# 3rd Edition – Time 4 Fractions – Problem #2 – Gardening

My daughter and I went on a 12 week- journey the past two years to explore Fractions. We are doing it again this Fall. I am updating the posts from our previous journeys,  in case you want to join us this year. Click here if you want to know more about the journey.

Hope you had fun with your child exploring Problem 1. Here is Problem #2, a second multiplication problem, before introducing division problems next week. Please remember that the goal of our journey is to provide children with plenty of opportunities to explore fractions through Equal Sharing problems (Empson & Levi, 2011), and solving multiplication and division problems will prepare them to do so (Empson & Levi, 2011, p 9).

Time 4 Fractions –  Problem #2 – Gardening

Level Yellow : Mr. Purple loves gardening. He planted 3 rows of pumpkin seeds. In each row, there were 2 seeds. How many pumpkin seeds did Mr. Purple plant?

Level Orange: Mr. Purple loves gardening. He planted 5 rows of pumpkin seeds. In each row, there were 4 seeds. How many pumpkin seeds did Mr. Purple plant?

Level Red : Mr. Purple loves gardening. He planted ____ rows of pumpkin seeds. In each row, there were ____ pumpkin seeds. How many pumpkin seeds did Mr. Purple plant ?

What to do as a parent ?

As with Problem #1, invite your child to solve one of the problems, and listen to his/her way of solving it. He/she can make sense of the problem while using small objects (such as buttons, marbles, etc, and small containers) or drawing a picture. He/she may write an equation. Each child should pick the problem that he/she feels like exploring. With Level Red, invite the child to pick numbers he/she feels like comfortable using. For instance, if your child picks 5 pumpkin seeds, he/she may end up counting the seeds by 5, or he/she may use from memory the 5s times table (i.e 5 x 12 if he/she picks 12 rows of seeds).

If your child calls out the answer right away, remind him/her that the answer is fine, but how it was obtained is even more important in this journey. How would he/she explain it to a younger child? Could he/she represent the problem with a drawing? a diagram? Using small objects ?

Sharing my experience

I thought it would be helpful this week to provide some work samples I gathered in the past 3 years from Rosie and the daughters of a dear friend of mine.  No teaching was involved, the girls were just invited to solve the problems in a way that made sense to them. It may give you an idea of strategies a child may use. Please remember that I am sharing these samples to help you see what a child may come up with, not as examples of what a child should come up with :-)

Level Yellow : Mr. Purple loves gardening. He planted 3 rows of pumpkin seeds. In each row, there were 2 seeds. How many pumpkin seeds did Mr. Purple plant?

• Making sense of the problem with a picture. The child wrote then both a repeated addition and a multiplication.

Level Orange – “Mr. Purple loves gardening. He planted 5 rows of pumpkin seeds. In each row, there were 4 seeds. How many pumpkin seeds did Mr. Purple plant?”

•  Making sense of the problem with marbles and paper.  The child counted the marbles by 1s’. Your child may count by 4s’ ?
•  Making sense of the problem with a picture representing the rows of pumpkin seeds. The child wrote, as an equation, a repeated addition. Your child may write a multiplication (4 x 5 = 20) instead?
• Making sense of the problem with a different visual representation, an array. The child wrote then both a repeated addition and a multiplication.

Level Red – “Mr. Purple loves gardening. He planted ____ rows of pumpkin seeds. In each row, there were _____  pumpkin seeds”.

• Making sense of the problem with Duplos® (5 rows, 8 seeds). The child counted the blocks by 1 up to 15, and noticed that she was counting by 5. She started over, counting by 5, and answered 40 pumpkin seeds. This sure was fun to watch a child, noticing a pattern of counting, changing her strategy to a more efficient one.

• Making sense of the problem with buttons. (7 rows, 5 seeds). The child also wrote, as an equation, a multiplication 7 x 5 = 35.
• Making sense of the problem  (8 rows, 5 seeds) with a picture representing the rows of pumpkin seeds, drawing the seeds in the first row, and writing the number of seeds instead on the next rows. Always interesting to see how a child may switch from a drawing to a more symbolic representation.

• Making sense of the problem  (2 rows, 4 seeds) with a picture representing the rows of pumpkin seeds, writing, as an equation, an addition.

Also, here are some examples of questions I asked to follow the child’s reasoning:

• Tell me about what you did.
• Could you tell me about the marbles you used ?
• I see you wrote the equation 4 + 4 + 4 + 4 + 4 = 20. Could you show me on your drawing where the 4 comes from? The 20 ? Why did you add 4 five times? What does the symbol “+” mean? And the symbol “=”?
• You said “20”. 20 what? Could you tell me the unit?

No video this week, as the problem is similar to the one explored last week.

Have fun, and see you next week for Problem #3 !

Reference:

Empson, S. E., and Levi, L. (2011). Extending Children’s Mathematics: Fractions and Decimals. Portsmouth, NH : Heinemann. ISBN-13: 978-0325030531.