Start. Draw two circles and three squares. Say, “There are two circles for every three squares. How many shapes are in one whole group?” Invite a learner to mark the whole five-shape group. Write circles:squares = 2:3, circles:all shapes = 2:5 and circles as a fraction of all shapes = 2/5. Ask which statement compares two parts and which compares a part with the whole.
The misconception to notice is “2:3 means two-thirds of the whole.” Have the learner point to all five shapes before changing their notation.
Route 21A · Build the whole
Draw a repeating pattern with 3 triangles and 4 dots in each group. Make two groups. Label triangles:dots, triangles:all marks and the fraction of marks that are dots. Then add a third group and explain which relationships stayed the same. Keep the groups visible; do not simplify the total counts before showing where they came from.
Teacher next move: expect 3:4, 3:7 and 4/7; with three groups, counts are 9 and 12, whole 21. Ask a learner who writes 4/3 to circle the denominator's whole.
Route 21B · Find and repair a claim
A fictional puzzle says: “A box has 4 plain tiles and 6 striped tiles. Plain tiles make up 4/6 of the box.” Draw the ten tiles, decide whether the sentence is correct, and write a corrected sentence. Find a different true comparison involving 4 and 6. Explain why both the ratio and the fraction need a named comparison.
Teacher next move: plain:striped = 4:6 = 2:3; plain fraction of the whole = 4/10 = 2/5. A ratio of quantities is not wrong simply because it is not a fraction of the whole.
Route 21C · Design a three-part pattern
Design a paper border with 2 circles, 1 square and 3 triangles per group. Draw two groups. State the three-part ratio, the fraction of all marks that are triangles, and the ratio of non-triangles to triangles. Then design another group with six marks that has a different triangle fraction. Name what changed.
Teacher next move: 2:1:3; triangles 3/6 = 1/2; non-triangles:triangles = 3:3 = 1:1. Accept any second design with six marks and a triangle count other than three, with its fraction explained.
Close: “A ratio compares ___ with ___. My denominator means ___.”