All Sorts of Denominators

10 min

Narrative

The purpose of this How Many Do You See? is for students to visualize a common denominator of two fractions. The diagram can be seen as showing 812+312\frac{8}{12} + \frac{3}{12}, but it also can be seen as showing 23+14\frac{2}{3} + \frac{1}{4}. The area diagram provides a way to visualize why the product of their denominators works as a common denominator for two fractions. 

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Launch

  • Groups of 2
  • “How many do you see? How do you see them?”  
  • Display the image.
  • 1 minute: quiet think time
Teacher Instructions
  • Display the image. 
  • “Discuss your thinking with your partner.”
  • 1 minute: partner discussion
  • Record responses.

Student Task

H​​​​ow many do you see? How do you see them?

Two diagrams, both squares with length and width, 1. Each partitioned into 3 rows of 4 of the same size rectangles. On left, 8 rectangles shaded. On right, 3 rectangles shaded.

Sample Response

Sample responses:

  • 812\frac{8}{12} and 23\frac{2}{3}: The shaded parts of the left square
  • 312\frac{3}{12} and 14\frac{1}{4}: The shaded parts of the right square
  • 1112\frac{11}{12}: 11 parts, each of which is 112\frac{1}{12}. 
Activity Synthesis (Teacher Notes)
  • “How does the diagram show 23+14\frac{2}{3} + \frac{1}{4}?” (23\frac{2}{3} of the left square and 14\frac{1}{4} of the right square are shaded.)
  • “What is the value of 23+14\frac{2}{3} + \frac{1}{4}? How do you know?” (1112\frac{11}{12}, because there are 11 shaded pieces and each is 112\frac{1}{12}.)
Standards
Addressing
  • 5.NF.1·Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. <em>For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)</em>
  • 5.NF.A.1·Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. <span>For example, <span class="math">\(2/3 + 5/4 = 8/12 + 15/12 = 23/12\)</span>. (In general, <span class="math">\(a/b + c/d = (ad + bc)/bd\)</span>.)</span>

15 min

20 min