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 M6A, 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.

M6A1. Students will understand the meaning of ratio and how to use it.

Percents, fractions, and ratios:

• percents, fractions, and ratios,
• percents, fractions, and ratios 2,

Comparing ratios, equivalent ratios:

• comparing drink mixtures, equivalent ratios for drink mixtures. Suggested use: after the children do 1 - 3, let them taste the three mixtures to see if their predictions were right. Red Hawaiian punch works well as the juice (for some juices, there won't be much difference between the first and third mixtures, so test your mixtures first -- use a juice with a strong taste).

M6A2. Students will consider relations between varying quantities.

a. Analyze and describe patterns arising from function rules, tables, and graphs.

b. Understand the meaning of direct proportion. Investigate its features by using mathematical expressions. Graph simple functions in the form y = kx.

c. Apply methods for solving proportions.

d. Write and solve equations in the form y = kx where x and y are whole numbers, decimals, or fractions.

e. Use proportional reasoning including percents to solve problems.

Calculating easy benchmark percentages (encourage doing these mentally):

• multiplying and dividing by 10; finding 10% by dividing by 10 and finding 50% by taking half,
• another multiplying and dividing by 10; finding 10% by dividing by 10 and finding 50% by taking half,
• finding 10% and then finding 5% by taking half of 10%,
• finding 10%, 5%, and 50% mentally,
• finding 15% by first finding 10%, then 5%, then adding, another one,
• finding 5%, 10%, 15%, and 50%,

Calculating a percentage of a number by first calculating 1% of the number : (With this method, you do not have to set up a proportion and cross-multiply and you can often avoid multiplying a decimal by a decimal. This is the method shown in the math textbooks used in Singapore, where decimal times a decimal is not taught until after 6th grade. Note that children in Singapore scored first in the world in math on the TIMSS assessment.)

• Finding 3% and 7% by first finding 1%,

Graphing points in the plane: Graphing ordered pairs,

More on equivalent ratios:

• equivalent ratios in ratio tables: paint mixtures, drink mixtures,
• graphing equivalent ratios in ratio tables, another example,

Solving ratio problems with Singapore-style strip diagrams: Note: the textbooks used in Singapore (whose children scored first in the world in math on the international TIMSS assessment) do not teach the standard US method of solving proportions by setting two fractions equal to each other and then cross-multiplying. Instead, story problems that give rise to proportions are solved either with a ratio table or, more commonly, with strip diagrams like the ones in the next problem sets. These methods used in Singapore are more transparent and less abstract.

• Ratio strip diagram problem set 1,
• Ratio strip diagram problem set 2,
• Ratio strip diagram problem set 3,
• Using ratio tables and strip diagrams,
• Using ratio tables to compare ratios, using strip diagrams,

Problems on Ratio, Rate, and Percent:

• Set 1,
• Set 2,
• Rate problems,

Radius, diameter, and circumference of circles and the number pi (might fit better in 5th grade GPS):

• wrapping a strip of paper around a circle to check circumference formula,
• estimating circumference,
• what a circle is (mathematical definition) and radius, diameter, circumference,
• zooming in on the number pi on number lines,
• zooming in on pi (easier version),

M6A3. Students will solve simple one-step equations.

REMARK: M6A. Use letters to represent quantities.

Writing and solving equations:

• Equations arising from scenarios,
• solving equations using a pan balance image
• solving equations using a pan balance image 2
• solving equations using a pan balance image 3

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