TEKS: Grade 6
Read an introduction to Texas Essential Knowledge and Skills charts.
TEKS  Examples  Commentary 
111.22. MATHEMATICS, GRADE 6 (A) compare and order nonnegative rational numbers; (B) generate equivalent forms of rational numbers including whole numbers, fractions, and decimals; (C) use integers to represent real life situations; (D) write prime factorizations using exponents; (E) identify factors of a positive integer, common factors, and the greatest common factor of a set of positive integers; and (F) identify multiples of a positive integer and common multiples and the least common multiple of a set of positive integers. 


6.2 Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve problems and justify solutions. (A) model addition and subtraction situations involving fractions with objects, pictures, words, and numbers; (B) use addition and subtraction to solve problems involving fractions and decimals; 


(C) use multiplication and division of whole numbers to solve problems including situations involving equivalent ratios and rates; 
Measure with string around a globe at the equator and call the length e. Place this string on a latitude line on the globe between the cities of Ahmadabad, India, and Cabo San Lucas, Mexico; call this distance d. Measure e and d in centimeters. Set up a proportion to find the actual distance in kilometers between the two cities using the following proportion: d (in cm) / e (in cm) ? (km) / 40,000km 
Extension: Have the students use an atlas to find the actual distance between the two cities. Estimate distances between other cities. Use different size globes and determine if the size of the globe makes a difference in distances. 
(D) estimate and round to approximate reasonable results and to solve problems where exact answers are not required; and (E) use order of operations to simplify whole number expressions (without exponents) in problem solving situations. 


6.3 Patterns, relationships, and algebraic thinking. The student solves problems involving direct proportional relationships. 


(A) use ratios to describe proportional situations; (B) represent ratios and percents with concrete models, fractions, and decimals; and 
Use a small bag of M&Ms to make inferences about the proportion of certain colors of M&Ms in a larger bag. Collect, organize, and analyze data; use ratios to compare quantities of different colors; use equivalent fractions to understand proportions; and make predictions from data using proportions. Have students make a chart showing the number of each color, the corresponding fraction, decimal, and percent. Draw a bar graph using the data in percent form. Compare among students or groups. Use a calculator to find out the number of degrees in a circle represented by the percent of M&Ms by color. Use this information to make a circle graph. 
Compare with M&M/Mars Co published proportions: Red 20%, green 10%, yellow 20%, orange 10%, brown 30%, blue 10% 
(C) use ratios to make predictions in proportional situations. 
Collect data in class such as how many students wear glasses. Based on the data, make a prediction for the entire grade level. Have several students count at lunch to determine the accuracy of the prediction. 
Extension: Discuss if the class is an unbiased sample of the grade level. Talk about different methods of sampling and ways of eliminating bias in sampling. 
6.4 Patterns, relationships, and algebraic thinking. The student uses letters as variables in mathematical expressions to describe how one quantity changes when a related quantity changes. 


(A) use tables and symbols to represent and describe proportional and other relationships such as those involving conversions, arithmetic sequences (with a constant rate of change), perimeter, and area; and 
Arrange 12 square tiles to form a rectangle. What is the length and width of your rectangle? a. Use tiles to form as many rectangles as you can with an area of 12 square tiles. Use this information to complete a table of values. b. Graph the data pairs (length, width) from your table on a coordinate plane. c. As the length of the rectangle gets larger, describe what happens to the width. d. Can a rectangle ever have a width of 0 units? Justify your answer. e. As the length of the rectangle gets smaller, describe what happens to the width. f. Can a rectangle ever have a length of 0 units? Justify your answer. g. Write an equation relating the length and width. Solve your equation for width. Explain how your equation supports your answer to part d and f. 

(B) use tables of data to generate formulas representing relationships involving perimeter, area, volume of a rectangular prism, etc. 
From a piece of paper that is 8.5 inches by 11 inches, cut a oneinch square out of each corner. Fold the paper into a box without a lid. What is the volume of the box? What would happen to the volume if you cut a twoinch square out of each corner instead of oneinch? What if you cut 3 inches? If you were designing a box using this piece of paper that would hold the most popcorn, what would be the dimensions of this box? Use a table and graph to help explain your answer. What would be a reasonable domain in this problem? Write an equation that would give the volume of the box for any given cutout square. 
This type of problem leads to a calculus topic, optimization. A similar problem is in the NCTM Addenda Series, Grades 58: Patterns and Functions, pages 6465 
6.5 Patterns, relationships, and algebraic thinking. The student uses letters to represent an unknown in an equation. formulate equations from problem situations described by linear relationships. 6.6 Geometry and spatial reasoning. The student uses geometric vocabulary to describe angles, polygons, and circles. (A) use angle measurements to classify angles as acute, obtuse, or right; (B) identify relationships involving angles in triangles and quadrilaterals; and (C) describe the relationship between radius diameter and circumference of a circle. 
Samuel checked out the inside of the family refrigerator. He saw 4 times as many apples as strawberries and half as many oranges as apples. Represent this as many ways as you can with a diagram, table, and number sentence. 

6.7 Geometry and spatial reasoning. The student uses coordinate geometry to identify a location in 2 dimensions. locate and name points on a coordinate plane using ordered pairs of nonnegative rational numbers. 6.8 Measurement. The student solves application problems involving estimation and measurement of length, area, time, temperature, volume, weight, and angles. (A) estimate measurements (including circumference) and evaluate reasonableness of results; 


(B) select and use appropriate units, tools, or formulas to measure and to solve problems involving length (including perimeter), area, time, temperature, volume, and weight; 
Daniel collects ducks and dachshunds. This morning he counted 20 heads and 56 legs. How many ducks and dachshunds does he have? Daniel plans to build a pet house to keep them in. He plans to follow the guidelines of the local animal shelter. They say for each dog the cage must be at least 27 cubic feet and for 3 ducks, the cage must be at least 16 cubic feet. If the pet house is 9 feet wide, 20 feet long, with an 8 feet ceiling, will he have enough space for all his pets? 

(C) measure angles; and (D) convert measures within the same measurement system (customary and metric) based on relationships between units. 


6.9 Probability and statistics. The student uses experimental and theoretical probability to make predictions. (A) construct sample spaces using lists and tree diagrams; and (B) find the probabilities of a simple event and its complement and describe the relationship between the two. 


6.10 Probability and statistics. The student uses statistical representations to analyze data. 


(A) select and use an appropriate representation for presenting and displaying different graphical representations of the same data including line plot, line graph, bar graph, and stem and leaf plot; 
Collect data of interest to the class. Use this data to make box plots, stem and leaf, scatter plots and number line plots. Examples of data:

Ideas from Exploring Data (1996); Dale Seymour Publications. This introduces the AP* Statistics concept of comparing distributions of univariate datadot plots, backtoback stemplots, and parallel boxplots. 
(B) identify mean (using concrete objects and pictorial models), median, mode, and range of a set of data; (C) sketch circle graphs to display data; and (D) solve problems by collecting, organizing, displaying, and interpreting data. 
In the second race, the tortoise traveled 5 meters per minute. The hare left 5 minutes after the tortoise and ran for 2 minutes, stopped for 14 minutes, then realized he was behind, and he continued to race. Did he overtake the hare? At what point? Draw a chart and graph of the race. 
This problem introduces the concepts of rate of change and systems of equations. It is also an example of a piecewise function. 
6.11 Underlying processes and mathematical tools. The student applies Grade 6 mathematics to solve problems connected to everyday experiences, investigations in other disciplines, and activities inside and outside of school. 

The underlying processes and mathematical tools are the same for grades 68. TEKS 7.13, 7.14, 7.15 and 8.14, 8.15, 8.16 are the same as 6.11, 6.12, and 6.13. Although the TEKS are the same, older students should show greater sophistication in their mathematical reasoning and communication. 
(A) identify and apply mathematics to everyday experiences, to activities in and outside of school, with other disciplines, and with other mathematics topics; (B) use a problemsolving model that incorporates understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness; (C) select or develop an appropriate problemsolving strategy from a variety of different types, including drawing a picture, looking for a pattern, systematic guessing and checking, acting it out, making a table, working a simpler problem, or working backwards to solve a problem; and (D) select tools such as real objects, manipulatives, paper/pencil, and technology or techniques such as mental math, estimation, and number sense to solve problems. 


6.12 Underlying processes and mathematical tools. The student communicates about Grade 6 mathematics through informal and mathematical language, representations, and models. (A) communicate mathematical ideas using language, efficient tools, appropriate units, and graphical, numerical, physical, or algebraic mathematical models; and (B) evaluate the effectiveness of different representations to communicate ideas. 


6.13 Underlying processes and mathematical tools. The student uses logical reasoning to make conjectures and verify conclusions. (A) make conjectures from patterns or sets of examples and nonexamples; and (B) validate his/her conclusions using mathematical properties and relationships. 

