Which gaps are equal? An additive and multiplicative axis worksheet

By Figran Team. Two neighbouring points can be equally far apart on a page while representing very different additions. To read that gap, a reader needs the axis rule as well as the labels. This worksheet compares an additive axis with a multiplicative axis using the same five invented values. We are associated with Figran, a commercial AI-assisted visual creation service. This original teaching activity was prepared with AI assistance and checked for internal arithmetic consistency. It is a paper exercise, not measured research data, a classroom trial, or evidence of product performance. No drawing software or account is required. 1. Define the source and both axis rules The source card assigns these five values in arbitrary exercise units. They are not observations of physical objects, experimental outcomes, or Figran account credits. Record; Assigned value Alder; 2 Birch; 4 Cedar; 8 Dune; 16 Elm; 32 Make two horizontal number lines. A position is a location along a line, not an additional source value. Axis A: additive positions. Put a value v at position p = v. Thus the positions of the five points are 2, 4, 8, 16, and 32. Each one-position step represents an addition of one exercise unit. Keep the spacing per position constant along the entire line. Axis B: multiplicative positions. Put value 2 at position 0, value 4 at position 1, value 8 at position 2, value 16 at position 3, and value 32 at position 4. Each one-position step represents multiplication by two. The rule is v = 2 × 2^p. It defines a logarithmic axis for positive values; the base and starting value are explicit here. Position 0 represents value 2, not value zero. Use those rules to draw points or reason from the following position table. Do not draw five equally spaced points for Axis A merely because there are five source records. Record; Value; Position on A; Position on B Alder; 2; 2; 0 Birch; 4; 4; 1 Cedar; 8; 8; 2 Dune; 16; 16; 3 Elm; 32; 32; 4 For the paper task, the two lines need not have the same physical length. Compare gaps within each line. A centimetre on one independently drawn line is not automatically comparable with a centimetre on the other. 2. Work through six questions Cover the answers below while attempting these tasks. For every answer, identify whether you used a value difference, a value ratio, or a position difference. 1. Compare Alder → Birch with Dune → Elm. On Axis A, are their gaps equal? How much is added in each transition? 2. Compare the same transitions on Axis B. Are their gaps equal? Does that mean the additions are equal? 3. Compare Alder → Cedar with Cedar → Elm. Each moves two positions on Axis B. What value ratio does each transition represent? What addition does each represent? 4. A point is placed halfway between Birch and Dune on each axis. What value belongs at that halfway position on Axis A? What value belongs there on Axis B? Explain why the two answers differ. 5. A reviewer says, “Cedar is at position 2 on Axis B, so its source value must be 2.” Diagnose the mistake and give a caption that prevents it. 6. The exercise gains a new record, Frost, assigned value 0. Can the original Axis B rule place it at position 0? Can it place it at any finite position? Suggest a repair that retains the zero in the source. These are questions about the declared mapping. A correct answer does not demonstrate that either axis is the best choice for every communication task. 3. Check the differences and ratios Question 1. On Axis A, Alder → Birch moves from position 2 to 4: a gap of 2 and an addition of 2 units. Dune → Elm moves from 16 to 32: a gap of 16 and an addition of 16 units. The second gap is eight times the first. Equal record-to-record steps in a list do not establish equal value differences. Question 2. On Axis B, both transitions move one position. Their ratios are 4/2 = 2 and 32/16 = 2. They both double the assigned value. Their additions remain 2 and 16 units. Changing the position rule has not changed either source value. Question 3. Alder → Cedar has ratio 8/2 = 4 and addition 8 − 2 = 6. Cedar → Elm has ratio 32/8 = 4 and addition 32 − 8 = 24. Two one-position doubling steps represent multiplication by four. Equal gaps on Axis B establish equal ratios under its rule, not equal additions. A useful review sentence is: “This gap comparison concerns ratios because the stated axis rule doubles the value per position.” Keep the arithmetic differences available in the companion source table when the reader also needs them. 4. Check the halfway point Question 4, Axis A. Birch and Dune are at positions 4 and 16. Their halfway position is (4 + 16)/2 = 10. Since p = v, it represents value 10. Its additions from the endpoints are equal: 10 − 4 = 6 and 16 − 10 = 6. Question 4, Axis B. Birch and Dune are at positions 1 and 3. Their halfway position is 2, already occupied by Cedar, whose assigned value is 8. Its ratios to the endpoints are equal: 8/4 = 2 and 16/8 = 2. Its additions are 4 and 8, which are unequal. The midpoint represents different values because the mappings are different. Do not move the source's Cedar value from 8 to 10 to make the drawings agree. That would edit the data rather than explain the axis. 5. Separate a position from a value, and retain zero Question 5. The reviewer has substituted a position for a source value. At p = 2, Axis B gives v = 2 × 2^2 = 8. A suitable caption is: “Assigned values on a multiplicative axis. Value 2 is at position 0; each one-position step doubles the value. Source values remain in the companion table.” The caption defines the map without claiming a scientific mechanism or an observed growth process. Question 6. Axis B cannot put Frost's zero at position 0: that position already represents value 2. It cannot represent zero at any finite position under v = 2 × 2^p, because that expression is positive. Moving left gives 1 at p = −1 and 0.5 at p = −2; neither is zero. Retain Frost = 0 in the source table. If all six records must be compared on one number line, use Axis A, which places zero at position 0. Another possible presentation is a positive-value Axis B with an explicit separate note, “Frost = 0 is retained in the source but cannot be positioned on this axis.” State that exclusion visibly. Do not replace zero with 1, silently delete Frost, or invent a small positive measurement to make the logarithmic map accept it. 6. Write a caption that names the comparison Choose the communication task before choosing the axis. For this exercise, Axis A supports comparing additions through position gaps. Axis B supports comparing ratios through position gaps. Neither drawing turns the assigned values into observations or validates a causal story. Write two short captions, one for each line. Each must name the axis rule, what equal gaps mean, and where the unchanged source values can be found. If Frost is included, explain how zero is shown or separately retained. Exchange the captions with another reader and ask them to solve Questions 1, 2, and 6 without your verbal explanation. Record ambiguities in the caption as feedback on this proposed activity. Do not report that exchange as evidence of improved learning or a completed classroom study. 7. Credit and reuse This is a tool-independent exercise in reading an explicit numeric mapping. It is not an independent review of Figran or guidance for validating research measurements. For a real chart, consult its actual source, scale definition, reporting requirements, and relevant subject expertise. Public contact: [email protected]. Our commercial relationship and AI assistance are disclosed above; no product account is needed for the activity.