Equal sharing using multiplication

1. Show the students the goat cards made from the goats copymaster. Note that there are different goats to make the cards more interesting. If this is distracting to your students you may prefer to only use identical goats. Alternatively, you might choose to change this context completely to something that is more relevant to your students (e.g. characters from a book, flowers in the school garden, sports gear).
   In this lesson we are going to put goats into pens. They have been running wild and we need to check them over to see they are well.
   Draw four pens on a large sheet of paper.
    
   I have 12 goats to share into the pens (count that many goat cards). I want the same number in each pen.
   How many goats should I put in each pen?
   Let students work out their prediction. Discuss how they predicted the correct number of goats for each pen.
    
2. Provide time for students to share their strategies. These might include equally sharing by ones, skip counting, addition, and multiplication.
   Can we use what we know about multiplication to work out the number of goats in each pen?
   If a student used a fact like 4 x 3 = 12 to solve the problem, invite them to explain how that fact is helpful. Make the link between multiplication and the problem solution transparent.
   Four goats are used up each time I put one in each pen.
   I need to ask myself, “how many fours make up 12?” That will tell me how many times I can put one goat in each pen.
   
    
    
3. Give the students a calculator and invite them to use the calculator to solve the problem.
   You could work through your “times four” facts until you get a product of 12. 
   Is there another way?
   Some students may know about division and share how the operation might be used. If not, guide students in the keys to press.
   12 ÷ 4 = 3
   What does 12 represent in our story? (Total number of goats)
   What does 4 represent? (Number of pens)
   What does 3 represent? (Number of goats per pen)
   What does ÷ represent in our story? (Equally shared into)
   Record the equations 3 x 4 = 12 and 12 ÷ 4 = 3 and ask students what connections they see.
   They should notice that the same three numbers are involved in each equation. The order of the numbers changes and there are different operations.
    
4. Pose similar problems using the goat cards, that involve equally sharing goats into pens. Model the problem physically with goat cards and pen diagrams but expect anticipation of the result before acting. Record the multiplication and division equation for each story and connect the equal sharing to the numbers in each equation. Use calculators to reinforce the link between division and equal sharing. 
   • I have 20 goats and 5 pens. How many goats should I put in each pen?
   • I have 30 goats and 10 pens. How many goats should I put in each pen?
   • I have 18 goats and 2 pens. How many goats should I put in each pen?
   • I have 15 goats and 3 pens. How many goats should I put in each pen?
      
5. Pose goat sharing problems that are connect by adding to known multiplication facts or by doubling. Ensure students are given opportunities to explain their thinking using multiple means of action and expression (e.g. spoken, written, paired explanations, drawings). Examples might be:
   • I have 40 goats and 5 pens. How many goats go in each pen? (5 x 8 = 40 and 40 ÷ 5 = 8)
     What changes if I increase the number of goats to 45 but still have 5 pens? (9 x 5 = 45 and 45 ÷ 5 = 9)
     Record the two division equations to see what students notice.
     40 ÷ 5 = 8 and 45 ÷ 5 = 9
     Students might notice that if the dividend (total number of goats) increases by five the quotient (number of goats per pen) increases by one. There is one more set of five goats, so each pen gets one more goat. That assumes the number of pens stays constant.
   • I have 30 goats and 6 pens. How many goats go in each pen? (5 x 6 = 30 and 30 ÷ 6 = 5)
     What changes if I double the number of goats to 60 but still have 6 pens? (10 x 6 = 60 and 60 ÷ 6 = 10)
     Record the two division equations to see what students notice.
     30 ÷ 6 = 5 and 60 ÷ 6 = 10
     Students might notice that if the dividend (Total number of goats) doubles the quotient (Number of goats per pen) doubles as well. That assumes the number of pens stays constant.

Next steps 

1. Challenge the students to link known multiplication facts to division facts using equations alone. For example, “if you know that 7 x 5 = 35 what division facts do you also know?”
    
2. Ask students to make up an equal sharing problem to match a division equation. For example, “make up a ‘goats in pens’ problem for the equation 40 ÷ 5 = 8.”
    
3. Explore using the calculator to solve equal sharing problems that are outside the students’ range of known facts. For example, “I have 100 goats and 4 pens. I want equal numbers in each pen. What operation should I do on my calculator to work out how many goats should go in each pen?”