Problems and tasks for 6th graders

by Sybilla Beckmann, Department of Mathematics, University of Georgia

The linked problems and tasks are ones I gave (or modifications of ones I gave) to the average class of 6th graders I taught every day during the 2004/2005 academic year at Clarke Middle School, a public school in Athens, Georgia. These problems and tasks address portions of the Georgia Performance Standard M6N1, which is shown below. All linked problems and tasks are copyright Sybilla Beckmann, 2004, 2005. Teachers may freely copy and use these problems and tasks with their students. The Georgia Department of Education may use these problems and tasks for the Georgia Performance Standards.

M6N1. Students will understand the meaning of the four arithmetic operations as related to positive rational numbers and percents using these concepts to solve problems.

a. Use factors and multiples.

b. Decompose numbers into their prime factorization (Fundamental Theorem of Arithmetic)

c. Determine the greatest common factor (GCF) and the least common multiple (LCM) for a set of numbers.

d. Multiply and divide fractions and mixed numbers.

e. Use fractions, decimals, and percents interchangeably.

f. Solve problems involving fractions, decimals, and percents and justify the process.

g. Evaluate expressions using order of operations including exponents.

REMARKS: M6N1d and M6N1e. Only terminating decimal fractions should be used.

Problem solving:

• Mixed Problems 1
• Mixed Problems 2
• Mixed Problems 3

Mathematical expressions:

• Mathematical expressions arising from a scenario
• Mathematical expressions arising from a design or picture: example 1, example 2, example 3, example 4, example 5, example 6, example 7, example 8, example 9, example 10
• example 11 ,example 12 ,
• Literacy day: Anno's Mysterious Multiplying Jar lesson plan

Multiplying multiples of 10:

• mental multiplication with multiples of 10,

Factors:

• Factors via rectangles

Decimals

• decimals on a number line: zooming in on a number line,
• decimals on a number line: examples 1, examples 2
• decimal addition, subtraction, multiplication, and division problems
• Understanding the long division procedure for dividing a decimal by a whole number, example 1, example 2, example 3

Fractions on number lines, equivalent fractions, comparing fractions, what fractions mean:

• Quilt fractions,
• Equivalent Fractions 1, Equivalent Fractions 2,
• Equivalent Fractions and Random Samples,
• Problems on Equivalent Fractions and Random Samples,
• Equivalent fractions and random samples,
• Fractions on Number Lines,
• Comparing Fractions,
• Fractions close to 0, to 1/2, or to 1, (fill the second "board" as part of a game)
• Some fraction story problems,

Mixed numbers and improper fractions:

• Understanding how to convert between mixed numbers and improper fractions by considering number lines: page 1, page 2, page 3,
• several more pages,

Adding and subtracting mixed numbers:

• story problems,
• game: game board (makes 2 game boards, use each for 2 players, one uses the top, the other uses the bottom), game pieces (enough for 2 game boards -- cut apart and put in a paper bag). To play the game, each student has 2 paper clips. To start, each student has a paper clip at 0. Students take turns drawing a game piece from the bag (without looking!). The student takes his or her 2nd paper clip and moves it to the correct location. The opponent checks if it's correct, and if so, the 1st paper clip is then removed. The first person to get to 5 or beyond wins.

Relating fractions and decimals:

• fractions as division cut out activity: copy PIES on 3 different colors of paper, copy one SHARES for each student, give each student 3 pies and ask them to divide them equally among 4 people, gluing onto the "shares" page. After a discussion all should then write a statement or an equation that 3 divided by 4 = 3/4.
• fractions as division and equivalent fractions with denominator a power of ten,
• fractions and decimals on number lines 1,
• fractions and decimals on number lines 2,

Percents, decimals, and fractions:

• percents and fractions on hundreds grids,
• percents and fractions on "strips",
• percents and fractions on strips 2,
• percents and fractions on strips 3,
• Mixed story problems,

Multiplying fractions (why we multiply fractions the way we do, when we multiply fractions and when we don't):

• Fraction multiplication as a part of a part,
• another fraction multiplication as part of a part,
• Canceling in fraction multiplication,
• When do we use fraction multiplication, when do we add, subtract or do something else?
• Another: when do we use fraction multiplication versus another operation?
• How to solve these fraction story problems?

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