# Journal: Integers and Fractions Negative numbers and fractions change how math works compared to whole numbers. This is because dividing by a fraction is the same as multiplying by its reciprocal (the fraction flipped upside down). Prompt: Explain why dividing a positive number by a proper fraction (a positive fraction less than 1) gives an answer bigger than the number you started with. Also explain why multiplying two negative integers gives a positive answer. Use a real-life example for each rule. ### Where to start: - Think of a case like sharing 6 pizzas into 1/2-pizza slices. Why does the answer (12) end up larger than 6? - What does "the number of fraction-sized groups that fit into a quantity" mean, in your own words? - Try a temperature or money example where two negatives multiply together. What happens, and why does that make sense? - How is the − symbol doing two different jobs when you write something like 5 − (−3)? ### What a strong explanation includes: - A real-life example that clearly shows why dividing by a proper fraction produces more, smaller groups, making the answer larger than the starting number. - A sentence explaining that students apply operations to positive and negative numbers. - Correct use of the words "groups," "fits into," or "size" when describing division by a fraction. - A sentence that describes operations applied to positive and negative numbers. ### Stretch: Write one expression that mixes integers, fractions, and more than one operation, such as 6 ÷ 1/2 + (−3) × 1/4 = 11.25, and explain the order of operations you'd use to evaluate it. --- *AILI prompt · language en · model claude-sonnet-5 · generated 2026-09-17 · id 2515b8be-99cf-4088-b21f-34db8a0cfd90* ### Sources - node:n1: Mathematics › Mathematics (7–9) › Grade 7 › Number: Numbers are organized into systems with unique notation to communicate quantities and to facilitate calculation… › How can multiplication and division be generalized? (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/20891) - node:n2: Mathematics › Mathematics (7–9) › Grade 7 › Number: Numbers are organized into systems with unique notation to communicate quantities and to facilitate calculation… › How can operations on positive and negative numbers be understood? (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/20871) - node:n3: Mathematics › Mathematics (7–9) › Grade 7 › Number: Numbers are organized into systems with unique notation to communicate quantities and to facilitate calculation… › How can multiplication and division be generalized? › Students interpret multiplication and division of positive fractions and of positive decimal numbers. (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/20897) - node:n4: Mathematics › Mathematics (7–9) › Grade 7 › Number: Numbers are organized into systems with unique notation to communicate quantities and to facilitate calculation… › How can multiplication and division be generalized? › Students interpret multiplication and division of positive fractions and of positive decimal numbers. (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/20898) - node:n5: Mathematics › Mathematics (7–9) › Grade 7 › Number: Numbers are organized into systems with unique notation to communicate quantities and to facilitate calculation… › How can multiplication and division be generalized? › Students interpret multiplication and division of positive fractions and of positive decimal numbers. (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/20899) - node:n6: Mathematics › Mathematics (7–9) › Grade 7 › Number: Numbers are organized into systems with unique notation to communicate quantities and to facilitate calculation… › How can multiplication and division be generalized? › Students interpret multiplication and division of positive fractions and of positive decimal numbers. (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/20902) - node:n7: Mathematics › Mathematics (7–9) › Grade 7 › Number: Numbers are organized into systems with unique notation to communicate quantities and to facilitate calculation… › How can multiplication and division be generalized? › Students interpret multiplication and division of positive fractions and of positive decimal numbers. (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/20901) - node:n8: Mathematics › Mathematics (7–9) › Grade 7 › Number: Numbers are organized into systems with unique notation to communicate quantities and to facilitate calculation… › How can multiplication and division be generalized? › Students interpret multiplication and division of positive fractions and of positive decimal numbers. (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/20903) - node:n9: Mathematics › Mathematics (7–9) › Grade 7 › Number: Numbers are organized into systems with unique notation to communicate quantities and to facilitate calculation… › How can multiplication and division be generalized? › Students interpret multiplication and division of positive fractions and of positive decimal numbers. (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/20895) - node:n10: Mathematics › Mathematics (7–9) › Grade 7 › Number: Numbers are organized into systems with unique notation to communicate quantities and to facilitate calculation… › How can multiplication and division be generalized? › Students interpret multiplication and division of positive fractions and of positive decimal numbers. (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/20893) - node:n11: Mathematics › Mathematics (7–9) › Grade 7 › Number: Numbers are organized into systems with unique notation to communicate quantities and to facilitate calculation… › How can multiplication and division be generalized? › Students interpret multiplication and division of positive fractions and of positive decimal numbers. (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/20894) - node:n12: Mathematics › Mathematics (7–9) › Grade 7 › Number: Numbers are organized into systems with unique notation to communicate quantities and to facilitate calculation… › How can multiplication and division be generalized? › Students interpret multiplication and division of positive fractions and of positive decimal numbers. (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/20905) - node:n13: Mathematics › Mathematics (7–9) › Grade 7 › Number: Numbers are organized into systems with unique notation to communicate quantities and to facilitate calculation… › How can multiplication and division be generalized? › Students interpret multiplication and division of positive fractions and of positive decimal numbers. (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/20906) - node:n14: Mathematics › Mathematics (7–9) › Grade 7 › Number: Numbers are organized into systems with unique notation to communicate quantities and to facilitate calculation… › How can multiplication and division be generalized? › Students interpret multiplication and division of positive fractions and of positive decimal numbers. (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/20907) - node:n15: Mathematics › Mathematics (7–9) › Grade 7 › Number: Numbers are organized into systems with unique notation to communicate quantities and to facilitate calculation… › How can operations on positive and negative numbers be understood? › Students apply operations to positive and negative numbers. (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/20873) - node:n16: Mathematics › Mathematics (7–9) › Grade 7 › Number: Numbers are organized into systems with unique notation to communicate quantities and to facilitate calculation… › How can operations on positive and negative numbers be understood? › Students apply operations to positive and negative numbers. (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/20875) - node:n17: Mathematics › Mathematics (7–9) › Grade 7 › Number: Numbers are organized into systems with unique notation to communicate quantities and to facilitate calculation… › How can operations on positive and negative numbers be understood? › Students apply operations to positive and negative numbers. (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/20874) - node:n18: Mathematics › Mathematics (7–9) › Grade 7 › Number: Numbers are organized into systems with unique notation to communicate quantities and to facilitate calculation… › How can operations on positive and negative numbers be understood? › Students apply operations to positive and negative numbers. (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/20879) - node:n19: Mathematics › Mathematics (7–9) › Grade 7 › Number: Numbers are organized into systems with unique notation to communicate quantities and to facilitate calculation… › How can operations on positive and negative numbers be understood? › Students apply operations to positive and negative numbers. (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/20878) - node:n20: Mathematics › Mathematics (7–9) › Grade 7 › Number: Numbers are organized into systems with unique notation to communicate quantities and to facilitate calculation… › How can operations on positive and negative numbers be understood? › Students apply operations to positive and negative numbers. (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/20877)