Grade 6 Practice Problems: Geometry
Examples from Standards Revision and GLEs
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.
Expectations & Examples of Geometry from the 2008 Math Standards Revision (draft) - Grade 6
Draw first-quadrant graphs from a contextual situation or a table of values. Identify the ratio of the circumference to the diameter of a circle as the constant p, and recognize 22/7 and 3.14 as common approximations of p. Verify area formulas for triangles, rectangles, and parallelograms using models, pictures, or verbal reasoning. Example:
Find the perimeter and area of triangles, rectangles, and parallelograms. Draw quadrilaterals and triangles from given information about sides and angles. Example:
Verify formulas for circumference and area of circles using models, pictures, or verbal reasoning. Find the circumference and area of circles. Find the perimeter and area of composite figures that can be divided into shapes, such as triangles, rectangles, and parts of circles. Students select relevant information and use appropriate strategies. Use the relationships among radius, diameter, circumference, and area of circles to solve problems. Measure angles; identify pairs of angles as complementary, supplementary, adjacent, or vertical; and use these relationships to find missing angle measures. Make and test conjectures about geometric figures and their properties. Example:
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Examples of Geometric Sense from the 2006 GLEs – Grade 6
Describe circles or rectangular prisms using geometric properties. Draw a figure given properties that describe a circle or rectangular prism. Explain lines of symmetry for 2 dimensional figures including circles. Describe the relationship between the diameter and the radius of a circle. Use, sort, classify, and label geometric figures in illustrations, nature, and art. Sort and classify 2 dimensional shapes and/or figures according to their properties including number of sides, number of vertices, types of angles, parallel sides, perpendicular sides, symmetry, and/or congruence. Combine polygons to create a figure. Find the missing angle given two angles of a triangle. Describe or draw lines of symmetry for angles and/or polygons. Identify, describe, or draw angles or polygons using geometric properties. Plot integers and non negative fractions and/or decimals on a number line. Locate the point of final destination given directions for movement on an integer number line. Determine and describe the distance between any two integers on a number line. Describe the relative location of points and objects on a number line with both positive and negative numbers. Locate objects on a number line based on given numeric locations. Identify or name the location of points on a number line using coordinates or labels. Describe a 90° or 180° rotation of a figure about its center or a vertex. Describe a rotation so that another person could draw it. Describe whether an object has been translated or rotated on a coordinated grid. Draw a design using a 90°, 180°, 270°, or 360° rotation of a shape or figure. Plot the points and write the coordinates of an object or figure that has been rotated 90°, 180°, or 270° about its center or a vertex on a coordinate grid. |