--- title: "Moving between representations of a relationship" description: "Move fluently between real-world situations, diagrams, coordinate grids, algebraic expressions, tables, and symbolic equations involving fractions, ratio, and algebra, explaining connections between r" canonical: https://lightmysky.com/learn/mathematics/moving-between-representations-of-a-relationship-mt_9KmVWCuh5_ source: https://lightmysky.com/learn/mathematics/moving-between-representations-of-a-relationship-mt_9KmVWCuh5_.md retrieved: 2026-09-02 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Moving between representations of a relationship Move fluently between real-world situations, diagrams, coordinate grids, algebraic expressions, tables, and symbolic equations involving fractions, ratio, and algebra, explaining connections between representations Subject: Mathematics · Area: Mathematical Thinking · Ages 10 to 11 Page: https://lightmysky.com/learn/mathematics/moving-between-representations-of-a-relationship-mt_9KmVWCuh5_ ## Ready when they can - Translate a word problem into an equation, with a bar or symbol standing for the unknown and also represent it on a bar model - Plot data from a table onto a coordinate grid and interpret the relationship - Explain how a pie chart, a fraction, and a percentage all represent the same proportion ## Lesson: Same idea, many pictures A math relationship can be told in more than one way. A story about sharing cookies, a picture split into equal parts, an equation with a fraction, a table of numbers, and a graph can all describe the exact same amount. Once you can move between them, math stops feeling like separate tricks and starts feeling like one connected idea. *(drawing: Three of the four equal slices are shaded. A pie chart slice is one more way to show a fraction.)* A fraction, a percent, and a pie chart slice can all name the same part of a whole. Percent means out of 100. To turn a fraction into a percent, rename it in hundredths: 3/5 is 60/100, which is 60%. To go back, read the percent as hundredths and simplify: 60/100 is 3/5. The picture changes, the amount does not. **Example.** Zara has $20 and spends 2/5 of it on a book. The equation is 2/5 x 20 = 8, so she spent $8 and has $12 left. A bar model splits the $20 into 5 equal strips: 2 strips ($8) spent, 3 strips ($12) left. Story, equation, and bar model all agree. A bar model also runs backwards. If you know what one strip is worth, multiply it by the number of equal strips to get the whole. A bar cut into 3 equal strips where one strip holds 11 pencils has 3 x 11 = 33 pencils in the whole bar. A part plus the number of equal parts is enough to rebuild the total. **Example.** A shop's price table shows: 1 shirt costs $10, 2 shirts cost $20, 3 shirts cost $30. Plot these as points on a coordinate grid, shirts on one side and cost on the other. The points line up in a straight line through zero, because the price per shirt never changes. **Tip.** When you switch between a story, a bar model, an equation, and a table, check that the numbers match in every version. If the story says 'a quarter', your equation should use 1/4, not 4/1. **Recap.** A fraction, a percent, a pie chart slice, a bar model, a table, and a graph can all describe the same relationship, just in different clothes. ## Practice 24 questions on this page, each with its working shown. ## Needs first - [Line graphs (age 10+)](https://lightmysky.com/learn/mathematics/line-graphs-age-10-mt_bESTSBB0wK) - [Moving between bar models, number lines and equations](https://lightmysky.com/learn/mathematics/moving-between-bar-models-number-lines-and-equations-mt_CUmjcE7W6c) - [Number Pattern Relationships](https://lightmysky.com/learn/mathematics/number-pattern-relationships-mt_fxPtngwUfz) ## Opens up - [Percentages as Fractions](https://lightmysky.com/learn/mathematics/percentages-as-fractions-mt_ALUrJpY0cZ)