Appendix: matching the Drum Loop to New York State’s mathematics standards

Use of the Drum Loop by math teachers could support the following requirements from the New York State Learning Standards and Core Curriculum for Mathematics.

Fractions—Grade 3

  • CCSS.Math.Content.3.NF.A.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
  • CCSS.Math.Content.3.NF.A.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
  • CCSS.Math.Content.3.NF.A.3b Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.

Fractions—Grade 4

  • CCSS.Math.Content.4.NF.B.3a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
  • CCSS.Math.Content.4.NF.B.3b Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
  • CCSS.Math.Content.4.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
  • CCSS.Math.Content.4.NF.B.4a Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).
  • CCSS.Math.Content.4.NF.B.4b Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

Ratios & Proportional Relationships—Grade 6

  • CCSS.Math.Content.6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
  • CCSS.Math.Content.6.RP.A.3b Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 34 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

Ratios & Proportional Relationships—Grade 7

  • CCSS.Math.Content.7.RP.A.2a Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
  • CCSS.Math.Content.7.RP.A.2b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

Representation Strand—Grade Eight

  • 8.R.1 Use physical objects, drawings, charts, tables, graphs, symbols, equations, or objects created using technology as representations
  • 8.R.2 Explain, describe, and defend mathematical ideas using representations
  • 8.R.3 Recognize, compare, and use an array of representational forms
  • 8.R.4 Explain how different representations express the same relationship
  • 8.R.5 Use standard and non-standard representations with accuracy and detail
  • 8.R.6 Use representations to explore problem situations
  • 8.R.9 Use mathematics to show and understand physical phenomena (e.g., make and interpret scale drawings of figures or scale models of objects)

Geometry

  • G.PS.3 Use multiple representations to represent and explain problem situations (e.g., spatial, geometric, verbal, numeric, algebraic, and graphical representations)
  • G.CM.2 Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, and diagrams
  • G.CN.1 Understand and make connections among multiple representations of the same mathematical idea
  • G.CN.3 Model situations mathematically, using representations to draw conclusions and formulate new situations
  • G.CN.6 Recognize and apply mathematics to situations in the outside world
  • G.R.1 Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts
  • G.R.2 Recognize, compare, and use an array of representational forms
  • G.R.3 Use representation as a tool for exploring and understanding mathematical ideas
  • G.R.5 Investigate relationships between different representations and their impact on a given problem
  • G.G.21 Investigate and apply the concurrence of medians, altitudes, angle bisectors, and perpendicular bisectors of triangles
  • G.G.54 Define, investigate, justify, and apply isometries in the plane (rotations, reflections, translations, glide reflections)
  • G.G.55 Investigate, justify, and apply the properties that remain invariant under translations, rotations, reflections, and glide reflections
  • G.G.56 Identify specific isometries by observing orientation, numbers of invariant points, and/or parallelism
  • G.G.57 Justify geometric relationships (perpendicularity, parallelism, congruence) using transformational techniques (translations, rotations, reflections)
  • G.G.60 Identify specific similarities by observing orientation, numbers of invariant points, and/or parallelism
  • G.G.61 Investigate, justify, and apply the analytical representations for translations, rotations about the origin of 90o and 180o, reflections over the lines, and dilations centered at the origin

Algebra 2 and Trigonometry

  • A2.R.6 Use mathematics to show and understand physical phenomena (e.g., investigate sound waves using the sine and cosine functions)
  • A2.A.56 Know the exact and approximate values of the sine, cosine, and tangent of 0o, 30o, 45o, 60o, 90o, 180o, and 270o angles
  • A2.A.60 Sketch the unit circle and represent angles in standard position
  • A2.A.61 Determine the length of an arc of a circle, given its radius and the measure of its central angle
  • A2.A.69 Determine amplitude, period, frequency, and phase shift, given the graph or equation of a periodic function
  • A2.M.1 Define radian measure
  • A2.M.2 Convert between radian and degree measures

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