Question
Does sugar make children hyperactive?
Answer
No.
Explanation
The idea that eating sugary foods makes children hyperactive is an old and deeply entrenched idea, but has not been shown to be true.
In 1975, William G. Crook found that parents reported sugar as the most frequent stimulus for uncontrolled behaviour. Studies into the validity of this connection were reviewed in the 1995 meta-analysis The Effect of Sugar on Behavior or Cognition in Children, covering 16 studies conducted between 1982 and 1994. These double-blinded studies compared behaviour of children after consuming sugars (raises blood sugar levels) versus artificial sweeteners (no immediate effect on blood sugar levels) and concluded that "sugar (mainly sucrose) does not affect the behavior or cognitive performance of children".
But I'm a parent and my child gets hyperactive after eating sugary snacks
A study by Daniel W. Hoover and Richard Milich into "the hypothesis that commonly reported negative effects of sugar on children's behavior may be due to parental expectancies" (with the study testing mothers and sons specifically) showed that while "the children's behavior was actually shown by actometer readings to be more subdued in the sugar expectancy group", "mothers who were told their children had ingested sugar rated their children as engaging in significantly more hyperactive behavior than did control mothers".
See also: Confirmation bias
But doesn't sugar give you energy?
Ingesting sugar raises the amount of glucose in your blood, which is then provided to your body's cells for fuel or stored as glycogen for future use. It doesn't automatically make you energetic, it allows your body's cells to use energy if needed. In persons experiencing an abnormally high or low level of blood sugar (hyperglycaemia and hypoglycaemia respectively), neither condition is associated with the symptom of hyperactivity.
Sources
Crook, W.G., Can your child read? Is he hyperactive? : a pediatrician's suggestions for helping the child with hyperactivity, behavior and learning problems (Jackson, TN, USA : Pedicenter, 1975)
Hoover, D.W., and R. Milich, Effects of Sugar Ingestion Expectancies on Mother-Child Interactions (Journal of Abnormal Child Psychology, Vol. 22, No. 4, 1994)
NHS, 'Hyperglycaemia (high blood sugar)', 8 August 2018, https://www.nhs.uk/conditions/high-blood-sugar-hyperglycaemia/, (accessed 6 January 2019).
NHS, 'Low blood sugar (hypoglycaemia)', 17 August 2017, https://www.nhs.uk/conditions/low-blood-sugar-hypoglycaemia/, (accessed 6 January 2019).
Nichol, A.D., M.J. Holle, and R. An, Glycemic impact of non-nutritive sweeteners: a systematic reviewand meta-analysis of randomized controlled trials (European Journal of Clinical Nutrition, 2018)
Wasserman, D.H., Four grams of glucose (American Journal of Physiology - Endocrinology and Metabolism, 2009)
Wolraich, M. L., D.B. Wilson, and J.W. White, The Effect of Sugar on Behavior or Cognition in Children (JAMA, 1995)