Mathematics · Foundation article
Moving between words, pictures and symbols
Use words, a picture, a table and a formula to describe one arrangement, then check that meaning survives changes.
Useful background article: Symbols, definitions and mathematical language
Back to MathematicsAfter reading, you can
- Connect a concrete situation with words, a schematic picture, a labelled table and a formula.
- Keep the quantities, fixed conditions and object labels consistent between representations.
- Check a translation in both directions, including a case with zero groups.
One situation can have several useful forms
Arrange full groups of four stones, with two loose stones outside. Keep four per group and two outside fixed. Let g count full groups and T count all stones. Here g is zero or a positive whole number. How many stones are there altogether?
Words explain conditions, pictures show structure, tables compare cases, and formulas describe relationships briefly. Changing representation should preserve which objects you count and what the answer means.
Read what a schematic picture represents
Draw one dot for each stone. Put four dots inside each outline representing a full group, and two dots outside all the outlines. For g = 2, the picture contains two groups of four plus two separate dots: ten stones. Count each dot once, including the two outside.
This schematic picture need not be drawn to scale. Equal dots need not mean equal stone sizes or masses. Outlines show grouping, not extra stones. When g = 0, draw no group outlines but keep two separate dots: no full groups does not mean no stones.
A table needs meaningful headings
Name the columns “Number of full groups, g” and “Total number of stones, T.” Rows g = 0, 1, 2, 3 pair with T = 2, 6, 10, 14. Read across one row: it describes one arrangement. Do not mix entries from different rows.
The headings distinguish groups from stones. A table of measured lengths would instead name the length unit, such as centimetres. Here neither column measures mass. Adding one full group adds four stones while the two loose stones remain. The table shows examples; the stated arrangement explains why that change occurs.
Connect each formula part to the objects
Write T = 4 × g + 2. The 4 × g counts grouped stones; + 2 includes the loose stones. To find a table entry, replace g with that row’s group count. At g = 3, the grouped stones number twelve, then adding two gives fourteen. At g = 0, the total is two.
T = 4 × (g + 2) describes four stones in each of g + 2 groups. The extra two now counts groups, not stones. At g = 2, that gives sixteen stones rather than ten. Parentheses change the structure; explain each formula part.
One arrangement, four ways to describe it
Each full group contains four stones, and two loose stones remain outside the groups. Change the number of groups. The words, picture, table and formula all describe the same arrangement.
In words
Four stones per full group. Full groups: 3. Loose stones: 2. Total: 14 stones.
As a picture
| Full groups, g | Total stones, T |
|---|---|
| 0 | 2 |
| 1 | 6 |
| 2 | 10 |
| 3 | 14 |
| 4 | 18 |
| 5 | 22 |
In symbols
T = 4 × g + 2
g counts full groups; T counts all stones, including the two loose ones.
g = 3. T = 4 × 3 + 2 = 14 stones.
The selected table row, drawing and calculation give the same total.
Read back and check in both directions
From words to picture: did you include all groups and both loose stones? From picture to table: do counts match headings? From table to formula: does each row fit the stated rule? Reverse the journey: explain the formula in words or draw one case.
Representations also leave things out. A total of ten alone does not tell you the arrangement: ten stones could all lie separately. Our table and formula do not describe their colours or positions either. Restore a picture using the given grouping conditions; do not invent extra information from a bare total.
Choose a helpful form and carry the idea forward
Use words to explain conditions, a picture to inspect grouping, a table to compare cases, and a formula to calculate without drawing every object. Switch forms when something is unclear. A picture’s clear meaning matters more than its decoration.
For trays containing three seedlings each and one separate seedling, let t count full trays and P count all seedlings. Write P = 3 × t + 1. The same structure applies, with new objects, definitions and fixed counts.
Worked examples
Check one case in four forms
Show three full groups of four stones and two loose stones using words, a picture, a table entry and a formula.
- Say: three groups of four, plus two loose stones. Sketch three outlines containing four dots each and two dots outside.
- The picture contains twelve grouped stones and two loose stones. Record the row g = 3, T = 14 under the named columns.
- Use the formula: 4 × 3 + 2 = 14. Read it back and check that the two extra stones are included once.
All four forms describe fourteen stones under the same grouping conditions.
Transfer the structure to seedlings
There are six full trays, each with three seedlings, and one separate seedling. Represent the total without drawing every seedling.
- Let t count full trays and P count all seedlings. Label each sketched tray “3 seedlings” and show one separate seedling.
- Write P = 3 × t + 1. A table would begin t = 0, 1, 2 with P = 1, 4, 7.
- For six trays, 3 × 6 + 1 = 19. The answer counts seedlings, not trays or their mass.
Nineteen seedlings. Each representation keeps three per full tray and one separate seedling fixed.
Try it yourself
Under T = 4 × g + 2, what should the picture and table show when g = 0?
Show the answer and reasoning
The picture shows two loose stones and no full groups. The table row is g = 0, T = 2.
Zero removes the grouped contribution, but the two fixed loose stones remain. The total is 4 × 0 + 2 = 2.
A table for the stone arrangement gives g = 2, T = 12. Check this row using a picture and correct it.
Show the answer and reasoning
Draw two groups of four and two loose stones. The corrected row is g = 2, T = 10.
The two groups contain eight stones, and the extras add two. Twelve does not match the given arrangement.
The stated stone arrangement has eighteen stones altogether. How many full groups should its picture show? Explain using the objects.
Show the answer and reasoning
Four full groups of four stones, plus the two loose stones.
Set aside the two loose stones. The remaining sixteen make four groups of four. Check forward: 4 × 4 + 2 = 18.
What to carry forward
- Different representations should preserve the same quantities and conditions.
- Label tables, explain pictures, and connect formula parts to their meaning.
- Translate back as a check, and notice what a representation does not tell you.
References
Sources for the concepts and teaching approaches in this article.