I love working with my friend and colleague Jonily. When I started teaching in my current school she was teaching at one of the other middle schools. My school district gave us many opportunities to work with teachers in the other buildings. 6+ years ago my district added instructional coaching positions at the elementary and middle school level. Jonily and I moved into the MS math positions so I had the opportunity to work even closer with her. She has really influenced how I think about teaching and learning. Kids always come first. She has an amazing ability to see how students build their understandings and connect mathematics. The coolest thing is that she is able to present things so that average people like me can also begin to see teaching and learning as she sees it.
Check out her website. You can sign up for a weekly newsletter where she shares some of her awesomeness. The latest newsletter is below.
To continue our theme of "Teaching with Questions," this week I will introduce the Candy Probem. I will refer to the Candy Problem all year and extend the amount of content that can be drawn from this single situation. The problem can be given to ALL ages from 1st grade through Algebra! Keep in mind that the questions we ask are the most important instructional pieces of any task or problem. These questions will guide the instructional process and will open the door for exploration of grade level content throughout the Common Core State Standards. The purpose of the questions we ask is to generate mathematical thinking. The more questions generated for each problem or task, the longer the problem can be extended (even while introducing additional situations, tasks and problems). Brace yourself! This email is a long one this week!!Begin by introducing the situation:
Hello from "The Math Girl"!
The Candy ProblemTwo boys share 80 candies in the ratio 2:3Next, ask students...
What math questions can you create for this situation?Give students some time to generate creative math questions. Collect questions as a class and either answer some now or save for later!Now ask students...
How many candies will each boy get? How do you know? To eliminate the discussion and instruction of "ratio" for students below grade 4, phrase the situation, Two boys share 80 candies with boy 1 getting 2 pieces and boy 2 getting 3 pieces. Emphasize that "sharing" is not always equal. Students can act out, use manipulatives and/or draw a picture of the situation. The pupose of this task for younger students is to begin building the ideas of ratio, proportional reasoning and algebraic thinking by doing and exploring mathematics.
Do not give this alternative form of the situation to students in grades 4 and higher. One of the purposes of having students answer the question "How many candies does each boy get?" is to use student responses as an assessment. Maybe even have students put their initial "guesses" on a post it or index card with their name to turn in.
Also for students in grades 4 and higher, do not initially define ratio as a class. Again, use this problem as an assessment of their knowledge of ratio.
**Look for students who initially say that each boy gets 40 pieces of candy. These students have limited knowledge of ratio. Document these students names and move on.
At this point, a variety of next steps could happen based on the mathematical understanding of the students in your class, or the grade level you are teaching.
1. Teach a mini-lesson on ratio using the example "What is the ratio of boys to girls in this classroom?" (Emphasis on MINI - - - the lesson should be no more than 10-15 minutes) Discuss equivalent forms of that ratio. Could extend the discussion, now or at a later time, to ask "If the ratio of boys to girls in our school was the same as the ratio in our classroom, how many total students are there if there are _______ boys? For most students, especially struggling students, encourage the use of a variety of strategies to figure this out - NOT setting up and solving a proportion!
2. Discuss the possibility of each boy getting 40 pieces. If the boys end up with the same amount of candy, then they will always have the same amount of candy. What ratios are equivalent to 40:40? At one point both boys had 20 pieces - the ratio 20:20. Get students to continue to move toward how many pieces each boy would have gotten to begin with (1:1). Point out that the ratio 1:1 is not the same as the ratio 2:3. Show with drawings, manipulative or even actual candy what the "Passing out of candy" looks like for the ratio 1:1 and then for the ratio 2:3.
3. Give students time to explore the problem by acting it out, using manipulatives or drawing a picture. Have many discussions about what is happening and what is the mathematics involved.
4. Eventually, have students create a TABLE of the information and look for patterns.
5. Relate the amount of candies each boy has at each "passing outing" to the multiples of that number.
1. What fraction of the candies does boy 1 get?
2. What fraction of the candies does boy 2 get?
3. What percent of the total candies does each boy end up with?
4. What is the difference between a "part-to-part" and a "part-to-whole" ratio?
5. How is a ratio the same as a rate? How are they different?
6. What algebraic equation can you write to solve this situation?
7. At what rate does each boy get candy?
8. How many candies does boy 1 have after the 7th passing outing?
9. How many candies will boy 2 have after the 50th passing outing?
10. If the two boys still share candies in the ratio 2:3, what total candies can there be so that no candies are left over after all passing outings?
11. If the two boys still share a total of 80 candies, what ratios are possible so that no candies are left over after all passing outings? Is the ratio 1:4 possible? Is the ratio 1:2 possible?
12. (Another variation) If 2 boys share 90 candies in the ratio 2:3, how many candies does each boy get? **Look for students who say 45 and 45. Again, these students have a limited understanding at this point. Document these students names and flag for intervention.
Do not worry about the levels of understanding of students!! Make sure to document levels of understanding and note which students need extention and which students need intervention, BUT just HAVE FUN engaging students in mathematics!! This will build the classroom culture and climate of THINKING and LEARNING!! Let students DO MATH themselves, not watch you do all of the math!As teachers, let's reflect on these questions:
What other math questions can be asked related to this situation?What other variations of the situaion are possible?What is all of the additional math content that can be drawn out from this situation?
"The Math Girl"