log is 12 feet long, and she wants to have it cut into four pieces that are each 3 feet long. If the lumber-yard charges $2 per cut, how much will she spend?  Write to help explain your best thinking using words, numbers, or pictures.

5G-12) Answer: 15 times

Joan went fishing. On the first cast she hooked a trout 80 feet from the boat. Each time she reeled in 10 feet of line the trout would take out 5 feet. How many times did she have to reel in to get the fish to the boat?  Write to help explain your best thinking using words, numbers, or pictures.

5G-11) Answer: 24 inches

 A square piece of paper is folded in half as shown and then cut into two rectangles along the fold.  The perimeter of each of the two rectangles is 18 inches.  What is the perimeter of the original square?  Write to help explain your best thinking using words, numbers, or pictures.

5G-10) Answer: top: Lana-Gerald-Lara; bottom: Meredith-Farley-Sam

Teresa built a 2-story apartment house for her hamsters: Gerald, Farley, Sam, Meredith, Lana, and Lara. She put each hamster in one of the 6 apartments. She put Gerald next to Lara and above Farley. Lana was at the other end from Lara. Sam was not directly below Lana. Where did Teresa put each of the hamsters?  Write to help explain your best thinking using words, numbers, or pictures.

5G-9) Answer: 246 square feet

A rectangular garden is 14 feet by 21 feet and is bordered by a concrete walk 3 feet wide.  How many square feet are in the surface area of just the concrete walk?  Write to help explain your best thinking using words, numbers, or pictures.

5G-8) Answer: 132

Muriel and Betty are stacking cartons in the middle of the room. They are stacking the cartons side by side, 2 rows of 9 cartons across and 6 cartons high. They have to mark each carton where there is a side facing out, except for the tops of the cartons. How many sides do they have to mark?  Write to help explain your best thinking using words, numbers, or pictures.

5G-7) Answer: 11 cages

Troy was feeding the animals at the zoo. He started at the monkey's cage. Then he walked down the hill 3 cages to feed the birds. The birds are in the first cage in the zoo. Then Troy went up the hill 7 cages to feed the seals. From there he went down the hill 5 cages to feed the bears. Next he went up the hill 8 cages to the elephants. They are in the last cage in the zoo. How many cages are there in the zoo?  Write to help explain your best thinking using words, numbers, or pictures.

5G-6) Answer: row l: records & tapes-books-toys row 2: t-shirts-jewelry-posters

Libby and Kay were setting out records and tapes, jewelry, toys, books, posters, and T-shirts for the White Elephant Sale at school. They were setting up six tables in two rows. They put the records and tapes across from T-shirts and next to the books. They put the posters at the Opposite end of the row from the T-shirts. The toys were not put next to the T-shirts. How did Libby and Kay set up the tables for the White Elephant Sale?  Write to help explain your best thinking using words, numbers, or pictures.

5G-5) Answer: 40 boxes

There are 4 separate large boxes, and inside each large box there are 3 separate small boxes, and inside each of these small boxes there are 2 separate smaller boxes. How many boxes, counting all sizes, are there altogether?

  Write to help explain your best thinking using words, numbers, or pictures.

5G-4) Answer: 28

Clayton is getting some friends together to go to a movie. He is getting his friends who live on his floor of the apartment building. First he walks down the hall 6 apartments to get Martha. Martha lives in the last apartment on the floor. Then Clayton goes up the hall 10 apartments to get William. Next he goes down the hall 3 apartments to find Rita. From here he goes up the hall 6 apartments to pick up James. He lives in the last apartment on the floor. If there are the same number of apartments on both sides of the hall, how many apartments are there altogether on Clayton's floor?  Write to help explain your best thinking using words, numbers, or pictures.

5G-3) Answer: 112 feet

Mr. Granger is building a fence. He places 15 fence posts 8 feet apart. What is the distance from the first fence post to the last?  Write to help explain your best thinking using words, numbers, or pictures.

5G-2) Answer: 75 seconds

Five flags are spaced evenly around a track. It took a runner 30 seconds to get from the first flag to the third flag. If the runner continues at the same speed, how long will it take her to get completely around the track?  Write to help explain your best thinking using words, numbers, or pictures.

5G-1) Answer:  144 rectangles

Rectangular cards, each 2 inches by 3 inches, are cut from a rectangular sheet 2 feet by 3 feet.  What is the greatest number of cards that can be cut from the sheet?  Write to help explain your best thinking using words, numbers, or pictures.

Math Practice Problems for Sixth Grade in Geometry

6G-23) Answer: 225 square feet

The area of a square is 25 square feet. What will the area be if the sides are made three times larger?  Explain in detail how you found your answer using words, numbers, and/or pictures.

6G-22) Answer: Make a drawing. Place the 30-foot by 8 foot mat along the 30-foot edge. This leaves 30 feet by 36 feet to cover. The nine mats can be positioned in a 3x3 array to fill the room.

A group of gymnasts are putting mats on the floor of the gymnasium prior to their exhibition. The room measures 30 feet by 44 feet. They have nine mats that measure 10 feet by 12 feet, and one mat that measures 8 feet by 30 feet. Show how the mats are placed in the room to cover the floor.  Explain in detail how you found your answer using words, numbers, and/or pictures.

6G-21) Answer: 16 posts

A rectangular field is 30 feet wide and 50 feet long. Fence posts are to be placed every 10 feet around the field. How many posts are needed?  Explain in detail how you found your answer using words, numbers, and/or pictures.

6G-20) Answer: 25 pepper plants

Alice is planning her garden in a square plot 6' x 6'. She wants to plant pepper plants one foot apart, in rows that are also one foot apart. She leaves a border one foot from each edge of the plot. How many pepper plants can she plant?  Explain in detail how you found your answer using words, numbers, and/or pictures.

6G-19) Answer: 16 tables

Eric is arranging the dining room in his restaurant to accommodate a party of 34 people. He is taking small, square tables that seat one person on each side, and is placing them end-to-end to make one long table. How many tables will Eric need?  Explain in detail how you found your answer using words, numbers, and/or pictures.

6G-18) Answer: 15 cuts

For Gilda's party, the Hoagie House prepared a huge sub sandwich on a 7-foot long hoagie roll. Gilda wants to feed 16 people. How many cuts must she make?  Explain in detail how you found your answer using words, numbers, and/or pictures.

6G-17) Answer: 3 hours

Kevin can mow a square lawn that is 30 yards on each side in 45 minutes. At the same rate, how long will it take him to mow a square lawn that is 60 yards on a side?  Explain in detail how you found your answer using words, numbers, and/or pictures.

6G-16) Answer: 26 posts

(Challenging problem!)  Mr. Johannsen and Mr. Yan have decided to separate their property by placing a fence along the property line. They ordered enough fence posts so that the fence would have posts placed 8 feet apart. However, five of the posts were not usable. They were still able to put the fence up by placing the remaining posts 10 feet apart. How many fence posts did they originally order?  Explain in detail how you found your answer using words, numbers, and/or pictures.

6G-15) Answer: 40 blocks away

On his morning walk, Jeremy leaves his home and walks 20 blocks. Then he turns right and proceeds for 10 blocks, turns left for another 20 blocks, and then another left turn and walks 10 blocks. How far is he from his home?  Explain in detail how you found your answer using words, numbers, and/or pictures.

6G-14) Answer: square

There is a big special at the local pizza shop! You can buy either a square pizza, 12 inches on a side, or a round pizza with a 12-inch diameter for the same price. Which is the better buy? Explain your answer.  Explain in detail how you found your answer using words, numbers, and/or pictures.

6G-13)

Use 8 cubes (2 red, 2 green, 2 yellow, 2 blue). Follow these clues to build a 2x2 cube: The two green cubes touch on a face. The red, yellow, and green cubes never share a face with a cube of the same color. One red cube and one yellow cube each touch a face of a green cube. Draw your solution.  Explain in detail how you found your answer using words, numbers, and/or pictures.

6G-12)

Use a geoboard or dot paper of geoboard arrays . Make the largest square. Record the largest octagon you can make inside that square. What fraction of the area of the square is outside the octagon?  Explain in detail how you found your answer using words, numbers, and/or pictures.

6G-11)

Use a geoboard or dot paper of geoboard arrays. How many different isosceles triangles can you make on a geoboard. Record your answers. Give the base, height, and area for each one.  Explain in detail how you found your answer using words, numbers, and/or pictures.

6G-10)

Use a geoboard or square dot paper. Can you form a scalene triangle touching yet inside an isosceles triangle?  Explain in detail how you found your answer using words, numbers, and/or pictures.

6G-9)

Use one set of tangram pieces. Using just four tangram pieces, can you form a figure with two acute angles? Can you form a figure with no more than three right angles? Justify your answer.  Explain in detail how you found your answer using words, numbers, and/or pictures.

6G-8)

Use a geoboard or square dot paper. Record each of the following triangles. Label each triangle: A right triangle (one angle that is 90° and two angles that are less than 90° ); An obtuse triangle (one angle that is greater than 90° and two angles that are less than 90° ; An acute triangle (three angles that are less than 90° )  Explain in detail how you found your answer using words, numbers, and/or pictures.

6G-7)

Use one set of tangrams. There are five triangles in a set of tangrams. Examine the triangles and answer these questions: Are any of the triangles congruent? If so, which ones? Explain why you believe they are congruent. Are any of the triangles similar? If so, which ones? Explain why you believe they are similar.  Explain in detail how you found your answer using words, numbers, and/or pictures.

6G-6)

Use square dot paper. A hexomino is a shape made of six squares connected by one or more entire sides. How many different hexomino shapes can you make that can by traced on paper and then folded into boxes? Record your different shapes.  Explain in detail how you found your answer using words, numbers, and/or pictures.

6G-5)

Which of the following statements is true, which are false and why? All rectangles are squares. All squares are rectangles. No square is a rectangle. No rectangle is a square.  Explain in detail how you found your answer using words, numbers, and/or pictures.

6G-4)

Symmetry means that a figure can be folded in half and have both sides match exactly.  The letter “A” is symmetrical.  Of the 26 capital letters, which are symmetrical?  (Some may have more than one way they can be folded.)  Explain in detail how you found your answer using words, numbers, and/or pictures.

6G-3)

Broadway is parallel to Girard Street. 44th Street is perpendicular to Denver Street. Denver is parallel to Girard. Is 44th parallel or perpendicular to Broadway?   Explain in detail how you found your answer using words, numbers, and/or pictures.

6G-2)

In a grid that is 3x3 remove four short segments and make a new shape that consists of five congruent squares. Draw your solution.  Explain in detail how you found your answer using words, numbers, and/or pictures.

6G-1)

Julia is making place cards for the homecoming dance. How many 1 1/ 2 x 3 inch cards can she cut from a 9 inch square of paper?   Explain in detail how you found your answer using words, numbers, and/or pictures.