Equivalent ratios in 6th grade

Line A matches the table with 1, 2, 3, 4 over 2, 4, 6, 8. Line B matches 3, 6, 9, 12 over 4, 8, 12, 16. Line C matches 2, 4, 6, 8 over 5, 10, 15, 20. The table with 1, 2, 3, 4 over 3, 6, 9, 12 is the extra table.

1. 1On Line A, the aligned nonzero pairs are 1 with 2, 2 with 4, 3 with 6, and 4 with 8. The table with those same two rows matches Line A.
2. 2On Line B, the aligned nonzero pairs are 3 with 4, 6 with 8, 9 with 12, and 12 with 16. The table with those rows matches Line B.
3. 3On Line C, the aligned nonzero pairs are 2 with 5, 4 with 10, 6 with 15, and 8 with 20. The table with those rows matches Line C.
4. 4The extra table has 1, 2, 3, 4 on top and 3, 6, 9, 12 on the bottom. Its first pair is 1 with 3, but no line shows that pair — Line A pairs 1 with 2, not 1 with 3.

Common mistake. A common mistake is looking only at the top row of a table. Tables 1 and 2 both start with 1 in the first column, so you must check the bottom row to tell them apart.

The idea underneath. Compare the four nonzero pairs on each number line with the table columns. Keep top values in the top row and bottom values in the bottom row. These tables leave out 0 with 0.

The answer is Ratio table because it lists several matching pairs, one pair per row, which is the job this question needs.

1. 1The question needs the cups of oats for 8, 12 and 20 cups of flour listed one pair per row.
2. 2A ratio table puts each pair on its own row. Each row multiplies both numbers of the starting pair, 4 and 3, by the same number, so every row keeps the same ratio. For example, 8 is 2 × 4, so the amount that goes with it is 2 × 3 = 6.
3. 3Other representations can also find the numbers in this problem, but each does a different job. A tape diagram shows a known total cut into equal parts. A double number line shows where amounts fall along a scale. An equation gives one rule that works for any amount. None of them lists several matching pairs, one pair per row, so the ratio table is the one that matches.

Common mistake. A common mistake is choosing Equation because the amounts can be calculated. An equation gives one rule that works for any amount, but this question needs the pairs listed together, one pair per row.

The idea underneath. Each representation does a different job. A ratio table lists several matching pairs, one pair per row. A tape diagram shows a known total cut into equal parts. A double number line shows where amounts fall along a scale. An equation gives one rule that works for any amount. Read what the question needs, then choose the representation that does that job.

The answer is 21 because 35 heart stickers split into 5 equal parts gives 7 stickers per part, and 3 parts make 21 star stickers.

1. 1The heart stickers row has 5 equal parts that together make 35 stickers.
2. 2Divide 35 by 5 to find how many stickers are in 1 equal part.
3. 3The star stickers row has 3 equal parts. Multiply 3 by 7 to find its total.
4. 4So the missing total for star stickers is 21.

Common mistake. A common mistake is to subtract 2 from 35 because the star row has 2 fewer parts than the heart row. In a ratio tape diagram, first divide the known total by its row's part count to find the value of 1 equal part, then multiply that value by the target row's part count.

The idea underneath. In a ratio tape diagram, every equal bar stands for the same-sized group, so once you know the value of 1 group, you can find any row total.

The correct choice is that the equation matches the model. The multiplier $\frac{3}{2}$ correctly reflects the 3 blue parts compared with the 2 red parts.

1. 1Look at the tape diagram. The red row has 2 equal parts of 4 for a total of 8, and the blue row has 3 equal parts of 4 for a total of 12.
2. 2The blue row has 3 parts and the red row has 2 parts, so the correct multiplier from red to blue is $\frac{3}{2}$. Multiplying 8 by $\frac{3}{2}$ gives 12.
3. 3Alex wrote $8 \times \frac{3}{2} = 12$. The multiplier $\frac{3}{2}$ correctly compares 3 blue parts to 2 red parts, and both totals are correct.
4. 4The equation matches the model.

Common mistake. A common mistake is to think the equation used the part label 4 as the multiplier. Here, Alex correctly used $\frac{3}{2}$, the ratio of 3 blue parts to 2 red parts, not the number written inside each segment.

The idea underneath. In a tape diagram, the multiplier between two quantities comes from comparing the whole rows, not from the number written inside one equal part.