Describe ratios and preserve the relationship when scaling

Learning goal: Describe a relationship between two quantities using ratio language, preserve its order and generate equivalent ratios. Use a table and coordinate graph to show the same relationship. A ratio compares specified quantities; its meaning depends on what each number counts.

A bead pattern uses two blue beads for every three gold beads. Blue to gold is 2:3, read “two to three.” Gold to blue is 3:2. These statements describe the same pattern in reversed order, but 2:3 and 3:2 do not give the same ordered comparison. Always name the quantities before using the numbers.

One complete pattern group contains five beads. Blue to all beads is 2:5, and gold to all beads is 3:5. The fraction of the group that is blue is 2/5, not 2/3. The denominator three in the blue-to-gold comparison counts only gold beads. A part-to-part ratio and a part-to-whole fraction answer different questions.

The diagram shows three identical groups. Each group contains two blue circles and three gold circles. There are six blue and nine gold beads altogether, so the blue-to-gold ratio is 6:9. Both counts were multiplied by three, preserving two blue for every three gold. The complete pictured collection has fifteen beads; no unselected bead disappears from its whole.

Equivalent ratios arise by scaling both quantities by the same positive factor. Starting from 2:3, multiplying both terms by two gives 4:6, by three gives 6:9, and by four gives 8:12. Dividing both counts in 6:9 by three returns 2:3. Scaling only one count changes the relationship.

Adding the same amount to both counts usually does not preserve a ratio. Adding one to 2:3 gives 3:4. The blue share changes from 2/5 to 3/7. To compare without approximating, consider matching gold counts: 2:3 scales to 8:12, while 3:4 scales to 9:12. With twelve gold beads, the second pattern needs one more blue bead.

Build a table with blue count first and gold count second: (2,3), (4,6), (6,9), (8,12). Plot these ordered pairs on a coordinate plane whose horizontal axis counts blue beads and vertical axis counts gold beads. The first point goes two units right and three units up. Reversing coordinates would represent a differently labeled graph.

Each table pair belongs to the same relationship. For example, twelve blue beads require eighteen gold beads because six groups of the original pattern use 6 × 2 and 6 × 3 beads. A row with ten blue and fifteen gold also fits, even if it was not one of the first four rows drawn.

In a whole-bead pattern built from complete groups, valid group counts are whole numbers. The straight-line relationship can help visualize scaling, but half a physical bead is not available in this construction. A context involving continuous amounts such as lengths can allow intermediate values. Interpret the allowable values using the situation rather than assuming every point on a line is a feasible object count.

Use tables to compare patterns. Pattern A uses two blue for three gold. Pattern B uses three blue for five gold. At fifteen gold beads, A requires ten blue and B requires nine blue. Pattern A has more blue relative to the same gold count. Comparing only two with three would ignore the different amounts of gold.

Worked missing value: A table row has fourteen blue beads and an unknown gold count. Fourteen is seven times two, so the gold count is seven times three, or twenty-one. The total is thirty-five beads, and 14/35 simplifies to the same blue fraction, 2/5.

Worked correction: “The diagram has six blue and nine gold, so six ninths of all beads are blue.” Nine counts gold alone. All beads total fifteen; the blue fraction is 6/15 = 2/5. The ratio 6:9 remains a correct blue-to-gold comparison.

Guided modeling: Build several complete 2:3 groups with two kinds of counters. Label blue, gold and total counts. Make a table and plot at least four corresponding pairs with labeled axes and a consistent scale. Explain how one group appears in every representation.

Independent performance: Design a different two-color pattern, state both ordered part-to-part ratios and each part-to-whole fraction, and generate a table of at least five equivalent pairs. Find a missing value and compare with another pattern by matching one quantity. Submit the model, graph and explanation of why equal addition to both ratio terms does not generally preserve the relationship.